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[ { "title": "ABSTRACT", "content": "Several independent measurements have confirmed the existence of fluctuations ( δF obs ≈ 0 . 1 nW / m 2 / sr at 3 . 6 µ m) up to degree angular scales in the source-subtracted Near InfraRed Background (NIRB) whose origin is unknown. By combining high resolution cosmological N-body/hydrodynamical simulations with an analytical model, and by matching galaxy Luminosity Functions (LFs) and the constraints on reionization simultaneously, we predict the NIRB absolute flux and fluctuation amplitude produced by highz ( z > 5) galaxies (some of which harboring Pop III stars, shown to provide a negligible contribution). This strategy also allows us to make an empirical determination of the evolution of ionizing photon escape fraction: we find f esc = 1 at z ≥ 11, decreasing to ≈ 0 . 05 at z = 5. In the wavelength range 1 . 0 -4 . 5 µ m, the predicted cumulative flux is F = 0 . 2 -0 . 04 nW / m 2 / sr. However, we find that the radiation from highz galaxies (including those undetected by current surveys) is insufficient to explain the amplitude of the observed fluctuations: at l = 2000, the fluctuation level due to z > 5 galaxies is δF = 0 . 01 -0 . 002 nW / m 2 / sr, with a relative wavelength-independent amplitude δF/F = 4%. The source of the missing power remains unknown. This might indicate that an unknown component/foreground, with a clustering signal very similar to that of highz galaxies, dominates the source-subtracted NIRB fluctuation signal. osmology: diffuse radiation-galaxies: high redshift-methods: numerical.", "pages": [ 1 ] }, { "title": "The contribution of high redshift galaxies to the Near-Infrared Background", "content": "Bin Yue 1 , 3 , 5 , Andrea Ferrara 1 , 6 , Ruben Salvaterra 2 , Xuelei Chen 3 , 4", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Observations of high redshift galaxies are essential to understand cosmic reionization. Although current surveys have reached redshifts ∼ 8 -10 (Bouwens et al. 2010, 2011a,b), it is generally believed that the sources detected so far, usually rare and bright galaxies, are not the dominant contributors to reionization (Choudhury & Ferrara 2007; Lorenzoni et al. 2011; Jaacks et al. 2012; Finkelstein et al. 2012). Instead, reionization is likely powered by the large number of galaxies that are still below the detection limit. Even without detecting these faint galaxies individually, their cumulative radiations may still tell us much about their properties. Indeed, the bulk of their emission, mostly in the band between the Lyman limit and the visible light, is redshifted into the near InfraRed (IR) at present time. Therefore, the Near InfraRed Background (NIRB) which is obtained after removing the contributions from the Solar system, the Milky Way and lowz galaxies, could provide a wealth of information on high redshift galaxies, such as their integrated emissivity and large scale clustering properties. The NIRB measurement has a history dating back to more than two decades (see the review by Kashlinsky 2005). Early measurements gave a NIRB flux > ∼ 10 nW / m 2 / sr (Dwek & Arendt 1998; Gorjian et al. 2000; Matsumoto et al. 2000; Cambr'esy et al. 2001; Matsumoto et al. 2005). These works showed that a non-zero residual remains after the foreground and the emission from known galaxies are removed (Totani et al. 2001; Matsumoto et al. 2005). Salvaterra & Ferrara (2003) and Santos et al. (2002) suggested that Pop III stars could possibly be the sources of such leftover signal. If true, this residual would be an exquisite tool to study Pop III stars. However, not long afterwards, Madau & Silk (2005) and Salvaterra & Ferrara (2006) found that this scenario needs a very high star formation efficiency and may overpredict the highz dropouts galaxies 1 . To solve this problem, we need either an alternative theoretical explanation (proved hard to be found), or a more accurate determination of the residual flux, or both. Due to the difficulties in foreground subtraction (Dwek et al. 2005), in recent observational works more attentions are paid to the angular fluctuations. In such observations, the influence of strong but smooth foregrounds, such as the zodiacal light, is reduced, and one can also infer the large scale clustering properties of the unknown sources (Kashlinsky et al. 2002, 2004; Magliocchetti et al. 2003; Cooray et al. 2004; Matsumoto et al. 2005; Salvaterra et al. 2006). The recent measurements (Kashlinsky et al. 2005, 2007a, 2012; Matsumoto et al. 2005, 2011; Thompson et al. 2007a,b; Cooray et al. 2012b) have obtained angular power spectra of the source-subtracted NIRB (i.e. all resolved galaxies have been removed) at wavelengths from 1 . 1 µ m to 8 µ m. These angular power spectrum measurements show that the sources have a large clustering signal up to degree scales. The source-subtracted NIRB fluctuations are found to be much higher than the theoretically predicted contribution from lowz faint galaxies. Although Kashlinsky et al. (2005, 2007b); Matsumoto et al. (2011); Kashlinsky et al. (2012) favored a scenario in which the observed fluctuations come from Pop III stars, Cooray et al. (2012a) showed that, to be consistent with the electron scattering optical depth measured by WMAP (Komatsu et al. 2011), the contribution from highz galaxies (including Pop III stars) must be smaller by at least an order of magnitude than what is observed. Instead, Cooray et al. (2012b) suggested recently that a large fraction of the observed NIRB fluctuations comes from the diffuse light of intrahalo stars at intermediate redshifts ( z ∼ 1 to 4). While an intriguing idea, this explanation relies on the poorly known fraction and spectral energy distribution of intrahalo stars. It also predicts, contrary to the faint distant galaxies hypothesis, that fluctuations induced by the much closer intrahalo stars should extend into the optical bands, where the light from first galaxies is blanketed by intervening intergalactic neutral hydrogen. Numerical simulations by Fernandez et al. (2010, 2012) stressed the importance of nonlinear effects in theoretical calculations as a possible way to reconcile the theory with data. There have also been proposals that the sources of these fluctuations are lower redshift galaxies (Thompson et al. 2007b; Cooray et al. 2007; Chary et al. 2008), but this possibility has become less attractive by now. Indeed, Helgason et al. (2012) recently reconstructed the emissivity history from the luminosity functions (LFs) of observed galaxies, and found that the fluctuations from the known galaxy population below the detection limit are unable to account for the observed clustering signal on sub-degree angular scales. To make further progress, it is essential to make more accurate predictions of the NIRB contributed by Pop III stars and the galaxies before reionization, using models which are consistent with all current observational constraints, including both the high redshift LFs and reionization. In this paper, we attempt to make the most detailed theoretical NIRB model developed so far, with predictions on both the absolute flux and the angular power spectrum contributed by highz galaxies. To do this, we used a simulation with detailed treatment of the relevant physics of star/galaxy formation, including gas dynamics, radiative cooling, supernova explosion, photoionization and heating, and especially a detailed treatment of chemical feedback (Tornatore et al. 2007a,b). The LFs of high redshift galaxies in the simulation match remarkably well with observations, this is the starting point of our NIRB model. The layout of the paper is as follows. In Section 2 we introduce the simulation, and describe the steps to calculate the NIRB absolute flux and the angular power spectrum. In Section 3 we present our results and compare them with observations. Conclusions are presented in Section 4. In Appendix A we compare the different approximate solutions for the analytical calculation of the emissivity. Throughout this paper, we use the same cosmological parameters as in Salvaterra et al. (2011): Ω m =0.26, Ω Λ =0.74, h =0.73, Ω b =0.041, n = 1 and σ 8 =0.8. The transfer function is from Eisenstein & Hu (1998). Magnitudes are given in the AB system.", "pages": [ 1, 2, 3 ] }, { "title": "2.1. The Absolute Flux", "content": "At z = 0, the cumulative flux of the NIRB observed at frequency ν 0 is the integrated contribution of sources whose emission is shifted into a band of central frequency ν 0 . Following Salvaterra et al. (2006), we write it as where r p is the proper distance, ν = (1 + z ) ν 0 is the rest frame frequency, /epsilon1 ( ν, z ) is the comoving specific emissivity, H ( z ) is the Hubble parameter given by H ( z ) = H 0 √ Ω m (1 + z ) 3 +Ω Λ in a flat ΛCDM cosmology, c is the speed of light. The effective optical depth of absorbers between redshift 0 and z , τ eff , is composed of two parts: the line absorption and the continuum absorption; we use the expressions in Salvaterra & Ferrara (2003). We calculate the emissivity (see also Appendix for further discussions on subtleties related to the various approximations used in the literature) from the results of the simulation presented in Salvaterra et al. (2011), which includes a detailed treatment of chemical enrichment developed by Tornatore et al. (2007a). In our model, both Pop II stars and Pop III stars are assumed to follow the Salpeter initial mass function (IMF) (Salpeter 1955), for Pop II stars the mass range is 0 . 1 -100 M /circledot , while for Pop III stars the mass range is set to be 100 -500 M /circledot . Some recent works indicate that Pop III stars may not be so massive as was predicted previously, but may be limited to /lessorsimilar 50 M /circledot (Hosokawa et al. 2011). Our choice then corresponds to an upper limit to the contribution of these sources. Using this simulation, Salvaterra et al. (2011) generated the LFs of galaxies down to the magnitude far below the current observation limits at high redshifts. In the redshift range 5 < z < 10, the simulated LFs match the observed ones almost perfectly in the overlapping luminosity range. Suppose the specific luminosity of the i -th galaxy in the simulation box is L i ν ( z ) at redshift z , the comoving specific emissivity is then 2 where V is the comoving volume of the simulation, N is the total number of galaxies in the simulation box at redshift z . In the emissivity calculation, we must correct for rare bright galaxies that are not caught by the simulation due to the finite box size (10 h -1 Mpc). We follow the steps in Salvaterra et al. (2011). We first calculate the absolute magnitude corresponding to the mean luminosity of the two brightest galaxies in the simulation box, M UV , up . The contribution (to be added to the numerator in Eq. (2)) from galaxies brighter than this magnitude is obtained by integration where L 1 ν ( z ) is the luminosity of the brightest galaxy in the simulation (we assume all rare bright galaxies have the same Spectral Energy Distribution (SED) of this one), L UV is the luminosity corresponding to the UV absolute magnitude M UV . The wavelength used to calculate the absolute UV magnitude in this paper is 1700 ˚ A, ν UV is the frequency corresponding to this wavelength. In observations, the selected wavelength corresponding to the UV absolute magnitude may be somewhat different in different measurements and at different redshifts (Bouwens et al. 2007; Oesch et al. 2010; Bouwens et al. 2010), however, our results are not sensitive to such differences. For the LF Φ( M UV , z ) in the redshift range 5 < z < 10, we use the Schechter formula (Schechter 1976) with the redshift-dependent parameters given by Bouwens et al. (2011b) (see their Sec. 7.5), who fitted the observed LFs in z ∼ 4 -8 and extrapolated them to higher redshifts. For redshifts above 10, we simply add an exponential tail normalized to the simulated LF amplitude at M UV , up . We find that this results only a small correction. As discussed below (see also bottom panel of Figure 3), ∼ 90% of the highz galaxy contribution to the NIRB flux comes from sources at 5 < z < 8 where the correction is at most 12%. This correction is also applied to the calculation of ionizing photons below. For each galaxy, the radiation comes from two different mechanisms: the stellar emission and the nebular emission. The former comes directly from the surface of stars, while the latter is generated by the ionized nebula around stars and depends on the fraction of ionizing photons that cannot escape into the intergalactic medium (IGM), i.e., 1 -f esc , where f esc is the escape fraction. Ionizing photons escaping from galaxies would ionize the IGM; such ionized gas could also produce the nebular emission. However, due to the very low recombination rate, as shown in, e.g., Nakamoto et al. (2001) and Cooray et al. (2012a), its emissivity is much weaker than the radiation from galaxies, so we ignore this contribution in this paper. The IGM contribution to the NIRB fluctuations is also negligible (Fernandez et al. 2010). To determine the escape fraction averaged over the galaxy populations present at a given redshift, we proceed as follows. First we compute the number of ionizing photons emitted per baryon in collapsed objects as: where q II H is the emission rate of ionizing photos from Pop II stars (this quantity depends on both the age and the metallicity of the stellar population) corresponding to a continuous star formation rate 1 M /circledot yr -1 . We derive this quantity from the Starburst99 templates 3 (Leitherer et al. 1999; V'azquez & Leitherer 2005; Leitherer et al. 2010) adopting the mean age, τ II ,i , and metallicity, Z i , of each simulated galaxy. q III H is the emission rate of ionizing photons for Pop III stars according to Schaerer (2002) 4 . M i gas is the gas content, ˙ M II ,i /star is the mean star formation rate of Pop II stars in this galaxy, while M III ,i /star is the cumulative mass of Pop III stars. We use a mean lifetime τ III = 2 . 5 × 10 6 yr for massive Pop III stars (Schaerer 2002; Salvaterra et al. 2011). We then compare the above quantity with the number of ionizing photons per baryon in collapsed objects, N ion , required by interpreting the observations as in Mitra et al. (2012) (the 'mean' value) to get the escape fraction, i.e., f esc = min( CN ion f /star N γ , 1 . 0), where C is the clumping factor. Throughout this paper we assume C = 1 to get the minimum f esc therefore the maximum contribution of the nebular emission to the NIRB. We note that the clumping factor could be higher than 1 even at high redshifts (Pawlik et al. 2009; Shull et al. 2012). For example, Shull et al. (2012) gives C ≈ 3 (1 . 7) at z = 5 (9) by numerical simulations. Our nebular emission is therefore reduced by about 10% to 60% from z = 5 to z = 9 if this clumping factor is adopted. However, for Pop II stars which are the dominant contributors to the NIRB, the nebular emission is much smaller than the stellar emission. So the final reduction in the NIRB would be much smaller. Furthermore, we will show later that the flux from high redshift galaxies and Pop III stars is unable to explain the observed fluctuations level, so that a reduction in the NIRB flux would in any case strengthen this conclusion. We plot the derived f esc as a function of redshift in Figure 1. There is a clear trend of an increasing escape fraction towards higher redshifts; it reaches 1 at z ≈ 11. At z = 5, the final redshift of the simulation, f esc ≈ 0 . 05. Although required by reionization data, an increasing trend of f esc ( z ) has not yet been fully understood theoretically in spite of the several, often conflicting, studies on this problem. Based on observations, Inoue et al. (2006) concluded that f esc > 0 . 1 when z > 4; by combining the observations of Lyman α absorption and UV LF, and also using N-body simulations and semi-analytical prescriptions to model the ionizing background, Srbinovsky & Wyithe (2010) found that for galaxies at z ∼ 5 . 5 -6, if the minimum mass of star forming galaxies corresponds to the hydrogen cooling threshold, f esc ∼ 0 . 05 -0 . 1; Wyithe et al. (2010) used the star formation rate derived from gamma-ray burst observations to conclude that in the redshift range 4 -8 . 5, f esc ∼ 0 . 05; Wise & Cen (2009), by radiation hydrodynamical simulations, found that at redshift 8 for galaxies with M vir < 10 7 . 5 M /circledot , f esc ∼ 0 . 05 -0 . 1, while for more massive galaxies f esc ∼ 0 . 4, if a normal IMF is adopted; also via simulations, Razoumov & Sommer-Larsen (2010) found f esc ∼ 0 . 8 when z = 10. The escape fraction derived by us is broadly consistent with these values. The important difference however is that our derivation of f esc matches both the LF and the reionization history simultaneously, i.e., a more phenomenological derivation, so that we can get around the detailed physical mechanisms of the escape fraction. Different from our approach, Mitra et al. (2013) computed the LFs of high redshift galaxies by means of semi-analytical models and derived the star formation efficiency f /star required to match the observed ones. They found f esc ≈ 0 . 07 at z = 6 and f esc ≈ 0 . 16 at z = 7, which are consistent with our f esc ≈ 0 . 06 (0 . 18) at those two redshifts. At higher redshifts, however, their escape fraction is somewhat lower than ours. As a final remark, we underline that when computing the escape fraction, we do not make a distinction between Pop III and Pop II stars. In the calculation of f /star N γ ionizing photons from both populations are accounted for, f esc can be regarded as a kind of 'effective' escape fraction averaged over the galaxy population. In principle, f esc for Pop III stars should be higher due to their harder spectrum. However, as we will see in Section 3, Pop III stars only contribute a negligible flux to the present-day NIRB, a more detailed modeling is then not necessary. The luminosity of the i -th galaxy, L i ν , is the sum of the contribution of Pop II and Pop III stars. For Pop II stars we use the age and metallicity dependent spectrum templates provided by the Starburst99 code. The nebular emission contribution has been renormalized by adopting the escape fraction computed above. In addition to the free-free, free-bound and two-photon emissions which have already been included in Starburst99 , we add the Lyman α emission to the template by using (Fernandez & Komatsu 2006) in which f α = 0 . 64 (Fernandez & Komatsu 2006), h p is the Plank constant, ν α = 2 . 47 × 10 15 Hz is the frequency of Lyman α photons. We use the line profile φ ( ν -ν α ) provided in Santos et al. (2002): where is the fitted form of results given in Loeb & Rybicki (1999). For the template of Pop III stars, l III ν , we still use the spectrum in Schaerer (2002), but renormalize the nebular emission part by the factor 1 -f esc . The luminosity of the i -th galaxy is then given by (Salvaterra et al. 2011) here the Lyman α emission in Eq. (5) has already been included in l II ν . With the luminosity for each galaxy given as above, we can then obtain the emissivity according to Eq. (2). As an example, we plot the ν/epsilon1 ( ν, z ) at redshifts 12.0, 9.0 and 6.0 respectively in Figure 2. At high redshifts, the escape fraction ≈ 1 . 0, yielding a very weak Ly α line, since such emission is produced by recombinations of the ionized nebula around stars. At lower redshifts, the escape fraction drops, while more ionizing photons are absorbed by the material around the stars, and producing more Ly α emission which is more clearly seen in the spectrum. The part of spectrum with energy below 10.2 eV is of the most interest to us, here the spectrum becomes increasingly flatter at later time. For example, at z = 12, the slope of ν/epsilon1 ( ν, z ) ∝ ν β with β ≈ 2, while at z = 5 β ≈ 1 . 2. This is clearly the result of an aging effect enhancing the rest frame optical/IR bands flux with respect to the UV ones. Since the NIRB from z > 5 galaxies is dominated by the lower redshift galaxies (5 < z < 8), we do not expect to have a very steep NIRB spectrum, as we will see in the results presented in Sec. 3.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "2.2. NIRB Fluctuations", "content": "Using the Limber approximation, the angular power spectrum of the fluctuations of the flux field is (Cooray et al. 2012a) where r ( z ) is the comoving distance and P gg ( k, z ) is the galaxy-galaxy power spectrum, k = l/r ( z ). In Eq. (9) we assume that the luminous properties of galaxies are independent of their locations, so that the only factor which determines their contribution to the NIRB fluctuations is their spatial fluctuations (see Shang et al. 2012 for an improved model). The 10 h -1 Mpc box size of Salvaterra et al. (2011) simulations is too small to provide us with the large scale correlation function of galaxies (for sources at z = 6, the comoving transverse separation corresponding to 1 · angular size is about 100 h -1 Mpc), so we use the halo model (Cooray & Sheth 2002; Cooray 2004) to calculate the galaxy-galaxy power spectrum. This power spectrum is composed of two parts, the one-halo term from the correlation of galaxies in the same halo (including the central galaxies and satellite galaxies), and the two-halo term from galaxies in different halos: Assuming that the distribution of galaxies in a halo traces the profile of dark matter, and the mean number of central galaxies and satellite galaxies in a halo with mass M are 〈 N sat 〉 and 〈 N cen 〉 , respectively, we have and In the above expressions, M min ( z ) is the minimum mass of halos that could host galaxies, and we set it to be the minimum mass of halos that contain stars in our simulations, which is ∼ (2 -8) × 10 7 M /circledot , depending on the redshift. M max ( z ) is the maximum mass contributing to the emissivity and the clustering, we will describe how to determine it later, dn/dM is the mass function (Sheth & Tormen 1999; Sheth et al. 2001), while u ( M,k ) is the normalized Fourier transform of the halo profile. For a NFW profile (Navarro et al. 1997), the analytical expression is given by Cooray & Sheth (2002), and we use the concentration parameter, c M , from Prada et al. (2012) which fits simulations well. However, we find that our results are insensitive to the use of different concentrations, as e.g., from Zehavi et al. (2011). Even if a very different concentration parameter is adopted, its impacts are non-negligible only in the one-halo term which dominates the signal at small scales, where galaxy clustering is well below the shot noise. As we are interested primarily in the large-scale ( > 1 ' ) clustering, our conclusions are unaffected by the adopted value of c M . Finally, for the halo bias b ( M,z ) we use the formula and fitted parameters given by Tinker et al. (2010), which is higher than Sheth et al. (2001) for massive halos, but is better fit to simulations. The linear matter power spectrum P lin ( k, z ) is taken from Eisenstein & Hu (1998). The mean number of central galaxies and satellites in a halo with mass M is modeled by the halo occupation distribution (HOD) model (Zheng et al. 2005), and We adopt the parameters M sat = 15 M min , σ log 10 M = 0 . 2 and α s = 1 . 0, which are from both simulations and semi-analytical models (Zheng et al. 2005), and observations (Zehavi et al. 2011). With the mean number of central and satellite galaxies in each halo, the galaxy number density is simply In addition to the above galaxy clustering, Poisson fluctuations in the number of galaxies would generate shot noise in observations, whose power spectrum dominates at small scales. If the redshift derivative of the number of sources with flux between S and S + dS is d 2 N dSdz , the angular power spectrum of such shot noise is where ∆Ω is the beam angle. Considering that and where L ν ( M ) is the luminosity of halos with mass M , the shot noise power spectrum is As we assume the luminous properties of galaxies are independent of their location, in the square bracket we can simply use an average light-to-mass ratio that is independent of the halo mass, We finally obtain the shot noise angular power spectrum where and the halo mass density is . In observations, the detected sources are generally removed down to a certain limiting magnitude, m lim , the residue is the source-subtracted NIRB fluctuations. To simulate this, we also remove bright galaxies in the simulation box and in the bright-end. In theoretical calculations, this limiting magnitude is determined by letting the predicted shot noise level match the values found in the measurements. The apparent limiting magnitude (at wavelength λ 0 ) is converted into the rest frame absolute magnitude (at wavelength λ 0 / (1 + z )) M λ 0 / (1+ z ) by where DM ( z ) is the distance modulus (Helgason et al. 2012). By using a light-to-mass ratio constructed from the simulation, we determine the maximum halo mass M max ( z ). Things are slightly more complicated when calculating the absolute flux and the spectrum of fluctuations, since they both depend on wavelength, while the limiting magnitude in observations at different wavelength is different. In this case we simply give the theoretical prediction without removing any sources in the simulation box and the bright-end, as shown in Figure 3 and Figure 7, corresponding to M max = ∞ . Then we discuss the effects of galaxy removal, i.e., in Figure 4. Throughout this paper we adopt z min = 5 and z max = 19 unless otherwise specified.", "pages": [ 8, 10, 11, 12 ] }, { "title": "3. RESULTS", "content": "We start by presenting the contribution of highz galaxies to the absolute flux of the NIRB observed at z = 0 in the (observer frame) wavelength range 0 . 3 -10 µ m. Figure 3 (top panel) shows the predicted cumulative flux when all sources with z > 5 are included; also shown separately are the contributions from Pop II and Pop III stars. The flux peak value is 0 . 2 nW / m 2 / sr at λ 0 = 0 . 9 µ m, and decreases to 0 . 04 nW / m 2 / sr at λ 0 = 4 . 5 µ m. The small bump on the left side of the peak is due to intergalactic Ly α absorption by intervening neutral hydrogen. We find that in our case, the Pop III contribution is almost negligible (it never exceeds 1%). This is not surprising, for in the simulation the Pop III star formation rate is about three orders of magnitude lower than that of Pop II stars at z = 10; the ratio is even smaller below this redshift (Tornatore et al. 2007b). Stated differently, halos with the highest Pop III stellar fraction are usually smaller and less luminous, and their contribution to the total luminosity is very low (Salvaterra et al. 2011). This means that it is very difficult to find Pop III signatures by means of NIRB observations. In the bottom panel of Figure 3, we plot the contributions from the sources above redshift 5.0, 8.0 and 12.0, respectively. From the figure, it is clear that the contributions from the sources at 5 < z < 8 dominate, providing about 90% of the flux from all sources with z > 5. Most of these sources are the low-luminosity galaxies which cannot be detected individually in current surveys, and they are believed to be the major contributors to reionization. In principle, then, the NIRB could be a perfect tool to study the reionization sources without detecting them individually. One way to approach the highz components is to remove bright sources in the field of view. We plot the flux before and after removal of bright galaxies in Figure 4 at wavelengths from 1.25 µ m to 4.5 µ m, corresponding to the J through M bands. The crosses refer to flux from all galaxies with z > 5 in our work, while the solid line corresponds to the flux from all galaxies at all redshifts in the 'default' model of Helgason et al. (2012). After removal of galaxies brighter than m lim = 28, the flux from z > 5 galaxies in our work is shown by diamonds, while flux from remaining galaxies in Helgason et al. (2012) is shown by the dashed line. The flux from all galaxies (solid line) is about 1-2 orders of magnitude larger than that from galaxies with z > 5 (crosses) in our work. Hence, without bright galaxy removal, z < 5 galaxies largely dominate the NIRB flux. However, if we remove the galaxies down to m lim = 28, the flux from the remaining galaxies at all redshifts in Helgason et al. (2012) (dashed line) is comparable to that from the remaining galaxies with z > 5 in our work. Even considering the uncertainties on the faint-end of LFs (the shaded regions), in the source-subtracted flux, galaxies at z > 5 still contribute at least ∼ 20% -30% of the flux from galaxies at all redshifts and fainter than m = 28. So at least in principle we can access the signal of reionization sources by subtracting the bright galaxies from the NIRB. Before moving to fluctuations, we emphasize that the expected contribution of highz galaxies (including Pop III stars) to the NIRB flux is very small compared with the residual flux in the measurement of Matsumoto et al. (2005). These sources largely fall short of accounting for such residual ( ∼ 60 -6 nW / m 2 / sr in the wavelength range 1 . 4 -4 µ m). It has to be reminded that the flux measured by Matsumoto et al. is likely to be still dominated by incomplete zodiacal light subtraction, as discussed by Thompson et al. (2007a), who concluded that no residual flux is present. Considering the difficulties in modeling the zodiacal light accurately, currently the residual flux measurements are not very useful to constrain models. Therefore, we will base all the conclusions in the present paper on the analysis of fluctuations only, see below. The fluctuations of the NIRB after subtracting galaxies down to the detection limits of observations, √ l ( l +1) C l / (2 π ), at λ 0 = 1 . 6 , 2 . 4 , 3 . 6 , 4 . 5 µ m are shown by the thick solid line in each panel of Figure 5. The contribution from z > 5 faint galaxies, which is studied in this work, and the contribution from z < 5 galaxies, which is calculated by following the reconstruction of Helgason et al. (2012), are shown by dashed line and dash-dotted line respectively. Unless the limiting magnitude is very faint so that the relative fraction of the contribution of highz galaxies is larger as in the upper left panel, the contribution of z > 5 galaxies is negligible compared with the z < 5 galaxies, i.e., the total amplitude (solid line) coincides with that of z < 5 galaxies (dash-dotted line). To account for the uncertainties of the faint-end of LFs, Helgason et al. (2012) considered two models of the faint end of LFs (adopted for lowz faint galaxies here) which are likely to bracket the real case. Considering this we show the range of the total power spectrum by shaded regions. We also plot observations at corresponding wavelength in each panel by filled circles with errorbars, which are from Thompson et al. (2007a) (1 . 6 µ m), Matsumoto et al. (2011) (2 . 4 µ m), Cooray et al. (2012b) (3.6 and 4.5 µ m) respectively. The measurements of Cooray et al. (2012b) agree well with observations of Kashlinsky et al. (2012) at the same wavelength, but extend to larger angular scales. In the theoretical predictions, we remove the bright sources by selecting a limiting magnitude at each wavelength to get a shot noise level of the remaining fainter galaxies (including both lowz and highz ones, but the latter is almost negligible) that matches each measurement, i.e., m lim = 26.7, 23.2, 23.9 and 23.8 respectively, the first two values are the same as Helgason et al. (2012). At small scales where the shot noise dominates, the model predictions should match the observations, as shown in the 3.6 and 4.5 µ m panels. In the 2.4 µ m panel, there is some discrepancy at small scales, this is because the suppression of the power by beam effects is not corrected in the observations data. For the 1.6 µ m case, a footnote in Helgason et al. (2012) noted that the images at other wavelength are used to subtract bright sources, so there would be spread on the limiting magnitudes. From the figure, we also see that the contributions of the z > 5 galaxies (dashed lines) exceed the shot noise level on large angular scales ( l < 10 4 ), this means that the NIRB fluctuations do have the potential to provide us information on the nature of the undetected reionization sources. However, the predicted amplitudes are only ∼ (2 -4) × 10 -3 nW / m 2 / sr, which is even much smaller than the contribution from lowz faint galaxies, and both the highz and lowz contributions are much smaller than the observed values, which are at the ∼ 0 . 1 nW / m 2 / sr level. Even considering the uncertainties about the faint-end of LFs, the difference is still quite significant, again indicates the existence of one or more unknown component(s) we are missing. Somewhat surprisingly but interestingly, the missing component has a clustering signal very similar to that of the high redshift galaxies, and extends to degree angular scales. Obviously, this component/foreground must be identified before we can make further progress and use the NIRB to study reionization sources. Next, we define the fluctuation amplitude δF = √ l ( l +1) C l / (2 π ), and plot its ratio to F = ν 0 I ν 0 in Figure 6. Such relative fluctuation is almost independent of λ 0 (see also Fernandez et al. 2010), δF/F ∼ 4% at l = 2000, with the only slight deviation of the 1 . 6 µ m band where it is somewhat lower than in redder bands, as a consequence of a deeper ( m lim ≈ 27) galaxy removal. Nicely, the relative fluctuation agrees with that found by Fernandez et al. (2010) and Cooray et al. (2012a). In addition, δF/F increases with z min , i.e., high redshift sources have higher relative fluctuations. For example, δF/F = 7% for z min = 8 . 0, while it reaches 12% if z min = 12 . 0 is adopted. It reflects the more biased spatial distributions of higher redshift sources. The relative fluctuation δF/F is only weakly dependent on the intrinsic properties of galaxies (Fernandez et al. 2010), but more so on the spatial clustering features. Thus, δF/F is a key indicator to identify NIRB sources; yet, in practice, it is hard to get an accurate absolute flux. The spectrum of the fluctuations, δF ( λ 0 ) from all galaxies with z > 5 at l = 2000, shown in Figure 7, has a slope λ p 0 , with p = -1 . 4 above 1 µ m. Such slope is essentially the same as that of the flux, reflecting the above mentioned wavelength independency of δF/F .", "pages": [ 12, 15, 17 ] }, { "title": "4. CONCLUSIONS", "content": "By combining high resolution cosmological N-body/hydrodynamical simulations and an analytical model, we predicted the contributions to the absolute flux and fluctuations of the NIRB by high redshift ( z > 5) galaxies, some of which harboring Pop III stars. This is the most robust and detailed theoretical calculation done so far, as we simultaneously match the LFs and reionization constraints. The simulations include the relevant physics of galaxy formation and a novel treatment of chemical feedback, by following the metallicity evolution and implementing the physics of Pop III/Pop II transition based on a critical metallicity criterion. It reproduces the observed UV LFs over the redshift range 5 < z < 10, and extend it to faint magnitudes far below the detection limit of current observations. We directly calculate the stellar emissivity from the simulations. We use Starburst99 to generate metallicity and age dependent SED templates, then calculate the luminosity for each galaxy according to its current star formation rate, stellar age and metallicity, instead of using a constant metallicity and average main sequence spectrum template. Except for the mass range of the IMF which has already been fixed in the simulation, there are no other free parameters in the calculation of the emissivity. By comparing the number of ionizing photons produced per baryon in collapsed objects, f /star N γ , in the simulation and the ionizing photon rate N ion ≈ f esc f /star N γ deduced from observationally constrained reionization models, we obtained the evolution history of the escape fraction of ionizing photons, f esc ( z ). We find f esc ≈ 1 at z > 11, decreasing to ≈ 0 . 05 at z = 5. This escape fraction is used to renormalize the nebular emission of Pop III and Pop II stars in the emissivity. Pop III stars are unlikely to be responsible for the observed NIRB residual, and their contribution is very small, making up < 1% of the total absolute flux in our calculation. This is the natural result of the much lower star formation rate of Pop III stars compared with Pop II stars in the simulation, since even metals from a single Pop III star could enrich above the critical metallicity a large amount of gas around it (Tornatore et al. 2007b). The formation of Pop III stars is regulated by such a chemical feedback mechanism, which limits their contribution to the NIRB. However, a rapid Pop III-Pop II transition brings also a little advantage in terms of integrated emissivity, due to the longer lifetime of Pop II stars (Cooray et al. 2012a). We predict that in the wavelength range 1 . 0 -4 . 5 µ m, the NIRB flux from z > 5 galaxies (and their Pop III stars) is ∼ 0 . 2 -0 . 04 nW / m 2 / sr, while the fluctuation strength is about δF = 0 . 01 -0 . 002 nW / m 2 / sr at l = 2000. If we remove galaxies down to m lim = 28, the above flux level is only slightly reduced; however, by comparing with Helgason et al. (2012), we find that the flux from z < 5 dramatically decreases and the remaining becomes comparable to the predicted signal of z > 5 galaxies. This implies that in principle it is possible to get the signal from reionization sources by subtracting galaxies down to a certain magnitude. The relative fluctuation amplitude, δF/F , at l = 2000 is ∼ 4%, almost independent of the wavelength. This ratio may be helpful to investigate the clustering features of the sources contribute to the NIRB, since the intrinsic properties of galaxies almost cancel out. Despite the difficulties in measuring the absolute flux accurately, it could be treated as a quality indicator in the data reduction process: if a much higher/lower ratio is obtained from the data, this might suggest that a more careful analysis work is required to extract the genuine contribution from reionization sources. In spite of being accurate and consistent with the observed LFs and reionization data, thus offering a robust prediction of the NIRB contribution from highz galaxies which likely reionized the universe, a puzzling question remains: the predicted fluctuations are considerably lower than the observed values, indicating that in addition to the contribution from the expected highz galaxy population (and Pop III stars), we should invoke some other - yet unknown - missing component(s) or foreground(s) which dominates the currently observed source-subtracted NIRB. Moreover, the angular clustering of this missing component must be very similar to that of the high redshift galaxies and extends to degree scales. Obviously, this component/foreground must be identified and removed before we are ready to exploit the NIRB to study reionization sources. On the other hand, sources located at 5 < z < 8 provide about 90% of the flux from all sources with z > 5 in our simulation; most of them are the faint galaxies currently undetected by deep surveys. Thus, if the above mentioned additional spurious sources/foregrounds can be removed reliably, the NIRB will become the primary tool to investigate the properties of the reionizing sources.", "pages": [ 19, 20 ] }, { "title": "ACKNOWLEDGMENTS", "content": "It is a pleasure to acknowledge intense discussions and data exchange with A. Cooray, K. Helgason, E. Komatsu, T. Matsumoto, R. Thompson, S. Kashlinsky, S. Mitra and T. Choudhury. AF thanks UT Austin for support and hospitality as a Centennial B. Tinsley Professor and the stimulating atmosphere of the NIRB Workshop organized by the Texas Cosmology Center. BY and XC also acknowledges the support of the NSFC grant 11073024, the MoST Project 863 grant 2012AA121701, and the Chinese Academy of Science Knowledge Innovation grant KJCX2-EWW01.", "pages": [ 20 ] }, { "title": "REFERENCES", "content": "Bouwens, R. J. et al. 2009, ApJ, 705, 936 Bouwens, R. J., Illingworth, G. D., Franx, M., & Ford, H. 2007, ApJ, 670, 928 Kashlinsky, A., Arendt, R. G., Ashby, M. L. N., Fazio, G. G., Mather, J., & Moseley, S. H. 2012, ApJ, 753, 63 Kashlinsky, A., Arendt, R. G., Mather, J., & Moseley, S. H. 2005, Nature, 438, 45 --. 2007a, ApJ, 654, L5 --. 2007b, ApJ, 654, L1 Kashlinsky, A., Odenwald, S., Mather, J., Skrutskie, M. F., & Cutri, R. M. 2002, ApJ, 579, L53 Komatsu, E. et al. 2011, ApJS, 192, 18 Leitherer, C., Ortiz Ot'alvaro, P. A., Bresolin, F., Kudritzki, R.-P., Lo Faro, B., Pauldrach, A. W. A., Pettini, M., & Rix, S. A. 2010, ApJS, 189, 309 Leitherer, C. et al. 1999, ApJS, 123, 3 Loeb, A., & Rybicki, G. 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AN ANALYTICAL DERIVATION OF THE EMISSIVITY", "content": "At redshift z , the comoving emissivity of stellar population at frequency ν is given by the following integral (Fernandez & Komatsu 2006): where L ν ( m ) is the specific luminosity of a star with mass m , m 1 is the minimum mass of stars while m 2 is the maximum mass, n /star ( m ) is the number density of shining stars between m and m + dm , which is written as where t ( z ) is the age of the universe at redshift z , τ ( m ) is the lifetime of a star with mass m . For Pop II stars with metallicity 1/50 Z /circledot , useful fitting formulae for these quantities as a function of m are collected in Fernandez & Komatsu (2006). Eq. (A2) means that only stars formed between t ( z ) -τ ( m ) and t ( z ) emit photons at time t ( z ). The formation rate of stars with mass between m and m + dm , ˙ n /star ( m,t ' ), is in which f ( m ) is the normalized stellar IMF, i.e., ∫ m 2 m 1 f ( m ) dm = 1 and m /star = ∫ m 2 m 1 mf ( m ) dm , while is the comoving star formation rate density in halos with mass above M min , provided a fraction f /star of baryons are converted into stars. Two approximate solutions can be found under particular circumstances. If the star formation rate is almost constant over the time interval τ ( m ), i.e., τ ( m ) < t SF ( z ), where the star formation time scale t SF ( z ) = [ ˙ ρ /star ( z ) ρ /star ] -1 , then we can make 'Approximation 1', i.e., and the emissivity is approximated as (Fernandez & Komatsu 2006; Fernandez et al. 2010) which is usually used for relative massive stars with short lifetime. On the other hand, if τ ( m ) is longer than the age of the universe (this is true for stars of smaller mass, and means that no stars die), then we can use 'Approximation 2', the emissivity becomes (Fernandez & Komatsu 2006) This also holds true if τ ( m ) is much longer than the star formation time scale t SF ( z ), i.e. the death of stars is less significant compared with the formation of new stars, so that and the emissivity could also be approximated as Eq. (A7). In reality, a galaxy is composed of stars with different mass; some of them may have lifetime longer than t SF , while others not. In this case a 'Hybrid' approximation could be used, where m t is the stellar mass determined by the condition τ ( m t ) = t SF . We compare the full analytical solution of Eqs. (A1-A4) with these three approximate solutions. Furthermore, we will also consider the emissivity obtained by adopting the Starburst99 template at Z = 1 / 50 Z /circledot instead of the simplified fitting formula given in Fernandez & Komatsu (2006). In this case the emissivity is given by where L ν, SB99 is the luminosity per unit mass (note that here for integration purposes we use the burst star formation model) from Starburst99 . In our work, the mass range of Pop II stars is 0 . 1 -100 M /circledot , while the fitted formula of the main sequence age used in Fernandez & Komatsu (2006), Fernandez et al. (2010) and Cooray et al. (2012a) (taken from Schaerer 2002) is based on data of massive stars. To avoid introducing more uncertainties, in this comparison we adopt a mass range 1 -100 M /circledot for Pop II stars. We checked that for Pop II stars with mass 1 M /circledot , the fitted main sequence age still agrees with Girardi et al. (2000). Since Pop II stars are found to contribute much more than Pop III stars to the NIRB (see Figure 3), and stellar emission is the dominant component, we neglect here the nebular emission. L ν can then be represented by a blackbody spectrum, and we truncate it at hν = 13 . 6 eV, (Fernandez & Komatsu 2006; Fernandez et al. 2010; Cooray et al. 2012a). We plot the emissivity at z = 10 calculated by different methods in Figure 8 for a star formation efficiency f /star = 0 . 01 and a minimum mass M min = 10 6 M /circledot . It is not surprising that 'Approximation 2' overestimates the emissivity, since it assumes that none of the stars die. However, we find that for the stellar mass range 1 -100 M /circledot , 'Approximation 1' also overestimates the emissivity when hν < 8 eV, because of the contribution of low mass stars whose lifetime is even longer than the age of the universe at that redshift, so that τ ( m ) < t SF is not fulfilled. However, high energy photons come mainly from massive stars, which satisfy τ ( m ) < t SF . So at high energies, 'Approximation 1' results agree well with the full analytical solution. The 'Hybrid' approximation however, is more accurate through the whole range of energy shown in Figure 8; yet, the results still deviate from the full analytical solution. However, the emissivity calculated from the template of Starburst99 is still higher than the results obtained from the full analytical solution of Eqs. (A1-A4) with fitting formulae. This is mainly due to the full stellar evolutionary tracks used by Starburst99 , which extend beyond the zero-age main sequence (ZAMS) stage. For example, the luminosity of a 7 M /circledot star with metallicity 1/50 Z /circledot at the end of the main sequence is three times as large as the ZAMS luminosity; at the end of its evolution, the luminosity is ∼ 10 × higher compared to ZAMS luminosity. The analogous value for a 100 M /circledot star of the same metallicity, is about 2 × the ZAMS luminosity.", "pages": [ 24, 25, 27 ] } ]

[ { "title": "ABSTRACT", "content": "We estimate the maximum temperature at which planets can form via gravitational instability (GI) in the outskirts of early circumstellar disks. We show that due to the temperature floor set by the cosmic microwave background, there is a maximum distance from their host stars beyond which gas giants cannot form via GI, which decreases with their present-day age. Furthermore, we show that planet formation via GI is not possible at metallicities < ∼ 10 -4 Z /circledot , due to the reduced cooling efficiency of low-metallicity gas. This critical metallicity for planet formation via GI implies a minimum distance from their host stars of ∼ 6 AU within which planets cannot form via GI; at higher metallicity, this minimum distance can be significantly larger, out to several tens of AU. We show that these maximum and minimum distances significantly constrain the number of observed planets to date that are likely to have formed via GI at their present locations. That said, the critical metallicity we find for GI is well below that for core accretion to operate; thus, the first planets may have formed via GI, although only within a narrow region of their host circumstellar disks. Key words: Planets and satellites: formation - Cosmology: theory", "pages": [ 1 ] }, { "title": "Jarrett L. Johnson /star and Hui Li", "content": "Los Alamos National Laboratory, Los Alamos, NM 87545, USA Nuclear and Particle Physics, Astrophysics and Cosmology Group (T-2) 28 August 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "When did the first planets form and what were their properties? The answers to these questions depend critically on the process by which the first planets formed. There are two main mechanisms of planet formation that are widely discussed (e.g. Papaloizou & Terquem 2006; Youdin & Kenyon 2012): core accretion, in which dust coagulates into larger and larger bodies which become the cores of planets (e.g. Pollack et al. 1996); and gravitational instability (GI), in which the self-gravity of a circumstellar disk triggers the fragmentation and collapse of gas into a gas giant planet directly (e.g. Boss 1997). In previous work, we have discussed the formation of the earliest planets via core accretion (see also Shchekinov et al. 2012 for related calculations). In this scenario, we estimated the minimum, or 'critical', metallicity to which the gas must be enriched before planet formation can begin (Johnson & Li 2012; hereafter Paper I). We found that the critical metallicity is a function of the distance from the host star, and that the minimum metallicity for the formation of Earth-like planets is likely to be ∼ 0.1 Z /circledot . Furthermore, we were able to show that our prediction of the critical metallicity was consistent with the data that were available on planetary systems, with no planets lying in the 'Forbidden zone' in which their metallicities imply formation times longer than the time available (i.e. the disk lifetime). Recently, however, Setiawan et al. (2012) have announced the discovery of a planet orbiting a star with a very low iron abundance of [Fe/H] /similarequal -2 (HIP 11952b) which appears to lie in the Forbidden zone for planet formation via core accretion (if this claim proves correct; see Desidera et al. 2013 in prep). This implies that this metal-poor planet likely formed via some other process, such as GI. † Furthermore, if it is true that planets can form via GI at metallicities below the critical value for core accretion, then the first planets to form may have been gas giants formed via GI. Terrestrial planet formation (and perhaps the emergence of life), which likely must instead occur via core accretion, ‡ may only occur at later stages of cosmic history when the process of metal enrichment has progressed further. This then raises † Alternatively, this planet may have initially formed closer to its host star via core accretion and then migrated outward. ‡ Gas giants can be formed via GI, as this process involves the gravitational collapse of gas fragments; terrestrial planets, however, are by definition not gas-dominated and so likely form via some other mechanism, such as core accretion (but see Boley et al. 2010; Nayakshin 2010 on how GI may yield terrestrial planets). the question of, instead of the critical metallicity for planet formation via core accretion, what conditions must be satisfied for the first planets to form via GI. Here we address two aspects of this question, namely the impacts of the cosmic microwave background (CMB) and of the reduced cooling efficiency of low-metallicity gas in regulating planet formation via GI. In the next Section, we review the conditions required for the formation of planets via GI. In Section 3, we consider the constraints placed on this model due to the temperature floor imposed at high redshifts by the CMB, and in Section 4 we estimate the minimum metallicity necessary for planet formation via GI. In Section 5 we compare our predictions for when the first planets form via GI to the available observational data. We give our conclusions in Section 6.", "pages": [ 1, 2 ] }, { "title": "2 CONDITIONS FOR PLANET FORMATION VIA GRAVITATIONAL INSTABILITY", "content": "We begin by reviewing the conditions required for GI in a circumstellar disk, which we will use to derive constraints on where planets can form in early disks via GI. The first condition for fragmentation to occur in a thin disk is that Q < ∼ 1, where (e.g. Toomre 1964; Boss 1998) § Here c s is the sound speed of a gas at temperature T with mean molecular weight µ = 2 (close to the value for fully molecular gas) and adiabatic index γ = 2, Ω is the Keplerian angular velocity (which here we assume is identical to the epicyclic frequency), Σ is the surface density of the disk, and r is the distance from the central host star. In the second equality we have assumed a disk temperature normalized to T = 100 K at r = 10 astronomical units (AU), and a disk surface density normalized to Σ = 10 2 g cm -2 at r = 10 AU. Finally, we have normalized the above formula to a central stellar mass of m ∗ = 1 M /circledot . According to equation (1), fragmentation is only possible in sufficiently dense and/or cold disks, and/or far out from the host star, especially if it is massive. We shall use this condition in Section 3 to derive constraints on planet formation via GI due to the temperature floor set by the CMB. The second requirement for planet formation via GI is that a circumstellar disk also cools sufficiently fast (see e.g. Gammie 2001; Nayakshin 2006; Levin 2007). Indeed, it is disk cooling which is thought to drive disks to Q /similarequal 1 (e.g. Goodman 2003; Thompson et al. 2005). We shall use this condition, along with the requirement that Q /similarequal 1, in Section 4 to explore the limits on first planet formation via GI due to the limited cooling efficiency of low-metallicity gas. § Instabilities leading to fragmentation can occur even for Q /similarequal 1.4 - 1.7 (e.g. Mayer et al. 2004; Durisen et al. 2007), but adopting such slightly higher values would not affect our results strongly.", "pages": [ 2 ] }, { "title": "3 CONSTRAINTS FROM THE CMB", "content": "Here we estimate the maximum temperature at which planet-mass fragments may form and we then use this to derive the limits on planet formation via GI due to the temperature floor set by the CMB.", "pages": [ 2 ] }, { "title": "3.1 Minimum Planet Mass", "content": "Here we define the minimum mass of a planet formed via GI, as a function of the properties of the disk. We follow Kratter et al. (2010; see also Rafikov 2005; Levin 2007; Cossins et al. 2009; Forgan & Rice 2011) who estimate the initial mass of fragments formed via GI (in a disk with Q /similarequal 1) as that on the scale of the most unstable wavelength. This yields the following for the minimum planet mass (in units of the mass M J of Jupiter): Simulations of disk fragmentation via GI also suggest that this is a reasonable estimate for the minimum mass of planets (e.g. Boley 2009; Stamatellos & Whitworth 2009). Indeed, given that this is the initial mass scale of fragments, it is very likely that the final mass they achieve via the continued accretion of gas will be much higher than this value. As noted by Kratter et al. (2010), if such fragments accreted enough gas to attain their isolation mass they will have greatly overshot the planet mass range and may end up instead as e.g. brown dwarfs. As noted by these authors, it appears that some mechanism must halt accretion in order for planet-mass objects to survive (see also e.g. D'Angelo et al. 2010; Boss 2011). One possibility is that the circumstellar disk is photoevaporated (e.g. Gorti & Hollenbach 2009; Ercolano & Clarke 2010) or otherwise disappears (e.g. Melis et al. 2012) before accretion to super-planet mass scales occurs. As we shall show next, our adoption of the minimum planet mass allows to estimate a maximum disk temperature at which planets may form via GI.", "pages": [ 2 ] }, { "title": "3.2 Maximum Disk Temperature for Planet Formation", "content": "To ensure that fragments which arise in the disk are not too large to be classified as planets we must have m min < ∼ 13 M J , which is the commonly adopted upper mass limit for planets - above this mass deuterium burning occurs and we assume the object to be a brown dwarf. We can combine equation (2) with Q = 1 in equation (1) to obtain the maximum disk temperature T max ( r ) from which a planet of mass m min can form from fragmentation of the disk: where we have normalized to the maximum planet mass of m min = 13 M J . If the temperature of the disk exceeds this value, then planet formation via GI may be impossible, either because fragmentation is suppressed (see equation 1), or if fragmentation occurs the fragment(s) form with super-planetary masses (becoming e.g. brown dwarfs instead of planets; see equation 2). While temperatures below this maximum value may be necessary for planet formation (and are found in simulations including radiative cooling; e.g. Nelson et al. 2000; Mej'ıa et al. 2005; Boley et al. 2006; Forgan et al. 2011), they are not alone sufficient. In addition, the surface density of the disk must also be high enough that Q < ∼ 1 (equation 1), and in the case of non-isothermal disks the cooling criterion of Gammie (2001) must also be satisfied. ¶", "pages": [ 2, 3 ] }, { "title": "3.3 The CMB temperature floor", "content": "It is well-known that gas cannot cool radiatively to temperatures lower than that of the CMB, given by T CMB = 2.73 K (1+ z ), where z is redshift. Therefore, planet formation will not be possible if it requires that the disk cools below T CMB . ‖ Following equation (3), this implies that at high redshifts planet formation can only occur relatively close to the host star. We can express this maximum radius r max for planet formation, as a function of the host star mass m ∗ and redshift z , by equating T CMB to the maximum disk temperature for planet formation given by equation (3). This yields Therefore, even for the highest mass stars that may have survived to the present day ( /similarequal 0.8 M /circledot ), planets can only form inside r < ∼ 40 AU at z > ∼ 10, roughly the epoch of the first metal-enriched star formation in the earliest galaxies (e.g. Bromm & Yoshida 2011). Thus, we would expect such old planets formed via GI to be found on relatively tight orbits around old, low-mass, metal-poor stars. In the next Section, we estimate how tight these orbits can be, given the limited cooling properties of metal-poor gas.", "pages": [ 3 ] }, { "title": "4 CONSTRAINTS AT LOW METALLICITY", "content": "To explore the effect of metallicity on the fragmentation properties of circumstellar disks, we impose the two conditions required for planet formation via GI described in Section 2. We follow the common approach of estimating the cooling rate of the disk based on its opacity (e.g. Rafikov 2005; Levin 2007; Kratter et al. 2010), which we assume to be proportional to the metallicity of the disk. Specifically, we follow exactly the calculation presented by Levin (2007), for four different metallicities: Z = 10 -6 , 10 -4 , 10 -2 , and 1 Z /circledot . We use the opacities for solar metallicity gas κ (Z /circledot ) given ¶ We note that the cooling criterion has been shown to always be met for isothermal disks and disks subjected to external irradiation (Kratter & Murray-Clay 2011). by equation (9) of Levin (2007; from Bell & Lin 1994), and we scale them with metallicity, such that κ ( Z ) = κ ( Z /circledot ) × ( Z /Z /circledot ). From this, we solve for the critical surface density Σ crit and temperature T crit at which the effective viscosity of the disk reaches the critical value of α crit = 0.3 (Gammie 2001) and the disk fragments. These are shown in Figure 1, for the various metallicities we consider. We then use these critical values for the surface density and temperature in equation (2) to find the minimum mass m min of fragments formed. The values we find for m min , as functions of the mass m ∗ of and distance r from the host star, are shown in Figure 2. For the case of solar metallicity, we successfully reproduce the results presented by Levin (2007), as expected. /star/star For the lower metallicity cases, the effect of the reduced opacity of the disk is that the critical temperature T crit and surface density Σ crit are significantly higher (lower) at large (small) radii, as shown in Fig. 1. In turn, this translates into larger minimum fragment masses at smaller radii, at lower metallicities, as shown in Fig. 2. At solar metallicity, our result for the minimum fragment mass is similar to that found by e.g. Rafikov (2005), and our result that fragmen- ation into planet-mass objects is still possible even at 10 -2 Z /circledot is similar to that found by Meru & Bate (2010), assuming a metallicity dependent opacity similar to what we have adopted (see also e.g. Boss 2002; Cai et al. 2006; Helled & Bodenheimer 2011 on the susceptibility of low-metallicity disks to GI) . Our calculation is also broadly consistent with previous work which has shown that planet formation via GI is difficult to achieve within tens of AU at metallicities near the solar value (Stamatellos & Whitworth 2008; Clarke & Lodato 2009; Rice & Armitage 2009; Rogers & Wadsley 2011; Kimura & Tsuribe 2012; Vazan & Helled 2012), as discussed by e.g. Boley et al. (2009) and Boss (2012),. At metallicities Z < ∼ 10 -4 Z /circledot there exists no region of the disk where m min < ∼ 13 M J , the maximum planet mass; therefore, we interpret this to be the critical metallicity for planet formation via GI. This critical metallicity for GI is well below that for core accretion (Paper I), which implies that the first planets may well have formed via GI in very low-metallicity circumstellar disks in the early universe. We note furthermore that this critical metallicity is below estimates of the metallicity to which the primordial gas is enriched by the first supernovae ( /similarequal 10 -3 Z /circledot ; e.g. Wise & Abel 2008; Greif et al. 2010), and this suggests that the first planets may have even formed around second generation stars via GI. As shown in Fig. 2, at succesively lower metallicity, the minimum fragment mass drops into the planetary regime at smaller radii. Thus, the critical metallicity also implies a minimum distance r min from their host stars at which planets can form via GI. From Fig. 2, we see that at the critical metallicity of /similarequal 10 -4 Z /circledot , this minimum distance is r min /similarequal 6 AU, for a m ∗ /similarequal 1 M /circledot host star. The exact value of r min changes slightly for different host stellar masses, with r min ∝ m 1 / 3 ∗ as shown on the x-axis in Fig. 2. At separations smaller than the r min shown in Fig. 2, the minimum fragment mass is super-planetary, regardless of the metallicity of the gas. We plot this value for r min in Figure 3 to show how this minimum distance and the r max we found in Section 3 bracket a region in which planet formation via GI is possible. We refer to the region inside r min as the 'metal cooling-prohibited' region, as it is ultimately the limited cooling efficiency of the gas at low metallicities that sets the critical metallicity corresponding to r min . From Fig. 3, we see that just ∼ 100 Myr after the Big Bang the CMB temperature drops to values low enough to make r min ≤ r max , and so for planet formation via GI to proceed. Thereafter, planets can form via GI in larger regions of their host circumstellar disks at later cosmic times. Another key point that is evident from Fig. 2 is that at higher metallicity r min becomes larger, further constraining the radii in metal-enriched circumstellar disks in which planet formation via GI is possible. In particular, at Z > ∼ 0.1 Z /circledot we find that r min > ∼ 50 AU, which is larger than r max set by the CMB at < ∼ 2 Gyr after the Big Bang (or at z > ∼ 3), as shown in Fig. 3. This suggests that planet formation via GI may be difficult to achieve at metallicities higher than just ∼ 10 percent of the solar value at these early times. While in Fig. 3 we present our results for the CMBprohibited regime and the metal cooling-prohibited regime independently, in principle they are most realistically considered together, since at high redshift the metallicity of circumstellar disks is likely lower and the CMB temperature is higher at the same time . It is for simplicity that we have treated these effects separately, in part because the spatially inhomogeneous nature of metal enrichment implies that there is no clear one-to-one mapping between redshift (or T CMB ) and metallicity. That said, we emphasize that selfconsistently including the effect of background irradiation in our calculations could result in somewhat larger fragments (see e.g. Levin 2007; Forgan & Rice 2013), which in turn would raise the critical metallicity that we find in Fig. 2. As shown in Fig. 3, at early times the temperature of the CMB may indeed be high enough to effect such a change.", "pages": [ 3, 4 ] }, { "title": "5 COMPARISON WITH DATA", "content": "Here we compare our theoretical predictions of r max set by the CMB and r min set by the critical metallicity for GI with the star-planet separations inferred from observations. This allows to test whether GI is a viable explanation for the formation of the oldest known planets. In Fig. 3, we make this comparison, plotting the semimajor axes and host stellar ages of planets compiled in Wright et al. (2011) †† with host stars having sub-solar iron abundance ([Fe/H] < 0), which we take as an indicator of old age. We have also included the four gas giants in our Solar System, as well as the metalpoor planetary systems reported by Sigurdsson et al. (2003) and Setiawan et al. (2012), and the wide orbit planets reported by Chauvin et al. (2004), Marois et al. (2008) and Lagrange et al. (2010). Here we have taken the Sigurdsson et al. (2003) host stellar age to be that of the globular cluster in which it was found, and the semimajor axis is taken to be its original one inferred from the modeling done by these authors. Also shown in Fig. 3 are the maximum radii of formation r max for planets orbiting stars of three different masses: 0.1, 0.3 and 1 M /circledot , following equation (4) with the maximum planet mass of m min = 13 M J . ‡‡ To facilitate the comparison with the ages of the observed planetary systems, we have converted from redshift z (in which r max is expressed in this equation) to the time elapsed since redshift z , following the formulae describing Hubble expansion in the standard ΛCMD cosmology presented in e.g. Barkana & Loeb (2001) and assuming a flat universe with the following cosmological paramenters: H 0 = 70.3 km s -1 Mpc -1 , Ω Λ = 0.73 and Ω M = 0.27 (Komatsu et al. 2011). We expect the CMB temperature floor to suppress the formation of planet-mass objects at radii > ∼ r max , in the upper shaded region of Fig. 3. We term this the 'CMB-prohibited' zone. ‡‡ We have chosen to plot the curves for this single maximal planet mass m min , since most of the data imply only a lower limit to their mass, meaning that such a high mass can not in general be ruled out. We emphasize, however, that the region in which planet formation is suppressed is larger for planets with lower masses (see equation 4 and Fig. 2). All of the planets shown in Fig. 3 appear to lie at radii much smaller than r max , in part because most are relatively young (e.g. < ∼ 10 Gyr old) and formed at times when the temperature of the CMB temperature was low. We also note that Boss (2011) argues that the formation of wide orbit gas giants, such as those shown at > ∼ 20 AU, may be best explained by GI, especially if they are formed around relatively massive stars, consistent with the curves in Fig. 3. There are additional candidate planets with very wide orbits that are not included in Fig. 1. These candidates, reported by Kalas et al. (2008) and Lafreni'ere et al. (2008; 2010), respectively, would lie at /similarequal 115 AU and /similarequal 150 AU from their host stars, which have masses of /similarequal 1.9 M /circledot and /similarequal 1 M /circledot , and ages of just /similarequal 0.4 Gyr and /similarequal 5 Myr (see also B'ejar et al. 2008; Bowler et al. 2011; and Ireland et al. 2011 for other very wide orbit /similarequal 14 M J companions). If veritable planets, they would lie just outside the CMBprohibited zone and so may have formed via GI at their present locations. Alternatively, they could have formed at smaller radii and migrated outward (e.g. Veras et al. 2009; but see Dodson-Robinson et al. 2009; Bowler et al. 2011) or originated as free-floating planets (Perets & Kouwenhoven 2012; Strigari et al. 2012). §§ While the planets shown in Fig. 3 lie well below the CMB-prohibited zone, there are only a few planets that are outside the metal cooling-prohibited zone, at r > ∼ 6 AU. Thus, unless they migrated inward from larger radii, it appears that there are only a handful of known planets that could have formed via GI. In particular, this is the case for the planets reported by Setiawan et al. (2012). While they orbit a star with [Fe/H] /similarequal -2, suggesting that they formed from gas with metallicity well above the critical metallicity for GI, they lie at r < ∼ 0.81 AU, well within r min /similarequal 6 AU. Importantly, however, given the old age of /similarequal 12.8 Gyr inferred for this planetary system, if there is indeed a larger r min of ∼ 25 AU for circumstellar disks at this metallicity (and for the mass m ∗ /similarequal 0.8 M /circledot inferred for its host star), as suggested by Fig. 2, then this would pose a strong challenge to GI as an explanation even in this case. Finally, we note that it has been suggested that planets currently on relatively tight orbits around their host stars may have formed from the collapse of significantly more massive (perhaps super-planetary) fragments at larger radii, which then migrated inward and lost mass due to tidal shear or stellar irradiation (Nayakshin 2010). If this process is indeed at play, then it is possible that some of the planets in Fig. 3 may have originated from GI, despite residing in the metal cooling-prohibited zone today. §§ Migration and/or capture by the host star are important caveats to consider with regard to conclusions drawn from comparison with data, which only reflect where the planets orbit their host stars today. In particular, we note that inward migration is especially likely for planets formed via GI (e.g. Baruteau et al. 2011), which could potentially place some of the planets in Fig. 1 in the CMB-prohibited zone at their formation, or place some of those currently within r min in between r min and r max at their formation.", "pages": [ 4, 5 ] }, { "title": "6 CONCLUSIONS", "content": "As an alternative to the core accretion model for the formation of the first planets (discussed in Paper I), we have considered here the formation of the earliest planets via GI. We have argued that there is a maximum circumstellar disk temperature only below which can planets form via GI. In turn, this implies a maximum distance from their host stars at which planets can form via GI due to the temperature floor set by the CMB. As the CMB temperature is higher at earlier times, planets may only form via GI at distances from their host stars which decerease with their present-day age. We have furthermore estimated the minimum metallicity required for the fragmentation of circumstellar disks into planet mass objects. We find that this critical metallicity for GI is Z crit /similarequal 10 -4 Z /circledot , well below that for core accretion. In turn, because planet formation via GI is possible at smaller distances from the host star at lower metallicities, this critical metallicity implies a minimum distance of a few AU at which planets can form via GI. These two limits together imply that, while planet formation via GI can take place at metallicities below those required for core accretion, it can only occur at metallicities > ∼ 0.1 Z /circledot at times > ∼ 2 Gyr after the Big Bang. In particular, this does not rule out that the first planets in the Universe may indeed have formed via GI at metallicities 10 -4 < ∼ Z < ∼ 10 -1 Z /circledot during the epoch of the first galaxies, ∼ 500 Myr after the Big Bang (e.g. Bromm & Yoshida 2011). That said, we find that there are only a handful of known planets which lie within the bounds of the metal cooling- and CMB-prohibited zones in which planets can form via GI. It may be, however, that some known planets could have migrated inward from their formation sites outside the metal cooling-prohibited zone. This may explain, in particular, the existence of the very low-metallicity planets reported by Setiawan et al. (2012), the formation of which is otherwise difficult to explain in the core accretion model.", "pages": [ 6 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "This work was supported by the U.S. Department of Energy through the LANL/LDRD Program. JLJ gratefully acknowledges the support of a Director's Postdoctoral Fellowship at Los Alamos National Laboratory. The authors thank the reviewers for constructive and cordial reports, as well as for encouraging us to explore the impact of low metallicity on planet formation via GI as we have done in Section 4.", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Barkana, R., Loeb, A. 2001, PhR, 349, 125 Baruteau, C., et al. 2011, MNRAS, 416, 1971 B´ejar, V. J. S., et al. 2008, ApJ, 673, 185 Bell, K. R., Lin, D. N. C. 1994, ApJ, 427, 987 Boley, A. C., et al. 2006, ApJ, 651, 517 Boley, A. C. 2009, ApJ, 695, L53 Boley, A. C., et al. 2010, Icar, 207, 509 Boss, A. P. 1997, Sci, 276, 1836 Boss, A. P. 1998, ApJ, 503, 923 Boss, A. 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[ { "title": "ABSTRACT", "content": "Spectroscopic analysis of the Galactic halo star SDSS J102915+172927 has shown it to have a very low heavy element abundance, Z < 7.4×10 -7 , with [Fe/H] = -4.89 - 0.10 and an upper limit on the C abundance of [C/H] < -4.5. The low C/Fe ratio distinguishes this object from most other extremely metal poor stars. The effective temperature and surface gravity have been determined to be Teff = 5811 - 150 K and log g = 4.0 - 0.5. The surface gravity estimate is problematical in that it places the star between the main sequence and the subgiants in the Hertzsprung-Russell diagram. If it is assumed that the star is on the main sequence, its mass and are estimated to be M = 0.72 ± 0.06 M /g1622 and L = 0.45 ± 0.10 L /g1622 , placing it at a distance of 1.35 ± 0.16 kpc. The upper limit on the lithium abundance, A (Li) < 0.9, is inconsistent with the star being a dwarf, assuming that mixing is due only to convection. In this paper, we propose that SJ102915 is a sub-giant that formed with significantly higher Z than currently observed, in agreement with theoretical predictions for the minimum C and/or O abundances needed for low mass star formation. In this scenario, extremely low Z and low Li abundance result from gravitational settling on the main sequence followed by incomplete convective dredge-up during subgiant evolution. The observed Fe abundance requires the initial Fe abundance to be enhanced compared to C and O, which we interpret as formation of SJ102915 occurring in the vicinity of a type Ia supernova. Key words: stars: evolution; stars: population III; stars: individual (SDSS J102915+172927)", "pages": [ 1 ] }, { "title": "J. MacDonald 1 , T.M. Lawlor 2 , N. Anilmis 1 and N.F. Rufo 2", "content": "1 University of Delaware, Department of Physics and Astronomy, Newark, DE 19716, USA 2 Pennsylvania State University, Brandywine Campus, Department of Physics, Media, PA 19063 USA accepted; received;", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Theoretical studies (Bromm & Loeb 2003; Schneider et al. 2003) indicate that low-mass stars cannot form until the mass fraction Z of heavy elements in the interstellar medium has been enriched to a critical value estimated to lie in the range 1.5×10 -8 to 1.5×10 -6 . Bromm & Loeb (2003) argue that the abundances of carbon and oxygen are the crucial factor in determining whether only massive stars can form (as in stellar population III) or both massive and low-mass stars can form (as in stellar populations II and I). In this scenario, the fine structure lines of ionized carbon and neutral oxygen provide efficient cooling of the protostellar clouds in the primitive interstellar medium. Frebel, Johnson, & Bromm (2007) define an 'observer friendly' transition discriminant [ ] [ ] ( ) C H O H 10 log 10 0.3 10 D = + × such that low-mass star formation is possible only if D > -3.5. Here the notation, [X/H] means log of the star's abundance ratio relative to the solar abundance ratio. Frebel et al (2007) make a prediction that any star with [Fe/H] /lessequivlnt - 4 will have enhanced C and/or O abundances. Support for this prediction comes from the discovery of stars with very low Fe abundances and relatively high C and/or O abundances such as HE 0107-5240 and HE1327-2326 (Christlieb et al. 2002). However, the very recent discovery of SDSS J102915+172927 (hereafter SJ102915), which is not carbon enhanced casts doubt on this picture. Caffau et al. (2012, hereafter Caf12) report that SJ102915 has [Fe/H] = - 4.89 - 0.10 and has no measurable enhancement of carbon or nitrogen. They estimate that Z /lessequivlnt 7.4×10 -7 . Another unusual feature of SJ102915 is the complete absence of the neutral lithium resonance doublet, a feature that is all but constant in other metal poor dwarf stars (Caffau et al. 2011). In this paper, we use our stellar evolution models to compare the observed properties of J102915 with main sequence and subgiant model predictions. In Section 2 we briefly describe our stellar evolution code, BRAHMA. In section 3, we consider models in which the effects of gravitational settling and element diffusion are neglected. We also discuss here the 'lithium problem'. In section 4, we show that including the effects of gravitational settling and element diffusion resolves the lithium problem provided that SJ102915 is a subgiant star. Our conclusions and discussion are given in section 5.", "pages": [ 1, 2 ] }, { "title": "2 THE EVOLUTION CODE", "content": "Here we briefly describe our evolution code, BRAHMA (Mullan & MacDonald 2010; Lawlor et al. 2008; Lawlor & MacDonald 2006). The code uses a relaxation method to simultaneously solve the stellar structure equations along with adaptive mesh and composition equations for the star as a whole. OPAL opacities (Iglesias & Rogers 1996) are used for temperatures greater than 6000 K with a smooth transition to the Ferguson et al. (2005) opacities at lower temperatures. Interpolation in the opacity tables is handled by using the subroutines of Arnold Boothroyd (which are obtainable from http://www.cita.utoronto.ca/~boothroy/kappa.html) . Convective energy transport is treated by mixing length theory as described by Mihalas (1978), which is the same as that of Böhm-Vitense (1958) but modified to include a correction to radiative losses from convective elements when they are optically thin. The nuclear reaction network explicitly follows the evolution of the isotopes 1 H, 2 H, 3 He, 4 He, 7 Li, 7 Be, 12 C, 13 C, 14 N, 16 O, 20 Ne, 24 Mg, 28 Si and 56 Fe. All the nuclear reaction rates relevant to hydrogen burning are from Angulo et al. (1999), except for 14 N(p, γ ) 15 O (Herwig & Austin 2004). The electron screening enhancement factor for a nuclear reaction is taken to be the lowest of the weak screening (Salpeter 1954), intermediate screening (Graboske et al. 1973) and strong screening factors (Itoh et al 1979; Itoh et al 1990). Composition changes due to convective mixing are treated by adding diffusion terms to the composition equations, with the diffusion coefficient consistent with mixing length theory. For the calculations in which we include composition changes due to element diffusion and gravitational settling, the diffusion velocities for all 14 species in the nuclear reaction network are calculated by numerically solving the multicomponent flow equations derived by Burgers (1969) and summarized by Muchmore (1984). Our earlier use of this approach for modeling element diffusion processes in white dwarf stars can be found in Iben & MacDonald (1985), Iben, Fujimoto & MacDonald (1992), MacDonald, Hernanz & Jose (1998). More recently, we have used the Burger's formulation in modeling the Sun (Mullan, MacDonald & Townsend 2007). We have not included radiative levitation in our diffusion calculations. Even at the relative low luminosities of the low mass stars considered here, radiative levitation of elements with very low abundances may be important. Seaton (1997) has addressed calculation of the radiative accelerations using OP data for a number of elements. From his figure 6, the maximum value of radiative acceleration on Fe at temperature 2×10 5 K at 1/100 the solar abundance is determined to be about 4000 times the radiative acceleration of free electrons. The radiative acceleration of Fe will be larger still at lower abundances. Exploratory calculations that include radiative of Fe are presented in appendix A.", "pages": [ 2, 3 ] }, { "title": "3 MODELS WITHOUT ELEMENT DIFFUSION OR GRAVITATIONAL SETTLING", "content": "We have evolved models of initial composition X = 0.765, Y = 0.235 and Z = 1.86×10 -6 (which corresponds to 10 -4 times the solar system heavy element abundance), for masses from M = 0.50 M /g1622 to 0.90 M /g1622/g481/g3/g139/g144/g3/g139/g144/g133/g148/g135/g143/g135/g144/g150/g149/g3/g145/g136/g3/g882/g484/g882/g887/g3 M /g1622 . We use our solar calibrated mixing length ratio α = 1.70. All models are evolved from the pre-main sequence to the age of the Universe (1.37×10 10 yr) or, for the more massive models, to the onset of the helium core flash. We do not consider masses higher than 0.90 M /g1622 for two reasons: 1) For more massive stars of this very low Z , the evolutionary paths in the log Teff - log g diagram get further away from the observations, and 2) more massive stars reach the helium flash in less than 10 10 years. Since the oldest globular clusters have age > 12 Gyr and Z ≥ 0.0002 (Jimenez et al. 1996; Salaris, degl'Innocenti & Weiss, 1997), it seems improbable that a star could form 10 10 years ago with Z ≈ 10 -6 . For comparison with the observational data, we give in Figure 1 evolutionary tracks in the log Teff - log g diagram. The corresponding Hertzsprung-Russell diagram is shown in figure 2. Because of the low heavy element abundance, we dismiss the possibility that SJ102915 is in the short-lived pre-main sequence phase of evolution. Figure 1 shows that main sequence models do not provide a good match to the surface gravity. However, sub-giant models are only marginally better. If we ignore the gravity estimate, inspection of figure 1 shows that the effective temperature constrains the mass to the range 0.66 - 0.78 M /g1622 , assuming that SJ102915 is a dwarf star. If we further assume that SJ102915 is old, e.g. has an age of 12 Gyr, then its mass is constrained to be 0.67 - 0.71 M /g1622 , in agreement with the mass found by Caf12 using unpublished Chieffi & Limongi models. As can be seen from figure 2, the corresponding luminosity range for our models is L = 0.35 - 0.54 L /g1622 ( Mbol = 5.41 - 5.88). To determine the predicted distance, we first estimate the apparent bolometric magnitude of SJ102915 by using the transformations between the Johnson-Cousins UBVRI and SDSS ugriz systems given by Jordi, Grebel & Ammon (2006). For g = 16.922 ± 0.004 and r = 16.542 ± 0.004, we obtain V = 16.686 ± 0.008. We determine the bolometric correction (BC) by using results from the NextGen atmosphere models (Hauschildt, Allard & Baron 1999). Using the tables given on the web site of France Allard (http://phoenix.ens-lyon.fr/Grids/NextGen/), we obtain for the main sequence model, BC = -0.26 ± 0.01. After correcting for extinction, we obtain mbol = 16.37 ± 0.02. The resulting distance is 1.19 - 1.51 kpc. Alternatively, if we assume that SJ102915 is a subgiant, then the spectroscopic Teff constrains the mass to be greater than 0.815 M /g1622 . For an assumed age of 12 Gyr, the mass must be close to 0.845 M /g1622 . Using a bolometric correction appropriate to a subgiant, the range in model luminosity, L = 8.7 - 9.7 L /g1622 ( Mbol = 2.27 - 2.39), implies a distance of 6.0 - 6.4 kpc. We note that one uncertain factor that may improve the log g fit is the mixing length ratio. We have also calculated models with α = 1.00 but find that the fit is not improved and the log g problem remains. Also our adopted helium abundance, from Peimbert, Peimbert & Ruiz (2000). is lower than more recent determinations of the primordial helium abundance, Yp = 0.2477 ± 0.0029 (Peimbert, Luridiana & Peimbert 2007), Yp = 0.2565 ± 0.0060 (Izotov & Thuan 2010). Models calculated with Y = 0.250 are in general hotter at a given log g than those for Y = 0.235. At a given Teff , the trend with increasing Y is that log g increases on the main sequence but decreases on the subgiant branch. Hence a higher initial Y value makes the fit between observations and theory poorer in both the main sequence and subgiant scenarios. The shift in log g on the subgiant branch is of order 0.04, which is small compared to the uncertainty in the observationally determined value of log g .", "pages": [ 3, 5, 6 ] }, { "title": "3.1 The lithium problem", "content": "In terms of A (Li) = 12 + log( N (Li) /N (H)), Caf12 find an upper limit on the lithium abundance of A (Li) < 0.9, far below the Spite 'plateau' value of A (Li) ≈ 2.2 (Spite & Spite 1982) and the big bang nucleosynthesis value, A (Li) = 2.72 /g115 0.05 (Cyburt et al. 2008). Figure 3 shows the time evolution of the 7 Li abundance for our models of stars in the mass range 0.50 - 0.80 M /g1622 . We see that the observed lithium depletion is inconsistent with the range in mass determined from Teff under the assumption that SJ102915 is a dwarf star. Figure 4 shows the lithium abundance evolution for models that reach the sub-giant phase of evolution. Although some reduction of the lithium abundance does occur during the dredge-up phase, it mainly occurs after Teff is below the lower limit found for SJ102915. Even then the lithium abundance is higher than the spectroscopic upper level. From this we conclude that standard stellar evolution models for SJ102915 that only include convective mixing are inconsistent with the observed 7 Li abundance limit. This suggests that other physical processes beyond standard convection are required.", "pages": [ 6, 7, 8 ] }, { "title": "4 MODELS WITH ELEMENT DIFFUSION AND GRAVITATIONAL SETTLING", "content": "Models of low mass main sequence stars with extremely low heavy element abundance have higher surface gravity than models of population I stars of the same mass. As a consequence they are hotter and have shallower surface convection zones. The higher gravity and shallower convection leads to faster settling of heavy elements out of the surface layers, and so gravitational settling will be able to modify the surface composition for main sequence stars of lower mass than for Pop I composition stars (assuming that there are no competing mechanisms other than convective mixing). We illustrate this point by showing in figure 5 the evolution of the photospheric carbon abundance for 0.8 M /g1622 models of different initial Z . /g491 /g3 We see that gravitational settling can reduce the 12 C abundance below the upper limit found by Caf12 for initial abundances as high as 10 -3 Z /g1622 . The effects of gravitational settling are even more pronounced at higher mass, but become negligible at lower mass because of the deeper surface convection zones. Since the subgiant model requires the mass to be greater than 0.8 M /g1622 , we consider the possibility that the initial heavy element abundances of SJ102915 were greater than what is observed today, and gravitational settling is responsible for reducing them to the observed levels. We first explore the constraints imposed by the 7 Li upper limit. In figure 6, we show how the Li abundance changes with Teff for 0.85 M /g1622 models of initial heavy element abundance 10 -4 and 10 -3 Z /g1622 for mixing length ratios α = 1.3 and 1.7. The general trend of the photospheric 7 Li abundance is that it decreases to very low values during the main sequence phase of evolution due to gravitational settling. The 7 Li abundance recovers to almost its initial value during the subgiant phase due to convective dredge-up, before it becomes slightly depleted during the red giant phase due to proton captures at the base of the surface convection zone. It is clear from figure 6 that the lithium abundance on the subgiant phase is sensitive to the mixing length ratio. For α = 1.7, the model 7 Li abundances at Teff values consistent with those inferred for SJ102915 are higher than the upper limit found by Caf12. In contrast, the model 7 Li abundances for α = 1.3 are consistent with the upper limit. Hence subgiant models that are consistent with the Li abundance can be found by reducing the mixing length ratio below the solar calibrated value. For the α = 1.3 models, the log g values on the part of the subgiant phase within the observe Teff limits are between 3.5 and 3.6, which are also consistent with observations. We now consider the constraints imposed by the other abundance determinations. Our approach is to determine the depletion with time or Teff of a species relative to its initial abundance. We then use the observed abundances of SJ102915 to constrain its initial abundances. Figure 7 shows the degree of depletion of the heavy elements for a 0.85 M /g1622 model of initial heavy element abundance 10 -3 Z /g1622 and mixing length ratio α = 1.3. We see that the degree of depletion during the subgiant phase for the spectroscopically determined temperature range is sensitive to the value of Teff . We also see that the depletions of Fe, Mg, and Si are greater than for C, N, and O. For this particular model, the Li abundance upper limit requires that Teff > 5740 K. The degree of depletions for Si and Fe require that their minimum initial abundances consistent with the observed abundance must be [Si/H] = - 1.49 and [Fe/H] = - 0.98. Required lower limits on initial abundances are given in Table 1. a values in italics assume that current abundances are equal to the upper bounds given by Caffau et al. (2012) The results in table 1 indicate that in the subgiant scenario the initial silicon and iron abundance must be enhanced relative to the other initial abundances. For elements other than Si and Fe, their abundances are consistent with initial values of about 1/300 solar.", "pages": [ 8, 9, 10, 11, 12 ] }, { "title": "5 CONCLUSIONS AND DISCUSSION", "content": "The Galactic halo star SDSS J102915+172927 is an unusual object in that it has a very low Z (< 7.4×10 -7 ), and a low C/Fe ratio that distinguishes it from most other extremely metal poor stars. Caffau et al. (2012, Caf12) determined effective temperature and surface gravity Teff = 5811 150 K and log g = 4.0 - 0.5. The surface gravity estimate is problematical in that it places the star between the main sequence and the subgiants in the Hertzsprung-Russell diagram. If we assume that the star is on the main sequence, we estimate from Teff alone that its mass and luminosity to be M = 0.72 ± 0.06 M /g1622 and L = 0.45 ± 0.10 L /g1622 , placing it at a distance of 1.35 ± 0.16 kpc. However the upper limit on the lithium abundance, A (Li) < 0.9, is inconsistent with the star being a dwarf, assuming that mixing is due only to convection. We, therefore, propose an alternative scenario in which SDSS J102915+172927 is currently in the subgiant phase of evolution. To reach the subgiant phase in the age of the Universe requires a mass > 0.815 M /g1622 . Since stars of this mass with low heavy element abundance have higher surface gravity and shallower surface convection zones compared to population I stars of the same mass, gravitational settling of heavy elements has a significantly larger effect on surface abundances. We show that, in the absence of mixing processes other than convection, gravitational settling reduces the surface lithium and heavy element abundances to essentially zero during the main sequence phase of evolution. Convective dredge-up during the subgiant phase restores the abundances to about their initial values. We have shown that the effective temperature at which this occurs is sensitive to adopted value for the mixing length ratio, α . In this scenario, the observed lithium depletion is a result of SDSS J102915 being at an evolutionary stage in which convective dredge-up has not yet completed. To obtain consistency between the constraints set by the upper limit on the lithium abundance and the range in Teff , we find that α must be less than about 1.5, which is lower than our solar calibrated value of α = 1.7. We further find that the initial abundances required to give the current epoch abundances are broadly consistent with [M/H] = -2.5, with the exceptions of silicon and iron which we find must have a significantly higher initial abundances of [Si/H] ~ -1.5 and [Fe/H] ~ -1. In the subgiant scenario, the spectroscopic Teff constrains the mass to be greater than 0.815 M /g1622 . If we assume an age of 12 Gyr, the mass must be close to 0.845 M /g1622 . The corresponding luminosity range, L = 9.2 ± 0.5 L /g1622 , implies a distance of 6.2 ± 0.2 kpc. Caf12 discuss the limits placed on the distance by interstellar absorptions of the Na I D-line doublet at 589.0 nm and the Ca II-K and H lines at 393.3 and 396.8 nm. By using Na I as a tracer of neutral hydrogen, they infer a hydrogen column density similar to that directly measured in ρ Leo, which has a Hipparcos parallax of 0.60 ± 0.18 mas, corresponding to a distance of 1.3 - 2.4 kpc. Caf12 interpret this as setting a lower limit to the distance of SDSS J102915+172927 of 1.3 kpc, in good agreement with the distance from main sequence fitting. However, Caf12 also point out that the Na I column density is similar to that observed towards η Leo, which has a Hipparcos parallax of 2.57 ± 0.16 mas, placing it at a distance of 0.37 - 0.41 kpc. Hence, we interpret the interstellar absorption measurements as indicating that the absorbing material is mainly in the Galactic plane, and limits the distance of SDSS J102915 only to being greater than ~ 0.4 kpc. This is also consistent with the larger distance required by the subgiant scenario. Another important aspect of the subgiant scenario is that allows SJ102915 to have formed from material with [C/H] ~ -2.5 in agreement with the theoretical result of Frebel, Johnson, & Bromm (2007A) that their 'observer friendly' transition discriminant D must be greater than -3.5 for low mass star formation to occur. In terms of mass fractions, the inferred initial abundances in our subgiant scenario are X ( 12 C) ~ 10 -5 , X ( 16 O) ~ 3×10 -5 , X ( 28 Si) ~ 4×10 -5 and X ( 56 Fe) ~ 1.5×10 -4 . The ratios of these mass fractions are in good agreement with those from simulations of type Ia supernovae (Iwamoto et al. 1999) but differ significantly from the nucleosynthetic yields of population III type II supernovae models (Tominaga, Umeda & Nomoto 2007) which give carbon abundances equal to or greater than the abundances of iron-peak elements. Hence we propose that SDSS J102915 formed from ISM material of heavy element composition dominated by ejecta from a type Ia supernova, but that most other extremely metal poor stars formed from a more homogeneously mixed ISM containing material from type II supernovae. In conclusion, we have shown that the observed abundances in the extremely metal poor star SDSS J102915+172927 are better explained by a model in which the star is in the subgiant phase of evolution rather than being a main sequence star. In our models, we included the heretofore neglected effects of gravitational settling and element diffusion which have significant impact on the surface abundances of extremely metal poor stars that are massive enough to evolve to the subgiant phase in times less than the age of the Universe. However, we have not included the radiative force on the individual elements. Exploratory calculations that include the radiative force on Fe indicate that radiative levitation may also play an important role in the evolution of the surface abundances of extremely metal poor stars. /g3", "pages": [ 12, 13 ] }, { "title": "ACKNOWLEDGMENTS", "content": "JM thanks John Gizis for valuable discussions. We also thank an anonymous referee for many useful suggestions for improving the manuscript and for pointing out that radiative levitation of Fe may be an important effect in stars of very low heavy element abundance.", "pages": [ 13 ] }, { "title": "REFERENCES", "content": "Angulo C. et al., 1999, Nuclear Physics A, 656, 3 Bromm V., Loeb A., 2003, Nature, 425, 812 Böhm-Vitense E., 1958, Z. Astrophys., 46, 108", "pages": [ 13 ] }, { "title": "APPENDIX A: RADIATIVE LEVITATION OF FE", "content": "In this appendix we present the results of some preliminary calculations that include radiative levitation of Fe. In these calculations only Fe is allowed to diffuse independently and all other elements are assumed to have the same diffusion velocity. To calculate the radiative acceleration of Fe we use Opacity Project (OP) routines (Seaton 2005) downloaded from http://cdsweb.ustrasbg.fr/topbase/TheOP.html. Specifically the routine accv.f is used to calculate the Rosseland mean opacity, , R κ and the dimensionless radiative acceleration parameter, , γ for a grid of values of temperature, T and electron number density, Ne . For simplicity, we use a mixture of H and Fe. At each ( T , Ne ) grid point, the accv.f routine provides R κ and γ for a specified range of values of an abundance multiplier, , χ which in the current context is a measure of the ratio of Fe to H number densities, with 1 χ= corresponding to the solar ratio. At each ( T , Ne ) grid point, we use a Gaussian function to fit R κ and γ as functions of . χ We then use bilinear interpolation in the grid to evaluate R κ and . γ The radiative acceleration, grad , of Fe is related to γ and other quantities by where F is the radiative flux, M is the mean atomic mass and M (Fe) is the mass of an atom of Fe. To avoid reference to the opacity, we find it convenient to relate the radiative acceleration to the gradient of the radiation pressure, We have evolved stars of masses 0.7 and 0.85 M /g1622 , heavy element abundance Z = 10 -4 Z /g1622 and mixing length ratio 1.7. α= The surface Fe abundance for the 0.7 M /g1622 model slowly declines during the main sequence phase, reaching 92% of its initial value at the time when Teff equals that of SDSS J102915. When radiative levitation is ignored the reduction is to 84%. We conclude that radiative levitation is likely to have only a small effect on the evolution of the elemental abundances in the main sequence scenario for SDSS J102915. The evolution of the surface Fe abundance is markedly different for the 0.85 M /g1622 model. During the main sequence phase, the Fe abundance increases significantly reaching a mass fraction of 0.0029 at the end of the main sequence phase. The large enhancement is due to the radiative force exceeding the force of gravity just below the convection zone during the later parts of the main sequence phase. As the star evolves on the main sequence, the mass of the convection zone decreases from an initial value of 10 -3 M /g1622 to 10 -8 M /g1622 at the end of the main sequence. Once the radiative force exceeds the force of gravity just below the convection zone, all of the Fe is 'trapped' in the convection which leads to the large surface Fe abundance increase. After the model leaves the main sequence, the convection zone mass increases and the surface Fe abundance decreases. When the model Teff on the subgiant branch equals that of SDSS J102915, the Fe abundance is about twice the initial abundance. Although this model is incomplete, e.g. it does not include the radiative force on other elements nor does it allow the other elements to diffuse independently, it does indicate that radiative levitation could be an important physical process in determining the evolution of the surface abundances of extremely metal poor stars.", "pages": [ 15, 16 ] } ]

[ { "title": "ABSTRACT", "content": "A model for pulsars with polar-cap magnetic flux density B antiparallel with spin Ω is described. It recognizes the significance of two elementary processes, proton production in electromagnetic showers and photoelectric transitions in ions accelerated through the blackbody radiation field, which must be present at the polar cap in the Ω · B < 0 case, but not for pulsars of the opposite spin direction. The two populations are likely to be indistinguishable observationally until curvature radiation pair creation ceases to be possible. The model generates, and provides a physically realistic framework for, the polar-cap potential fluctuations and their time-scales that can produce modechanges and nulls. The RRATs are then no more than an extreme case of the more commonly observed nulls. The model is also able to support the basic features of subpulse drift and to some extent the null-memory phenomenon that is associated with it. Unfortunately, it appears that the most important neutron-star parameter for quantitative predictive purposes is the whole-surface temperature T s , a quantity which is not readily observable at the neutron-star ages concerned. Key words: instabilities - plasma - stars:neutron - pulsars:general", "pages": [ 1 ] }, { "title": "An incomplete model of RRATs and of nulls mode-changes and subpulses", "content": "P. B. Jones /star Department of Physics, University of Oxford, Denys Wilkinson Building, Keble road, Oxford OX1 3RH, England", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The discovery within the last decade of further complex pulsar phenomena, particularly the Rotating Radio Transients (RRATs; McLaughlin et al 2006), has extended the problem of understanding these systems by introducing new timescales. In a number of previous papers (Jones 2010a, 2011, 2012a, 2012b; hereafter Papers I-IV) physical processes at the polar-cap surface in pulsars with spin Ω and magnetic magnetic flux density B such that Ω · B < 0 have been examined to see if they are relevant. The processes that introduce important time-scales are those associated with the formation of electromagnetic showers by reverse-electrons from either electron-positron pair formation or from photoelectric transitions in accelerated ions. They are of no significance in Ω · B > 0 pulsars with Goldreich-Julian charge density ρ GJ < 0 and outward electron acceleration. The work has assumed that the space-charge-limited flow (SCLF) boundary condition E · B = 0 is satisfied on the polar-cap surface at all times. The aim of these papers has been to determine the composition and energy distribution of the accelerated particle flux and to look for instabilities that might be relevant to the phenomena which are observed in the radio-frequency spectrum. These properties can then be compared with those needed to produce the observed emission spectra. This has a large bandwidth, of the order of 10 9 Hz, and efficient conversion of kinetic energy to radio frequencies. For example, the peak power of the brightest pulses in PSR B0656+14 (Weltevrede et al 2006a), is of the order of 7 × 10 27 erg s -1 , which is equivalent to approximately 10 2 -3 MeV per unit charge accelerated at the polar caps. Growth of a collective mode able to transfer energy at this rate to the radiation field constrains the longitudinal effective mass, m i γ 3 i , of beam particles with mass m i and Lorentz factor γ i . For a secondary electron-positron pair plasma, it is well known that this requires a low energy, γ e ∼ 100, which is also of the same order as that needed if coherent curvature radiation, in a dipole-field, were the source of the observed radiation. Papers III and IV showed that, under the SCLF boundary condition, the creation of a reverse flux of electrons by photoelectric transitions in the accelerated ions limits the acceleration potential, analogously with the effect of electron-positron pair creation. Thus the outward particle flux has two principal components: protons formed in the electromagnetic showers and ions with Lorentz factors γ p ≈ 2 γ A,Z which are relativistic, but not ultra-relativistic as they would be in the absence of photoelectric transitions. They can have longitudinal effective masses that allow the rapid growth of the quasi-longitudinal Langmuir mode considered by Asseo, Pelletier & Sol (1990). As noted by Asseo et al, the quasi-longitudinal mode couples directly with the radiation field and so introduces, in principle, a second source of coherent radio-frequency emission which is not present in Ω · B > 0 pulsars. Electromagnetic shower theory (see Landau & Rumer 1938, Nordheim & Hebb 1939) makes it possible to calculate the total photon track length per unit interval of photon frequency. It is an almost linear function of primary electron energy in the ultra-relativistic limit. Then known partial cross-sections for the formation and decay of the giant dipole resonance enable W p , the number of protons formed per unit incident electron energy, to be estimated with adequate reliability for the present work. Proton formation is concentrated at shower depths of the order of 10 radiation lengths and it was shown in Paper I that the time τ p for diffusion to the top of the polar-cap atmosphere introduces the likelihood of instability in the composition of the accelerated particle flux. Estimates in Paper II, under more appropriate assumptions about the depth of the atmosphere, were that τ p would be of the order of 10 -1 -10 0 s. With this result in mind, it was evident that the photoelectric transitions considered in Papers III and IV would introduce large-scale fluctuations in the the acceleration potential above the polar cap. The time-scale τ p is sufficiently long that the electric field is simply given by solving Poisson's equation for the charge densities instantaneously present above the polar cap. It follows that the conditions necessary for coherent radio emission can appear or disappear with time-scales related to τ p . /negationslash Before proceeding further, it is appropriate to place this paper in the context of recent studies of the pulsar magnetosphere based on computational techniques in numerical plasma kinetics. It is true that the inductance per unit length of the tube of open magnetic flux lines varies little with altitude above the polar cap so that the magnetosphere as a whole limits current-density time derivatives. The further hypothesis that the actual current densities are determined by the whole magnetosphere rather than by considerations specific to the small region immediately above the polar cap appeared in a paper by Mestel et al (1985) and was later extended by Beloborodov (2008). Solutions for the magnetic flux density external to a sphere, typically of radius ∼ 0 . 2 R LC , where R LC is the light-cylinder radius, have been obtained by Kalapotharakos & Contopoulos (2009) and by Bai & Spitkovsky (2010) under the assumption of a forcefree magnetosphere. Current densities can then be derived from the B -distribution as functions of basic parameters; the dipole moment, the rotation period P , and the angle ψ subtended by Ω and B . Those found by Bai & Spitkovsky deviate considerably from ρ GJ c , the Goldreich-Julian value, and were followed by one-dimensional time-dependent numerical studies of the polar cap with the E · B = 0 surface boundary condition (Timokhin 2010) and later with the SCLF boundary condition (Chen & Beloborodov 2013, Timokhin & Arons 2013) assuming fixed values of the time-averaged current density that are either larger than ρ GJ c or of the opposite sign. These produced evidence for the development of high potential differences and microsecond bursts of pair creation. All authors assumed the case Ω · B > 0 which is not the subject of the present paper. Some care will be needed in translating the results obtained by Chen & Beloborodov and by Timokhin & Arons to the Ω · B < 0 case. For this, the ba- it of length in the description of the one-dimensional current and charge density, derived from the plasma frequency, is almost two orders of magnitude larger than for electrons. The one-dimensional potential must form a continuous function with the three-dimensional potential derived from the Lense-Thirring effect (see Muslimov & Tsygan 1992). In this case, owing to the larger scale-length, there is no possibility of the particle backflow noted by Beloborodov (2008) and Chen & Beloborodov (2013) so that only the stable zero-temperature solution they refer to is realistic. The present paper does not accept the assumption of a precisely force-free magnetosphere on which the computational work is based. Instead, it retains the assumption that the current density is determined in the polar-cap region and that it flows toward the light cylinder through a magnetosphere that is not precisely force-free. Paper IV attempted to describe, qualitatively, the relevance of physical processes at the surface of the polar cap to the complex phenomena that are observed in radio pulsars, but a number of numerical estimates made earlier, in Papers I and II, have been superseded by later work. Thus the aim of the present paper is to introduce an elementary mathematical model of the polar cap which embodies those results obtained in Papers I-IV that remain relevant. The model itself and the information which has been derived from it are described in Section 3. Section 2 summarizes the physical properties of the polar-cap surface that are essential for its construction and also considers the non-static aspects of charge-to-mass ratio fractionation of atmospheric composition which were neglected in the earlier papers. It is the scale of the acceleration potential fluctuations which is the most significant feature displayed by the model. We relate this with the observed properties of the RRATs and nulls in Section 4 and with the phenomena of subpulse drift and null memory in Section 5. Our conclusions are summarized in Section 6 along with a brief discussion of observed phenomena on which, at the present time, our model appears to have no impact.", "pages": [ 1, 2 ] }, { "title": "2 ESSENTIAL FEATURES OF THE POLAR CAP", "content": "It is assumed, unlike Papers I and II, that the actual polarcap magnetic flux density is of the same order of magnitude as that inferred from the spin-down rate, which is usually less than the critical field m 2 c 3 /e ¯ h = 4 . 41 × 10 13 G. Thus the ion separation energy (Medin & Lai 2006) is small enough for the mass of the atmosphere at a polar-cap temperature T pc ∼ 10 6 K to be significant. This atmosphere is very compact and in local thermodynamic equilibrium (LTE); its scale height is of the order of 10 -1 cm. The total mass is poorly known owing to its exponential dependence on the ion separation energy, but is possibly in an interval equivalent to 10 -1 to 10 1 radiation lengths. Thus it may contain the whole or some part of the electromagnetic showers formed by inward accelerated electrons. The extent of a shower is itself uncertain because the Landau-Pomeranchuk-Migdal effect is present at the energies, densities and magnetic fields concerned (see Jones 2010b). At depths immediately below the atmosphere, the state of matter is uncertain and could be either liquid or solid (see Paper II). The nuclear charge is not known, but we assume the canonical value Z = 26. Electromagnetic showers reduce this to a mean Z s and mass number A at the top of the atmosphere with an LTE ion charge ˜ Z .", "pages": [ 2, 3 ] }, { "title": "2.1 Atmospheric fractionation", "content": "There is a fractionation of ion charge-to-mass ratio with the largest values at the top of the atmosphere. The relatively small number of protons created in showers are nowhere in equilibrium within the LTE ion atmosphere and, under the influence of the small electric field E present, move outward and are either accelerated or, if their flux exceeds the Goldreich-Julian current density ρ GJ c , form an atmosphere above the ions. Its scale height is 2 k B T/m p g , at local temperature T and gravitational acceleration g . The chemical potential gradient that causes their motion within the ion sector of the atmosphere is initially mostly an entropy gradient, but at lower densities changes to that derived from the electric field. This fractionation is of central importance to the model described in this paper, but we have not previously considered the adequacy of our elementary static treatment of the problem. Also, the presence of partial ionization means that the possibility of convective instability has to be considered (for a simple explanation see, for example, Rast 2001). Bearing in mind the functioning of a laboratory high-vacuum diffusion pump, it is also necessary to ask if the upward flux of protons is likely to carry with it sufficient ions to interfere with fractionation. In order to examine this, we require two transport relaxation times. We employ approximate expressions using lowest-order zero-field perturbation theory, satisfactory here because the proton cyclotron energy quantum is small compared with the polar-cap thermal energy k B T pc . The first, for upward movement of a proton in the ion atmosphere at ion number density N Z is, in which the function F p is, Here, κ Dp is the Debye wavenumber for the ions, and, The equivalent ion relaxation time for movement relative to the proton atmosphere above the ions, if it exists, is also required. At proton number density N p it is, in which F Z is given by equation (2) with the substitutions, It is typically several orders of magnitude longer than τ tr p . The proton drift time from a depth z = 0 with ion number density N Z to the top of the atmosphere is, under the influence of a chemical potential gradient fixed by the ions, equivalent to an upward directed force. The upward velocity is given by, and the scale height of the atmosphere is, for A = 20, ˜ Z = 6, g = 2 × 10 14 cm s -2 and T = T pc = 10 6 K. Evaluation for N Z = 10 24 cm -3 gives 1 /τ tr p = 2 . 5 × 10 15 s -1 and a drift time of 1 . 0 s. This should be regarded only as a tentative order of magnitude because equations (1), (3) and (4) are strictly valid only at number densities much lower than 10 24 cm -3 . The position of the change of phase from gas to liquid or solid is also uncertain. The proton sector number densities are also sufficient for it to form an LTE atmosphere, but during phases of proton emission into the magnetosphere it has the GoldreichJulian outward flux. Thus as a first approximation, its kinetic distribution function is isotropic in a frame of reference moving outward with a velocity ρ GJ c/N p e until the proton number density N p becomes so small that the LTE condition breaks down. If this is not to interfere with fractionation, the force exerted on an ion by the upward flux of protons must be small compared with its chemical potential gradient in the static proton LTE atmosphere, which is, A proton atmosphere able to maintain a typical GoldreichJulian flux for one second would have a density at base of ∼ 10 22 cm -3 at which its ion transport relaxation time would be 1 /τ tr Z = 5 × 10 10 s -1 giving a force several orders of magnitude smaller than equation (10) and of small effect on fractionation. The existence of convective instability at any density within the atmosphere is unlikely because the lateral motion implicit in a convective cell is strongly suppressed by magnetic fields of the order of 10 12 G. We refer to Miralles, Urpin & Van Riper (1997) for a full treatment of this problem. But even if convective cells exist, the presumption must be that the electric field E , reflecting the internal equilibrium of the adiabatically-moving volume, is still present within it. Thus the velocity of the protons, averaged over many circulations, would remain as calculated above and there would be no interference with fractionation.", "pages": [ 3 ] }, { "title": "2.2 Acceleration potential", "content": "The polar-cap radius, as in Papers III and IV, denotes the division between open and closed magnetic flux lines, and is that given by Harding & Muslimov (2001), where P is the rotation period. We assume a neutron-star mass 1 . 4 M /circledot and radius R = 1 . 2 × 10 6 cm, for which f (1) = 1 . 368. Our approximation for the electrostatic potential Φ is based on the Lense-Thirring effect described by Muslimov & Tsygan (1992), in cylindrical polar coordinates, at altitude z and radius u ( z ), for the specific case of a charge density ρ ( z ) that is independent of u and is only a slowly varying function of z . (It is almost identical with the potential that would be present given a time-independent outward flow of electrons under SCLF boundary conditions in Ω · B > 0 pulsars.) It also assumes approximate forms, valid at altitudes well within the light-cylinder radius, for the more precise charge densities which were given by Harding & Muslimov (2001). These are independent of u ( z ) as required by equation (12) and are, and in which ψ is the angle between Ω and B , κ = 0 . 15 is the dimensionless Lense-Thirring factor, and η = (1 + z/R ). Equation (12) is certainly a satisfactory approximation to the true potential at altitudes z /greatermuch u 0 , but for lower values of z , we must assume that the H-M potential changes continuously to the one-dimensional potential that exists at z /lessmuch u 0 , which was originally described by Michel (1974) and then further investigated by Mestel et al (1985) and by Beloborodov (2008). We have already noted, in Section 1, that the length scale associated with the form of the one-dimensional potential is large, in the Ω · B < 0 case. Thus equation (12) is a fair representation of the true potential, which would remove any possibility of the backflow described by Beloborodov. Under the assumption that equation (12) is valid, the condition ρ (0) = ρ GJ (0) ensures that the SCLF condition E · B = 0 is satisfied at all u on the polar-cap surface. The maximum potential available for acceleration above the polar cap occurs if ρ ( z ) is independent of altitude as would be the case if the particles were ultra-relativistic and there were no charge-separating interactions. Expressed in convenient energy units, it is, per unit charge on a flux line with radial coordinate u .", "pages": [ 4 ] }, { "title": "2.3 Photoelectric transitions", "content": "The blackbody temperatures responsible for photoelectric transitions are those of the polar cap, T pc , and of the whole surface, T s . The polar cap is not significant at altitudes z /greatermuch u 0 (0) because the photon Lorentz transformation to the ion rest frame becomes unfavourable. There is a little ionization at z < h ≈ 0 . 05 R to a mean charge Z h but the ion Lorentz factors there are small, given the SCLF boundary conditions, as is the contribution to the total reverseelectron energy per ion. We assume a fixed value for this component of /epsilon1 h = 20 GeV. Lorentz transformations of whole-surface photons are much more favourable at higher altitudes and produce the major part of the total reverseelectron energy per ion accelerated, /epsilon1 = /epsilon1 h + /epsilon1 s . Ionization may or may not be complete and we define the mean final charge as Z ∞ . We have shown previously that the flux of shower-produced protons reaching the top of the neutronstar atmosphere at any point u is given by, in terms of the ion flux J z (see equation (20) of Paper IV). The first component J p cannot exceed the Goldreich-Julian flux; the remainder ˜ J p accumulates at the top of the atmosphere. Paper I assumed the neutron-star atmosphere to be of negligible depth so that f p was assumed to be the standard diffusion function. But for the depth considered in Paper IV and here, an expression based on the drift time given by equation (7) is a better approximation. If the proton atmosphere is exhausted, ion emission commences extremely rapidly in order to satisfy the SCLF condition E · B = 0 at the polar-cap surface. In general, the screening of any electric field in the Ω · B < 0 case for which a positive charge density is needed occurs preferentially through ion or proton emission in a single relativistic-particle transit time, whereas the process of electron-positron pair multiplication requires many transit times. Protons accelerated to γ p ∼ 10 3 have only a very small probability of creating electron-positron pairs through interaction with blackbody photons and, with ions of the same energy per unit charge, would have only negligible growth rates for the quasi-longitudinal Langmuir mode. There are three ways in which particle beams capable of giving strong coherent radio-frequency emission might be produced. The two considered here in the first instance are as follows. (a) V max is so large that self-sustaining curvatureradiation (CR) electron-positron pair production occurs. A permanent proton atmosphere exists over much of the polar cap giving a primary current density of protons and positrons which, at least for times long compared with u 0 /c , may be in a steady state. A plasma of low-energy secondary electrons and positrons forms. (b) The potential is so reduced from V max by photoelectron backflow that ion and proton Lorentz factors are either in a region that allows rapid growth of the quasilongitudinal mode or are of magnitude such that downward fluctuations to the necessary values can occur. In this case, the Lorentz factors are such that the mode wave-vector component perpendicular to B is unlikely to be negligible so that coherent radio-frequency emission need not be exactly parallel with local flux lines. The third set of processes are more obscure in the case of Ω · B < 0 pulsars and concern inverse Compton scattering (ICS) of blackbody photons above the polar cap. Pair production by the conversion of outward-moving high-energy ICS photons is known to be significant in Ω · B > 0 pulsars (see Hibschman & Arons 2001, Harding & Muslimov 2002) even if dipole-field geometry is assumed. Also, deviations from such a field, to the extent that they exist, can greatly enhance pair densities (Harding & Muslimov 2011). But in Ω · B < 0 pulsars, high-energy ICS photons are directed inward and the sources of outward-moving positrons to scatter blackbody photons are more obscure. Clearly, a fraction of the photons may have sufficient momentum transverse to B to produce pairs at low altitudes. A further possible source of low-altitude pairs is the conversion of neutron-capture γ -rays at the top of the atmosphere. Showers produce approximately the same numbers of neutrons and protons, but reliable calculation of the probability that a neutron would reach the top of the atmosphere is not easy. It is also not obvious how the density of electron-positron pair production via these processes would respond to fluctuations in the acceleration potential. However, although the model described in Section 3 has been considered in the context of either lowenergy ion-proton beams or CR pair production, it is necessary to bear the more obscure pair-production processes in mind.", "pages": [ 4, 5 ] }, { "title": "3 THE POLAR-CAP MODEL AND ITS PROPERTIES", "content": "The previous Section summarized those specific physical properties of the polar cap that are important for the model, but we shall also give a brief description of its framework. This is addressed primarily to the typical radio pulsar with an age of the order of 1 Myr, or older, and not to the smallP case in which continuous pair production is anticipated. Photoelectric transitions in accelerated ions are a source of electrons which flow inward to the polar-cap surface and partially screen the potential given by equation (12). Data presented in Table 1 of Paper IV, for specific values of B and T s , gave Z ∞ , the final charge of an accelerated ion, and the energy of its reverse electrons /epsilon1 s , as functions of the acceleration potential V ( u , ∞ ) experienced by that ion. Then with the inclusion of the small energy /epsilon1 h arising from photoelectric transitions at altitudes z < h , as described in Section 2.3, the total energy /epsilon1 = /epsilon1 h + /epsilon1 s gives the number of protons created in the electron showers. This is the approximately linear function /epsilon1W p = KZ ∞ (see Papers I and II). It is convenient here to define K as the number of protons formed per unit ion charge. (In Papers I and II, it was defined as the number of protons per unit nuclear charge of the ion.) The model described here uses an elementary form of equation (16) which describes the temporal distribution of the protons reaching the top of the LTE atmosphere. But the distribution of electron back-flow and the values of Z ∞ over the polar-cap also define through solution of Poisson's equation, at any instant, the actual electrostatic potential which, of course, is not identical with equation (12) because ρ is not independent of u . The model then finds by relaxation procedures finite-element approximations to the self-consistent functions Z ∞ ( u , t ) and V ( u , ∞ , t ) for the polar cap. This is circular, with radius given by equation (11), which assumption is not essential but is made for ease of calculation.", "pages": [ 5 ] }, { "title": "3.1 The model", "content": "The polar cap is divided into n s = n 2 elements of equal area by first defining n annuli of outer radius iu 0 /n , where i = 1 ....n and then further dividing each of these into 2 i -1 equal azimuthal elements. Within each annulus, the elements are displaced, by a random azimuthal angle, with respect to the axis φ = 0. It is assumed that the charge density is independent of u within any element. Equation (16) is much simplified by the model assumption f p ( t ) = δ ( t -t ' -τ p ) giving, within any element, so that the state of the element alternates between proton and ion emission. The total number of protons produced at time t on unit area of surface at the top of the atmosphere following an ion-emission interval of length τ p is, The ion current density is J z ( t ) = N Z ( t ) cZ ∞ ( t ) e and the Goldreich-Julian charge density is ρ GJ ( h ) = N Z ( t )(2 Z h -Z ∞ ( t )) e . Therefore, if we define the length of the proton phase by ˜ Kτ p = q p /ρ GJ c , An event is defined as a change of state in any element from proton to ion emission or vice-versa. The length of an ion phase is τ p , which is assumed constant. The length of the following proton phase is ˜ Kτ p , and is a function of the state of the whole polar cap within the integration time over the preceding ion phase. Table 1 of Paper IV gave values of /epsilon1 s and of Z ∞ as functions of a cut-off potential V c for different values of the surface magnetic flux density B 12 , in units of 10 12 G, and the whole-surface temperature T s . For reasons which we outline later, we assume a surface nuclear charge Z s = 10. The cutoff potential is a representation of the effect of photo-electric transitions on the maximum possible acceleration potential difference V max given by equation (15). From Table 1, with some additional values, we have been able to tabulate the functions /epsilon1 s ( V ) and Z ∞ ( V ) for the interval 0 < V ( ∞ ) < V max . They are not strongly dependent on u and we can assume that they are functions only of the magnitude of V ( u , ∞ ), the potential at the upper end of the polar-cap acceleration zone which is represented here by a cut-off at η = 4. In the model, it is assumed to be the potential on the central axis of an element. The excess charge density within an element during the ion phase is, and is the source of a potential deviation downward from V max ( u , ∞ ). In order to calculate this with some economy, we use the approximation on which equation (12) is based, specifically that at an altitude z >> u 0 , a section of the long narrow open magnetosphere can be approximated locally by a cylinder of constant radius u 0 ( z ). Then the Green function satisfying the SCLF boundary condition for a line source at cylindrical polar coordinates ( u, φ ) is, The potential on the axis of an element derived from the charge excess within that element is the most important factor and is obtained by numerical integration using the Green function. The potential generated on its axis by any other ion-phase element is found directly from the Green function by using a line approximation for the excess charge. The potential within a proton-phase element is of no interest and is not calculated. After each event, the Z ∞ in all ion-phase elements, also referred to as ion-zones, are recalculated by a relaxation procedure so that there is consistency between the Poisson equation solution for V ( u , ∞ ) in terms of all the Z ∞ and the tabulated function Z ∞ ( V ) obtained from Table 1 of Paper IV which is based purely on the photoelectric transition rates. This also gives the function /epsilon1 s and hence K which, with the values of Z ∞ and their times of duration, then allow calculation of the integral for ˜ K and the duration of the subsequent proton phase. The initial state of the system has all elements in the proton phase. All n s elements, selected in random order, are then assigned sequential times for transition to the ion phase. The interval between any two adjacent times is xτ p , where x is a random number in the interval 0 < x < x max . Our initial choice was x max = 0 . 5. Throughout the calculation, a list is maintained, in temporal order, of the time of the next event in each of the n s elements. This procedure has the advantage that the calculation of every ion phase is entirely self-contained so that cumulative errors are not carried forward. The model has been run for intervals equivalent to real times of the order of ten days.", "pages": [ 5, 6 ] }, { "title": "3.2 Model results", "content": "For a given random initial state, the subsequent states of the model polar cap are not, of course, random but are chaotic, and variation of x max appears not to affect their character. The proton production parameter W p is a slowly varying function of B but we have chosen the conservative value W p = 0 . 2 GeV -1 and an ion charge Z h = 6 as in Paper IV. The numbers of elements have been in the interval 10 2 /lessorequalslant n s /lessorequalslant 4 × 10 2 . The usual procedure has been to let the model run for a time 10 4 τ p before sampling its state at intervals of τ p . The presence of the instability described in Paper I has been confirmed. The system has shown no sign of settling down to other than a chaotic state. Its main characteristics are as follows. (iii) Peripheral ion-emission elements have a significant autocorrelation function in the angle φ , unaffected by running time, indicating the formation of clusters. At this stage, we should note that the model as precisely defined by equations (17) - (19) does not produce, at any point on the polar-cap surface, the mixture of protons and ions which is required for growth of the quasi-longitudinal Langmuir mode. But the growth rate is only a slowly varying function of the proton-ion ratio (see Papers III and IV) and we do not doubt that, given a physically realistic form of the diffusion function f p in equation (16), there will be a sufficient interval of time, of the order of τ p , within which the current density contains both components. The potential fluctuations given in the Table are, of course, dependent on n s . The autocorrelation function in φ indicates that a smaller of n s should be preferred and, within the limited framework of our model, would be a better approximation to reality. But n s = 100 is the smallest value for which our approximation for the interaction between two elements can be reasonably adequate. Table 1 therefore underestimates the true scale of fluctuation. The values of B 12 and P have been chosen to be representative of the typical pulsar exhibiting nulls as listed by Wang, Manchester & Johnston (2007). The scale of the downward fluctuations in V is, as might be anticipated, very strongly dependent on T s and hence, presumably on pulsar age. At T s = 10 5 K, significant downward fluctuations would be present only for very large values of B 12 P -2 , the parameter that scales V max . With increasing rotation period, downward fluctuations also increase. The most significant feature of the model is the size of the potential fluctuations that occur for larger values of T s . But this is superimposed on a further instability, with medium time-scale, described in Paper II. Showers reduce the atomic number from Z , possibly Z = 26, to Z s at the top of the atmosphere. Provided we ignore capture of shower-produced neutrons and subsequent β -decay of the neutron-rich nuclei formed, this liberates Z -Z s protons which diffuse with time-scale τ p . At any instant, there is naturally a distribution of Z s values, but over a time long compared with the ablation time (the interval in which one radiation length of matter is removed from the polar cap at the Goldreich-Julian current density) the average quantities must satisfy This means that Z s and ˜ K defined by equation (19) cannot be independent variables when considered over medium time-scales of the order of τ rl . The reason is that large values of K within some interval of time imply reduction of Z at the shower maximum to very small Z s so producing nuclei able to move to the top of the atmosphere either by diffusion on a time-scale possibly some orders of magnitude larger than τ p , or by Rayleigh-Taylor instability. In particular, values smaller than Z s ∼ 5 have a high probability of being completely ionized in the LTE atmosphere and the consequent absence of a reverse-electron flux stops shower and proton formation. The condition of these surface layers is obviously not well understood, but in Paper II a case was advanced that it would be one of instability rather than a steady-state value of Z s . This is the case for our model assumption of a compromise value Z s = 10 and is also the basis for the observed phenomena, mode-changes and long nulls, with time-scales of the order of τ rl .", "pages": [ 6, 7 ] }, { "title": "4 RRATS NULLS AND POTENTIAL FLUCTUATIONS", "content": "Our proposal is that the potential fluctuations found in the model provide a basis for understanding why the extent of null lengths and fractions seen in both pulsars and the RRATs occur quite naturally.", "pages": [ 7 ] }, { "title": "4.1 The RRATs", "content": "We refer to Burke-Spolaor & Bailes (2010), Keane et al (2011) and Keane & McLaughlin (2011) for reviews of the observational data on these sources. Values of f on , the fraction of periods in which a pulse is detected, are broadly in the interval 10 -4 < f on < 10 -1 within which their distribution is approximately uniform in log f on . The bursts of emission are short, usually one but occasionally several periods, and are consistent with being of the same order of magnitude as τ p . The RRATs listed in Table 3 of Keane et al are all quite distant with the exception of J1840-1419 which also has a period P = 6 . 6 s and a value of the acceleration potential parameter B 12 P -2 = 0 . 15, that is, below the cut-off value 0 . 22 which can be found from the ATNF catalogue (Manchester et al 2005). Thus Weltevrede et al (2006a) proposed that the RRAT pulses are simply analogues of the giant pulses seen in the nearby pulsar B0656+14 and that their emission, apart from the giant pulses, is unobservable owing to distance. However, there is a difference whose significance becomes obvious in view of the model described here. With the exception of J1554-5209, the 14 RRAT listed with surface magnetic fields by Keane et al are well below the threshold for CR secondary pair production as defined by Harding & Muslimov (2002). From Fig.1 of their paper, this condition can be represented approximately in terms of the parameter X = B 12 P -1 . 6 : the threshold is given by the critical value X c = 6 . 5. PSR 0656+14 lies well above this threshold, which assumes dipole-field flux-line curvature, as does the exceptional RRAT J1554-5209, and it is reasonable to suppose that these pulsars are capable of supporting CR secondary pair creation over at least a fraction of their polar caps. Therefore, we suggest that the giant pulses of B0656+14 are produced by process (a) through a rare upward fluctuation in V toward V max and its more general emission by process (b). Some RRATs may, of course, have sufficient non-dipole flux-line curvature to enable process (a), but discounting this possibility, the RRATs listed by Keane et al are so far below the Harding-Muslimov threshold that we suggest their isolated pulses are a consequence of rare but large downward fluctuations in V . It might be argued that, owing to the boundary condition satisfied by equation (12), Lorentz factors γ A,Z small enough for growth of the quasi-longitudinal Langmuir mode always exist, in principle, at u close to u 0 so that there should be some coherent emission at all times. But in this case it is not obvious that the lateral (perpendicular to B and to u ) depth of the particle beam would be suitably large in relation to the wavelength of the mode to allow the necessary growth rate. There is an obvious need for some analogue of the relativistic Penrose condition (see Buschauer & Benford 1977) extended to allow for a beam with lateral velocity gradients and limited lateral depths. The question of the conditions necessary for observable emission will be further considered in Section 5.1 in relation to the structure and drift of subpulses. The ages and surface magnetic fields of RRATs given by Keane et al can be compared with those of the nulling pulsars listed in Tables 1 and 2 of Wang, Manchester & Johnston (2007). It is immediately seen that RRAT fields are typically an order of magnitude larger than those given in the ATNF catalogue (Manchester et al 2005) for the Wang et al list. But the distributions of the acceleration parameter B 12 P -2 are very similar because the RRAT have long periods. The age distributions are also similar though both are very wide. The relation between age and proper-frame surface temperature T s is, unfortunately, obscure in the region assumed in Table 1, which is well within the photon-cooling epoch. Potekhin & Yakovlev (2001; Fig.7) have shown that increasing surface fields accelerates cooling, but an even more important factor would be the presence of lowZ elements in the outer crust, possibly from fall-back, which have the same effect (Potekhin et al 2003).", "pages": [ 7 ] }, { "title": "4.2 Null lengths and surface temperature", "content": "With reference to Table 1, it is possible to see qualitatively how nulls change with age during the cooling from highT s to lowT s . We consider pulsars which lie below the paircreation threshold. Initially, values of ˜ V are usually small enough to give the mode growth rates necessary for emission by process (b) (see Section 2.3). Upward fluctuations of ˜ V increase the ion and proton Lorentz factors and, owing to the exponential dependence of amplitude growth on them, can produce short nulls. Upward fluctuations become more frequent as cooling proceeds, giving higher null fractions 1 -f on . Eventually, cooling reaches the stage at which the mean ˜ V approaches unity and downward fluctuations are required to produce the growth rates necessary for observable emission. Null fractions are then near unity and, as the limit is approached, there is no observable emission. We emphasize that these fluctuations are superimposed on medium time-scale fluctuations (see Section 3.2) in the surface atomic number Z s which are unlikely to be uniform over the whole polar-cap surface. Values smaller than Z s ∼ 5 produce a negligible reverse-electron energy flux and hence, if present over a substantial area of the polar cap, much reduced deviations of ˜ V from unity. It is proposed here that these are responsible for null and burst lengths more nearly of the order of τ rl rather than τ p . Isolated Neutron Stars (INS) are a small group of radioquiet thermal X-ray emitting sources positioned close to the RRAT in the P -˙ P plane (see Keane et al 2011) but at slightly greater periods. The six sources for which timing solutions exist have been listed by Zhu et al (2011). Given their kinematic ages, the observer-frame temperatures are large ( ∼ 10 6 K) compared with those considered in Table 1 but, with the exception of the anomalous J0420-5022, their B 12 P -2 are typically close to the general cut-off value of 0.22 found from the ATNF catalogue (Manchester et al 2005). If they form part of the Ω · B > 0 population, they should support pair formation by the ICS mechanism; for Ω · B < 0, extrapolation from Table 1 shows that small ˜ V would be expected satisfying the condition necessary for rapid growth of the ion-proton beam instability. However, the fact that no radio emission has been observed from this small number of sources is, perhaps, not surprising in view of their anticipated small beaming fractions compared with the 4 π -observability of the whole-surface X-rays.", "pages": [ 7, 8 ] }, { "title": "5 SUBPULSE DRIFT AND NULL MEMORY", "content": "Subpulse drift, and the null memory which, in some pulsars, is observed with it, is obviously an important diagnostic of polar-cap physics. Unfortunately, our model is incomplete in that it does not have the capacity to spontaneously exhibit this phenomenon following the randomly-constructed initial state described in Section 3. Its finite-element structure and the very elementary approximation represented by equation (17) are the reasons for this. However, the phenomenological model developed by Deshpande & Rankin (1999) from the classic Ruderman & Sutherland (1975) model has proved so useful that any physical polar-cap model must be able to support it.", "pages": [ 8 ] }, { "title": "5.1 Nulls and null memory", "content": "It was noted in Section 3 that the model system settles down to a chaotic state, but with ion-emission elements most likely to be near the periphery u 0 and an autocorrelation function in the azimuthal angle φ indicating the formation of clusters. This is unsurprising, because ion-emission elements at smaller u have larger reverse-electron energy fluxes and so lead to longer phases of proton emission. Thus our model generates quite naturally the conal structure that is required for the Deshpande-Rankin carousel model and we can therefore proceed to examine the extent to which it is capable of supporting its organized subpulse motion. For convenience, we shall consider a circular path of radius u on the polar cap and associate a moving ion zone with the formation of a subpulse. Then it is possible to envisage organized circular motion in which an ion-emission zone of angular width δφ i ( u ) is followed by a proton-emission zone of ˜ Kδφ i ( u ). Thus at any instant, with i = 1 ....n in which n is here the number of elements on the carousel and each ˜ K is determined according to equation (19) by the preceding ion-zone. Comparison with measured values of the longitudinal subpulse separation P 2 indicates that the order of magnitude of model values should not be large, ˜ K i ∼ 3, but this order of magnitude is associated with values of the acceleration potential, on the ion-zone flux lines, that are unlikely to enable significant electron-positron pair creation. This is then consistent with mechanism (b), based on protons and ions, for the production of coherent radio emission. It is easy to see that a steady uniform state of equation (23) with δφ = τ p ˙ φ and constant ˙ φ that is independent of u does not, in general, exist. However, we have to bear in mind the limitations of our model based on equation (16), which is local in u , and on its very elementary approximation given by equation (17). Thus in a model with a more realistic diffusion function f p and without the finite element structure, we would expect that at constant φ , ˜ K would increase as a function of u , starting from a negligible value at u 0 and reaching a maximum before declining as a protonemission zone is entered. The presence of this extremum is a possible basis for the existence of short-term quasi-steadystate solutions of equation (23) within a finite interval of u at u c which also coincides with conditions suitable for the growth of the instability giving observable emission. It may be that the tendency to form a cluster, present in our model, indicates that a more realistic model would counteract the dispersion inherent in equation (23) and maintain subpulse shape, but it is not possible to assert that it would be so. However, it is possible to envisage this motion on a polar cap of any plausible geometrical shape, elliptical or approximately semi-circular. The motion is, of course, independent of E × B drift and could be in either direction, as is seen in the survey of Weltevrede, Edwards & Stappers (2006b), or may even be bi-directional at some instant on different areas of the polar cap. There are many ways in which potential fluctuations could disrupt this organized motion. We have seen in the previous Section that a whole-surface temperature T s large enough to give low values of ˜ V favours the occurrence of nulls through upward potential fluctuations. The most simple fluctuation is one that increases the potential over much of the polar cap and in particular at u = u c . The functions /epsilon1 s ( V ), and hence ˜ K , increase quite rapidly with V . There are two consequences. Firstly, the exponent in the expression for the mode growth rate (Paper IV, equation 17) is dependent on γ -3 / 2 A,Z so that the conditions necessary for observable coherent radio emission are not reached and a null occurs. Secondly, the (unobserved) motion of the ion zones is substantially changed. It is possible to see the nature of these changes by considering how two variables, the proton density q ( φ, t ) at the top of the atmosphere and the ion current density J z satisfying equations (17) - (19) on a section of the path at u = u c , evolve over a sequence of times. In doing this, we assume that the ion-zone is able to move continuously and abandon the fixed finite-element calculation of the model described in Section 3.1 The ion-zone motion during a null shows how a form of null memory occurs. It is convenient, for this problem, to represent the density q in units of ρ GJ (0) cτ p and J z in units of ρ GJ (0) c . The ion-zones at time t have J z ∼ 1, width δφ (1) and move with velocity ˙ φ (1) = δφ (1) /τ p . Each generates a proton density q ( φ, t ) at the top of the atmosphere which is depleted by the proton-zone current density J p = 1. Thus at an instant t , and as a function of φ , the proton density following an ion zone rises linearly to a maximum of ˜ K (1) -1 in a distance δφ (1) and falls linearly to zero in a further distance ( ˜ K (1) -1) δφ (1) . The velocity of an ion zone is therefore determined by the gradient ∂q/∂φ of the preceding proton zone. This is the quasi-steady-state motion at time t , at which instant, an upward potential fluctuation increases proton production to ˜ K (2) . The increase in ion Lorentz factor reduces the quasi-longitudinal mode growth rate to a sub-critical value and so initiates the null. A new maximum proton density q = ˜ K (2) -1 is reached at a later time t +2 τ p but the system velocity remains ˙ φ (1) until t + ˜ K (1) τ p . The width of an ion zone is then reduced. In detail, and with reference to equations (17) - (19), this happens because the velocity of its leading edge is reduced whilst that of the rear edge remains at ˙ φ (1) until t +( ˜ K (1) +1) τ p at which time the ion-zone width and velocity are both reduced by a factor of ( ˜ K (2) -˜ K (1) + 1). The ion-zone velocity then tends to its new steady-state value ˙ φ (2) = ˙ φ (1) (1 + ˜ K (1) ) / (1 + ˜ K (2) ). Hence subpulse drift does not stop in the model, but is much reduced after a null time interval of ˜ K (1) τ p . Thus for short nulls, there is a subpulse memory, but it does not correspond precisely with the conservation of subpulse longitude over a short null that has been observed, for example in PSR B0809+74, by van Leeuwen et al (2002). We are not unduly disturbed by this, because there are many possible spatial distributions of potential fluctuation over the polar cap other than the uniform case considered above. It is also not clear that the precise form of memory seen in B0809+74 is universally observed. The upward potential fluctuation that caused this could reverse at any time during or after the sequence described in the previous paragraph. In relation to null memory, the question then is, when does observable coherent radio emission recommence? The problem arises owing to the very elementary nature of our model defined by equations (17) - (19) and is that δφ (2) in the model can be very small. The acceleration potential within a very small interval of φ may be too large to permit adequate growth of the mode and the conditions discussed briefly in Section 4.1 may not be satisfied.", "pages": [ 8, 9 ] }, { "title": "5.2 Subpulse drift", "content": "Published observations on mode-changes and subpulse drift demonstrate the extremely heterogeneous nature of these phenomena. Some features are quite distinct in a small number of pulsars, but much less distinct or not present at all in others. For this reason, we have used the results of the extensive survey made by Weltevrede et al (2006b) based on observations of 187 pulsars selected only by signal-to-noise ratio, and list some of their conclusions below. Conclusions (i) and (ii) are quite naturally consistent with the model. We emphasize again that E × B drift is not involved and there is no reason why, given the chaotic state of the polar cap, there should not be reversals or even bi-directionality. The nature of the chaotic state also means that it is unrealistic to suppose that these phenomena can be predictable. Conclusion (iii) is also a natural consequence of medium time-scale instability in the value of the surface atomic number Z s and hence in the potential as mentioned in Section 4.2. The existence of a polar-cap area emitting nuclei with Z s too small to produce a significant reverseelectron flux is obviously associated with an upward displacement of ˜ V for times very broadly of the order of τ rl . This is quite consistent with the observed time intervals of 10 3 -4 s for mode-changes. Non-linear drift bands (iv) are simply a consequence of non-uniformity of ˜ K i values in equation (23). Band separation P 3 was discussed in Paper II. It is given by P 3 = ( ˜ K +1) τ p . In this expression, τ p is dependent only on the properties of the LTE atmosphere at the polarcap surface, but ˜ K may have some dependence on rotation period and magnetic flux density; also on whole-surface temperature T s and hence pulsar age. It was noted in Paper II that the distribution of P 3 is quite compact, most of the values listed by Weltevrede et al (2006b) being within the interval 1 < P 3 < 10 s. The fact that the carousel path is restricted to being near the polar-cap periphery u 0 may well act as a constraint on the values of ˜ K and so P 3 that occur. Finally, Kloumann & Rankin (2010) have observed randomly distributed pseudo-nulls in B1944+17 of length less than 7 P with weak emission, but well above noise. They interpreted them as a state of no subpulse on that band of polar-cap flux lines from which photons can enter the line of sight. These occur naturally in the model. The final column of Table 1 gives p vn , the probability that the whole polar cap is in the proton phase at any instant. There would then be no possible way of producing particle beams satisfying conditions (a) or (b) of Section 2.3 and capable of generating coherent radio emission. This contributes to the total null fraction that is observed. The probability p sn > p vn that there will be no ion-zones merely in the part of the polar cap which is observable along the line of sight is therefore a natural parameter of the model.", "pages": [ 9 ] }, { "title": "6 CONCLUSIONS", "content": "/negationslash Papers I - IV, followed by the present paper, have attempted to predict some of the consequences of there being two populations of isolated pulsars having opposite spin directions. In many important respects, the physical state of the polar-cap has been found to be dependent on the sign of the GoldreichJulian charge density. In the Ω · B < 0 case considered in these papers, SCLF boundary conditions have been assumed rather than the E · B = 0 condition of the original Ruderman & Sutherland (1975) model. Then the state of the polar cap is unstable on time-scales of the order of τ p and of the longer polar-cap ablation time τ rl . These are both many orders of magnitude longer than the characteristic polar-cap time-scale of u 0 /c and there can be little doubt that the system should be able to allow the movements of charge necessary to maintain the SCLF boundary conditions for them. We therefore need to look at published observational data to see if there is any real evidence for the existence of two populations. The Weltevrede et al (2006b) survey is an obvious starting point. It lists 187 pulsars selected only by signal-to-noise ratio and examines both subpulse modulation (the wide variations of intensity at a fixed longitude in a sequence of observed pulses) and subpulse drift. Subpulse modulation is an almost universal characteristic and we assume it to be a direct consequence of the plasma turbulence that itself is now widely believed to be the source of the radio emisssion (see Asseo & Porzio 2006). Subpulse drift with measurable values of both longitudinal separation of successive subpulses P 2 and band separation P 3 was detected in 72 pulsars and these can be compared with a set of 113 which do not show detectable subpulse drift. The distributions of both sets as a function of age are wide and are broadly similar except that only 7 of the 72 have an age less than 1 Myr as opposed to 31 of the set of 113. The distributions as functions of the parameter X whose critical value X c = 6 . 5 defines the CR pair creation threshold are also wide but the set of 113 has an excess at X > X c . There are 35 pulsars with X > X c as opposed to only 7 of the 72. This is consistent with, but does not prove, the existence of two populations which separate as X moves with age to values below X c . We propose that the Ω · B < 0 set show the phenomena of mode-changes, nulls and subpulse drift as they age through growth of the ion-proton beam quasi-longitudinal mode (mechanism (b) of Section 2.3 but see also the comments there on more obscure processes of pair creation for pulsars of this spin direction). Pair creation by ICS photons in the Ω · B > 0 case has been very fully investigated by Hibschman & Arons (2001) and by Harding & Muslimov (2002, 2011) and our assumption is that this is the source of coherent emission in the population that has no nulls or subpulse drift. Having divided the Weltevrede et al list into two populations, we should compare these with the extensive survey of nulls made by Wang et al (2007). No pulsar in Table 1 of Wang et al appears in the earlier Weltevrede et al list, presumably because they were not known or did not satisfy the selection criteria. Of the 46 pulsars with measured null fractions given in Table 2 of Wang et al, 18 also do not appear in Weltevrede et al, but 20 are listed with a measured value of P 3 and 8 have no detected P 3 . A smaller but more recent list of nulling pulsars in the paper of Gajjar, Joshi & Kramer (2012) includes a further 8 that were not considered by Wang et al, of which 2 have measured values of P 3 but 6 do not appear in the paper of of Weltevrede et al. Of the 8 pulsars that have nulls but no detected P 3 , 4 have null fractions 1 -f on < 0 . 01 and B0656+14 is the special case described in Section 4.1. There are only 3 pulsars having substantial null fractions but no detected P 3 . These are the otherwise unremarkable B1112+50, B2315+21 and B2327-20. But it is not necessarily correct to regard them as anomalous because, as noted by Weltevrede et al, drift bands are often indistinct and detectable only by two-dimensional spectral analysis. We suggest that the results of this comparison are by no means inconsistent with our division of the Weltevrede et al pulsars into two populations. A different, but perhaps more anomalous set are the three pulsars known to exhibit nulls of very long duration, of the order of 10 d. The first of these to be found (B1931+24; Kramer et al 2006) enabled spin-down rate measurements to be made separately in both on and off states of emission. This was followed by similar measurements for J1832+0029 (Lorimer et al 2012) and J1841-0500 (Camilo et al 2012). In each case, the off-state spin-down rate was about half that of the on state. But these pulsars have quite large values X = 3 . 6, 2 . 5 and 6 . 6 respectively, close to the critical value X c = 6 . 5, and may quite possibly support self-sustaining CR pair creation during the on but not the off state. The consequent difference in the flux and nature of particles passing through the light cylinder appears to be the only physically plausible mechanism for a spin-down torque change of this magnitude. But the 10 6 s time-scale for both on and off states of emission appears a little too long compared with times of the order of 10 τ rl found from equation (22). The question of quasi-periodicity with such time-scales should also be addressed, as in the case of the RRATs (Palliyaguru et al 2011). We have stressed that our model is not random, but deterministic. Quasi-periodicites are therefore not impossible in principle, but remain quite difficult to explain. Changes in the polar-cap acceleration potential have little direct effect on the spin-down torque. Changes in the proton-ion composition occurring in the process (b) nulls described in section 4.2 would lead to a change in the mean charge to mass ratio of particles crossing the light cylinder, but it must be doubted whether mechanism (b) could explain the large observed difference in spin-down torque. However, it is not yet clear that this is a universal feature of nulls. The only other pulsar for which measurements have been made (PSR B0823+26: Young et al 2012) has a fractional upper limit of 0 . 06 for the change in spin-down torque. It has a value X = 2 . 6, rather below the putative critical value X c = 6 . 5 denoting the pair creation threshold, so that the relatively small change in torque could be consistent either with relatively weak pair production or with process (b). In the absence of electron and positron densities large enough to adjust and so cancel an electric field, as in mechanism (a), we have to remember that acceleration or deceleration remains at higher altitudes beyond η ∼ 10 as a consequence of natural flux-line curvature. But at this stage, growth of the quasi-longitudinal mode has already occurred so that further acceleration would have negligible effect on the coherent radio emission. This is a further distinction between the two populations. In the Ω · B < 0 case, the polar cap of open magnetic flux lines may well have an approximately semi-circular shape. In this case, the observed drift of subpulses would be along a diameter instead of an arc of a circle. Although all the above problems remain, the model Ω · B < 0 polar cap described here has some positive features. It does not require that neutron-star magnetic fields lie in a particular interval. This is important because radio pulsar inferred polar fields can vary by up to six orders of magnitude. The physical processes in electromagnetic shower development or in the photoelectric transitions exist in the zero-field limit and do not change in any qualitative way with increasing field. The model is deterministic, but chaotic, but is incomplete in that its use of finite elements for the reason stated at the end of Section 3.1 and the very elementary nature of the approximation made in equation (17) appear to preclude the spontaneous appearance of subpulse drift from the random initial state which is used. It is possible to do no more than assert that the model can support subpulse drift in a quasi-stable way. It is unfortunate that quantitative model predictions depend so much on surface atomic number and particularly on whole-surface temperature T s , parameters which are not well-known. Cooling calculations (see the review of Yakovlev & Pethick 2004) show that T s falls steeply at ages greater than 1 Myr. Bearing in mind that, for a neutronstar with the mass and radius assumed here, the observerframe temperature is T ∞ s ≈ 0 . 8 T s , we can see that the temperatures assumed in Table 1 fall well below the values currently observable. Although cooling in this interval is photon-dominated, the whole-surface temperature must be regarded as very uncertain. But having made these reservations, the model does represent a physically-realistic framework for understanding the reasons why RRATs and the varied phenomena of modechanges, nulls and subpulse drift appear during neutron-star aging. It may be that further observations at frequencies below 100 MHz will provide evidence for the existence of a population emitting by process (b) and having spectra biassed towards lower frequencies than those for the process (a) population.", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "This paper has been typeset from a T E X/ L A T E Xfile prepared by the author.", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "Cosmological parameters from WMAP 7 year data are re-analyzed by substituting a pixel-based likelihood estimator to the one delivered publicly by the WMAP team. Our pixel based estimator handles exactly intensity and polarization in a joint manner, allowing to use low-resolution maps and noise covariance matrices in T, Q, U at the same resolution, which in this work is 3.6 · . We describe the features and the performances of the code implementing our pixel-based likelihood estimator. We perform a battery of tests on the application of our pixel based likelihood routine to WMAP publicly available low resolution foreground cleaned products, in combination with the WMAP high/lscript likelihood, reporting the differences on cosmological parameters evaluated by the full WMAP likelihood public package. The differences are not only due to the treatment of polarization, but also to the marginalization over monopole and dipole uncertainties present in the WMAP pixel likelihood code for temperature. The credible central value for the cosmological parameters change below the 1 σ level with respect to the evaluation by the full WMAP 7 year likelihood code, with the largest difference in a shift to smaller values of the scalar spectral index n S . Key words: Cosmology: cosmic microwave background, cosmological parameters", "pages": [ 1 ] }, { "title": "F. Finelli 1 , 2 /star , A. De Rosa 1 † , A. Gruppuso 1 , 2 ‡ , D. Paoletti 1 , 2 §", "content": "1 INAF-IASF Bologna, Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna Istituto Nazionale di Astrofisica, via Gobetti 101, I-40129 Bologna, Italy 2 INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy 12 November 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The anisotropy pattern of the cosmic microwave background (CMB) is a treasure for understanding the costituents of our Universe and how it evolved from the Big Bang. Under the assumption of isotropy and Gaussianity of CMB fluctuations, the power spectra of intensity and polarization anisotropies include all the compressed information on our Universe through the determination of the cosmological parameters. There has been a tremendous improvement in the estimate of cosmological parameters driven by the increasingly better quality of CMB data, mainly due to the full sky observations in temperature and polarization by the Wilkinson Microwave Anisotropy Probe (WMAP), (see Larson et al. (2010); Komatsu et al. (2010) and references therein) and to the small angular scales measurements by QUaD in polarization (Brown et al. 2009), by the South Pole Telescope (Lueker et al. 2010; Keisler et al. 2011; Reichardt et al. 2012) and the Atacama Cosmology Telescope (Das et al. 2011; Dunkley et al. 2011) in temperature. Planck will lead to a drastic improvement of CMB full sky maps in temperature and polarization, leading to an eagerly expected improvement in cosmological parameters with uncertainties at the percent level (Planck Collaboration 2005). A joint likelihood analysis in temperature and polarization is one of the accepted methods in securing the scientific expectations of observational achievements in terms of cosmological parameters. Although the likelihood could be written exactly in the map domain under the Gaussian hypothesis, its computation is almost prohibitive already at the resolution of 2 degrees, whereas cosmological information is encoded in the temperature and polarization power spectra up to the angular scales of the order of few arcminutes, where the Silk damping suppress the CMB primary anisotropy spectrum. It is now commonly accepted to use an hybrid approach which combines a pixel approach at low resolution with an approximated likelihood based on power spectrum estimates at high multipoles (see Bond, Jaffe and Knox (2000); Verde et al. (2003); Hamimeche and Lewis (2008) for some of these approximations). Since the three year release of the full polarization information, the WMAP team adopted such a hybrid scheme approach, which has been suggested independently in Efstathiou (2004); Slosar, Seljak and Makarov (2004); O'Dwyer et al. (2004); Efstathiou (2006). At a first appearance of the three year data, the WMAP team adopted a pixel approach on HEALPIX (Gorski et al. 2005) resolution N side = 8 1 temperature and polarization maps, and considered the high/lscript approximated likelihood to start at /lscript = 13 in temperature and /lscript = 24 in polarization and temperaturepolarization cross-correlation for the determination of cosmological parameters in Spergel et al. (2007). The WMAP team treats separately temperature and polarization as explained in Page et al. (2007) and Hinshaw et al. (2007), by using the approximation that the noise in temperature is negligible. As a consequence, the WMAP likelihood code includes either ( Q,U ) and the temperature-polarization crosscorrelation in the same sub-matrix. It was then shown by Eriksen et al. (2007) that by increasing the resolution of the temperature map to HEALPIX N side = 16 and therefore the multipole of transition to high/lscript approximated likelihood in temperature from /lscript = 12 to /lscript = 30, the mean value for the scalar spectral index n s shifted to higher values by a 0.4 σ . The asymmetric handling of the low-resolution temperature map at N side = 16 and polarization at N side = 8, became the final treatment of the three year data release. This low/lscript likelihood aspect in the WMAP hybrid approach has not changed since the final release of the WMAP 3 year data to the current WMAP 7 year one. In this paper we wish to perform an alternative determination of the cosmological parameters from WMAP 7 public data, substituting the WMAP low/lscript likelihood approach with a pixel based likelihood code which treats T, Q, U at the same HEALPIX resolution N side = 16 connected to the standard WMAP high/lscript package. In this analysis we therefore increase the resolution of polarization products digested by the pixel base likelihood from N side = 8 to N side = 16, in analogy with what done by Eriksen et al. (2007) for temperature only. The WMAP 7 year foreground cleaned ( Q,U ) maps, covariance matrices and masks at the resolution N side = 16 are also publicly available at http://lambda.gsfc.nasa.gov: therefore, all data used in this paper are made available by the WMAP team. The paper is organized as follows. In Section II we briefly describe the WMAP hybrid approach to the likelihood, with particular care to the low multipole part. In Section III we describe our pixel approach, implemented in the BoPix code. We then present in Section IV the cosmological parameters obtained by using our alternative pixel approach in place of the WMAP one for a ΛCDM scenario. In Section V we extend our investigations to other cosmological models. In Section VI we draw our conclusions.", "pages": [ 1, 2 ] }, { "title": "2 A BRIEF DESCRIPTION OF THE WMAP HYBRID LIKELIHOOD ANALYSIS", "content": "In the map domain, the likelihood as function of the cosmological parameters { θ } where the data, d = s + n , is a CMB fully polarized map, considered as a vector combining T , Q and U foreground reduced maps, the sum of signal s and noise n ; the quantity C = S + N is the total covariance matrix, the sum of the CMB signal covariance matrix S ( θ ), and the noise matrix N . The signal covariance matrix is constructed by the power spectra C XY /lscript , where X,Y are any of T, E, B (Zaldarriaga and Seljak 1997) as given in Tegmark and de Oliveira-Costa (2001): if not otherwise stated, the sum over multipoles starts from /lscript = 2. The WMAP low/lscript likelihood is described in the Appendix of Page et al. (2007) and we report here the essentials. The WMAP approach is based on the assumption to ignore the noise in temperature, which leads to a simplification of the likelihood, useful from the numerical computation perspective. By assuming that the noise in temperature is negligible at low multipoles, the WMAP approach consists in rewriting Eq. (1) as: where S T is the temperature signal sub-matrix, the new polarization data vector is ˜ d P = ˜ s P + ˜ n P , with ˜ s P = ( ˜ Q, ˜ U ) given by with ˜ S P ( ˜ N P ) is the signal (noise) covariance matrix for the new polarization vector (Page et al. 2007). The noise covariance matrix for ( ˜ Q, ˜ U ) equals the original one for ( Q,U ) when the noise in temperature is zero (Page et al. 2007). As temperature a TT /lscriptm , the full-sky internal linear combination (ILC) map is used (Hinshaw et al. 2007). According to Page et al. (2007), Eq. (1) and Eq. (2) are mathematically equivalent when the temperature noise is ignored. With this assumption, the new form, Eq. (2), allows the WMAP approach to factorize the likelihood of temperature and polarization, with the information in their cross-correlation, C TE /lscript , retained in the polarization submatrix. As already mentioned in the introduction, temperature is considered at the HEALPIX resolution N side = 16 and smoothed with a Gaussian beam of 9.1285 · , whereas polarization is considered at N side = 8 and not smoothed. The range of multipoles used in the polarization sub-matrix is up to the Nyquist limit at N side = 8, i.e. /lscript P = 23. Two computation options are available for the temperature likelihood, Gibbs sampling (Jewell, Levin and Anderson 2004; Wandelt, Larson and Lakshminarayanan 2004; Eriksen et al. 2004) with a range of multipole considered up to /lscript T = 32 and direct pixel evaluation, with /lscript T = 30 2 . All the computations by the WMAP low/lscript likelihood reported here are performed with the option ifore =2 for temperature (we have checked that differences are minimal with respect to the alternative options ifore =0 and 1) and without considering marginalization over foreground uncertainties in polarization. The high/lscript likelihood, described in Larson et al. (2010) and in Verde et al. (2003), has been updated to beam/point sources uncertainties through the various subsequent WMAP releases (Hinshaw et al. 2007; Nolta et al. 2009). The high/lscript TT likelihood takes into account multipoles from /lscript = 31 ( /lscript = 33) when connected with the pixel (Gibbs) likelihood evaluation of the low resolution temperature data up to /lscript = 1200; the high/lscript TE (and TB when used) likelihood takes into account multipoles from /lscript = 24 (Page et al. 2007) to /lscript = 800. High/lscript EE and BB data have not used so far in the various relases of the WMAP likelihood code.", "pages": [ 2, 3 ] }, { "title": "3 BOPIX", "content": "BoPix computes the likelihood function in Eq. (1) for the parameter space { θ } which the C XY /lscript ( { θ } ) depend on, without any approximation and with the same resolution in temperature and polarization . BoPix is a multithreaded OpenMP Fortran90 library which can be connected to a sampler - to CosmoMC Lewis and Bridle (2002) in this work. The computation of the likelihood given in Eq. (1) requires an environment initialization, in which BoPix calculates the geometrical functions dependent on the cosine of the angle between two pixels and reads the noise covariance matrix (C-binary format). BoPix then starts to compute the signal covariance matrix S for a given C XY /lscript ( { θ } ) with a OpenMP routine with a high intrinsic level of parallel architecture, to which the noise covariance matrix N is summed. The full covariance matrix is then Cholesky decomposed. The computation of the determinant is obtained from the properties of the Cholesky decomposed matrix L : det C = (det L ) 2 . The term C -1 d is computed as the solution for the variable x (vector with dimension 3 N pix ) of the equation Cx = d . The matrix manipulations are implemented on LAPACK and BLAS mathematical libraries (as nag, essl, acml and mkl). There is an effort to improve the BoPix capabilities and performances (in terms of run time and memory) to make the direct likelihood evaluation at low resolution for cosmological parameters extraction as fast as possible, in particular by reducing the time spent for the Cholesky decomposition, and optimizing the combined scalability in memory and CPU time of this code; indeed, the resources required by BoPix are larger than those for the WMAP low/lscript likelihood code since the polarization sector is treated at higher resolution. At present, BoPix can handle maps and full noise covariances up to HEALPIX N side = 32 resolution. On IBM Power6 (4.2GHz) architecture, available at CINECA (http://www.cineca.it), with 64 threads on 64 logical CPUs (32 cores) BoPix can calculate the likelihood in about 0.3 seconds at N side = 16, and in about 15 seconds at N side = 32. At N side = 16 on the same IBM Power6, a good trade off between computation time and memory required is obtained for 2 sec with 8 cores. More details about performances and comparison among different platforms will be provided in De Rosa (2013).", "pages": [ 3 ] }, { "title": "4 DATA SET FOR BOPIX", "content": "We use the temperature ILC map smoothed at 9 . 1285 degrees and reconstructed at HealPix (Gorski et al. 2005) resolution N side = 16, the foreground cleaned (unsmoothed) low resolution maps and the noise covariance matrix in ( Q,U ) publicly available at the LAMBDA website http://lambda.gsfc.nasa.gov/ for the frequency channels Ka (23 GHz), Q (41GHz) and V (61 GHz) as considered by Larson et al. (2010) for the low /lscript analysis. These frequency channels have been co-added by inverse noise covariance weigthing accordingly to the WMAP team (Jarosik et al. 2007) where d i , c i are the foreground reduced polarization maps and covariances, respectively (for i=Ka, Q and V). The total foreground reduced inverse noise covariance matrix is therefore: This polarization data set has been extended to temperature considering the ILC map with an extra noise term, as suggested in Dunkley et al. (2009). We have therefore added to the temperature map a random noise realization with variance of σ 2 TT = 1 µK 2 and consistently, the noise covariance matrix for TT is taken to be diagonal with variance equal to 1 µK 2 . The total noise covariance N for WMAP 7 yr data is therefore: Let us note that this prescription of the noise in the temperature ILC map added to mitigate the uncertainties due to foreground cleaning violates the assumption that the noise in temperature is vanishing, used to obtain Eqs. (2,3,4) from Eq. (1). Two masks are considered: KQ85y7 for T and P06 for (Q, U). Monopole and dipole have been subtracted from the observed ILC map through the HealPix routine removedipole (Gorski et al. 2005). The same data set has been used for the WMAP 7 yr power spectrum re-analysis by the Quadratic Maximum Likelihood (QML) estimator BolPol in Gruppuso et al. (2011) (similar data set for WMAP 5 yr data were previously used in Gruppuso et al. (2009); Paci et al. (2010)).", "pages": [ 3 ] }, { "title": "5 COSMOLOGICAL PARAMETERS EXTRACTION", "content": "We use CosmoMC (Lewis and Bridle 2002) in order to compute the Bayesian probability distribution of model parameters. The pivot scale of the primordial scalar and tensor power spectra was set to k ∗ = 0 . 017 Mpc -1 , as recommended by Cortes, Liddle and Mukherjee (2009). We vary the physical baryon density Ω b h 2 , the physical cold dark matter density Ω c h 2 , the ratio of the sound horizon to the angular diameter distance at decoupling θ , the reionisation optical depth τ , the amplitude and spectral index of curvature perturbations n S and log 10 [10 10 A s ]. We assume a flat universe, and so the cosmological constant for each model is given by the combination Ω Λ = 1 -Ω b -Ω c . We set the CMB temperature T CMB = 2 . 725 K (Mather et al. 1999) and the primordial helium fraction to y He = 0 . 24. We assume three neutrinos with a negligible mass. In order to fit WMAP data, we use the lensed CMB and we follow the method implemented in CosmoMC consisting in varying a nuisance parameter A SZ which accounts for the unknown amplitude of the thermal SZ contribution to the small-scale CMB data points assuming the model of Komatsu and Seljak (2002). We use CAMB (Lewis, Challinor and Lasenby 2000) with accuracy setting of 1. We sample the posterior using the Metropolis-Hastings algorithm (Hastings 1970) at a temperature T = 1, generating four parallel chains and imposing a conservative Gelman-Rubin convergence criterion (Gelman and Rubin 1992) of R -1 < 0 . 005. With the settings specified above we extract cosmological parameters with the WMAP likelihood code (version v4p1) available at http://lambda.gsfc.nasa.gov/ as benchmarks. We prefer to not quote the estimates for the cosmological parameters performed by the WMAP team since the conventions and the CAMB version might differ from those used in Larson et al. (2010); Komatsu et al. (2010). We then extract cosmological parameters by substituting the WMAP low/lscript likelihood approach with BoPix. In doing this we implicitly use the WMAP inputs in polarization at N side = 16 as described in Section III and not those contained in the WMAP likelihood routine publicly available. Since temperature and polarization are treated at the same resolution by BoPix, we include the WMAP high /lscript likelihood starting at /lscript = 31 both in temperature and temperature-polarization cross-correlation when using BoPix, unless otherwise stated. Unless otherwise stated, in BoPix we vary the C /lscript up to /lscript = 30 and we use the publicly available file test cls v4.dat as a fiducial power spectrum to complete the full covariance at low resolution from /lscript = 31 to /lscript = 64, as done for temperature only by the WMAP pixel likelihood. We find small differences in the estimate of the cosmological parameters by substituting BoPix to the WMAP low/lscript likelihood, as reported in Table I. 3 The main difference between the estimate of the cosmological parameters derived by our alternative low/lscript likelihood code and the one obtained with the WMAP approach is in the spectral index n s : we obtain a value for n s which is 0.86 σ lower than the WMAP one. This change would lead to quantitative differences in the evidence against the Harrison-Zeldovich of the WMAP 7 yr data. However, also the other directly sampled 3 Note that the small differences of our results with the full WMAP 7 year likelihood with respect to the results reported by Larson et al. (2010) or Komatsu et al. (2010) might be ascribed to the different version of RECFAST used, different tools for extracting cosmological parameters or different conventions, such as the pivot scale k ∗ . s cosmological parameters differ from the WMAP estimate in about 0 . 5 σ , pointing towards values higher for the physical CDM abundance Ω c h 2 and the amplitude of scalar perturbations A S and smaller for the baryon physical content Ω b h 2 and optical depth τ . As a derived parameters, we have a higher value for the matter content Ω M and σ 8 , smaller for the present Hubble rate H 0 . We show more details about these different estimates in the two-dimensional plots of Fig. 1. These differences seems robust to the change in the multipole transition to the high likelihood approximation and to the change of the fiducial model to complete the covariance at low resolution. Special mention should be made for the case in which we do not consider /lscript T = /lscript P , but we adopt the same /lscript T = 30 and /lscript P = 23 adopted by the WMAP team, but with BoPix for low resolution: the differences with respect to the estimates by the full WMAP yr likelihood are slightly smaller than in the case of /lscript T = /lscript P = 30, as can be seen in Table I. This means that differences we find are not fully due to the different threshold multipoles for polarization adopted in the two low/lscript likelihood approaches. No appreciable differences are noticed by constructing the signal covariance matrix up to 3 N side instead up to 4 N side . This can be understood since this different prescription in constructing the signal covariance matrix is damped by the Gaussian smoothing in intensity and is much below the noise in polarization. We have performed a further test excluding A SZ , just for code comparison. We find a smaller discrepancy between the estimates for the cosmological parameters and the bestfits from the two likelihood approaches when the nuisance parameter A SZ is omitted (i.e. fixed to zero). This additional foreground parameter A SZ is not well constrained by WMAP,but it contributes to the shape of the final likelihood and to the marginalized values of the parameters (shifting slightly the value of n s , for instance). We have checked that the different realizations of the µ K rms noise added to the ILC temperature map in the WMAP and BoPix likelihood lead to much smaller differences than those reported. Most of these small differences reported in the estimate of the cosmological parameters interfere destructively because of the cosmic confusion (Efstathiou and Bond 1998) and the best-fits C /lscript from the two likelihood analysis agree /negationslash very well. We present the CMB bestfit C /lscript in temperature and lensing (the latter not entering in the likelihood evaluation) obtained by BoPix in combination with the WMAP 7 high/lscript likelihood in comparison with those obtained by the full WMAP 7 likelihood in Fig. 2. The difference in the best-fit C /lscript in temperature is consistent with the different central values for the cosmological parameters displayed in Table I. Note how the relative difference in the lensing is slightly larger than the one in temperature and does not decrease at high multipoles. Differences in polarization and temperature-polarization cross-correlation are smaller than the ones shown here. We have checked that the best-fit C /lscript obtained in this work by the full WMAP 7 likelihood has ∆( -2 log L WMAP ) = -7 . 42 with respect to the reference WMAP 7 test cls v4.dat ; the best-fit C /lscript obtained in this work by BoPix in combination with the high/lscript WMAP 7 likelihood provides a better fit, with ∆( -2log L WMAP ) = -7 . 75 with respect to the reference WMAP 7 test cls v4.dat . We have then tested BoPix against the WMAP likelihood within the same range of multipole, i.e. up to /lscript = 30: BoPix has been run on the low-resolution WMAP 7 yr N side = 16 products varying C TT /lscript , C EE /lscript , C TE /lscript up to /lscript = 30 and compared to the likelihood obtained by the WMAP 7 yr pixel based routine plus the high/lscript likelihood value for TE from /lscript = 24 to /lscript = 30. In this way we subtract the same high/lscript likelihood information from hybrid runs presented in Table I. By assuming Ω b h 2 = 0 . 02246, Ω c h 2 = 0 . 1117 and sound horizon θ = 1 . 03965, we obtain results quite consistent with the hybrid ones: a slight smaller value in the estimate of τ and n S and a larger one for A S , as shown in Fig. 3. As already mentioned, one important aspect of the WMAP 7 year low/lscript likelihood is to use two different resolution for temperature and polarization; the polarization information at HEALPIX resolution N side = 8 is used up to the Nyquist multipole, i.e. /lscript P = 23. We run the two low/lscript likelihoods with /lscript T = /lscript P = 16 to make sure that the differences are not due mainly to a mismatch in the polarization data sets. As reported in Table 2, the differences in the estimates of the parameters decrease, as expected, but do not disappear. Another important difference between BoPix and the WMAP7yr likelihood routine is the treatment of monopole and dipole for the temperature map. In the ILC temperature map with the additional noise of 1 µK rms used in BoPix, the monopole and dipole in the masked sky are removed; no monopole and dipole terms are considered in the construction of the covariance matrix. The WMAP 7 yr temperature pixel routine instead does not subtract the monopole and dipole in the masked sky; in the observed sky with the KQ85y7 mask, the ILC temperature map has an offset of -0.07 µK and a dipole C 1 = 4 . 6 µK 2 . To take into account monopole and dipole residuals, the full sky signal covariance matrix is modified according to Slosar, Seljak and Makarov (2004): where P 0 (cos θ ) = 1 and P 1 (cos θ ) = cos θ are the Legendre polynomials associated to monopole and dipole, respectively. The fixed amplitude of the monopole and dipole terms is taken to be equal to the quadrupole of the fiducial ΛCDM model, i.e. λ = 1262 µK 2 . The subtraction of monopole and dipole in the masked ILC map has a little impact on the esti- 0 mate of cosmological parameters. Cosmological parameters instead have a strong dependence on the amplitude λ of the monopole and dipole terms which contribute to the signal covariance matrix, as shown in Fig. 4. The results obtained by subtracting monopole and dipole in the ILC temperature map used by the WMAP 7 yr temperature pixel likelihood routine and setting λ = 0 in the construction of the temperature covariance matrix do not match with those obtained by BoPix, as shown in Fig. 4. Viceversa, by tuning the amplitude of the monopole and dipole term to 0 . 17 µK 2 the results of the WMAP 7 yr likelihood routine agrees with those by BoPix. We conclude that part, but not all, of the discrepancy between BoPix and WMAP 7 yr likelihood is due to the monopole and dipole marginalization in Eq. 7.", "pages": [ 3, 4, 5, 6 ] }, { "title": "6 OTHER EXTENDED COSMOLOGICAL MODELS", "content": "We now consider few cosmological models beyond the ΛCDM model which can be constrained by WMAP 7 year data only. We consider only the baseline l trans = 30 and all the other settings consistently with the previous section, unless otherwise stated. Gravitational Waves. We consider all inflationary models which can be described by the primordial perturbation parameters consisting of the scalar amplitude and spectral index ( A S , n S ), and the tensorto-scalar ratio r . In canonical single-field inflation, in the slow-roll limit, the tensor spectrum shape is not independent of the scalar one. We will consider a tensor spectrum with a tilt n T = -r/ 8, as predicted for canonical single-field inflation at first-order in slow-roll. Our marginalised 68%-credible interval for the scalar spectral index is given by n S = 0 . 977 +0 . 020 -0 . 021 , half a sigma redder than the result we obtain by the full WMAP 7 year likelihood 0 . 987 ± 0 . 020. At 95% confidence level, our result for the tensor-toscalar ratio is r < 0 . 36, fully consistent with the result we obtain from the full WMAP 7 year likelihood, i.e. r < 0 . 34. Let us note that, differently from the WMAP low/lscript likelihood code, BoPix include BB polarization in the construction of the covariance at low resolution. Estimates of the cosmological parameters including tensor modes are compared in Table 3. The differences in the ( n S , r ) are shown in Fig. 5 and are mainly due to a shift of the constraints at smaller values for n S , as occurs for the standard ΛCDM model discussed in the previous section. Theoretical predictions of few popular inflationary models (including reheating uncertainties where appropriate) are displayed. One of the phenomenological differences from the different constraints would be a minor tension for a massless self-interacting inflaton model with WMAP 7 year data only (see Komatsu et al. (2010); Finelli et al. (2010) as examples for an higher tension of the λφ 4 potential with observations when additional cosmological data sets are added to WMAP). Running of the scalar spectral index. s In this subsection we consider the variation of the scalar spectral index with wavelength, i.e. we allow n run to vary in the range [ -0 . 2 , 0 . 2]. Our marginalised 95%-credible interval for the scalar spectral index is given by -0 . 065 < n run < 0 . 042, which has to be compared with the result we obtain by the full WMAP 7 year likelihood -0 . 074 < n run < 0 . 030. The results, shown in Table 4 and Fig. 6, are both consistent with the hypothesis of no wavelength dependence of the scalar spectral index. Neutrino Mass. In this subsection we constrain the total mass of neutrinos ∑ m ν = 94Ω ν h 2 eV, allowing to vary the fraction of massive neutrino energy density relative to the total dark matter one f ν = Ω ν / Ω DM . At 95% confidence level, our result for the fraction of massive neutrinos is f ν < 0 . 113, whereas we obtain f ν < 0 . 094 from the full WMAP 7 year likelihood. The resulting neutrino mass bound at 95% confidence level is ∑ m ν < 1 . 4 eV, compared to 1 . 1 eV obtained r s from the full WMAP 7 year likelihood. The results are shown in Table 5 and Fig. 7. Cosmological Birefringence. Since one of the main differences between the WMAP low resolution likelihood code and BoPix is the treatment of the polarization sector, we now wish to analyze an extended cosmological model different from ΛCDM only in ( Q,U ) and the relative cross-correlation with the temperature. Cosmological birefringence refers to a non-vanishing interaction ∝ φF µν ˜ F µν between photon and a cosmological evolving pseudo-scalar φ , which would generate non-vanishing TB and EB correlations (Lue, Wang and Kamionkowski 1999) through a rotation α of the polarization plane of CMB photons along their path from the last scattering surface to the observer. The resulting polarization and cross temperature-polarization spectra would encode the particular redshift dependence of the parity violation interaction (Liu, Lee and Ng 2006; Finelli and Galaverni 2009). However, a phenomenological shortcut exists, commonly used in the literature and also adopted by the WMAP team, and consists to neglect the redshift dependence of α and simple ν predict the power spectra as Lue, Wang and Kamionkowski (1999): The above formulae are valid when the primordial B-mode polarization is negligible, which is assumed in this paper. We have therefore sampled α in radiants with a flat prior [ -0 . 5 0 . 5] plus the other six cosmological parameters of the ΛCDM model by inserting Eqs. (8). As shown in Table 6, our marginalised 68% (95%)-credible interval for α is α = -1 · . 3 +0 · . 6 +2 · . 3 -0 · . 7 -2 · . 3 in agreement with the full WMAP 7 year likelihood result which we find α = -1 · . 0 +0 · . 7 +2 · . 4 -0 · . 6 -2 · . 3 Either the result using BoPix or the one based on the full WMAP 7 year likelihood are consistent with vanishing cosmological birefringence at 95% CL just by assuming the statistical uncertainty, and the agreement increases by using the systematic uncertainty, which is estimated as 1 · . 4 by the WMAP team Komatsu et al. (2010). Since the weight of the high/lscript TB likelihood plays a relevant role in these constraints we have also considered the case in which this is not taken into account. Such setting which emphasizes the role of polarization on large angular scales would be relevant to show clearly the potential differences between BoPix and the WMAP pixel likelihood code. On using only low resolution products to constrain cosmological birefringence, by using BoPix on N side = 16 resolution Q,U maps and matrices we obtain α = -4 · . 2 +1 · . 9 +10 · . 2 -3 · . 1 -7 · . 5 , still in agreement with the values we find by the WMAP 7 likelihood on N side = 8 resolution Q,U maps and matrices α = -0 · . 2 +3 · . 6 +10 · . 0 -3 · . 6 -9 · . 9 Although with larger uncertainties, our results agree with vanishing cosmological birefringence at 95% CL, without invoking systematic uncertainties. Note also that our result agrees with the analysis on large angular scales by Gruppuso et al. (2012), where much tighter constraints are given probably because all the cosmological parameters except α are kept fixed. α τ The full posterior likelihood for α and its two dimensional contour in combination with the optical depth τ are shown in Fig. 8, which shows that no degeneracy between τ and α is observed in WMAP 7 yr data. Note that the slight preference at 68% CL for negative values of α when using only BoPix on low resolution products is consistent with the WMAP 7 yr TB and EB power spectra QML estimates at /lscript < 30 and presented in Gruppuso et al. (2011, 2012).", "pages": [ 7, 8, 9 ] }, { "title": "7 CONCLUSIONS", "content": "We have performed an alternative estimate of the cosmological parameters from WMAP 7 year public data, by substituting the WMAP 7 low/lscript likelihood with a pixel likelihood code which treats ( T, Q, U ) at the same resolution without any approximation. We have used this code at the HEALPIX resolution N side = 16 on foreground cleaned pub- lic data, therefore increasing the resolution of the pixel based polarization products used in our extraction of the cosmological parameters with respect to the WMAP standard one. We have consistently increased the transition multipole from /lscript = 24 to /lscript = 31 for the high/lscript WMAP 7 year temperaturepolarization cross-correlation likelihood and included the marginalization over the nuisance parameter A SZ . With this setting we have found estimates for the cosmological parameters consistent with those obtained by the full WMAP 7 year likelihood package, although for some parameters the differences are of half σ or more. These differences between the two low/lscript likelihood treatments we find are larger than the WMAP 7 yr likelihood uncertainties from tests on simulations reported in Larson et al. (2010); however, we need to keep in mind that our differences between two likelihood treatments are reported for real data, with WMAP 7 year beam/points source corrections and various marginalizations taken fully into account, differently from the simulation analysis performed in Larson et al. (2010). The difference between the two best-fit C TT /lscript for ΛCDM found by the two alternative likelihood treatments show a maximum of 4% around at /lscript ∼ 10 and oscillate with an amplitude below 1% for /lscript > 100 4 . A 5% percent difference is found in the two best-fits for the lensing power spectrum, whereas smaller differences are found for temperaturepolarization cross-correlation and polarization power spectra. We have shown how part of the discrepancy, but not all, can be ascribed to the monopole/dipole marginalization used in the WMAP temperature likelihood and described in Slosar, Seljak and Makarov (2004). On restricting to the ΛCDM model the most important difference is for the scalar spectral index n S , which decrease to 0.956 from the value 0.968 we obtain with the full WMAP 7 yr likelihood code, i.e. a decrease of 0.86 σ . This different value for n S would increase the evidence against the Harrison-Zeldovich spectrum from WMAP 7 yr data. This difference for n S is consistent with the one between the two best-fit C /lscript and depend only partially from the threshold multipole from which the high/lscript TE likelihood starts. Other previous alternative likelihood treatments also reported the most important discrepancy for the scalar spectral index (Eriksen et al. 2007; Rudjord et al. 2009). A smaller value for n S with respect to the estimate by the full WMAP 7 year likelihood code, always within 1 σ , is then seen in all the extension of ΛCDM considered here. No major changes are found for the 95 % credible intervals for the tensor to scalar ratio and for the running of the scalar spectral index. A slight degradation has been found for the 95 % credible interval on the neutrino mass. The case of cosmological birefringence has been taken as a sensitive test for the two alternative likelihoods, whose most relevant difference is the treatment of polarization on large scales. A slight difference on the posterior of the polarization angle α has been found 4 We have checked that either the difference between the two best-fit C /lscript or between the estimates of the cosmological parameters decrease when the nuisance parameter A SZ is set to zero in both alternative likelihood treatments. The net effect of the variation of this foreground parameter, which is unconstrained by the data, is to increase the differences between the estimates of the cosmological parameters from the two likelihood treatments for the ΛCDM model. when only low resolution data are used, whereas the results are fully consistent when the high/lscript TB data are added to both likelihoods.", "pages": [ 9, 10 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank Paolo Natoli for comments on the manuscript and for help in the generation of the data set used in Gruppuso et al. (2012), also used here, and Eiichiro Komatsu for useful comments. We thank Loris Colombo for comparison of our code BoPix with his independent pixel base code BFlike (Rocha et al. 2010). We thank the Planck CTP and C2 working groups for stimulating and fruitful interactions. We wish to thank Matteo Galaverni for useful discussion on cosmological birefringence, Luca Pagano for useful comments on the WMAP likelihood code and Jan Hamann for useful comments on the manuscript. We acknowledge the use of the SP6 at CINECA under the agreement LFI/CINECA and of the IASF Bologna cluster. We acknowledge use of the HEALPix (Gorski et al. 2005) software and analysis package for deriving the results in this paper. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. Work supported by ASI through ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2 and by MIUR through PRIN 2009 (grant n. 2009XZ54H2).", "pages": [ 10 ] }, { "title": "REFERENCES", "content": "Bond J. R., Jaffe A. H., Knox L. E., 2000, ApJ, 533 , 19. Brown M. L. et al. [QUaD Collaboration], 2009, ApJ, 705 , 978. Cortes M., Liddle A. R., and Mukherjee P., 2007, Phys. Rev. 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[ { "title": "ABSTRACT", "content": "We present the results of a proper motion wide-field near-infrared survey of the entire Upper Sco (USco) association ( ∼ 160 square degrees) released as part of the UKIRT Infrared Deep Sky (UKIDSS) Galactic Clusters Survey (GCS) Data Release 10 (DR10). We have identified a sample of ∼ 400 astrometric and photometric member candidates combining proper motions and photometry in five near-infrared passbands and another 286 with HK photometry and 2MASS/GCS proper motions. We also provide revised membership for all previously published USco low-mass stars and substellar members based on our selection and identify new candidates, including in regions affected by extinction. We find negligible variability between the two K -band epochs, below the 0.06 mag rms level. We estimate an upper limit of 2.2% for wide common proper motions with projected physical separations less than ∼ 15000 au. We derive a disk frequency for USco low-mass stars and brown dwarfs between 26 and 37%, in agreement with estimates in IC 348 and σ Ori. We derive the mass function of the association and find it consistent with the (system) mass function of the solar neighbourhood and other clusters surveyed by the GCS in the 0.2-0.03 M /circledot mass range. We confirm the possible excess of brown dwarfs in USco. Key words: Techniques: photometric - stars: low-mass, brown dwarfs; stars: luminosity function, mass function - galaxy: open clusters and associations: individual (Upper Sco) infrared: stars - methos: observational", "pages": [ 1 ] }, { "title": "N. Lodieu 1 , 2 †", "content": "1 Instituto de Astrof'ısica de Canarias (IAC), V'ıa L'actea s /n, E-38205 La Laguna, Tenerife, Spain 2 Departamento de Astrof'ısica, Universidad de La Laguna (ULL), E-38205 La Laguna, Tenerife, Spain Accepted 11 June 2021. Received 11 June 2021; in original form 11 June 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The knowledge of the number of stars and brown dwarfs as a function of mass in open clusters and star-forming regions is important to address the question of the universality of the initial mass function (Salpeter 1955; Miller & Scalo 1979; Scalo 1986; Kroupa 2002; Chabrier 2003; Kroupa et al. 2011). The advent of large optical and near-infrared detectors has shed light on the properties of low-mass stars and substellar objects in a variety of environments and enabled an in-depth study of the mass function well below the hydrogen-burning limit (see review by Bastian et al. 2010, and references therein). However, many surveys in young regions lack homogeneity in the multi-band photometric coverage and accurate proper motions for brown dwarf members, making interpretation of their mass spectrum sometimes difficult. The UKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al. 2007) 1 is a deep large-scale infrared survey conducted with the UKIRT Wide field CAMera (WFCAM; Casali et al. 2007) equipped with five infrared filters ( ZY JHK ; Hewett et al. 2006). All data are pipeline-processed at the Cambridge Astronomical Survey Unit Irwin et al. (CASU; 2004, Irwin et al. in preparation) 2 , processed and archived in Edinburgh, and later released to the community through the WFCAM Science Archive (WSA; Hambly et al. 2008) 3 . One of its components, the Galactic Clusters Survey (hereafter GCS) imaged ∼ 1000 square degrees homogeneously in ten star-forming regions and open clusters down to 0.03-0.01 M /circledot (depending on the age and distance of each region) to investigate the universality of the initial mass function. In addition to the photometry, the latest releases of the GCS provide proper motions measured from the different epochs, with accuracies of about five per year (mas/yr). The USco region is part of the nearest OB association to the Sun, Scorpius Centaurus, located at 145 pc (de Bruijne et al. 1997). Its precise age is currently under debate (Song et al. 2012): earlier studies using isochrone fitting and dynamical studies derived an age of 5 ± 2 Myr (Preibisch & Zinnecker 2002) in agreement with deep surveys (Slesnick et al. 2006; Lodieu et al. 2008) but recently challenged by Pecaut et al. (2012) who quoted 11 ± 2 Myr from a spectroscopic study of F stars at optical wavelengths. The association was targeted at multiple wavelengths, starting off in X rays (Walter et al. 1994; Kunkel 1999; Preibisch et al. 1998), but also astrometrically with Hipparcos (de Bruijne et al. 1997; de Zeeuw et al. 1999), and more recently in the optical (Preibisch et al. 2001; Preibisch & Zinnecker 2002; Ardila et al. 2000; Mart'ın et al. 2004; Slesnick et al. 2006) and in the near-infrared (Lodieu et al. 2006, 2007; Dawson et al. 2011; Lodieu et al. 2011; Dawson et al. 2012). Tens of brown dwarfs have now been confirmed spectroscopically as USco members (Mart'ın et al. 2004; Slesnick et al. 2006; Lodieu et al. 2006; Slesnick et al. 2008; Lodieu et al. 2008; Mart'ın et al. 2010; Dawson et al. 2011; Lodieu et al. 2011) and the mass function of this population determined well into the substellar regime (Slesnick et al. 2008; Lodieu et al. 2011). Three independent studies noticed that USco may harbour an excess of brown dwarfs (Preibisch et al. 2001; Lodieu et al. 2007; Slesnick et al. 2008). Five T-type candidates reported by Lodieu et al. (2011) have been rejected as astrometric members of the association (Lodieu et al. 2013, in press). In this paper we present a photometric and proper motionbased study of ∼ 50 square degrees in USco released as part of the UKIDSS GCS DR10 (14 January 2013) along with a revised analysis of the GCS Science Verification (SV) data (6.7 square degrees). This study is complemented by HK imaging and proper motion from the 2MASS/GCS cross-match for the remaining area of the association. Our work improves on previous studies by selecting members based on accurate proper motions provided by the GCS down to masses as low as 0.01 M /circledot and identifying candidates in regions previously unstudied and affected by heavy extinction. In Section 2 we present the photometric and astrometric dataset employed to extract USco member candidates. In Section 3 we review the list of previously published USco members recovered by our analysis and revise their membership. In Section 4 we identify new stellar and substellar member candidates based on five-band photometry and astrometry. In Section 5 we investigate the level of K -band variability for USco low-mass stars and brown dwarfs. In Section 7 we derive the cluster luminosity and (system) mass functions and compare it to earlier estimates for this cluster and others, along with that of the field population. This work is in line with our recent studies of the Pleiades (Lodieu et al. 2012), α Per (Lodieu et al. 2012), and Praesepe (Boudreault et al. 2012) clusters.", "pages": [ 1, 2 ] }, { "title": "2 THE SAMPLE", "content": "We selected point sources in the full USco region, defined by RA=230-252 degrees and declinations between -32 and -16 degrees (Fig. 1). We retrieved the catalogue using a Structure Query Language (SQL) query similar to our earlier studies of the Pleiades (Lodieu et al. 2012), α Per (Lodieu et al. 2012), and Praesepe (Boudreault et al. 2012). Briefly, we selected high quality point sources with JHK photometry, allowing for Z and Y non detections. The query returned a total number of 2,943,321 sources. We refer to this sample as the ' ZY JHK -PM'sample throughout the paper. Below we distinguish the region free of extinction and the one affected by reddening although we will show that the same photometric and astrometric criteria can be applied to provide a clean sample of member candidates. Proper motion measurements are available in the WFCAM Science Archive for UKIDSS data releases from DR9 for all the wide/shallow surveys with multiple epoch coverage in each field (i.e. the LAS, GCS and GPS). Details of the procedure are in Collins & Hambly (2012) and summarised in Lodieu et al. (2012) for the purpose of the Pleiades. The typical error bars on the GCS proper motions in USco are 4 mas/yr and 6 mas/yr, down to Z =19 mag and 20 mag, respectively (Fig. 3). First, we applied a crude photometric selection to work with a subsample of the entire catalogue. We selected all sources located to the right of the line running from (0.5,12) to (2.2,21.5) in the ( Z -J , Z ) colour-magnitude diagram (Fig. 4). We made sure that this line allowed us to recover known spectroscopic members (see Section 3). This sample contains 29,382 sources, divided into 9351 in the region free of extinction and 20,031 in the parts affected by reddening (Fig. 1; Table 1). The rest of the USco association is not covered with enough epochs to measure proper motions based only on GCS data. This is the case for the GCS SV (Table 1) and the area covered in HK only. In the case of the GCS SV area, we have ZY JHK photometry and proper motions measured from the 2MASS/GCS cross-match. Lodieu et al. (2007) identified member candidates in this part of the association (although with a slightly smaller area released at that time) and confirmed a large number as spectroscopic members (Lodieu et al. 2011). Dawson et al. (2012) also included this region in their study of the disk properties of USco low-mass and brown dwarfs, as did Riaz et al. (2012). We have 430 sources in the GCS SVregion, after applying the crude photometric selection described above. We added to those samples the full coverage of the GCS DR10, only imaged in the H and K passbands (hereafter the HK sample). We also measured proper motions from the 2MASS/GCS correlation. Our query returned a total of 7,328,848 to which we should remove the GCS SV and GCS DR10 samples as well as the region most affected by reddening (defined by R.A. = 245-249.5 degrees and dec between -25.6 and -19 degrees, see Table 1). We applied a crude photometric selection in the ( H -K , H ) colourmagnitude diagram, keeping only sources to the right of a line running from ( H -K , H ) = (0.2,12) to (0.7,18). We are left with 39,450 sources to investigate astrometrically (Sect. 4.1).", "pages": [ 3 ] }, { "title": "3 CROSS-MATCH WITH PREVIOUS SURVEYS", "content": "We compiled a list of USco members published over the past decades by various groups (Preibisch et al. 1998, 2001; Preibisch & Zinnecker 2002; Ardila et al. 2000; Mart'ın et al. 2004; Slesnick et al. 2006; Lodieu et al. 2006, 2007, 2008; Dawson et al. 2011; Lodieu et al. 2011; Dawson et al. 2012; Luhman & Mamajek 2012) to update their membership status with the photometry and astrometry provided by the GCS DR10 (Table 2). This list will serve as starting point to identify new member candidates in the GCS,estimate the mean (relative) proper motion of USco members, and derive the cluster luminosity and mass functions. We compiled a list of 2079 candidates, reduced to 1566 after removing multiple pairs. We cross-correlated this list of 1566 known member candidates with the ZY JHK -PMcatalogue using a matching radius of three arcsec and found 125 sources in common (red open squares in Fig. 3; Table 2). We repeated the same process with the GCS SV and HK -only areas, yielding 73 and 651 member candidates in common, respectively (Table 2). The number of known member candidates recovered in GCS DR10 is generally low because most surveys focussed on the northern area with right ascensions between 240 and 245 degrees and declinations above -25 · (see list of sources in Luhman & Mamajek 2012). Below we provide a few comments on the recovery rate for the early studies listed in Table 2.", "pages": [ 3, 4 ] }, { "title": "4.1 Astrometric selection", "content": "After the original photometric selection and the recovery of known member candidates, we plotted them as red open squares in Fig. 3. We observe two groups of objects, one centered on (0,0) made of field objects, and another one depicting the position of USco. We measured mean (relative) proper motions of -8.6 and -19.6 mas/yr in right ascension and declination, respectively, compared to the absolute values of -11 and -25 mas/yr from Hipparcos (de Bruijne et al. 1997; de Zeeuw et al. 1999). We noticed the same effect in the Pleiades (Lodieu et al. 2012), α Per (Lodieu et al. 2012), and Praesepe (Boudreault et al. 2012) because the GCS provides relative motions rather than absolute motions as in the case of Hipparcos. We applied a 3 σ astrometric selection using the error bars for each source from the GCS in both directions, leaving 87 of the 186 known member candidates in the ZY JHK -PM sample. We applied the same 3 σ astrometric selection to the full sample of point sources towards USco, both for the sample free of extinction and the one affected by reddening. In the former case, we are left with 700 of the original 9351 sources, and, in the latter, 1357 of the 20,031 objects (grey crosses in Fig. 3). We plot these proper motion member candidates as grey crosses in the colourmagnitude diagrams displayed in Fig. 4. We tested the influence of our choice of the relative proper motion values in right ascension and declination by adding and removing 1 mas/yr in both directions (about 20% of the mean error bars on the proper motions). We found that the numbers of candidates would change by less than 5.4% (677 and 738 candidates in the worst cases compared to 700) and 8.3% (1253 and 1468 compared to 1357) in the case of the ZY JHK -PM samples without and with reddening, respectively. For the remaining areas of the association, we measured the proper motions from the 2MASS/GCS cross-match, whose accuracy is about twice worse than the GCS proper motions (10 mas/yr down to J =15.5 mag). The mean proper motion of known spectroscopic members in the SV area is -8.5 and -19.2 mas/yr in right ascension and declination, respectively. As described above, those values differ from the absolute mean proper motion of USco but we used them for our 2MASS/GCS astrometric selection. We note that these values are very similar to the mean proper motions derived from the two GCS epochs. We are left with 242 sources out of the 430 original photometric candidates in the GCS SV area, after applying a 2 σ selection (Table C1). Changing the mean values of the proper motions in each direction by ± 1 mas/yr results in a number of member candidates that differs by less than 3.75%. For the HK -only area, the situation is worse than for the SV region because we have only two bands available ( H + K ) where the cluster sequence is not so well separated from the field stars along the line of the association as in the ( Z -J , Z ) colour- magnitude diagram. Hence, our final HK sample will be significantly more contaminated than the aforementioned samples. First, we applied a conservative photometric selection in the ( H -K , H ) diagram by considering only point sources to the right of a line running from ( H -K , H ) = (0.2,12) to (0.7,18). After this first step, we are left with 63,520 candidates. Second, we applied a 2 σ astrometric selection (i.e. 2 × 10 mas/yr or 95.4% completeness) in the H =12.5-15 mag interval (corresponding to masses ranging from (99% completeness) and 3 σ (99.9% completeness) would lead to 360 and 412 candidates, respectively.", "pages": [ 4, 5, 6 ] }, { "title": "4.2 Photometric selection", "content": "To further refine our list of USco member candidates we applied additional photometric cuts in various colour-magnitude diagrams for the ZY JHK -PMsample, defined as follows: The numbers of candidates afer the Z -K , Y -J , and J -K photometric selections are 279, 257, and 252, respectively, demonstrating that the most influencial criteria are the astrometric and Z -K selections. Indeed, the additional Y -J and J -K selection remove small numbers of cluster member candidates. We stress that these cuts were chosen to recover known spectroscopic members from earlier surveys (see Section 3). These photometric selections returned 252 candidates in the region free of extinction (filled black dots in Fig. 4) and 396 in the region affected by reddening (filled black dots in Fig. 5). Similarly, we identified 84 member candidates in the SV region of the GCS. We tested the influence of the choice of our selection lines on the final numbers of candidates, in the specific case of the ZY JHK -PM sample without extinction. We shifted each selection line to the left and to the right by 0.1 mag, which corresponds roughly to the error on the colour (i.e. 1 σ ) at the faint end of the sequence at J =18 mag. This shift corresponds to 2.5 σ at J =17 mag. We found that the shift to the blue of the six selection lines enumerated above yields roughly 20-30% more cluster member candidates. Similarly, a shift to the red gives about 20% less candidates in the ( Z -K , Z ) diagram and 41-43% less in the other two diagrams. We know that the level of contamination will be high in the region affected by reddening. Hence, we applied an additional photometric criterion in the ( Z -K , J -K ) two-colour diagram (Fig. 6) to remove giants and reddened stars based on the location of previously-known spectroscopic members (red open squares; Lodieu et al. 2011). We selected sources satisfying the criteria: This selection returned 201 and 120 sources (corresponding to 80% and 30% of the candidates left after the astrometric and the first three photometric selections) in the region with and without reddening, respectively. Only two candidates (or 2.5%) were rejected in the SV sample after applying this additional criterion . We list the coordinates, photometry, and proper motions of the candidates identified in ZY JHK -PM(including known members published by other groups) in Tables A1 and B1 in the Appendix for the regions with and without reddening, respectively. We display their distribution in Fig. 1. We provide the list of member candidates within the SV area in Table C1. We tested the influence of the choice of the crude selection in the ( Z -J , Z ) colour-magnitude diagram by choosing a line shifted by 0.1 mag to the left of the original choice. We applied again the same aforementionned criteria and arrived at the same numbers of candidates for the two ZY JHK -PM samples and the SV-only sample.", "pages": [ 6, 7 ] }, { "title": "4.3 Catalogue summary", "content": "To summarise, we have identified a total of 688 sources in four different regions within USco: 195 and 111 in the ZY JHK -PM regions without reddening and with extinction, 79 in the SV area, and 276 in the HK -only region. We found 11 sources in common among them, leaving 677 USco member candidates. We crossmatched this list with itself and found 4, 12, and 15 sources within 10, 50, and 100 arcsec of each other, pointing towards binary fractions for wide common proper motions of 0.6%, 1.8%, and 2.2% for projected separations of 1450 au, 7250 au, and 14500 au, respectively.", "pages": [ 7 ] }, { "title": "5 VARIABILITY AT YOUNG AGES", "content": "We investigate the variability of low-mass stars and brown dwarfs in USco using the two K -band epochs provided by the GCS. Figure 7 shows the ( K 1K 2) vs K 2 diagram for USco member candidates in the ZY JHK -PM sample. This analysis is not possible for the SV sample because no second K -band epoch is available. The brightening in the K 1 = 10.5-11.5 mag range is due to the difference in saturation between the first and second epoch, of the order of 0.5 mag both in the saturation and completeness limit. This is understandable because the exposure times have been doubled for the second epoch with relaxed constraints on the seeing requirement and weather conditions. We excluded those objects from our variability study. Overall, the sequence indicates consistent photometry between the two K epochs with very few objects being variable. At first glance, we spotted four potential variables in Figure 7 (Table 3). The GCS images do not show anything anomalous so we did not proceed further. We selected variable objects by looking at the standard deviation, defined as 1.48 × the median absolute deviation which is the median of the sorted set of absolute values of deviation from the central value of the K 1 -K 2 colour. We identified one potential variable object with a difference of 0.37 mag in the 11.5-12 mag range whereas the other three fainter candidates with differences between 0.1 and 0.16 mag lie just below the 3 σ of the median absolute deviations of 0.05-0.06 mag. These small variations of the order of 0.1-0.15 mag can be interpreted by the presence of cool spots in low-mass stars (e.g. Scholz et al. 2009). We conclude that the level of K -band variability at 5-10 Myr is small, with standard deviations of the order of 0.06 mag range, suggesting that it cannot account for the dispersion in the cluster sequence. We arrived at the same conclusions in the case of the Pleiades (Lodieu et al. 2012), α Per (Lodieu et al. 2012), and Praesepe (Boudreault et al. 2012) although these clusters are older (85, 120, and 590 Myr, respectively).", "pages": [ 8 ] }, { "title": "6 DISK FREQUENCY IN USCO", "content": "We cross-correlated our lists of candidates with the Wide Field Infrared Survey Explorer (WISE) All-sky release (Wright et al. 2010) using a matching radius of three arcsec. We found a total of 660 matches, divided into 194 in the ZY JHK -PM, 111 in the ZY JHK -PM region with extinction, 79 objects in the SVonly area, and 276 counterparts in the HK -only region, respectively (Table E1). We looked at the WISE images and found that all WISE counterparts to the GCS objects are detected in w 1 and w 2 . We classified the sources listed in Table E1 in two categories: 224 (33.9%) source detections in w 3 but not in w 4 and 58 (8.8%) objects detected in w 3 and w 4 marked as 1110 (open triangles in Figure 8) and 1111 (open squares in Figure 8), respectively. The latter objects detected in all WISE bands are unambiguous diskbearing low-mass stars and brown dwarfs. We considered two different methods to estimate the disk frequency of USco members with masses below 0.2 M /circledot , according to the BT-Settl models (Allard et al. 2012). We plot in Figure 8 the ( w 1 -w 2 , K ) and ( K -w 1 , w 1 -w 3 ) diagrams. The former represents a good discriminant to separate disk-less and disk-bearing objects according to Dawson et al. (2012). We note that we chose K as the infrared band rather than J as in Dawson et al. (2012) because all targets in our four samples have K -band photometry. This diagram is efficient in Taurus but may not be the best criterion for USco where many transition disks are found (Riaz et al. 2012). The ( K -w 1 , w 1 -w 3 ) colour-colour diagram, however, clearly separates brown dwarf and M dwarf disks (Pe˜na Ram'ırez et al. 2012) as well as primordial disks in USco (Figure 5 of Riaz et al. 2012). Weobserve three sequences in the ( w 1 -w 2 , K ) diagram (lefthand side panel of Figure 8): one sequence to the left likely made of non-members mainly from the ZY JHK -PMregion with extinction and the HK -only area where our contamination is expected to be higher than in the other two areas ( ZY JHK -PMwithout reddening and SV-only). Optical spectroscopy is needed to confirm our claims though. Broadly, we have the same number of sources with disks (16% or 107 sources with w 1 -w 2 /greaterorequalslant 0.4 mag) as potential contaminants (100 sources or 15% wit w 1 -w 2 /lessorequalslant 0.1 mag), implying that the frequency if disk-bearing USco members lies between 107/600 = 16.2% and 107/(660 -100) = 19.1%, following the arguments of Dawson et al. (2012). However, these numbers are lower limits because 17/58 (29.3%) w 3 + w 4 detections have w 1 + w 2 colours bluer than 0.4 mag i.e. lie in the middle sequence, yielding corrected disk fractions of 18.8 and 22.1%. We are certainly missing some disk-bearing members hiding among w 3 detections but it is harder to quantify without a cautious fitting of the spectral energy distributions (beyond the scope of this paper), as discussed in details in Riaz et al. (2009) and Dawson et al. (2012). Nonetheless, we can place an upper limit of (224 + 58)/660 = 42.7% on the overall disk fraction for USco low-mass stars and brown dwarfs. Similarly, we can set a lower limit of 6/58 = 10% based on the six unambiguous disk objects with w 1 + w 2 /greaterorequalslant 0.8 among sources detected in all four WISE bands. We observe two groups of objects in the ( K -w 1 , w 1 -w 3 ) diagram (right-hand side panel of Figure 8) depicting photospheric sources ( w 1 -w 3 /lessorequalslant 1.2 mag) and disk-bearing members ( w 1 -w 3 /greaterorequalslant 1.5 mag). Objects in the middle may be good candidates to transition disks (Riaz et al. 2012). Among the 282 USco member candidates detected in three or four WISE bands, we have 89 sources with w 1 -w 3 /greaterorequalslant 1.5 mag, implying a disk fraction of 31.5% (most likely range of 26.6-37.1%). This fraction would increase by 5% if we include the potential transition disks. The disk frequencies derived by both methods are consistent within the error bars although the second one is on average higher. Our values should be compared with the 23 ± 5% of Dawson et al. (2012) for USco brown dwarfs and 25 ± 3% of Luhman & Mamajek (2012) for M4-L2 members. For earliertype members the disk fractions range from 10% for K0-M0 dwarfs (Luhman & Mamajek 2012) to 19% for K0-M5 dwarfs (Carpenter et al. 2006). Our disk fraction for USco substellar members is consistent with the 42 ± 12% and 36 ± 8% disk frequencies for brown dwarfs in IC 348 (1-3 Myr; Luhman et al. 2005) and σ Orionis (1-8 Myr; Pe˜na Ram'ırez et al. 2012).", "pages": [ 8, 9 ] }, { "title": "7 THE INITIAL MASS FUNCTION", "content": "We derive the cluster luminosity and system mass functions from our astrometric and photometric sample of ∼ 320 USco member candidates distributed over ∼ 50 square degrees (the ZY JHK -PM sample). We did not attempt to correct the mass function for binaries for two reasons. First, the presence of disks around USco low-mass stars and brown dwarfs tend to displace these sources to the right-hand side of the sequence, implying that disentangling binaries from disk-bearing members is harder than in the case of more mature clusters like the Pleiades (Lodieu et al. 2012) and Praesepe (Boudreault et al. 2012). Second, a recent highresolution imaging survey of 20 USco spectroscopic members fainter than J =15 mag by Biller et al. (2011) resolved only one binary, pointing towards a binary frequency lower than 10%. This fraction is lower than the uncertainty on the number of sources per mass bin at the low-mass end, assuming Gehrels error bars.", "pages": [ 9 ] }, { "title": "7.1 The age of USco", "content": "de Zeeuw & Brand (1985) and de Geus et al. (1989) derived an age of 5-6 Myr comparing sets of photometric data with theoretical isochrones available at that time. Later, Preibisch & Zinnecker (1999) confirmed that USco is likely 5 Myr-old with a small dispersion on the age by placing about 100 spectroscopic members on the Hertzsprung-Russel diagram. This age estimate was later supported by Slesnick et al. (2008) from a wide-field optical study of low-mass stars and brown dwarfs with spectroscopic membership: these authors derived a mean age of 5 Myr with an uncertainty of 3 Myr taking into account all possible sources of uncertainties. More recently, Pecaut et al. (2012) argued for an age older (11 ± 2 Myr) from an analysis of the F-type members of the full ScorpiusCentaurus association. They determined older ages for all three subgroups forming Scorpius-Centaurus, including ages of 16 and 17 Myr for Upper Centaurus-Lupus and Lower Centaurus-Crux, which are older than the ages derived by Song et al. (2012) from the abundances of lithium in the spectra of F/K members of the association. In this work, we adopt an age of 5 Myr for USco but we will also investigate the influence of age on the shape of the mass function (Section 7.3).", "pages": [ 9 ] }, { "title": "7.2 The cluster luminosity function", "content": "We construct the cluster luminosity function from our astrometric and photometric ZY JHK -PMsample of 320 candidate members spanning J =11-17.5 mag. We display the luminosity function divided into bins of 0.5 mag in the left panel of Fig. 9 with Gehrels error bars (Gehrels 1986). We note that the first and last bins are lower limits because of incompleteness at the bright and faint ends. We observe that the number of objects decreases with fainter magnitudes i.e. cooler temperatures.", "pages": [ 10 ] }, { "title": "7.3 The cluster mass function", "content": "Weadopt the logarithmic form of the Initial Mass Function as originally proposed by Salpeter (1955): ξ ( log 10 m ) = d n /d log 10 ( m ) ∝ m -α . We converted the luminosity into a mass using the BT-Settl models (Allard et al. 2012) and the J -band filter for all sources. We assumed a distance of 145 pc (van Leeuwen 2009) and an age of 5 Myr for USco (Table 4). We also considered an age of 10 Myr to compute the mass function (Table 5), keeping the same magnitude bins as a starting point. We compare in Fig. 9 the mass function derived from our ZY JHK -PM sample with the previous sample of spectroscopic members presented in Lodieu et al. (2011). We emphasise that the first and last bins are incomplete because of saturation on the bright side and incompleteness on the faint side. Both mass functions were computed assuming an age of 5 Myr but the earlier determination made use of the NextGen (Baraffe et al. 1998) and DUSTY (Chabrier et al. 2000) whereas we use the BT-Settl (Allard et al. 2012) with the new dataset presented here. Nonethless, we observe that both mass functions are similar, with a possible excess of brown dwarfs below ∼ 0.03 M /circledot as originally claimed by Preibisch et al. (2001) and Slesnick et al. (2008). This excess of substellar objects is enhanced if we consider an age of 10 Myr although it may be the result of the mass-luminosity relation, which does not reproduce the M7/M8 gap proposed by Dobbie et al. (2002) occurring around 0.015-0.02 M /circledot in USco (left panel in Fig. 9). Masses of brown dwarfs at 10 Myr are higher than at 5 Myr, making the number of substellar objects piling up at higher masses, causing an enhanced bump in the mass function. We computed the mass function using ages of 3 Myr and 1 Myr and found that the excess of brown dwarfs seems to disappear only if the association is 1 Myr, which is lower than any age estimate from previous studies . We observe that the high-mass part ( > 0.03 M /circledot ) of the USco mass function is well reproduced by the log-normal form of the field mass function (Chabrier 2003, 2005), independently of the age chosen for USco (see Section 7.1). We investigated the role of the bin size on the shape of the mass function, assuming an age of 5 Myr and a distance of 145 pc. We considered two options: on the one hand, we binned the mass function by a factor of two i.e. six bins instead of 12 (open triangles in Fig. 9), and, on the other hand, we employed four bins (open diamonds in Fig. 9) after removing the first and last bins which are incomplete (Table 4). We conclude that, overall, the bin size does not affect the shape of the mass function and the possible presence of the excess of low-mass brown dwarfs.", "pages": [ 10 ] }, { "title": "7.4 Comparison with other clusters", "content": "comes from a photometric and astrometric selection using GCS DR9 in the same manner as this work in USco. Both mass functions are very comparable in the interval where they overlap, from ∼ 0.2 M /circledot down to ∼ 0.03 M /circledot and match the log-normal form of the field mass function (Chabrier 2005). We note that the Pleiades mass function is comparable to the mass functions in α Per (Lodieu et al. 2012) and Praesepe (Boudreault et al. 2012) using the same GCS DR9database in an homogeneous manner. This result is in line with the numerous mass functions plotted in Figure 3 of Bastian et al. (2010), demonstrating the similarities between mass functions in many clusters over a broad range of masses. We plot in Figure 10 the mass function for the young (1-8 Myr) σ Ori cluster derived from a pure photometric study using the fourth data release of the GCS (blue open squares; Lodieu et al. 2009). The shape of the σ Ori mass function agrees with the field mass function in the 0.2-0.01 M /circledot mass range, as noted by independent studies of the cluster (B'ejar et al. 2001; Caballero et al. 2007; Bihain et al. 2009; B'ejar et al. 2011; Pe˜na Ram'ırez et al. 2 012).", "pages": [ 11 ] }, { "title": "8 SUMMARY", "content": "We have presented the outcome of a deep and wide photometric and proper motion survey in the USco association as part of the UKIDSS GCS DR10. The main results of our analysis are: ted their membership with the proper motion and photometry available from GCS DR10. This paper provides a full catalogue of photometric and astrometric members from 0.2 M /circledot down to ∼ 0.01 M /circledot in the southern part of the USco association. This catalogue is complemented by a complete census of the stellar and substellar populations in full association USco down to 0.02-0.015 M /circledot . This work will represent a reference for many years to come. We foresee further improvement when a second epoch will be obtained for the northern part of the association as part of the VISTA hemisphere survey (Emerson et al. 2004; Dalton et al. 2006). Our study provides a legacy sample that can be used to study the disk properties of low-mass stars and brown dwarfs in USco (Carpenter et al. 2006; Scholz et al. 2007; Riaz et al. 2009; Dawson et al. 2012; Luhman & Mamajek 2012), their binary properties (Biller et al. 2011), and their distribution in the association as a function of spectral type once a complete spectroscopic follow-up is available.", "pages": [ 11, 12 ] }, { "title": "ACKNOWLEDGMENTS", "content": "NL is funded by the Ram'on y Cajal fellowship number 08-30301-02 and the national program AYA2010-19136 funded by the Spanish ministry of science and innovation. I thank the anonymous referee for her/his constructive and quick report. This work is based in part on data obtained as part of the UKIRT Infrared Deep Sky Survey (UKIDSS). The UKIDSS project is defined in Lawrence et al. (2007). UKIDSS uses the UKIRT Wide Field Camera (WFCAM; Casali et al. 2007). The photometric system is described in Hewett et al. (2006), and the calibration is described in Hodgkin et al. (2009). The pipeline processing and science archive are described in Irwin et al. (2004) and Hambly et al. (2008), respectively. We thank our colleagues at the UK Astronomy Technology Centre, the Joint Astronomy Centre in Hawaii, the Cambridge Astronomical Survey and Edinburgh Wide Field Astronomy Units for building and operating WFCAM and its associated data flow system. We are grateful to France Allard for placing her latest BT-Settl models on a free webpage for the community. This research has made use of the Simbad database, operated at the Centre de Donn'ees Astronomiques de Strasbourg (CDS), and of NASA's Astrophysics Data System Bibliographic Services (ADS). This publication makes use of data products from the Two Micron All Sky Survey (2MASS), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.", "pages": [ 12 ] }, { "title": "REFERENCES", "content": "Bihain G., et al. 2009, A&A, 506, 1169 Casali M., et al. 2007, A&A, 467, 777 Dalton G. B., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 6269 of Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, The VISTA infrared camera Dawson P., Scholz A., Ray T. P., 2011, A&A Dobbie P. D., Pinfield D. J., Jameson R. F., Hodgkin S. T., 2002, MNRAS, 335, L79 Emerson J. P., Sutherland W. J., McPherson A. M., Craig S. C., Dalton G. B., Ward A. K., 2004, The Messenger, 117, 27 Gehrels N., 1986, ApJ, 303, 336 Hodgkin S. T., Irwin M. J., Hewett P. C., Warren S. J., 2009, MNRAS, 394, 675 Irwin M. J., et al. 2004, eds, Optimizing Scientific Return for Astronomy through Information Technologies. Edited by Quinn, APPENDIX A: USCO MEMBER CANDIDATES IN THE ZY JHK -PMREGIONFREE OF REDDENING APPENDIX B: USCO MEMBER CANDIDATES IN THE ZY JHK -PMREGIONAFFECTED BY REDDENING APPENDIX C: USCO MEMBER CANDIDATES IN THE GCS SV APPENDIX D: USCO MEMBER CANDIDATES IN THE HK SAMPLE APPENDIX E: USCO MEMBER CANDIDATES WITH WISE PHOTOMETRY", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "Estimating the uncertainty on the matter power spectrum internally (i.e. directly from the data) is made challenging by the simple fact that galaxy surveys o ff er at most a few independent samples. In addition, surveys have non-trivial geometries, which make the interpretation of the observations even trickier, but the uncertainty can nevertheless be worked out within the Gaussian approximation. With the recent realization that Gaussian treatments of the power spectrum lead to biased error bars about the dilation of the baryonic acoustic oscillation scale, e ff orts are being directed towards developing non-Gaussian analyses, mainly from N-body simulations so far. Unfortunately, there is currently no way to tell how the non-Gaussian features observed in the simulations compare to those of the real Universe, and it is generally hard to tell at what level of accuracy the N-body simulations can model complicated nonlinear e ff ects such as mode coupling and galaxy bias. We propose in this paper a novel method that aims at measuring non-Gaussian error bars on the matter power spectrum directly from galaxy survey data. We utilize known symmetries of the 4-point function, Wiener filtering and principal component analysis to estimate the full covariance matrix from only four independent fields with minimal prior assumptions. We assess the quality of the estimated covariance matrix with a measurement of the Fisher information content in the amplitude of the power spectrum. With the noise filtering techniques and only four fields, we are able to recover the results obtained from a large N = 200 sample to within 20 per cent, for k /lessorequalslant 1 . 0 h Mpc -1 . We further provide error bars on Fisher information and on the best-fitting parameters, and identify which parts of the non-Gaussian features are the hardest to extract. Finally, we provide a prescription to extract a noise-filtered, non-Gaussian, covariance matrix from a handful of fields in the presence of a survey selection function. Key words: Large scale structure of Universe - Dark matter - Distance Scale - Cosmology : Observations - Methods: data analysis", "pages": [ 1 ] }, { "title": "Joachim Harnois-D'eraps 1 , 2 /star and Ue-Li Pen 1 † 1 Canadian Institute for Theoretical Astrophysics, University of Toronto, M5S 3H8, Canada", "content": "26 September 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The matter power spectrum contains a wealth of information about a number of cosmological parameters, and measuring its amplitude with per cent level precision has become one of the main task of modern cosmology (York et al. 2000; Colless et al. 2003; Schlegel & others. 2007; Drinkwater et al. 2010; LSST Dark Energy Science Collaboration 2012; Ben'ıtez et al. 2009, Pan-STARRS 1 , DES 2 ). Cosmologists are especially interested in the detection of the Baryonic Acoustic Oscillation (BAO) scale, which allows to measure the evolution of the dark energy equation of state w ( z ) (Eisenstein et al. 2005; Hutsi 2006; Tegmark et al. 2006; Percival et al. 2007; Blake et al. 2011; Anderson et al. 2012). Estimating the mean power spectrum from a galaxy survey is a challenging task, as one needs to incorporate the survey mask, model the redshift distortions, estimate the galaxy bias, etc. For this purpose, many data analyses follow the prescriptions of Feldman et al. (1994) (FKP) or the Pseudo Karhunen-Lo'eve (Vogeley & Szalay 1996)(PKL hereafter), which provide unbiased estimates of the underlying power spectrum, as long as the observe field is Gaussian in nature. When these methods are applied on a non-Gaussian field, however, the power spectrum estimator is no longer optimal, and the error about it is biased (Tegmark et al. 2006). As first discussed in Meiksin & White (1999) and Rimes & Hamilton (2005), Non-Gaussian e ff ects on power spectrum measurements can be quite large; for instance, the Fisher information content in the matter power spectrum saturates in the trans-linear regime, which causes the number of degrees of freedom to deviate from the Gaussian prediction by up to three orders of magnitude. The obvious questions to ask, then, are 'How do non-Gaussian features propagate on to physical parameters, like the BAO scale, redshift space distortions, neutrino masses, etc.?' and 'How large is this bias in actual data analyses, as opposed to simulations?' As a partial answer to the first question, it was shown from a large ensemble of N-body simulations that the Gaussian estimator, acting on non-Gaussian fields, produces error bars on the BAO dilation scale (Padmanabhan & White 2008) that di ff er from full Gaussian case by up to 15 per cent (Ngan et al. 2012). The first paper of this series, Harnois-D'eraps & Pen (2012) (hereafter HDP1) addresses the second issue, and measure how large is the bias in the presence of a selection function. Starting with the 2dFGRS survey selection function as a study case, and modelling the non-Gaussian features from 200 N-body simulations, it was found that the di ff erence between Gaussian and non-Gaussian error bars on the power spectrum is enhanced by the presence of a non-trivial survey geometry 3 . The 15 per cent bias observed by Ngan et al. (2012) is, in that sense, a lower bound on the actual bias that exists in current treatments of the data. At this point, one could object that the sizes of the biases we are discussing here are very small, and that analyses with error bars robust to with 20 per cent are still in excellent shape and rather robust. However, the story reads di ff erently in the context of dark energy, where the final goal of the global international e ff ort is to minimize the error bars about w ( z ) by performing a succession of experiments with increasing accuracy and resolution. In the end, a 20 per cent bias on the BAO scale has a quite large impact on the dark energy 'figure-of-merit', and removing this e ff ect is the main goal of this series of paper. Generally, the onset of non-Gaussianities can be understood from asymmetries that develops in the matter fields subjected to gravitation, starting at the smallest scales and working their way up to larger scales (Bernardeau et al. 2002). In Fourier space, Gaussian fields can be completely described by their power spectrum, whereas non-Gaussian fields also store information in higher moments. For instance, the non-linear dynamics that describe the scales with k > 0 . 5 h Mpc -1 tend to couple the Fourier modes of the power spectrum, which e ff ectively correlates the measurements. This correlation was indeed found from very large samples of Nbody simulations (Takahashi et al. 2009), and act as to lower the number of degrees of freedom in a power spectrum measurement. One approach that was thought to minimizes these complications consists in excluding most of the non-linear scales, as proposed in Seo & Eisenstein (2003). However, this cuts out some of the BAO wiggles, thereby reducing our accuracy on the measured BAO scale. In addition, it is plausible that non-Gaussian features due to the non-linear dynamics, mask, and using simple Gaussian estimators on non-linear fields interact such as to impact scales as large as k ∼ 0 . 2 h Mpc, as hinted by HDP1. Optimal analyses must therefore probe the signal that resides in the trans-linear and nonlinear scales, and construct the power spectrum estimators based on known non-Gaussian properties of the measured fields. Characterization of the non-Gaussian features in the uncertainty about the matter power spectrum was recently attempted in HDP1, in which deviations from Gaussian calculations were parameterized by simple fitting functions, whose best fitting parameters were found from sampling 200 simulations of dark matter particles. This was a first step in closing the gap that exists between data analyses and simulations, but is incomplete in many aspects. First, it does not address the questions of cosmology and redshift dependence, halo bias nor of shot noise, which surely affect the best-fitting parameters, and completely overlooks the fact that observations are performed in redshift space. All these effects are crucial to understand in order to develop a self-consistent prescription with which we can perform non-Gaussian analyses in data. Second, and perhaps more importantly, this approach assumes that the non-Gaussian features observed in N-body simulations are unbiased representations of those that exist in our Universe. Such a correspondence is assumed in any external error analyses, which involve mock catalogues typically constructed either from N-body simulations or from semi-analytical techniques such as Log-Normal transforms (Coles & Jones 1991) or PThalos (Scoccimarro & Sheth 2002). In that aspect, the FKP and PKL prescriptions are advantageous since they provide internal estimates of the error bars, hence avoid this issue completely. In this second paper (HDP2 hereafter), we set out to determine how well can we possibly measure these non-Gaussian features internally from a galaxy survey, including simple cases of survey masks, with minimal prior assumption. In that way, we are avoiding the complications caused by the cosmology and redshift dependence of the non-Gaussian features, and that of using incorrect non-Gaussian modelling of the fields. In many aspects, this question boils down to the problem of estimating error bars using a small number of measurements, with minimal external assumptions. In such low statistics environment, it has been shown that the shrinkage estimation method (Pope & Szapudi 2008) can minimize the variance between a measured covariance matrix and a target , or reference matrix. Also important to mention is the attempt by Hamilton et al. (2006) to improve the numerical convergence of small samples by bootstrap resampling and re-weighting subvolumes of each realizations, an approach that was unfortunately mined down by the e ff ect of beat coupling (Rimes & Hamilton 2006). With the parameterization proposed in HDP1, our approach has the advantage to provide physical insights on the non-Gaussian dynamics. In addition, it turns out that the basic symmetries of the four-point function of the density field found in HDP1 allow us to increase the number of internal independent subsamples by a large amount, with only a few quantifiable and generic Bayesian priors. In particular, it was shown that the contributions to the power spectrum covariance matrix consist of two parts: the Gaussian zero-lag part, plus a broad, smooth, non-local, non-Gaussian component. In this picture, both the Gaussian and non-Gaussian contributions can be accurately estimated even within a single density field, because many independent configurations contribute. However, any attempt to estimate them simultaneously , i.e. without di ff erentiating their distinct singular nature, results in large sample variance. This paper takes advantage of this reduced variance to optimize the measurement of the covariance matrix from only four Nbody realizations. In such low statistics, the noise becomes dominant over a large dynamical range, hence we propose and compare a number of noise reduction techniques. In order to gain physical insights, we pay special attention to provide error bars on each of the non-Gaussian parameters and therefore track down the dominant source of noise in non-Gaussian fields. To quantify the accuracy of the method, we compare and calibrate our results with a larger sample of 200 N-body realizations, and use the Fisher information about the power spectrum amplitude as a metric of the performance of each noise reduction techniques. The issue of accuracy is entangled with a major aspect common to most non-Gaussian analyses: they require a measurement of the full power spectrum covariance matrix, which is noisy by nature. For instance, an accurate measurement of N 2 matrix elements generally requires much more then N realizations; it is arguable that N 2 independent measurements could be enough, but this statement generally depends on the final measurement to be carried and on the required level of accuracy that is sought. Early numerical calculations were performed on 400 realizations (Rimes & Hamilton 2005), while Takahashi et al. (2009) have performed as many as 5000 full N-body simulations. Ngan et al. (2012) have opted for fewer realizations, but complemented the measurements with noise reduction techniques, basically a principal component decomposition (see Norberg et al. 2009, for another example). In any case, it is often unclear how many simulations are required to reach convergence; in this paper, we deploy strategies such as bootstrap resampling to assess the degree of precision of our measurements. We first review in section 2 the formalism and the theoretical background relevant for the estimations of the matter power spectrum and its covariance matrix, with special emphases on the dual nature and symmetries of the four point function, and on the dominant source of noise in our non-Gaussian parameterization. We then explain how to improve the measurement of C ( k , k ' ) in section 3 with three noise-reduction techniques, and compare the results with a straightforward bootstrap resampling. In section 4 we measure the Fisher information of the noise filtered covariance matrices, and compare our results against the large sample. Finally, we describe in section 5 a recipe to extract the non-Gaussian features of the power spectrum covariance in the presence of a survey selection function, in a low statistics environment, thus completely merging the techniques developed here with those of HDP1. This takes us significantly closer to a stage where we can perform nonGaussian analyses of actual data. Conclusions and discussions are presented in the last section.", "pages": [ 1, 2, 3 ] }, { "title": "2.1 Power spectrum estimator", "content": "In the ideal situation that exists only in N-body simulations - periodic boundary conditions and no survey selection function e ff ect - the matter power spectrum P ( k ) can be obtained in an unbiased way from Fourier transforms of the density contrast. The latter is defined as δ ( x ) ≡ ρ ( x ) / ¯ ρ -1, where ρ ( x ) is the density field and ¯ ρ its mean. Namely, The Dirac delta function enforces the two scales to be identical, and the bracket corresponds to a volume (or ensemble) average. We refer the reader to HDP1 for details on our simulation suite, and briefly mention here that each of the N = 200 realization is obtained by evolving 256 3 particles with CUBEP 3 M (Harnois-D'eraps et al. 2012) down to a redshift of z = 0 . 5, with cosmological parameters consistent with the WMAP + BAO + SN five years data release (Komatsu et al. 2009). Simulated dark matter particles are assigned onto a grid following a 'cloud in cell' interpolation scheme (Hockney & Eastwood 1981), which is deconvolved in the power spectrum estimation. The isotropic power spectrum P ( k ) is finally obtained by taking the average over the solid angle. Observations depart from this ideal environment: they must incorporate a survey selection function and zero pad the boundary of the survey when constructing the power spectrum estimator, plus the galaxy positions are obtained in redshift space. We shall return to this framework in section 5, but for now, let focus our attention on simulation results. To quantify the accuracy of our measurements of P ( k ) and its covariance matrix in the context of low statistics, we construct a small sample by randomly selecting N = 4 realizations among the 200; we hereafter refer to these two samples as the N = 200 and the N = 4 samples. Fig. 1 shows the power spectrum as measured in these two samples, with a comparison to the non-linear predictions of HALOFIT (Smith et al. 2003). We present the results in the dimensionless form, defined as ∆ 2 ( k ) = k 3 P ( k ) / (2 π 2 ) in order to expose the scales that are in the trans-linear regime (loosely defined as the scales where 0 . 1 < ∆ 2 ( k ) < 1 . 0) and those in the non-linear regime (with ∆ 2 ( k ) > 1 . 0). The simulations show a ∼ 5 per cent positive bias compared to the predictions, a known systematic e ff ect of the N-body code that happens when the simulator is started at very early redshifts. Starting later would remove this bias, but the covariance matrix, which is the focus of the current paper, becomes less accurate. This is an unfortunate trade-o ff that will be avoided in future production runs with the advent of an initial condition generator based on second order perturbation theory. The plotted error bars are smaller in the N = 4 sample, a clear example of bias on the error bars: the estimate on the variance is poorly determined, as expected from such low statistics, and the error on the N = 4 error bar is quite large. The mean P ( k ) measured in the two samples, however, agree at the few per cent level. Following HDP1, we extract from this figure the regime of confidence, inside of which the simulated structures are well resolved, and exclude any information coming from k -modes beyond; in this case, scale with k > 2 . 34 h Mpc -1 are cut out.", "pages": [ 3 ] }, { "title": "2.2 Covariance matrix estimator", "content": "The complete description of the uncertainty about P ( k ) is contained in a covariance matrix, defined as : where ∆ P ( k ) refers to the fluctuations of the power spectrum about the mean. If one has access to a large number of realizations, this matrix can be computed straightforwardly, and convergence can be assessed with bootstrap resampling of the realizations. In actual surveys, however, only a handful of sky patches are observed, and a full covariance matrix extracted from these is expected to be singular. To illustrate this, we present in the top panels of Fig. 2 the covariance matrices, normalized to unity on the diagonal, estimated from the N = 200 and N = 4 samples. While the former is overall smooth, we see that the latter is, in comparison, very noisy, especially in the large scales (lowk ) regions. This is not surprising since these large scales are measured from very few Fourier modes, hence intrinsically have a much larger sample variance. It is clear why performing non-Gaussian analyses from the data is a challenging task: the covariance matrix is singular, plus the error bar about each element is large. This is nevertheless what we attempt to do in this paper, and in order to overcome the singular nature of the matrix, we need to approach the measurement from a slightly di ff erent angle. It was shown in HDP1 that there is an alternative way to estimate this matrix, which first requires a measurement of C ( k , k ' , θ ), where θ is the angle between the pair of Fourier modes ( k , k ' ). We summarize here the properties of C ( k , k ' , θ ), and refer the reader to HDP1 for more details: As discussed in section 5 . 3 and in the Appendix of HDP1, C ( k , k ' ) can be obtained from an integration over the angular dependence of C ( k , k ' , θ ): where G is the first term of the second line, and NG is the remaining two terms. In the above expressions, C ( k , k ' , µ = 1) is the zero-lag point, with µ = cos θ . The second line is simply in the above equation obtained by turning the integral into a sum, and splitting apart the contribution from the zero-lag term in the case where k ' = k . For instance, in the case where k ' /nequal k , only the third term contributes to NG , which now includes the θ = 0 point 4 . When comparing [Eq. 2] and [Eq. 3] numerically, slightly di ff erent residual noise create per cent level di ff erences, at least in the trans-linear and non-linear regimes 5 .", "pages": [ 4 ] }, { "title": "2.3 k = k ' case", "content": "The break down of the covariance matrix proposed in [Eq. 3] opens up the possibility to explore which of the two terms is the easiest to measure, which is noisier, and eventually organize the measurement such as to take full advantage of these properties. The first term on the right hand side, G ( k , k ' ), corresponds at the few per cent level - at least in the trans-linear and non-linear regimes - to the Gaussian contribution Cg ( k ) = 2 P 2 ( k ) / N ( k ), where N ( k ) is the number of Fourier modes in the k -shell. This comparison is shown in Fig. 3, where the error bars are estimated from bootstrap resampling. In the N = 200 case, we randomly pick 200 realizations 500 times, and calculate the standard deviation across the measurements. In the N = 4 case, we randomly select 4 realizations 500 times, always from the N = 200 sample. In the lowk regime, the simulations seem to underestimate the Gaussian predictions by up to 40 per cent. This is not too surprising since the discretization effect is large there, and the angle between grid cell, d µ , are less accurate. However, it appears that a substitution G ( k , k ' ) → Cg would improve the results by correcting for this lowk bias. We next look at the interplay between G and NG on the diagonal of the covariance matrix (for the case where k = k ' ). We know from Fig. 3 that the Gaussian term is well measured and has a rapid convergence about Cg ( k ). The top and middle panels of Fig. 4 present the diagonal components of the covariance matrix, di- ded by the Gaussian prediction, i.e. ( G ( k ) + NG ( k , k )) / Cg ( k ), for the N = 200 and N = 4 samples respectively. The linear regime agrees well with the Gaussian statistics, then we observe strong deviations about unity as the scales become smaller. In this figure, the error is again estimated from bootstrap resampling, even though the G + NG break down allows for more sophisticated error estimates (see section 3). An important observation is that the shape of the ratio is similar for both the large and small samples, which leads us to the conclusions that 1- departures from Gaussianities are clearly seen even in only four fields, and 2- both samples can be parameterized the same way. Following HDP1, we express the diagonal part of NG ( k , k ' ) and C ( k , k ' ) as 6 : In this parameterization, k 0 informs us about the scale at which non-Gaussian departure become important, and α is related to the strength of the departure as we progress deeper in the non-linear regime. The best-fitting parameters are presented in Table 1. As seen from Fig. 4, this simple power law form seems to model the ratio up to k ∼ 1 . 0 h Mpc -1 , beyond which the signal drops under the fit. Without a thorough check with higher resolution simulations, it is not clear whether this shortfall is a physical or a resolution e ff ect. In the former case, we could modify the fitting formula to include the flattening observed at k ∼ 1 . 0 h Mpc -1 , but this would require extra parameters, the number of which we are trying to minimize. We thus opt for the simple, conservative approach that consists in tightening our confidence region and exclude modes with k > 1 . 0 h Mpc -1 , even though the simulations resolve smaller scales. The two sets of parameters are consistent within 1 σ , which means that the N = 4 sample has enough information to extract the pair ( α , k 0), and therefore attempt non-Gaussian estimates of C ( k , k ' ). The fractional error on both parameters is of the order of a few per cent in the large sample, and about 20-50 per cent in the small sample. A second important observation is that the fractional error about α is about twice smaller than that of k 0, which means that α is the easiest non-Gaussian parameter to extract. We are now in a position to ask which of G ( k , k ' ) or NG ( k , k ' ) has the largest contribution to the error on C ( k , k ). We present in the bottom panel of Fig. 4 the fractional error on both terms, in both samples. We scale the bootstrap error by √ N to show the sampling error on individual measurements, and observe that in both cases, the non-Gaussian term dominates the error budget by more than an order of magnitude.", "pages": [ 4, 5, 6 ] }, { "title": "2.4 k /nequal k ' case", "content": "We now turn our attention to the o ff -diagonal part of the covariance matrix, whose sole contribution comes from the non-Gaussian term. For this reason, and because there are many more elements to measure from the same data ( N 2 vs. N ), it is expected to be much noisier that the diagonal part. It is exactly this noise that makes C ( k , k ' ) singular, but luckily we can filter out a large part of it from a principal component analysis based on the Eigenvector decomposition of the crosscorrelation coe ffi cient matrix r ( k , k ' ) ≡ C ( k , k ' ) / √ C ( k , k ) C ( k ' , k ' ). As discussed in Ngan et al. (2012), this method improves the accuracy of both the covariance matrix and of its inverse. HDP1 further provides a fitting function for the Eigenvector, and we explore here 6 The notation is slightly di ff erent than in HDP1, which expressed C ( k , k ) = 2 P 2 ( k ) N ( k ) ( 1 + ( k α ) β ) ≡ 2 P 2 ( k ) N ( k ) V ( k ). Note the correspondance ( α, β ) → ( k 0 , α ). how well the best-fitting parameters can be found in a low statistics environment. This decomposition is an iterative process that factorizes the cross-correlation coe ffi cient matrix into a purely diagonal component and a smooth symmetric o ff -diagonal part. The latter is further Eigen-decomposed, and we keep only the Eigenvector U ( k ) that corresponds to the largest Eigenvalue λ . In that case, it is convenient to absorb the Eigenvalue in the definition of the Eigenvector, i.e. √ λ U ( k ) → U ( k ), such that we can write the r ( k , k ' ) ∼ U ( k ) U ( k ' ) directly. Since the diagonal elements are unity by construction, the exact expression is r ( k , k ' ) = U ( k ) U ( k ' ) + δ kk ' [1 -U 2 ( k )]. This effectively puts a prior on the shape of the covariance matrix, since any part of the signal that does not fit this shape is considered as noise and excluded. As shown in HDP1, this decomposition is accurate at the few per cent level in the dynamical range of interest, for the N = 200 sample. We present in the top panel of Fig. 5 the Eigenvector extracted from both samples, against a fitting function of the form 7 : In this parameterization, A represents the overall strength of the non-Gaussian features, while k 1 is related to the scale where crosscorrelation becomes significant in our measurement. If A = 0, then we recover r ( k , k ' ) = δ kk ' , which corresponds to 'no mode crosscorrelation'. The best fitting parameters are shown in Table 1. We observe that the error bar on A is at the few per cent level, and that the measurements in both samples are both very close to unity. On the other hand, k 1 is much harder to constrain: in the N = 200 sample, the fractional error bar is about 30 per cent, and in the N = 4, the error bar is three times larger than its mean. In other words, k 1 is consistent with zero within 1 σ in the small sample, which corresponds to a constant Eigenvector. The non-Gaussian noise is thus mostly concentrated here, and improving the measurement on k 1 is one of the main tasks of this paper. If we model the N = 4 covariance matrix with this fit to the Eigenvector, the matrix is no longer singular, as shown in the bottom left panel of Fig. 2. It does exhibit a stronger correlation at the largest scales, compared to the matrix estimated from the large sample; this di ff erence roots in the fact that the N = 4 Eigenvector remains high at the largest scales, whereas the N = 200 vector drops. There is thus an overestimate of the amount of correlation between the largest scales, which biases the uncertainty estimate on the high side. In any case, these large scales are given such a low weight in the calculation of Fisher information - they contain a small number of independent Fourier modes - that their contribution to the Fisher information is tiny, as will become clear in section 4. The bottom left panel of Fig. 2 also shows that we do recover the region where the cross-correlation coe ffi cient is 60-70 per cent, also seen in the top left panel of the figure. It is thus a significant step forward in the accuracy of the error estimate compared to the Gaussian approach. We show in sections 3.1 and 3.2 that noise filtering techniques are able to reduce this bias down to a minor e ff ect, even when working exclusively with the same four fields. To contrast the pipeline presented in this section (of the form [ N = 4 → PCA → fit ]) to those presented in future sections, we hereafter refer to this approach as the ' N = 4 naive' way.", "pages": [ 6, 7 ] }, { "title": "3 OPTIMAL ESTIMATION OF C ( K , K ' ) : BEYOND BOOTSTRAP", "content": "In the calculations of section 2, we show that even in low statistics, we can extract four non-Gaussian parameters with the help of noise filtering techniques that assume a minimal number of priors on the non-Gaussian features. As mentioned therein, all the error bars are obtained from bootstrap resampling, which is generally thought to be a faithful representation of the underlying variance only in the large sample limit. How, then, can we trust the significance of our results in the N = 4 sample, which emulates an actual galaxy sur- vey? More importantly, can we do better than bootstrap? In this section, we expose new procedures that optimize the estimate of both the mean and the error on C ( k , k ' ), and we quantify the improvements on all four non-Gaussian parameters ( α , A , k 0 and k 1).", "pages": [ 7 ] }, { "title": "3.1 Wiener filtering", "content": "Let us recall that in the bootstrap estimate of the error on C ( k , k ' ), we resample the measured C ( k , k ' , θ ), integrate over the angle θ and add up the uncertainty from di ff erent angles - including the zerolag point - in quadrature. We propose here a di ff erent approach, based on our knowledge that C ( k , k ' , θ ) is larger for smaller angles. At the same time, we take advantage of the fact that the noise on G ( k ) is much smaller than that on NG ( k , k ' ). Since the mean and error should be given more weight in regions where the signal is cleaner, we replace the quadrature by a noise-weighted sum in the angular integration of C ( k , k ' , θ ). This replacement reduces the error on NG by an order of magnitude or so, depending on the scale (see Appendix A for details). At this stage, we now have accurate estimates of the error on the covariance C ( k , k ' ) in the N = 4 sample, as well as accurate measurements of the signal and noise of the underlying covariance matrix from the N = 200 sample, which we treat as a template . The technique we describe here is a Wiener filtering approach that uses known noise properties of the system in order to extract a signal that is closer to the template. The error on G ( k ) is obtained from bootstrap resampling the zero-lag point, while the error on NG ( k , k ' ) comes from the noise-weighted approach mentioned above and discussed in Appendix A. We first apply the filter on both quantities separately, and then combine the results afterward. Namely, we define our Wiener filters as with and Note that the errors that appear in the above two expressions correspond to the estimates from the N = 200 and N = 4 samples on G and NG respectively. Wepresent in Fig. 6 the Wiener filtered variance on P ( k ), compared to the N = 200 and N = 4 samples. We observe in the range 0 . 3 < k < 2 . 0 h Mpc -1 that the filter decreases the size of the fluctuations about the N = 200 sample, compared to the original N = 4 sample. For larger scales, it is not clear that the e ff ect of the filter represents a gain in accuracy: while the variance on the fundamental mode is 3 times closer to the template's measurement, the variance about the second largest mode does 3 times worst, while the change in others large modes seems to have no gain. However, we recall that very little weight is given to these lowk modes in the calculation of the Fisher information about the dark matter power spectrum. Hence slightly degrading the accuracy of the variance about the two largest modes is a mild cost if we can improve the range that ultimately matters, i.e. 0 . 1 < k < 1 . 0 h Mpc -1 . In addition, our parameterization of the non-Gaussian features assign a smaller weight to the large scales, which are smoothly forced towards the analytical Gaussian predictions. There is a positive bias of about 15 per cent and up that appears for k > 1 . 0 h Mpc -1 , both in the original N = 4 sample and in the Wiener filtered product. The bias in the unfiltered fields is simply a statistical fluctuation, since we know it does converge to the large sample by increasing the number of fields. It does, however, propagate in a rather complicated way through the Wiener filter, causing a sharp increase at k > 1 . 0 h Mpc -1 . Because the details of how and why this happens are rather unclear, we decided to exclude this region from the analysis, based on suspicion of systematics. The full Wiener filtered cross-correlation matrix is presented in the bottom right panel of Fig. 2 and shows that some of the noise has been filtered out: the regime k > 0 . 5 h Mpc -1 is smoother than the original N = 4 measurement, and except for the largest two modes, the matrix is brought closer to the N = 200 sample. The next steps consist in computing the new Eigenvector U ( k ) constructed with CWF and to find the new best-fitting parameters ( α , k 0, A , k 1). The bottom panel of Fig. 5 presents U ( k ) and shows that most of the benefits are seen for 0 . 08 < k < 0 . 2 h Mpc -1 , which is the range we targeted to start with, and the best-fitting parameters are tabulated in Table 1. Overall, the improvement provided by the Wiener filter is still hard to gauge by eye from Fig. 2, because the Eigenvector is still very noisy. This is to be expected: the method is mostly e ffi cient on the diagonal part, where we can take advantage of the low noise level of the Gaussian term.", "pages": [ 7, 8, 9 ] }, { "title": "3.2 Noise-weighted Eigenvector decomposition", "content": "In this section, we describe a last noise filtering technique that utilizes known properties of the noise about the Eigenvectors and their Eigenvalues to improve the way we perform the principal component decomposition in the N = 4 sample. It is a general strategy that could be combined with others techniques described in the preceding sections, however, for the sake of clarity, we only present here the standalone e ff ect. In the Eigen-decomposition, not all Eigenvalues are measured with the same precision. For instance, most of the covariance matrix can be described by the first Eigenvector U ( k ), hence we expect the signal-to-noise ratio about its associated Eigenvalue to be the largest. Considering again the N = 200 sample as our template of the underlying covariance, the error on each λ can be obtained by bootstrap resampling the 200 realizations. We present this measurement in Fig. 7, where we observe that the first Eigenvalue is more than an order of magnitude larger than the others, which are also noisier. Since general Eigenvector decompositions are independent of rotations, our strategy is to rotate the N = 4 cross-correlation matrix into a state T where it is brought closer to the template, then apply a signal-to-noise weight before the Eigenvector decomposition. More precisely, we apply the following algorithm, which we refer to as the 'T-rotation' method (for rotation into T -space) in the rest of this paper: The rotation in step (i) is defined as: and, by construction, reduces to the diagonal Eigenvalues matrix in the case where ρ is the template cross-correlation coe ffi cient matrix. The weighting in step (ii) is performed in two parts 8 : 1- we scale each matrix element Rij by 1 / √ λ i λ j , and 2- we weight the result by the signal to noise ratio of each λ . Combining, we define 9 : As seen in Fig. 7, the Eigenvalues drop rapidly, and we expect only the first few to contribute to the final result. In fact, our results present very small variations if we keep anywhere between two and six Eigenvalues and exclude the others. Since it was shown in HDP1 that in some occasions, we need up to four Eigenvectors to describe the observed C ( k , k ' ) matrix, we choose to keep four Eigenvalues as well, and cut out the contributions from λ i > 4. In step (iii) the resulting matrix D is decomposed into Eigenvectors S : Then, in step (iv), these Eigenvectors are weighted back by absorbing the weights directly: If no cut is applied on the Eigenvalues, this operation essentially does nothing to the matrix, as the equivalence between ˜ ρ and ρ is exact: every rotation and weights that are applied are removed, and we get U ( k ) ≡ ˜ T 1( k ). However, the cut, combined with the rotation and weighting, acts as to improve the measurement of the Eigenvector. Physically, this is enforcing on the measured matrix a set of priors, corresponding to the Eigenvectors of the template, with a strength that is weighted by the known precision about the underlying Eigenvalue. For a Gaussian covariance matrix normalized to 1 on the diagonal, all frames are equivalent, and any rotation from any prior has no impact on the accuracy. When the covariance is not diagonal, however, some frames are better than others, and the Eigenframe is among the best, as long as the simulation Eigenframe is similar to that of the actual data. If these frames were unrelated, this T-rotation would generally be neutral. We present in the bottom panel of Fig. 5 the e ff ect of this Trotation on the original N = 4 Eigenvector. We observe that it traces remarkably well the N = 200 vector, to within 20 per cent even at the low k -modes, and outperforms the other techniques presented in this paper in its extraction of U ( k ). The best fitting parameters corresponding to ˜ T 1( k ) are summarized in Table 1. We discuss in section 5 how, in practice, one can us these techniques in a real survey.", "pages": [ 9, 10 ] }, { "title": "4 IMPACT ON FISHER INFORMATION", "content": "In analyses based on measurements of P ( k ), the uncertainty typically propagates to cosmological parameters within the formalism of Fisher matrices (Tegmark 1997). The Fisher information content in the amplitude of the power spectrum, defined as e ff ectively counts the number density of degrees of freedom in a power spectrum measurement. In the above expression, Cnorm is simply given by C ( k , k ' ) / [ P ( k ) P ( k ' )]. The Gaussian case is the simplest, since the covariance is given by Cg ( k ) = 2 P 2 ( k ) / N ( k ), where N ( k ) is the number of cells in the k -shell. We recall that the factor of two comes in because the P ( -k ) = P ( k ) symmetry, which reduces the number of independent elements by a factor of two. Also, Cnorm reduces to 2 / N ( k ), and I ( kmax ) = k N ( k ) / 2 for Gaussian fields. ∑ We see how I ( k ) is an important intermediate step to the full Fisher matrix calculation, as it tells whether we can expect an improvement on the Fisher information from a given increase in survey resolution. It was first shown by Rimes & Hamilton (2005) that the number of degrees of freedom increases in the linear regime, following closely the Gaussian prescription, but then reaches a trans-linear plateau, followed by a second increase at even smaller scales. This plateau was later interpreted as a transition between the two-haloes and the one-halo term (Neyrinck et al. 2006), and corresponds to a regime where the new information is degenerate with that of larger scales. By comparison, the Gaussian estimator predicts ten times more degrees of freedom by k ∼ 0 . 3 h Mpc -1 . What stops the data analyses from performing fully nonGaussian uncertainty calculations is that the Fisher information requires an accurate measurement of the inverse of a covariance matrix similar to that seen in Fig. 2, which is singular. With the noise reduction techniques described in this paper, however, the covariance matrix is no longer singular, such that the inversion is finally possible. To recapitulate, these techniques are: (i) The 'Naive N = 4 way' : straight Eigenvector decomposition + fit of the N = 4 sample (section 2.4) In this section, we assume that there are no survey selection function e ff ects, and that the universe is periodic. We discuss more realistic cases in section 5. We present in the top panel of Fig. 8 the Fisher information content for each technique, compared to that of the template and the analytical Gaussian calculation. We also show the results for the N = 200 sample after the Eigen-decomposition, which is our best estimator of the underlying information (Ngan et al. 2012). The agreement between this and the original information content in the N = 200 sample is at the few per cent level for k < 1 . 0 h Mpc -1 anyway. We do not show the results from the N = 4 sample, nor that after the Eigenvector decomposed only, as the curve quickly diverges. It actually is the fitting procedure, summarized in Table 1, that clean up enough noise to make the inversion possible. In all these calculations, the error bars are obtained from bootstrap resampling. The bottom panel represents the fractional error between the di ff erent curves and our best estimator. As first found by Rimes & Hamilton (2005), the deviation from Gaussian calculations reaches an order of magnitude by k ∼ 0 . 3 h / Mpc, and increases even more at smaller scales. With the fit to the Eigenvector (technique (i) in the list above mentioned), we are able to recover a Fisher information content much closer to the template; it underestimates the template by less than 20 per cent for k < 0 . 3 h Mpc -1 , and then overestimates the template by less than 60 per cent away for 0 . 3 < k < 1 . 0 h Mpc -1 . The fit to the analytical substitution of G ( k ) (technique (ii)) has even better performances, with maximum deviations of 20 per cent over the whole range. The fit to the Wiener filter (iii) is not as performant, but still improves over the naive N = 4 way, with the maximal deviation reduced to less than 45 per cent. Finally, the fit to the T-rotated (iv) Eigenvectors also performs well, with deviations by less than 20 per cent. Most of the residual deviations can be traced back to the fact that in the linear regime, the covariance matrix exhibited a large noise that we could not completely remove. This extra correlation translates into a loss of degrees of freedom in the linear regime, a cumulative e ff ect that biases the Fisher information content on the low side. Better noise reduction techniques that focus in the large scales cleaning could outperform the current information measurement. In any case, this represents a significant step forward for non-Gaussian data analyses since estimates of I ( k ) can be made accurate to within 20 per cent over the whole BAO range, even from only four fields, with minimal prior assumptions.", "pages": [ 10 ] }, { "title": "5 IN THE PRESENCE OF A SURVEY SELECTION FUNCTION", "content": "The results from section 4 demonstrate that it is possible to extract a non-Gaussian covariance matrix internally, from a handful of observation patches, and that with noise filtering techniques, we can recover, to within 20 per cent, the Fisher information content in the amplitude of the power spectrum of (our best estimate of) the underlying field. The catch is that these are derived from an idealized environment that exist only in N-body simulations, and the objective of this section is to understand how, in practice, can we apply the techniques in actual data analyses. 10 We explore a few simple cases that illustrates how the noise reduction techniques can be ap- plied, and how the non-Gaussian parameters can be extracted in the presence of a survey selection function W ( x ).", "pages": [ 11, 12 ] }, { "title": "5.1 Assuming deconvolution of W ( k )", "content": "The first case we consider is the simplest realistic scenario one can think of, in which the observation patches are well separated, and for each of these the survey selection functions can be successfully deconvolved from the underlying fields. For simplicity, we also assume that each patch is assigned onto a cubical grid with constant volume, resolution and redshift, such that the observations combine essentially the same way as the N = 4 sample presented in this paper. Once the grid is chosen, one then needs to produce a large sample of realizations from N-body simulations, with the same volume, redshift and accuracy, and construct the equivalent of our N = 200 sample. (ii) To construct the Wiener filter described in section 3.1, one needs to use the methodology of HDP1 and compute C ( k , k ' , θ ) from the density fields of the data and the large simulated sample, and finally extract G ( k ) and NG ( k , k ' ) from [Eq. 3] to construct the filter. At that stage, it is also trivial to try the semi-analytical substitution G ( k ) → 2 P 2 ( k ) / N ( k ) and reduce the noise even more. After that, we need to compute the ratio and U ( k ) as described above, find the best-fitting parameters, and reconstruct the covariance matrix. (iii) To make use of the T-rotation technique, one needs to compute, from the large simulated sample, the Eigenvectors that describe the cross-correlation coe ffi cient matrix, plus the noise about each Eigenvalue, which can be obtained from bootstrap resampling the simulated realizations. The weights wi can then be computed, and the rest of the technique follows directly from section 3.2, such that we end up with a better estimate of U ( k ) - to be fitted as well. This technique improves only the estimate of ρ , so that one then has some freedom regarding which estimate of the diagonal element to choose (fit to the raw data, the Wiener filtered, etc.).", "pages": [ 12 ] }, { "title": "5.2 Assuming no deconvolution of W ( k )", "content": "The second case is a scenario in which the observation mask was not deconvolved from the underlying field. This set up introduces many extra challenges, as the mask tends to enhance the nonGaussian features, hence, for simplicity, we assume that there is a unique selection function W ( k ) that covers all the patches in which the power spectra are measured. Let us first recall that in presence of a selection function, the observed power spectrum of a patch ' i ' is related to the underlying one via a convolution with the Fourier transform of the mask, namely: The first paper of this series describes a general extension to the FKP calculation in which the underlying covariance matrix C ( k , k ' ) is non-Gaussian. Specifically, the 'observed' covari- ce matrix Cobs ( k , k ' ) is related to the underlying one via a sixdimensional convolution: In HDP1, C ( k , k ' ) is calculated purely from N-body simulations, therefore it is known a priori ; only P ( k ) and W ( x ) are extracted from the survey. It is in that sense that the technique of HDP1 provides an external estimate of the error. What needs to be done in internal estimates is to walk these steps backward: given a selection function and a noisy covariance matrix, how can we extract the non-Gaussian parameters of Table 1? Ideally, we would like to deconvolve this matrix from the selection function, but the high dimensionality of the integral in [Eq. 16] makes the brute force approach numerically not realistic. There is a solution, however, which exploits the fact that many of the terms involved in the forward convolution are linear. It is thus possible to perform a least square fit for some of the non-Gaussian parameters, knowing C obs and W ( x ). We start by casting the underlying non-Gaussian covariance matrix into its parameterized form, which expresses each of its Legendre multipole matrices C i j /lscript into a diagonal and a set of Eigenvectors (see [Eq. 51-54] and Tables 1 and 2 of HDP1 11 ). We recall that only the /lscript = 0 , 2 , 4 multipoles contain a significant departure from the Gaussian prescription; the complete matrix depends on 6 parameters to characterize the diagonal components, plus 30 others to characterize the Eigenvectors. In principle, these 36 parameters could be found all at once by a direct least square fit approach. However, some of them have more importance than other, and, as seen in this paper, some of them are easier to measure, therefore we should focus our attention on them first. In particular, it was shown in figure 22 of HDP1 that most of the non-Gaussian deviations come from C 0, hence we start by solving only for its associated parameters. To simplify the picture even more, we decompose the problem one step further and focus exclusively on the diagonal component. In this case, we get, with the notation of the current paper: The 'tilde' symbol serves to remind us that this is not the complete /lscript = 0 multipole but only its diagonal. The term with the δµ 1 is the Gaussian contribution, and yields to ([Eq. 56] of HDP1): For the second term, the δ kk ' allows us to get rid of one of the radial integral in [Eq. 16], and the remaining part is isotropic, therefore the angular integrals only a ff ect the selection functions. These integrals can be precomputed as the X ( k , k '' ) function in [Eq. 57] of HDP1, with w ( θ '' , φ '' ) = 1, and we get where the angle brackets refer to an average over the angular dependence. At the end of this calculation, we obtain, for each ( k , k ' ) 11 In the following discussion, we refer substantially to sections 7.2, 8 and 8.1 of HDP1, and we try to avoid unnecessary repetitions of lengthy equations here. Also, as discussed in section 2.4, some of the parameters in Table 2 of HDP1 are degenerate, and in fact only 30 are necessary. For each multipole, the main Eigenvector requires only A and k 1, and we can merge λ into α for the others. pair, a value for α and β , and all that is left is to find the parameter values that minimize the variance. The next step is to include the o ff -diagonal terms of the /lscript = 0 multipole, in which case the full C 0 is modelled. The underlying covariance matrix is now parameterized as where and In this case, [Eq. 19] is modified and we get There are two new best-fitting parameters that need to be found from H 0, namely A and k 1, and we can use our previous results on α and k 0 as initial values in our parameter search. Since the X functions only depend on W ( k ), we can still solve these N 2 equations with a non-linear least square fit algorithm and extract these four parameters. Including higher multipoles can be done with the same strategy, i.e. progressively finding new parameters from least square fits, using the precedent results as priors for the higher dimensional search. One should keep in mind that the convolution with C 2 and C 4 becomes much more involved, since the number of distinct X functions increases rapidly, as seen in Table 4 of HDP1. In the end, all of the 36 parameters can be extracted out of N 2 matrix elements, in which case we have fully characterized the non-Gaussian properties of the covariance matrix from the data only. The matrix obtained this way is expected to have very little noise, as this procedure invokes the fitting functions, which smooth out the fluctuations. In principle, assuming that the operation was loss-less, the recovered covariance matrix would be completely equivalent to the naive N = 4 way, had the selection function been deconvolved first. It could be possible to improve the estimate even further by attempting the T-rotation on the output, as explained is section 5.1, but since we do not have access to the underlying density fields, the Wiener filter technique is not available in this scenario.", "pages": [ 12, 13 ] }, { "title": "6 DISCUSSION", "content": "With the recent realization that the Gaussian estimator of error bars on the BAO scale is biased by at least 15 per cent (Ngan et al. 2012; Harnois-D'eraps & Pen 2012), e ff orts must be placed towards incorporating non-Gaussian features about the matter power spectrum in data analyses pipeline. This implies that one needs to estimate accurately the full non-Gaussian covariance matrix C ( k , k ' ) and, even more challenging, its inverse. The strategy of this series of paper is to address this bias issue, via an extension of the FKP formalism that allows for departure from Gaussian statistics in the estimate of C ( k , k ' ). The goal of this paper is to develop a method to extract directly from the data the parameters that describe the non-Gaussian features. This way, the method is free of the biases that a ff ect external non-Gaussian error estimates (wrong cosmology, incorrect modelling of the non-Gaussian features, etc.). We emulate a typical observation with a subset of only N = 4 N-body simulations, and validate our results against a larger N = 200 sample. The estimate of C ( k , k ' ) obtained with such low statistics is very noisy by nature, and we develop a series of independent techniques that improve the signal extraction: These techniques are exploiting known properties about the noise, and assume a minimal number of priors about the signal. We quantify their performances by comparing their estimate of the Fisher information content in the matter power spectrum to that of the large N = 200 sample. We find that in some cases, we can recover the signal within less then 20 per cent for k < 1 . 0 h Mpc -1 . We also provide error bars about the Fisher information whenever possible. By comparison, the Gaussian approximation deviates by more than two orders of magnitude at that scale. Wefindthat the diagonal component of NG ( k , k ' ) is well modelled by a simple power law, and that in the N = 4 sample, the slope α and the amplitude k 0 can be measured with a signal-tonoise ratio of 4 . 2 and 2 . 2 respectively. The o ff -diagonal elements of NG ( k , k ' ) are parameterized by fitting the principal Eigenvector of the cross-correlation coe ffi cient matrix with two other parameters: an amplitude A , which has a signal-to-noise ratio of 32 . 1, and a turnaround scale k 1, which is measured with a ratio of 0 . 30 and is thus the hardest parameter to extract. Even in the large N = 200 sample, k 1 is measured with a ratio of 3 . 1, which is five to ten times smaller than the other parameters. This help us to better understand the parts of the non-Gaussian signals that are the noisiest, such that we can focus our e ff orts accordingly. Wethen propose a strategy to extract these parameters directly from surveys, in the presence of a selection function. We explore two simple cases, in which a handle of observation patches of equal size, resolution and redshift are combined, with and without a deconvolution of the survey selection function. The first case is the easiest to solve, since the deconvolved density fields are in many respect similar to the simulated N = 4 sample that is described in this paper. We show that it is possible to apply all of the four noise filtering techniques described above to optimize the estimate of the underlying covariance matrix. In the second case, we explore what happens when deconvolution is not possible. We propose a strategy to solve for the non-Gaussian parameters with a least square fit method, knowing C obs ( k , k ' ) and W ( k ). The approach is iterative in the sense that it first focuses on the parameters that contribute the most to the non-Gaussian features, then source the results as priors into more complete searches. The main missing ingredient from our non-Gaussian parameterization is the inclusion of redshift space distortions, which will be the focus of the next paper of this series. To give overview of the challenge that faces us, the approach is to expand the redshift space power spectrum into a Legendre series, and to compute the covariance matrix term by term, again in a low statistics environment. Namely, we start with P ( k , µ ) = P 0( k ) + P 2( k ) P 2( µ ) + P 4( k ) P 4( µ ) + ... where Pi ( k ) are the multipoles and P /lscript ( µ ) the Legendre polynomials, and compute the nine terms in C ( k , µ, k ' , µ ' ) = 〈 P 0( k ) P 0( k ' ) 〉 + 〈 P 0( k ) P 2( k ' ) 〉P 2( µ ) + ... Wesee here why a complete analysis needs to include the auto- and cross-correlations between each of these three multipoles, even without a survey selection function. As mentioned in the introduction, many other challenges in our quest for optimal and unbiased non-Gaussian error bars are not resolved yet. For instance, many results, including all of the fitting functions from HDP1, were obtained from simulated particles, whereas actual observations are performed from galaxies. It is thus important to repeat the analysis with simulated haloes, in order to understand any di ff erences that might exist in the non-Gaussian properties of the two matter tracers. In addition, simulations in both HDP1 and in the current paper were performed under a specific cosmology, and only the z = 0 . 5 particle dump was analyzed. Although we expect higher lower redshift and higher Ω m cosmologies to show stronger departures form Gaussianities - clustering is stronger - we do not know how exactly this impact each of the non-Gaussian parameters. To summarize, we have developed techniques that allow for measurements of non-Gaussian features in the power spectrum uncertainty for galaxy surveys. We separate the contributions to the total correlation matrix into two kinds: diagonal and o ff -diagonal. The o ff -diagonal is accurately captured in a small number of Eigenmodes, and we have used the Eigenframes of the N-body simulations to optimize the measurements of these o ff -diagonal nonGaussian features. We show that the rotation into this space allows for an e ffi cient identification of non-Gaussian features from only four survey fields. The method is completely general, and with a large enough sample, we always reproduce the full power spectrum covariance matrix. We finally describe a strategy to perform such measurements in the presence of survey selection functions.", "pages": [ 13, 14 ] }, { "title": "ACKNOWLEDGMENTS", "content": "The authors would like to thank Chris Blake for reading the manuscript and providing helpful comments concerning the connections with analyses of galaxy surveys. UP also acknowledges the financial support provided by the NSERC of Canada. The simulations and computations were performed on the Sunnyvale cluster at CITA.", "pages": [ 14 ] }, { "title": "APPENDIX A: BOOTSTRAP VS NOISE-WEIGHTED INTEGRATION", "content": "This appendix describes a technique that improves the calculation of the uncertainty on NG ( k , k ' ) compared to the bootstrap approach. Recall that bootstrap combines the error from di ff erent angles in quadrature, even though the signal and noise strength vary for different angles. To perform the noise-weighted angular integration, we first normalize each realization Ci ( k , k ' , θ ) by the mean of the distribution: where σ C is the standard deviation in the sampling of C ( k , k ' , θ ). As seen in Fig. A1, the individual distributions of Di ( k , k ' , θ ) are relatively flat, with a slight tilt towards smaller angles. The two panels in this figure correspond to scales k = k ' = 2 . 1 h Mpc -1 and k = k ' = 0 . 31 h Mpc -1 respectively. In addition, the error bars σ D get significantly smaller towards θ = 0 (or µ = 1). It is thus a good approximation to replace each fluctuation by its noise-weighted mean: The measurement of a matrix element Ci ( k , k ' ) from a given realization becomes: where we have used the substitution of [Eq. A2] in the last step. Note that in the above expressions, σ C , D depend on the variables ( k , k ' , θ ), while σ T depends on ( k , k ' ). We have chosen not to write these dependencies explicitly in our equations to alleviate the notation. The mean value of C ( k , k ' ) computed with this method is identical to the bootstrap approach of [Eq. 3], since the realization average of ˜ Di ( k , k ' ) is equal to unity by construction. However, this method has the direct advantage to reduce significantly the error on NG and, consequently, on C . We show in Fig. A2 a comparison between the bootstrap sampling error bars on the C ( k , k ' ) matrix, and our proposed noiseweighted scheme. In the left panel, we hold k ' = k , whereas in the right panel, we keep k = 0 . 628 h Mpc and vary k ' . The error bars achieved in the noise-weighted scheme are up to two orders of magnitude smaller than the bootstrap errors, and the estimate of the error from the small sample is already accurate. We also observe that the bootstrap fractional error on C is scale invariant, whereas that from the noise-weighted method drops roughly as k -2 and thereby yields much tighter constraints on the measured matrix elements. This comes from the fact that as we go to larger k -modes, the signal becomes stronger for small angles, which improves the weighting. In both samples, by the time we have reach k = 0 . 1 h Mpc -1 , the improvement is about an order of magnitude, and at k = 1 . 0 h Mpc -1 , the improvement is almost two orders of magnitude. With a fractional error that small, the covariance matrix is precisely measured even with a handful of realizations, a claim that bootstrap approach can not support.", "pages": [ 14 ] }, { "title": "REFERENCES", "content": "Anderson, L., et al. 2012, ArXiv e-prints, 1203.6594 Ben´ıtez, N., et al. 2009, ApJ, 691, 241 Bernardeau, F., Colombi, S., Gazta˜naga, E., & Scoccimarro, R. 2002, Phys. 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[ { "title": "ABSTRACT", "content": "Studies of the peculiar velocity bulk flow based on different tools and datasets have been consistent so far in their estimation of the direction of the flow, which also happens to lie in close proximity to several features identified in the cosmic microwave background, providing motivation to use new compilations of type-Ia supernovae measurements to pinpoint it with better accuracy and up to higher redshift. Unfortunately, the peculiar velocity field estimated from the most recent Union2.1 compilation suffers from large individual errors, poor sky coverage and low redshift-volume density. We show that as a result, any naive attempt to calculate the best-fit bulk flow and its significance will be severely biased. Instead, we introduce an iterative method which calculates the amplitude and the scatter of the direction of the best-fit bulk flow as deviants are successively removed and take into account the sparsity of the data when estimating the significance of the result. Using 200 supernovae up to a redshift of z=0.2, we find that while the amplitude of the bulk flow is marginally consistent with the value expected in a ΛCDM universe given the large bias, the scatter of the direction is significantly low (at /greaterorsimilar 99 . 5% C.L.) when compared to random simulations, supporting the quest for a cosmological origin. Key words: peculiar velocities - bulk flow - supernovae: type-Ia", "pages": [ 1 ] }, { "title": "Ben Rathaus 1 , Ely D. Kovetz 2 and Nissan Itzhaki 1", "content": "1 Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, 69978, Israel 2 Theory Group, Department of Physics and Texas Cosmology Center, The University of Texas at Austin, TX, 78712, USA E-mail: [email protected] E-mail: [email protected] In original form: 1 October 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "In the last couple of decades a considerable effort has been devoted to the analysis of the peculiar velocity field in search for an overall bulk flow (BF) on ever increasing scales, lately reaching as high as ∼ 100 Mpc /h using galaxy surveys [116] and type-Ia supernovae (SNe) [17-23] and even an order of magnitude higher, based on measurements of the kinetic Sunyaev-Zeldovich effect in the cosmic microwave background (CMB) [24-29]. While there have been conflicting claims regarding the amplitude of the dipole moment of this field and its tension with the expected value in a ΛCDM universe, the vast majority of these surveys have been consistent in their findings for the direction of the dipole 1 . Meanwhile, several features in the CMB temperature maps from the COBE DMR [31] and WMAP [32] experiments have been identified in roughly the same region of the sky, from the dipole moment [33] to several reported anomalies, including the alignment between the quadrupole and octupole [34, 35], mirror parity [36-38] and giant rings [39]. This coincidence provides further motivation to search for a unified cosmological explanation [40]. Over the years a number of cosmological scenarios have been suggested as possible sources for a peculiar velocity BF, such as a great attractor [1, 2], a super-horizon tilt [41], over-dense regions resulting from bubble collisions [42] or induced by cosmic defects 2 [43, 44], etc. In an attempt to test these hypotheses and distinguish between them, any knowledge regarding the redshift dependence of the BF can be a crucial discriminator. Type-Ia SNe, whose simple scaling relations provide empirical distance measurements and which have been detected up to redshifts z /greaterorsimilar 1, provide a unique tool to estimate the peculiar velocity BF and study its direction and redshift extent. However, this approach also contains certain caveats. First, datasets from typical type-Ia SNe surveys are orders of magnitude smaller than those from galaxy surveys and their sky coverage and redshift-volume density are extremely poor. Secondly, different SNe compilations often use different light-curve fitters, involving different nuisance parameters. Currently, the most promising candidate for a large scale BF search is the Union2.1 compilation [45] (see also [46, 47]), comprising of 19 different surveys which are all analyzed with a single light-curve fitter (SALT2 [48]) The purpose of this work is to investigate the peculiar velocity field extracted from the Union2.1 data and given its limitations determine which conclusions can be reliably made as to the BF in the inferred radial peculiar velocity field, placing an emphasis on its direction and redshift extent. Accounting for the substantial bias due to the sparsity of the data and using a dedicated algorithm to iteratively remove outlying data points from the analysis, we test the amplitude and the scatter of the direction of the BF and estimate the significance of the results using Monte Carlo simulations. The paper is organized as follows. In Section 2 we describe the initial filtering of the data and the method for extracting the individual radial components of the peculiar velocities, as well as how we generate random simulations of data with the same spatial distribution. In Section 3 we address the inevitable bias due to sparsity in both the amplitude and direction in naive best-fit methods used to detect an overall BF. We introduce our method in Section 4 and define a score which measures the scatter of the best-fit direction in successive iterations. We demonstrate that this score is effective in identifying simulated datasets with an artificially inserted BF and discuss how the significance of its findings can be estimated. In Section 5 we consider both the full dataset and the application of the scatter-based iterative method to the data and present the results. We conclude in Section 6.", "pages": [ 1, 2 ] }, { "title": "2.1 Data filtering", "content": "We use the recent type-Ia SNe compilation Union2.1 [45], which contains 580 filtered SNe at redshifts 0 . 015 0 . 5c). In order to avoid these pathologies while still retaining the ability to examine the behavior at distances larger than those accessible with galaxy surveys ( /lessorsimilar 100 Mpc /h ), we apply an initial cutoff in redshift and remove all points with z > 0 . 2 (corresponding to /lessorsimilar 550 Mpc /h ) from our dataset. 0 . 5c). In order to avoid these pathologies while still retaining the ability to examine the behavior at distances larger than those accessible with galaxy surveys ( /lessorsimilar 100 Mpc /h ), we apply an initial cutoff in redshift and remove all points with z > 0 . 2 (corresponding to /lessorsimilar 550 Mpc /h ) from our dataset. In Fig. 1 we plot the spatial distribution of the Union2.1 dataset with the remaining 200 SNe marked in blue. It is apparent that even outside the galactic plane the sky coverage is quite poor and the three-dimensional distribution of the remaining data is significantly sparse and inhomogeneous. The implications of this will be discussed in the next section.", "pages": [ 2 ] }, { "title": "2.2 Peculiar velocities and best-fit bulk flow", "content": "The Union2.1 dataset specifies for each SN its measured redshift z , the inferred distance modulus µ obs and the error ∆ µ obs . The relation between the cosmological (' true ') redshift ¯ z and the distance modulus is given by where d L is the luminosity distance, which in a flat universe with matter density Ω M , a cosmological constant Ω Λ and a current Hubble parameter H 0 , is Due to the peculiar velocity, both the observed redshift and distance modulus (through the luminosity distance) will differ from their true cosmological values [49]. To first order in v · ˆ n , where ˆ n is the direction pointing to a SN with peculiar velocity v , we get In order to extract the radial peculiar velocity v r of the SNe in our dataset, we follow the first order expansion in [20, 49] where H ( z ) is the Hubble parameter at redshift z , and d A ( z ) = d L ( z ) / (1 + z ) 2 is the observed angular diameter distance to the SN. We then find the best-fit BF velocity v BF in our set of N SNe, each with a radial velocity amplitude v i r in a direction ˆ n i , by minimizing with respect to the direction and amplitude of v BF , where ∆ v i r are the individual errors obtained from the measurement errors in the distance moduli ∆ µ i obs using Eq. (4).", "pages": [ 2, 3 ] }, { "title": "2.3 Monte Carlo simulations", "content": "Our Monte Carlo simulations consist of random permutations of the sky locations of the SNe in our dataset, after removing the initial BF velocity v BF init from the entire set by subtracting its corresponding component from the individual velocities The new dataset will have the same spatial distribution and its own initial random BF with a typical v rms amplitude. In order to simulate a random realization with a specifically chosen cosmological BF (up to statistical noise), for the purposes of testing our method, we simply add the chosen non-random BF contribution to the individual velocities after the permutation. For the analysis in this paper we use 16 , 000 random realizations (spatial permutations) of the Union2.1 data with no artificially inserted BF in order to test against the null hypothesis. To examine the detection capabilities of our method, we use 6 , 000 different random realizations for each inserted BF amplitude in the range | v BF | = { 50 , 100 . . . 450 km / s } , all in the direction ( l, b ) = (295 · , 5 · ), which is the direction of the best-fit BF on the full dataset (for the purposes of estimating the significance of our results, we have verified that this specific choice of direction has no effect).", "pages": [ 3 ] }, { "title": "3 SPARSITY BIAS", "content": "As mentioned above, the spatial distribution of the SNe dataset is inhomogeneous and sparse across the sky and in redshift depth. As a consequence, any search for an overall BF will be severely biased. Such a bias must be taken into account when evaluating the significance of a measured bestfit BF vs. the expectation from a ΛCDM universe. We now examine this bias separately in terms of the direction and amplitude of the BF. In the first subsection we show that the sparsity of our dataset causes a preference for a flow in directions within the galactic plane. In the second subsection we show that the root-mean-square (rms) velocity typically used under the ΛCDM hypothesis is inappropriate for a significance estimation of the BF amplitude in a sparse dataset such as ours.", "pages": [ 3 ] }, { "title": "3.1 Bulk flow - direction", "content": "In a dense homogeneous dataset (which has no preferred direction), if we perform many random mixings of the sky locations of the SNe, the best-fit BF direction will be distributed uniformly over the 4 π area of the sky. In a histogram of the measured directions, inside a circle of radius α around any sky coordinate we expect to find a fraction of the results. To demonstrate the bias induced by the sparsity of our dataset, we plot in Fig. 2 the ratios between the measured fraction and its expected value f meas . /f for a uniformly distributed set ( Left ) as well as for our dataset ( Right ), using α = 20 · . We see that the result for our dataset is far from isotropic. Its spatial distribution, regardless of the observed magnitudes, is biased towards a specific portion of the sky, namely the region surrounding the galactic plane.", "pages": [ 3 ] }, { "title": "3.2 Bulk flow - amplitude", "content": "Another implication of the sparsity in the data is that the random component of the BF does not follow the expected ΛCDM behavior. We must therefore quantify the difference between the expected ΛCDM rms velocity and the rms velocity we expect when dealing with a sparse dataset such as Union2.1.", "pages": [ 3 ] }, { "title": "3.2.1 Velocity rms in Λ CDM", "content": "In ΛCDM, the expected value for the BF amplitude is zero 〈 v 〉 = 0 while its variance satisfies [3] where f = Ω 0 . 55 m is the dimensionless linear growth rate, P ( k ) is the matter power spectrum, W ( kR ) is the Fourier transform of a window function with characteristic scale R and the angle brackets 〈 .. 〉 denote an ensemble average. Since ΛCDM is isotropic, this means that for each primary direction i ∈ { x, y, z } in a Cartesian coordinate system the BF amplitude may be described using a normal distribution To estimate the significance of a non-vanishing BF measured in a given survey, a common approach is to tweak the frame of reference so that the BF points exactly in one of the primary directions, e.g. ˆ e y , and compare the 'single component' measured BF to σ Λ / √ 3. However, this ignores the fact that the BF amplitude in the other two directions vanishes due to this particular choice of frame and would lead to an overestimated significance of the BF amplitude. To resolve this, we use the fact that the BF amplitude is a square-root of a sum of three normally distributed variables | v | 2 = (∑ i v 2 i ) , and so in ΛCDM it satisfies That is, it follows an 'unnormalized' χ distribution with 3 degrees of freedom (a Maxwell-Boltzmann distribution) [50]. Eq. (10) represents the probability density function (PDF) of the BF amplitude inside some volume, that is modulated by the same window function as in Eq. (8), in an unbiased way.", "pages": [ 3 ] }, { "title": "3.2.2 Velocity rms in Union2.1", "content": "In order for the right-hand side of Eq. (10) to describe the observed BF | v obs | appropriately, one needs to measure the peculiar velocity in many spatial locations, so that the typical separation between any two nearest neighbors that were measured will be much smaller than the coherence scale. This is clearly not satisfied for the sparse Union2.1 dataset. Therefore we should replace the window function with a sum of N delta functions, each centered on the location R i of a single SN However, since in Fourier space the new window function term will consist of ∼ N 2 interference terms that are no longer spherically symmetric. Therefore using Eq. (11) to evaluate the sparse-case equivalent of Eq. (10) is unfeasible. Instead we use the amplitudes of the best-fit BF of 16 , 000 random spatial permutations of our dataset, as described in § 2.3, as an approximation of the PDF for the BF amplitude inside a sphere of radius z = 0 . 2. The difference between this approximation and the isotropic ΛCDM scenario will be encoded in a best-fit σ fit (instead of σ Λ ) which describes the observed distribution In Fig. 3 we plot the approximated PDF for the BF amplitude and the corresponding best-fit χ 3 distribution according to Eq. (13). We find as opposed to σ Λ = 43 km/s calculated directly from Eq. (8) for a top-hat window function of size R = 550 Mpc / h. We see from Fig. 3 that a naive estimation of the significance of a measured BF amplitude in a sparse survey would be highly overestimated.", "pages": [ 3, 4 ] }, { "title": "4 SCATTER-BASED METHOD", "content": "We present a method based on an iterative process of repeatedly fitting a BF to the peculiar velocity field after the removal of the datapoint with the highest deviation from the previous fit. If there is a significant BF in the full dataset, the compactness of the scatter in the directions identified for the best-fit BF in each iteration can be used as an efficient estimator of the significance of the original flow. The stronger the flow in the full dataset, the smaller the scatter we will measure in the iterations.", "pages": [ 4 ] }, { "title": "4.1 Iterative algorithm", "content": "Heuristically, the algorithm can be sketched as follows After calculating the best-fit BF of the complete set according to Eq. (5), we examine the residual velocities of the different SNe in order to identify the one with the strongest deviation from the bulk. In each iteration, we find the bestfit v iter BF and then identify the point i with the largest contribution and remove it from the dataset before the next iteration. By iteratively removing these deviants, we can also verify that our results are not dominated by a small subset of the data with some common characteristic such as low redshift or a specific location on the sky.", "pages": [ 4, 5 ] }, { "title": "4.2 Scatter score", "content": "To measure the scatter, we assign a scatter score to the data, defined by which is a cumulative sum of the consecutive and total shifts, i.e. the sum of the distances from the direction of the best-fit BF in the current iteration ˆ n j to the one in the last iteration ˆ n j -1 and to that of the first iteration ˆ n 1 . This measures both the tightness and the extent of the scatter of the measured directions throughout the iterative process. In Fig. 4 we demonstrate the results for the scatter score S by plotting the directions of the best-fit BF at each iteration for a random realization with just a random BF and with increasing artificially-added BF amplitudes in the direction ( l, b ) = (295 · , 5 · ), which is the direction of the bestfit BF of our dataset (this choice of inserted direction allows a straightforward comparison with the data and accounts for a possible bias, as mentioned in § 3.1, but we have verified that it has no effect on our significance estimation). The results are consistent with the expected behavior: as the inserted BF amplitude is increased, the scatter becomes smaller and converges to a region closer to the inserted BF direction.", "pages": [ 5 ] }, { "title": "4.3 Significance estimation", "content": "We compare S data with the mean value S evaluated using 6 , 000 random simulations for each inserted BF amplitude | v BF | and infer the significance of the data in terms of the probability that a random ΛCDM realization will get a lower score than the data where P ( | v | )d | v | is the probability that a ΛCDM realization of the data will have a BF of amplitude between | v | and | v | + d | v | given the sparsity of the data according to § 3.2, and P ( S < S data | | v | ) is the conditional probability that a random simulation will result in S < S data given a BF amplitude | v | .", "pages": [ 5 ] }, { "title": "5.1 Full dataset", "content": "Before applying our scatter-based method described in the last section, we note that using a naive best-fit, the overall BF in our dataset has an amplitude | v BF | = 260 km/s and points in the direction ( l, b ) = (295 · , 5 · ), which is in agreement with results reported elsewhere [5-7, 9, 10, 17, 1923, 26, 27, 29]. This direction lies in proximity to features in the CMB (most of all to the giant rings reported in [39]), but is also close to the galactic plane, as might have been expected given the sparsity bias shown in Fig. 2. In addition, referring back to Fig. 3 we see that when comparing with the expected rms amplitude in a finite-size survey with the same spatial distribution, this amplitude, although high, is consistent with ΛCDM at the 95% C.L. (naively using the unbiased ΛCDM expectation, one might have assigned a much larger significance to this result). Thus, using the full dataset, we conclude that no claim can be made as to the existence of a cosmological BF in the Union2.1 type-Ia SNe data up to redshift z = 0 . 2 given the significant bias induced by the poor sky coverage and redshift-volume density of this dataset.", "pages": [ 5 ] }, { "title": "5.2 Sifting iteratively through the data", "content": "The scatter of the best-fit BF direction measured in the iterative process described in § 4 is plotted in the left panel of Fig. 5. In the right panel we plot the luminosity distance to the excluded SN as a function of the iteration number, and demonstrate that our results are not dominated by a subset of nearby SNe (SNe at distances /greaterorsimilar 500 Mpc /h remain in the dataset until the final iterations). Comparing Figs. 4 ( Left ) and 5 ( Left ) we see that the scatter of the data is much smaller than that of the realizations with | v BF | ≤ 300 km/s, and is comparable in size to a | v BF | /greaterorsimilar 450 km/s realization. This is also shown quantitatively in Fig. 6 ( Left ), where we plot the significance of the compactness of the scatter with respect to ΛCDM, as described by Eq. (17), as a function of the total number of iterations N iter . For N iter =110 (chosen arbitrarily), we show in Fig. 6 ( Right ) a few percentiles of the results of the normalized score evaluated for random realizations of the data according to (17), along with the data result. We see that the score for our dataset is outside the 95% C.L. for any initial (i.e. for the whole dataset) BF amplitude smaller than 300 km / sec. Integrating over all possible initial BF amplitudes, we see that S data is surprisingly low with respect to the expectation from a ΛCDM universe: the overall probability that a single ΛCDM realization would get a score that is as low as the score of the real data is < 0 . 5% for any N iter > 30, and gets as low as 0 . 1% for some choices of N iter . Thus we conclude that the scatter of the best fit BF direction is significantly low, at a /greaterorsimilar 99 . 5% C.L.", "pages": [ 5 ] }, { "title": "6 CONCLUSIONS", "content": "The goal of this work was to use the most recent compilation of type-Ia SNe measurements in order to test the claims of a peculiar velocity BF in different studies. After truncating the Union2.1 catalogue at redshift z = 0 . 2 and extracting the radial peculiar velocity field, we showed that a naive attempt to measure a best-fit BF in this field ignores a significant bias due to its sparse spatial distribution and renders inconclusive results for the amplitude and direction of the best-fit flow. This sparsity bias was discussed in detail above along with the difficulty in determining the correct ΛCDM prediction with which any result should be compared. We presented a prescription for estimating this value in a finite survey of given redshift extent and spatial distribution, and concluded that the BF amplitude measured in the Union2.1 data up to redshift z = 2 is consistent with the 95% C.L. limits. Given the consistency in the reports from a wide spectrum of analyses regarding the direction of the measured BF and the alignment between the reported values and certain CMB features, we focused on the direction and introduced a method which measures the scatter in the best-fit BF direction as outlying points are removed in iterations. We were careful to use realistic expectations for a BF amplitude in a sparse dataset and used Monte Carlo simulations with similar sparsity to estimate the significance of our findings. Our results suggest that the Union2.1 data up to redshift z = 0 . 2 contains an anomalous BF at a 99 . 5% C.L. compared to random simulations with the same sparsity as the data. In the future, as more data is collected, the method used in this work will become more and more robust and enable the measurement of the BF in consecutive redshift bins to yield a better analysis of the redshift dependence of the measured result. In addition, it might be possible to focus on measurements from a single survey and thus reduce the errors stemming from combining several surveys with different characteristics. If the reports of a BF which is inconsistent with ΛCDM are verified by future observations, it shall serve as a promising lead for theoretical research exploring areas beyond the concordance cosmological model. The full potential of typeIa SNe data to settle this issue is yet to be realized.", "pages": [ 6, 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "BR thanks A. Nusser and D. Poznanski for useful discussions. We also thank A. Nusser for comments on the first version of this work. EDK was supported by the National Science Foundation under Grant Number PHY-0969020 and by the Texas Cosmology Center.", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "I adopt a formalism previously developed by Catelan and Theuns (CT) in order to estimate the impact of primordial non-Gaussianity on the quasi-linear spin growth of cold dark matter protostructures. A variety of bispectrum shapes are considered, spanning the currently most popular early Universe models for the occurrence of non-Gaussian density fluctuations. In their original work, CT considered several other shapes, and suggested that only for one of those does the impact of non-Gaussianity seem to be perturbatively tractable. For that model, and on galactic scales, the next-to-linear non-Gaussian contribution to the angular momentum variance has an upper limit of ∼ 10% with respect to the linear one. I find that all the new models considered in this work can also be seemingly described via perturbation theory. Considering current bounds on f NL for inflationary non-Gaussianity leads to the quasi-linear contribution being ∼ 10 -20% of the linear one. This result motivates the systematic study of higher-order non-Gaussian corrections, in order to attain a comprehensive picture of how structure gravitational dynamics descends from the physics of the primordial Universe. Key words: large-scale structure of the Universe", "pages": [ 1 ] }, { "title": "C. Fedeli", "content": "Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611 ( [email protected]) 27 February 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Gravitational instability and hierarchical growth of Cold Dark Matter (CDM) density perturbations provide an elegant description for the acquisition of angular momentum by protostructures. Accordingly, patches of matter are spun up by tidal torques exerted by the surrounding Large-Scale Structure (LSS, see Peebles 1969, 1971; Doroshkevich 1970; White 1984; Heavens & Peacock 1988). At the linear level the spin growth is given by the coupling of the firstorder deformation tensor and the inertia tensor of the patch, and it agrees quite well with more accurate numerical results (see the review by Schafer 2009 and the references therein). Acquisition of angular momentum beyond the simple linear description has been tackled by means of Lagrangian perturbation theory in a series of papers by Catelan and Theuns (Catelan 1995; Catelan & Theuns 1996a,b, 1997, CT henceforth) almost two decades ago. In particular, they found that the next-to-linear correction to the growth of ensemble-averaged spin is non-vanishing only if primordial density fluctuations are non-Gaussian. CT explored several non-Gaussian models, and concluded that only for one of these does the angular momentum acquisition appear perturbatively tractable: the log-normal model for the gravitational potential of Moscardini et al. (1991). For this template, and adopting a representative mass scale of M ∼ 10 12 h -1 M /circledot (with h = 0 . 5 and Gaussian filtering), CT found an upper limit of ∼ 24% for the quasilinear non-Gaussian contribution to the spin variance. This figure translates to a ∼ 10% value when rescaled to a smaller and more typical galactic mass of M = 10 10 h -1 M /circledot (with h = 0 . 7 and a real- top-hat filter). For other templates the non-Gaussian contribution is comparable to, or larger than, the linear term, suggesting the impossibility of a perturbative expansion. Since then, the issue of angular momentum growth in nonGaussian cosmologies has not been investigated further. On the contrary, new and more general models of primordial nonGaussianity exist nowadays and, most importantly, constraints on the level of primordial non-Gaussianity coming from the Cosmic Microwave Background (CMB) and the LSS have dramatically improved over the last decade. Given the cosmological relevance of primordial non-Gaussianity (see Bartolo et al. 2004; Chen 2010 for recent reviews) and the significance of the CDM halo angular momentum acquisition for the formation and evolution of galaxies, it is important to update this topic. Specifically, it is interesting to explore the amplitudes and behaviors of the quasi-linear contributions to spin growth given by non-Gaussian models that are popular nowadays. This is the scope of the present letter. The rest of the manuscript is organized as follows. In Section 2 I review the linear and next-to-linear contributions to the ensemble-averaged spin growth of matter patches. In Section 3 I summarize the non-Gaussian cosmologies explored here. In Section 4 results are displayed and in Section 5 conclusions are drawn. Where needed, I adopted the following cosmological parameters: Ω m , 0 = 0 . 272, Ω Λ , 0 = 1 -Ω m , 0, Ω b , 0 = 0 . 046, H 0 = 100 h km s -1 Mpc -1 with h = 0 . 704, σ 8 = 0 . 809, and n s = 1.", "pages": [ 1 ] }, { "title": "2.1 Lagrangian displacement", "content": "In Lagrangian theory the comoving position x of a mass element at time τ can be written in terms of its initial position q and a displacement vector field S , as Following CT here I used a time variable τ that is related to the standard cosmic time t by d τ = dt / a 2 (Shandarin 1980), where a is the scale factor. Perturbative approximations to this exact expression can be found by expanding the displacement field in a series, where S 1 corresponds to the Zel'dovich approximation S 1( q , τ ) = D ( τ ) ∇ ψ 1( q ). Here D ( τ ) is the growth factor of linear density perturbations, which in a Einstein-de Sitter universe reads D ( τ ) = τ -2 . The function ψ 1 is the first order (Zel'dovich) displacement potential (Zel'dovich 1970), related to the linear density perturbation field by the Poisson equation ∆ ψ 1( q ) = δ ( q ), so that in Fourier space ˆ ψ 1( p ) = ˆ δ ( p ) / p 2 . The second-order term of the displacement field can also be separated in time and space, according to S 2( q , τ ) = E ( τ ) ∇ ψ 2( q ). The growth factor E ( τ ) reads E ( τ ) = -3 τ -4 / 7 in an Einstein-de Sitter universe, while for its more general expression I refer to CT. The second-order displacement potential can be related to its firstorder counterpart in Fourier space by In the previous Equation K ( p 1 , p 2 ) is a symmetric integration kernel defined as where µ is the cosine of the angle between the two wavevectors p 1 and p 2 .", "pages": [ 2 ] }, { "title": "2.2 Spin growth", "content": "The angular momentum of the matter initially contained in a comoving Lagrangian patch Γ of the Universe at time τ can be written as an integral over Γ , By considering the series expansion of the Lagrangian displacement field S introduced in Eq. (2), the angular momentum of the patch can be similarly written as The first-order term of the angular momentum series takes the form By expanding the Zel'dovich potential around the center of mass of the patch (assumed to be, without loss of generality, the origin of the reference frame) up to the second order, the previous equation takes the compact form In the previous equation εαβγ is the fully antisymmetric Levi-Civita tensor, D 1 ,βσ is the Zel'dovich deformation tensor, while I σγ is the inertia tensor of the patch, Summation over repeated indices is implicit. Likewise, the second-order term in the series expansion of the angular momentum reads which, under a second-order Taylor expansion of the displacement potential takes the form It can be shown that the second-order deformation tensor in Fourier space reads in terms of the Zel'dovich potential.", "pages": [ 2 ] }, { "title": "2.3 Ensemble averages", "content": "In order to simplify the previous results, it is meaningful to consider the ensemble average of the square of the angular momentum. It then follows that, up to the next-to-linear order, where In the previous Equation σ M is the mean deviation of the matter density field smoothed on a scale corresponding to mass M , while ν 1 and ν 2 are the first and second invariant of the inertia tensor, respectively. To be more precise, if λ 1, λ 2, and λ 3 are the three eigenvalues of the inertia tensor, then ν 1 ≡ λ 1 + λ 2 + λ 3 and ν 2 ≡ λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3. The next term is non-vanishing only if density fluctuations are non-Gaussian, and reads where In the previous Equation B ψ 1 represents the bispectrum of the Zel'dovich potential, which is now explicitly smoothed on a scale R = (2 GM / Ω m , 0 H 2 0 ) 1 / 3 . I assumed the standard real-space top-hat smoothing. The Zel'dovich potential can be related to the standard gravitational potential ϕ by making use of the Poisson equation, where T ( p ) is the cold dark matter transfer function (Bardeen et al. 1986; Sugiyama 1995). The integral in Eq. (17) has to be solved numerically for realistic bispectrum shapes. Fortunately, the bispectrum usually depends only on the magnitude of its three arguments, so that the above six-dimensional integral reduces to a threedimensional one.", "pages": [ 2, 3 ] }, { "title": "3 NON-GAUSSIAN SHAPES", "content": "I considered five different shapes for the primordial bispectrum, that are briefly described below. The first four are motivated by inflationary physics, and the amplitude of non-Gaussianity is given by the parameter f NL (assumed to be constant). The fifth is non-inflationary in nature, and hence independent on f NL. See Fedeli et al. (2011) and references therein.", "pages": [ 3 ] }, { "title": "3.1 Local shape", "content": "This bispectrum shape arises when a light scalar field, additional to the inflaton, contributes to the curvature perturbations (Bernardeau & Uzan 2002; Babich, Creminelli, & Zaldarriaga 2004; Sasaki, Valiviita, & Wands 2006). It is the same shape produced by the standard model of inflation (Falk, Rangarajan, & Srednicki 1993) but in this case the amplitude can be arbitrary. The potential bispectrum takes the simple form and it is maximized for squeezed configurations. The quantity A is the spectral amplitude of the potential (given by σ 8), while n s is the spectral slope.", "pages": [ 3 ] }, { "title": "3.2 Equilateral shape", "content": "This shape is a consequence of the inflaton Lagrangian being non-standard, and containing higher-order derivatives of the field (Alishahiha, Silverstein, & Tong 2004; Arkani-Hamed et al. 2004; Li, Wang, & Wang 2008). The resulting bispectrum is maximized for equilateral configurations. A template for the equilateral bispectrum can be found in Creminelli et al. (2007), however the expression is rather cumbersome and I did not report it here. The same applies to the following shapes.", "pages": [ 3 ] }, { "title": "3.3 Enfolded shape", "content": "The enfolded shape results from primordial non-Gaussianity being evaluated without the regular Bunch-Davies vacuum hypothesis (Chen et al. 2007; Holman & Tolley 2008). In this case the bispectrum is maximized for squashed configurations. A template for such a bispectrum is reported in Meerburg, van der Schaar, & Corasaniti (2009).", "pages": [ 3 ] }, { "title": "3.4 Orthogonal shape", "content": "This shape is defined as being orthogonal (with respect to a suitably defined scalar product) to both the local and equilateral forms. The resulting bispectrum is maximized for both equilateral and squashed configurations, and a template can be found in Senatore, Smith, & Zaldarriaga (2010).", "pages": [ 3 ] }, { "title": "3.5 Matter bounce shape", "content": "This configuration is the consequence of a model universe without inflation, but with a scale factor that bounces in a non-singular way (Brandenberger 2009; Cai et al. 2009). The matter bounce leads to a scale-invariant spectrum of density fluctuations, and to a bispectrum whose shape is similar to the local shape. Being noninflationary in origin, the non-Gaussianity induced by a matter bounce model has no dependence on f NL. It can instead be shown that the matter bounce bispectrum is comparable to the local one with a fixed f NL = -35 / 8. I considered explicitly the matter bounce because it is in principle possible that a weighted integral of the bispectrum, such as the one in Eq. (17), would magnify its differences with respect to the local model. As I show below this is actually not the case.", "pages": [ 3 ] }, { "title": "4 RESULTS", "content": "In Figure 1 I show ω M as a function of the mass scale for the five non-Gaussian cosmologies considered in this letter. In all cases, except for the matter bounce, I selected f NL = 1 in order to purely highlight the effect of the bispectrum shape. It is however easy to see that ω M is simply proportional to f NL. All curves decrease with increasing mass, implying that larger matter patches acquire lower amounts of angular momentum than smaller ones. This behavior is similar to the mass dependence of the linear term given in Eq. (15). Also, curves referring to different models are rather similar in shape. Besides the matter bounce model, the local model is the one having the largest effect, while the orthogonal model has the lowest. This is different from the behavior of, e.g., the halo bias, for which the equilateral model displays the smallest effect. The function ω M for the matter bounce is virtually identical to that for the local model when assuming f NL = -35 / 8, in agreement with the previous discussion. As can be seen from the structure of Eqs. (15) and (16), the relative importance of the non-Gaussian contribution with respect to the linear one is The acquisition of angular momentum by protostructures happens at high redshift where, under the assumption of flat spatial geometry, the Universe is well approximated by an Einstein-de Sitter model. Corrections due to the presence of a cosmological constant can be considered to be negligible. If this is the case, then Note that the negative sign cancels with the negative sign in the definition of ω M , so that the non-Gaussian contribution is positive (increases the spin growth) for a model with positive skewness and negative otherwise. This has already been noticed by CT. The previous equation also shows that the non-Gaussian contribution to spin acquisition grows faster than the linear one. If the matter patch under consideration is an overdense region, it is reasonable to assume that the spin growth induced by tidal torques occurs until the overdensity detaches from the overall expansion of the Universe and collapses into a bound structure. This moment τ ∗ can be naively identified as D ( τ ∗ ) σ M = 1. For an Einstein-de Sitter cosmological model this implies τ 2 ∗ = σ M , and thus In Figure 2 I show the mass dependence of the function Υ M ( τ ∗ ) for the various non-Gaussian cosmologies that have been considered in this work. For models with inflationary non-Gaussianity I assumed f NL = 1. As can be seen the mass dependence is in all cases relatively weak. In the local and matter bounce models Υ M ( τ ∗ ) is basically unchanged for masses ranging between M = 10 8 h -1 M /circledot and M = 10 15 h -1 M /circledot . For the equilateral and enfolded models Υ M ( τ ∗ ) increases by ∼ 50% over the same interval, while for the orthogonal case it decreases by a factor of ∼ 3. Hence, despite the fact that the linear and non-Gaussian contributions to the spin growth both decrease with mass, their relative importance remains relatively unchanged. The only exception is represented by the orthogonal model. Next, I selected a reference mass scale of M = 10 10 h -1 M /circledot and computed the dependence of Υ M ( τ ∗ ) on f NL, shown in Figure 3. As previously mentioned, this dependence is always linear, however the Figure is important in order to understand for what value of f NL a certain non-Gaussian model provides a given contribution to the total angular momentum variance. The matter bounce nonGaussianity is independent of f NL, hence its contribution is always at the percent (negative) level compared to the linear one. As for the other models, in order for the non-Gaussian contribution to be comparable to the linear one, primordial non-Gaussianity would need to be at the unrealistic level of f NL ∼ 400 for the local case, and substantially larger than that for other bispectrum shapes. Figure 3 allows one to determine the non-Gaussian contribution, in units of the linear contribution, for the current bounds on f NL. Constraints from the CMB (Komatsu et al. 2011) imply -13 < f NL < 96 at 95% Confidence Level (CL) for the local shape 1 , meaning that the non-Gaussian contribution can be at most ∼ 19% of the linear one. The same CMB data constrain -278 < f NL < 346 for the equilateral shape, implying a ∼ 18% relative importance. The tighter constraints on the level of non-Gaussianity for the enfolded shape come from the LSS (Xia et al. 2011), corresponding to -16 < f NL < 465 at 2 σ confidence level. This means that the nonGaussian contribution is at most ∼ 16% of the linear one. Finally, for the orthogonal shape the CMB data by Komatsu et al. (2011) bear -533 < f NL < 8, implying a maximum ∼ 10% (negative) relative strength. These numbers can be appreciated also by looking at the positions of the filled circles in Figure 3. For comparison, the black dotted line shows the upper limit to the non-Gaussian contribution found by CT, after assuming a log-normal distribution for the primordial gravitational potential and after rescaling it to the scale M = 10 10 h -1 M /circledot . I stress the fact that, while CT adopted a Gaussian window function and h = 0 . 5, I used a real-space top-hat filter and h = 0 . 704. Moreover, while CT calibrated the level of primordial non-Gaussianity using a value SR = 4 for the skewness of the matter density field on a scale R = 8 h -1 Mpc, I adopted, conservatively, SR = 0 . 1. This value results from the large-scale skewness per unit f NL for local non-Gaussianity ( ∼ 10 -3 , e.g., Figure 1 of Fedeli et al. 2011) multiplied by the most recent upper limit on the level of non-Gaussianity for the same shape ( f NL ∼ 100).", "pages": [ 3, 4 ] }, { "title": "5 CONCLUSIONS", "content": "I reconsidered the impact of primordial non-Gaussianity on the acquisition of angular momentum by CDM protostructures. NonGaussian initial conditions provide a next-to-linear correction to the spin growth that is absent when density fluctuations are normally distributed. Previous results, obtained by CT after assuming a log-normal primordial gravitational potential, resulted in a contribution to the spin variance of ∼ 10% with respect to the linear one. This value holds for a scale M = 10 10 h -1 M /circledot (with the cosmology of this work) and is based on a matter skewness of SR ∼ 0 . 1, as deduced in the previous Section. Other models turned out to give a very large quasi-linear effect, suggesting that Lagrangian perturbation theory could not be successfully applied in those cases. I found that for several current models of non-Gaussian initial conditions, the contribution to the galactic spin variance during the mildly non-linear regime is similar to what predicted assuming a log-normal primordial gravitational potential. Considering the upper limits to the current constraints on the level of inflationary primordial non-Gaussianity returns a next-to-linear contribution at the level of ∼ 10 -20%. These results imply that the spin growth induced by inflationary non-Gaussianity seems to be generically tractable via perturbation theory. CT also demonstrated that higher-order contributions in the case of Gaussian density fluctuations provide a correction to the angular momentum variance equal to ∼ 60% of the linear term. This means that the next-to-linear non-Gaussian contribution estimated here has a significant impact on the spin acquisition by protostructures. Such an impact could potentially be even larger, because higher-order non-Gaussian corrections, that have not been considered here, depend on the trispectrum of the Zel'dovich potential, and hence also react to primordial non-Gaussianity. The results presented in this letter motivate the study of these higher-order contributions, and show how it is possible to consistently describe the dynamics of protostructures based on largely general cosmological initial conditions.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "I thank the University of Florida for support through the Theoretical Astrophysics Fellowship. I credit L. Moscardini for insightful comments on the manuscript and I am deeply indebted to the anonymous referee for help in substantially improving this work.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Alishahiha, M., Silverstein, E., & Tong, D. 2004, Phys. Rev. D, 70, 123505 Arkani-Hamed, N., Creminelli, P., Mukohyama, S., & Zaldarriaga, M. 2004, Journal of Cosmology and Astro-Particle Physics, 4, 1 Babich, D., Creminelli, P., & Zaldarriaga, M. 2004, Journal of Cosmology and Astro-Particle Physics, 8, 9 Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. 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[ { "title": "ABSTRACT", "content": "Nearby gamma-ray bursts (GRBs) are likely to have represented a significant threat to life on the Earth. Recent observations suggest that a significant source of such bursts is compact binary mergers in globular clusters. This link between globular clusters and GRBs o GLYPH<11> ers the possibility to find time intervals in the past with higher probabilities of a nearby burst, by tracing globular cluster orbits back in time. Here we show that the expected flux from such bursts is not flat over the past 550 Myr but rather exhibits three broad peaks, at 70, 180 and 340 Myr ago. The main source for nearby GRBs for all three time intervals is the globular cluster 47 Tuc, a consequence of its large mass and high stellar encounter rate, as well as the fact that it is one of the globular clusters which comes quite close to the Sun. Mass extinction events indeed coincide with all three time intervals found in this study, although a chance coincidence is quite likely. Nevertheless, the identified time intervals can be used as a guide to search for specific signatures of GRBs in the geological record around these times. Key words: globular clusters: general - Gamma-ray burst: general - Galaxy: kinematics and dynamics - Astrobiology", "pages": [ 1 ] }, { "title": "W. Domainko, 1 C.A.L. Bailer-Jones, 2 F. Feng, 2", "content": "1 Max-Planck-Institut fur Kernphysik, P.O. Box 103980, D-69029 Heidelberg, Germany 2 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117 Heidelberg, Germany Xxxxx XX", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Globular clusters, densely packed groups of old stars, can e GLYPH<14> -ciently produce close stellar binaries by dynamical interactions of their member stars. Examples for such dynamically formed binaries include low-mass X-ray binaries (e.g. Clark 1975; Katz 1975), cataclysmic variables (Pooley & Hut 2006) and milli-second pulsars (msPSRs, Ransom 2008; Abdo et al. 2010). The most extreme binaries found in globular clusters consist of two neutron stars (Anderson et al. 1990). Mergers of such binaries are believed to be the central engine of short gamma-ray bursts (GRBs) (Grindlay et al. 2006; Dado et al. 2009; Lee et al. 2010), that produce brief, intense flashes of ionising radiation. In contrast to short bursts, long bursts are believed to originate from the death of short-lived massive stars (see Gehrels et al. 2009, for a review on long and short bursts). It has been argued that the rate of short GRBs in the local universe is dominated by the merger of neutron star binaries formed in globular clusters (Salvaterra et al. 2008; Guetta & Stella 2009). A link between globular clusters and short GRBs is further supported by the presence of a short GRB remnant candidate in the Galactic globular cluster Terzan 5 (Domainko 2011a), observed in the very-high energy gamma-ray (Abramowski et al. 2011, 2013), X-ray (Eger et al. 2010; Eger & Domainko 2012) and radio wave band (Clapson et al. 2011). Additional evidence for the GRB - globular cluster connection comes from spatial o GLYPH<11> sets of short GRBs from their host galaxies (Berger 2010; Salvaterra et al. 2010; Church et al. 2011) and the redshift distribution of such events (Salvaterra et al. 2008; Guetta & Stella 2009). Since globular clusters follow well-defined orbits around the Galaxy (Domainko 2011b), their coupling with GRBs allows us to examine the long-standing question of the past history of gammaray flux on the Earth. (A similar approach for supernovae exploding in star clusters has been used in Svensmark (2012)). Numerous studies have shown that gamma rays from supernovae (SNe) or GRBs could, in principle, have had a significant impact on the Earth's atmosphere and biosphere, potentially even contributing to mass extinctions (see Thorsett (1995); Scalo & Wheeler (2002); Melott et al. (2004); Thomas et al. (2005) for the a GLYPH<11> ect of GRBs in general, and Dar et al. (1998); Melott & Thomas (2011) for the a GLYPH<11> ect of merger-induced bursts). However, demonstrating that SNe or GRBs may in fact have played some role first requires identifying that sources could have come near enough to the Earth at some point. Some previous studies have attempted to make a connection between the solar motion relative to the Galactic plane or spiral arms, on the assumption that the gamma ray flux incident on the Earth is larger in these regions of enhanced massive star formation rate and / or increased stellar density (see Bailer-Jones (2009) for a review). However, a recent study shows that the flux from these sources as modulated by the plausible solar motion over the past 550 Myr has a poor correlation with the variation of the extinction rate on the Earth (Feng & Bailer-Jones, submitted). Indeed, it seems that astronomical phenomena alone are unlikely to be the dominant driver of biological evolution or the cause of all (or even most) mass extinctions. Nonetheless, if a GRB were to explode near to the Earth, its consequences could be catastrophic, and globular clusters are presumably a significant source of GRBs. The goal of this paper is to reconstruct the orbits of globular clusters relative to the Sun in order to calculate the GRB flux at the Earth as a function of time, and thereby to identify potential candidate clusters. The data for this orbital reconstruction comes from the positions, distance, proper motion and radial velocity catalogues of globular clusters of (Dinescu et al. 1997, 1999a,b, 2003), from which we obtain the current Galactic coordinates and space velocities. By sampling over the (often significant) uncertainties in the reconstructed orbits of the globular clusters and the Sun, we infer the expected GRB flux as a function of time. This allows us to identify the most probable intervals in the Earth's history of a significantly increased gamma ray flux, which may (or may not) be associated with times of higher extinction rate. In section 2.1 we describe the orbital reconstruction method, and in section 2.2 we explain how we derive from this the probability distribution over the past cluster-Sun separation and the expected gamma ray flux at the Earth. This takes into account the di GLYPH<11> erent GRB rates in the clusters, which is derived in section 2.3. We give our results in section 3 where we also identify some past extinction events. We conclude in section 4 with an outlook on how to further this work.", "pages": [ 1, 2 ] }, { "title": "2.1 Reconstructing Galactic orbits", "content": "We trace the orbits of the Sun and the globular clusters back in time by integrating the equations of motion through the Galactic potential. In a purely gravitational system there is no dissipation of energy, so the dynamics are reversible. We adopt an analytic, three component, axisymmetric potential, GLYPH<8> , comprising the Galactic bulge, halo and disk The bulge and halo are represented with a Plummer distribution in which the characteristic length scales are bb = 0 : 35 kpc for the bulge and bh = 24 : 0 kpc for the halo, and the bulge and halo masses are Mb = 1 : 40 GLYPH<2> 10 10 M GLYPH<12> and Mh = 6 : 98 GLYPH<2> 10 11 M GLYPH<12> respectively. R is the radial coordinate perpendicular to the axis, and z is the distance from the Galactic plane. For the disk we use the potential from Miyamoto & Nagai (1975) with the values Md = 7 : 91 GLYPH<2> 10 10 M GLYPH<12> for the disk mass, and ad = 3 : 55 kpc and bd = 0 : 25 kpc for the scale length and scale height of the disk, respectively (after Garc'ıa-S'anchez et al. (2001)). The integration is performed numerically from the present back to 550 Myr BP (before present). This time limit is chosen because it corresponds to the beginning of the Phanerozoic eon, a time from which the fossil record becomes more indicative of biodiversity variations. The globular clusters (and Sun) are treated as massless. The initial conditions for the integration are the current phase space coordinates (three position and three velocity components) of the globular clusters (and Sun). These of course have significant uncertainties, each represented as a Gaussian with known mean (measured coordinate) and standard deviation (estimated uncertainty). These come from Dana Casetti-Dinescu's catalogue for globular cluster's three-dimensional space velocities (2012 version) 1 for the globular clusters, and from Hipparcos data by (Dehnen & Binney 1998) for the Sun. We further use a distance of the Sun to the Galactic center obtained from astrometric and spectroscopic observations of the stars near the supermassive black hole of the Galaxy (Eisenhauer et al. 2003) and the displacement of the Sun from the Galactic plane is calculated from the photometric observations of classical Cepheids by Majaess et al. (2003). Rather than just performing a single integration for each object (cluster or Sun), we Monte Carlo sample its initial conditions from the uncertainty distribution in order to build up a large sample of orbits. Figure 1 shows an example of such sample orbits for one globular cluster, 47 Tuc, by plotting the distance of the cluster from the Sun over time. (We sample over the possible orbits of the Sun too.) We do not take into account the (possibly significant) uncertainties in the Galactic potential. In principle we could adopt an uncertainty model for these parameters and marginalize over them also. But we choose to omit this in this first investigation. Finally we have to note that compact binaries may be ejected from their parent cluster before they merge and produce a GRB (e.g. Phinney & Sigurdsson 1991; Ivanova et al. 2008). This e GLYPH<11> ect will smear out the distribution of compact binaries around the producing cluster. The typical escape velocities for massive globular clusters are about 50 km s GLYPH<0> 1 , which is comparable to the present uncertainties of the globular cluster velocity. Although over time the orbit of the ejected binary could deviate considerably from its parent cluster, the uncertainty in its orbit is comparable to the uncertainty for its parent cluster, which we take into account. We therefore choose to omit the issue of ejected GRB progenitors for this first investigation. Furthermore, more massive clusters are better able to retain their binaries, and these are the clusters that preferentially produce GRBs (see Sec. 2.3).", "pages": [ 2 ] }, { "title": "2.2 The probability distribution over globular cluster distances and the expected GRB flux at the Earth", "content": "For a given globular cluster, c , we convert the set of (thousands of) relative orbits into a two-dimensional density distribution over time, t , and separation, r , using kernel density estimation. We interpret the resulting distribution as a probability distribution of the Sun-cluster separation over time, fc ( r ; t ), which is normalized such that R r fc ( r ; t ) dr = 1 for all t and for each cluster. This is shown in Figure 2 for 47 Tuc, in which the probability density is plotted as a grey scale. At any given time, the darker the band, the more concentrated the probability is around a smaller range of distances. The width of the distribution at any time is determined by how the uncertainties in the present coordinates of both globular cluster and Sun propagate back in time. The density estimates for some other globular clusters are shown in Figures 3-7. The flux of a gamma ray burst at the Sun is proportional to 1 = r 2 . Multiplying fc ( r ; t ) by 1 = r 2 , and assuming that gamma ray bursts occur at random times 2 , we get a 2D distribution which is proportional to the expected GRB flux from distance r at time t . If we integrate this (at a time t ) over all distances then we get a quantity, R r 1 r 2 fc ( r ; t ) dr , which is proportional to the expected GRB flux from that globular (at time t ). The important thing about this quantity is that it takes into account the uncertainties in the reconstructed globular cluster and solar orbits. We now extend this concept to the complete set of globular clusters. Each cluster has a di GLYPH<11> erent probability per unit time of producing a GRB, proportional to the factor wc , defined in section 2.3. We can then see that the quantity is proportional to the expected GRB flux at the Sun at time t from any globular cluster. In principle we integrate up to r max = 1 , but in practice we can truncate it to a few kpc. Indeed, if there is a minimum flux threshold below which the gamma ray flux is too small to have any significant a GLYPH<11> ect on the Earth's biosphere or climate, then truncation is appropriate. Note that the absolute scale of GLYPH<9> ( t ) is not calibrated: only relative values are meaningful.", "pages": [ 3 ] }, { "title": "2.3 Weighting individual globular clusters", "content": "Observationally, the frequency of occurrence of GRBs in individual globular cluster is not known. The dynamical formation of compact binaries, proposed progenitors of such events, is rather complex, involving at least two stellar encounters (see Ivanova et al. 2008, 2010). However, the rate of GRBs in each globular cluster is expected to be linked to the cluster properties. Several authors have already investigated the dependence of the compact binary formation rate on the characteristics of the clusters. Ivanova et al. (2008) found that the formation of close double neutron star binaries depends on the square of the cluster density, and that the number of retained neutron stars increases as the escape velocity (and thus cluster mass) increases. Grindlay et al. (2006) used a model where the formation of double neutron star binaries scales linearly with the neutron star number density, the velocity dispersion (and thus mass of the cluster) and the number of potential progenitor systems (binaries containing one neutron star). Both models find that massive clusters with a high concentration of stars strongly favour the formation of prospective GRB progenitor systems. Here we adopt a similar approach to these previous works and scale the expected GRBrate with quantities that are known for a large sample of globular clusters. Specifically, assuming that GRBs are caused by neutron star encounters, then the GRB rate will depend on the number of neutron stars in the cluster and their encounter rate. We assume that the number of neutron stars scales linearly with the mass of the globular cluster, m c, and thus linearly also with the cluster luminosity. The total encounter rate, GLYPH<0> c , is given as GLYPH<0> c / GLYPH<26> 1 : 5 0 r 2 core (Pooley & Hut 2006), where GLYPH<26> 0 is the central stellar number density and r core is the core radius of the globular cluster. Values for these parameters for our sample of clusters we obtained from Harris (1996, 2010 edition) 3 . Combining these two factors we get a quantity w c = m c GLYPH<0> c , which is proportional to the frequency of gamma rays bursts in the clusters, and is used as the weighting factor in section 2.2. Accordingly, and as already noted in the beginning of this section, massive clusters with high concentrations of stars at their center have a large GRB rate. We investigated the uncertainties of our approach by applying an alternative weighting scheme for individual globular clusters. We followed Ivanova et al. (2008) and adopted weights proportional to GLYPH<26> 2 0 m c. With this approach we found that the typical uncertainties for the leading clusters is a factor of a few, with a few notable exceptions (see Sec. 3). For the results in Sec. 3 we use the weights w c as defined earlier in this section. Having calculated the indivdual weights, w c, they are then normalised such that the sum of all weights equals 1. Here we used 141 clusters from Harris (1996, 2010 edition) where all necessary parameters are known. This, in principle, further allows us to estimate the expected absolute GRB rates for individual globular clusters by defining that a weight of 1 corresponds to the Galactic rate of GRBs launched in globular clusters. This galactic GRB rate can be calculated from the short GRB rate in the local Universe of 8 + 5 GLYPH<0> 3 Gpc GLYPH<0> 3 yr GLYPH<0> 1 (Coward et al. 2012) and the density of Milky Way-type galaxies of 0.01 Mpc GLYPH<0> 3 (Cole et al. 2001). This rate is obtained for GRBs beamed towards Earth and is thus independent of the degree of collimation of the events. If it is assumed that the occurrence of short GRBs in the local Universe is dominated by bursts launched in globular clusters (Salvaterra et al. 2008; Guetta &Stella 2009), then the combined GRB rate of all globular clusters is 10 GLYPH<0> 6 year GLYPH<0> 1 . This estimate is also consistent with the theoretically expected rate of short GRB production in these clusters (Lee et al. 2010).", "pages": [ 3, 4 ] }, { "title": "3 RESULTS", "content": "Figure 8 shows the expected GRB flux, GLYPH<9> ( t ), for the case r max = 5 kpc. This distance threshold covers 95% of all hazardous GRBs if a log-normal GRB luminosity distribution with log E GLYPH<13>; iso = 50 : 81 GLYPH<6> 0 : 74 erg (Racusin et al. 2011) and a critical fluence at Earth for a significant a GLYPH<11> ect on the biosphere or climate of 10 7 erg cm GLYPH<0> 2 (Melott & Thomas 2011) is assumed. (The profile of GLYPH<9> ( t ) has very Y similar shape for other values of r max, the di GLYPH<11> erence being that the 'background' level is higher for larger values of r max, and lower for smaller values.) We see a significant variation. There are three broad peaks, at 70, 180 and 340 Myr. These correspond to times in the Earth's history when - within the limitations of our orbital reconstruction and assumptions made - we would expect a significantly higher level of GRB flux than the average over the past 550 Myr. Examining the plots of fc ( r ; t ) for all clusters, we can identify those clusters which make the biggest contribution to GLYPH<9> ( t ) in each peak: The prominence of 47 Tuc is a consequence both of its high weight, wc , and the fact that it is one of the globular clusters which comes quite close to the Sun. All the main contributors are massive clusters that contain significant populations of dynamically formed stellar binaries. Specifically: As mentioned earlier, GRBs are of course discrete, rare events. Indeed, our calculations suggest that only about 10 GRBs will have occurred within 5kpc of the Sun over the course of the Phanerozoic. Thus the true distribution of flux with time would comprise of a series of narrow peaks of various heights. Fig. 8 shows the expected flux at time (times a constant), so is the best single estimate of that distribution. By way of comparison we overplot in Figure 8 the times of 18 mass extinction events on the Earth revealed by the fossil record, as compiled by (Bambach 2006). One may be tempted to draw a causal connection between one of these events and one of the peaks in GLYPH<9> ( t ), although clearly there is a reasonable chance that one of these 18 events could coincide with a peak just by chance. 4 It is nonetheless worthwhile identifying those events nearest to the three peaks. These are Whether or not a globular cluster GRB is implicated in any of these extinctions remains a subject for future work.", "pages": [ 4, 5 ] }, { "title": "4 OUTLOOK", "content": "In this paper we have traced globular cluster orbits back to the beginning of the Phanerozoic eon in order to identify time intervals where a high flux of ionizing radiation caused by a nearby GRB is more likely. We found that the probability for such an event is far from flat with time during the Earth's history. It instead exhibits several distinct peaks, the most prominent ones being around 70, 180 and 340 Myr BP. The main source of GRBs in all cases is 47 Tuc. All three time intervals can in principle be associated with a mass extinction event, although a chance coincidence is likely. Therefore, to establish a link between a nearby GRB and an impact on the Earth and its biota, supporting geological signatures are needed. Geological signatures could comprise radiation damage of crystals (e.g. fossil cosmic ray tracks (Fleischer et al. 1967) or color shifts (Ashbuugh 1988)), deposition of radioactive isotopes (Dar et al. 1998) or elevated rates of bone cancer (Rothschild et al. 2003). The time intervals identified in this paper can be used as a guideline to search for such signatures in the geological record. Finally, the current orbital parameters of globular clusters and the solar system are subject to considerable uncertainties. (These were taken into account in our analysis, and contribute to smearing out the probability curve.) This situation will be substantially improved in the near future with the launch of the Gaia satellite, which will determine the dynamics of the Galaxy with unprecedented accuracy. With better determined orbital parameters we will be able to constrain the past orbits more tightly, and so repeat this study to give results of higher confidence.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Abdo, A. A. et al. (Fermi-LAT collaboration) 2010, A&A, 524, 75 Abramowski, A. et al. (H.E.S.S. collaboration) 2011, A&A, 531, L18 W. Domainko et al.", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "In this work we present for the first time an analytic framework for calculating the individual and joint distributions of the n -th most massive or n -th highest redshift galaxy cluster for a given survey characteristic allowing to formulate Λ CDM exclusion criteria. We show that the cumulative distribution functions steepen with increasing order, giving them a higher constraining power with respect to the extreme value statistics. Additionally, we find that the order statistics in mass (being dominated by clusters at lower redshifts) is sensitive to the matter density and the normalisation of the matter fluctuations, whereas the order statistics in redshift is particularly sensitive to the geometric evolution of the Universe. For a fixed cosmology, both order statistics are e ffi cient probes of the functional shape of the mass function at the high mass end. To allow a quick assessment of both order statistics, we provide fits as a function of the survey area that allow percentile estimation with an accuracy better than two per cent. Furthermore, we discuss the joint distributions in the two-dimensional case and find that for the combination of the largest and the second largest observation, it is most likely to find them to be realised with similar values with a broadly peaked distribution. When combining the largest observation with higher orders, it is more likely to find a larger gap between the observations and when combining higher orders in general, the joint pdf peaks more strongly. Having introduced the theory, we apply the order statistical analysis to the SPT massive cluster sample and MCXC catalogue and find that the ten most massive clusters in the sample are consistent with Λ CDMandtheTinkermass function.For the order statistics in redshift, we find a discrepancy between the data and the theoretical distributions, which could in principle indicate a deviation from the standard cosmology. However, we attribute this deviation to the uncertainty in the modelling of the SPT survey selection function. In turn, by assuming the Λ CDMreference cosmology, order statistics can also be utilised for consistency checks of the completeness of the observed sample and of the modelling of the survey selection function. Key words: methods: statistical - galaxies: clusters: general - cosmology: miscellaneous.", "pages": [ 1 ] }, { "title": "J.-C. Waizmann 1 , 2 , 3 /star , S. Ettori 2 , 3 and M. Bartelmann 4", "content": "1 Dipartimento di Fisica e Astronomia, Universit'a di Bologna, viale Berti Pichat 6 / 2, I-40127 Bologna, Italy 2 INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy 3 INFN, Sezione di Bologna, viale Berti Pichat 6 / 2, 40127 Bologna, Italy 4 Zentrum fur Astronomie der Universitat Heidelberg, Institut fur Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany Received 2011", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Clusters of galaxies represent the top of the hierarchy of gravitationally bound structures in the Universe and can be considered as tracers of the rarest peaks of the initial density field. This feature renders their abundance across the cosmic history a valuable probe of cosmology (for an overview of cluster cosmology see e.g. Voit 2005; Allen et al. 2011, and references therein). The recent years brought significant advances to the field from an observational point of view. Past and present surveys, like e.g. the ROSAT All Sky Survey (RASS; Voges et al. 1999), the Massive Cluster Survey (MACS; Ebeling et al. 2001) and the Southpole Telescope (SPT; Carlstrom et al. 2011), provided rich data for a multitude of massive clusters ( > 10 15 M /circledot ). In the near future, cluster data will be drastically extended in terms of completeness, coverage and depth by surveys like for instance PLANCK (Tauber, J. A. et al. 2010), eROSITA (Cappelluti et al. 2011) and EUCLID (Laureijs et al. 2011), allowing for statistical analyses of the samples with increasing quality. A particular form of statistical analysis that recently entered focus are falsification experiments of the concordance Λ CDM cosmology, based on the discovery of a single (or a number) of cluster(s) being so massive that it (they) could not have formed in the standard picture (Hotchkiss 2011; Hoyle et al. 2011; Mortonson et al. 2011; Harrison & Coles 2012; Harrison & Hotchkiss 2012; Holz & Perlmutter 2012; Waizmann et al. 2012a,b). These studies were triggered by the discovery of massive clusters at high redshift (see e.g. Mullis et al. 2005; Jee et al. 2009; Rosati et al. 2009; Foley et al. 2011; Menanteau et al. 2012; Stalder et al. 2012). However, the usage of a single observation for such falsification experiments requires statistical care since several subtleties have to be taken into account. From the theoretical point of view, it is necessary to include the Eddington bias (Eddington 1913) in mass, as discussed in Mortonson et al. (2011) and the bias that stems from the a posteriori choice of the redshift interval for the analysis (Hotchkiss 2011). From the observational point of view, it might, particularly for very high redshift systems, be di ffi cult to define the survey area and selection function that are appropriate for the statistical analysis. Combining all of these e ff ects, recent studies (Hotchkiss 2011; Harrison & Coles 2012; Harrison & Hotchkiss 2012; Waizmann et al. 2012a,b) converge to the finding that, when taken alone, none of the single most massive known clusters can be considered in tension with the concordance Λ CDMcosmology. Conceptually, inference based on a single observation is not desirable, because by nature the extreme value might not be representative for the underlying distribution from which it is supposedly drawn. Thus, it is advised to incorporate statistical information from the sample of the most massive high redshift clusters, which in turn are also particularly sensitive to the underlying cosmological model since they probe the exponentially suppressed tail of the mass function. In this work, we introduce order statistics as a tool for analytically deriving distribution functions for all members of the mass and redshift hierarchy ordered by magnitude. By dividing our analysis in the observables mass and redshift, we avoid the bias due to an a posteriori definition of redshift intervals (Hotchkiss 2011) and avoid as well the arbitrariness of an a priori choice that had been necessary in our previous works based on the extreme value statistics. Furthermore, the formalism also allows for the formulation of joint probabilities of the order statistics. In the second part of this work, we compare our individual and joint analytic distributions to observed samples of massive galaxy clusters. This paper is structured according to the following scheme. In Sect. 2, we introduce the statistical branch of order statistics by discussing the basic mathematical relations in Sect. 2.1 and by applying the formalism to the distribution of massive galaxy clusters in mass and redshift in Sect. 2.2. This is followed by a discussion of how the order statistics of haloes in mass and redshift depends on cosmological parameters in Sect. 3. In order to compare our analytic results to observations, we prepare observed cluster samples for the analysis in Sect. 4. Afterwards, we discuss the results of the comparison for the case of the individual order statistic in Sect. 5 and for the joint case in Sect. 6. Then, we summarise our findings in Sect. 7 and draw our conclusions in Sect. 8. In Appendix A we give a more detailed overview of order statistics and in Appendix B fitting formulae for the order statistics in mass and redshift are presented. Throughout this work, unless stated otherwise, we adopt the Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) parameters ( Ω m0 , ΩΛ 0 , Ω b0 , h , σ 8) = (0 . 727 , 0 . 273 , 0 . 0455 , 0 . 704 , 0 . 811) (Komatsu et al. 2011).", "pages": [ 1, 2 ] }, { "title": "2 ORDER STATISTICS", "content": "Order statistics (for an introduction, see e.g. Arnold et al. 1992; David & Nagaraja 2003) is the study of the statistics of ordered (sorted by magnitude) random variates. In this section, the basic mathematical relations and the connection to cosmology are introduced as they will be needed in remainder of this work.", "pages": [ 2 ] }, { "title": "2.1 Mathematical prerequisites", "content": "Let X 1 , X 2 , . . . , Xn be a random sample of a continuous population with the probability density function (pdf), f ( x ), and the corresponding cumulative distribution function (cdf), F ( x ). Further, let X (1) /lessorequalslant X (2) /lessorequalslant · · · /lessorequalslant X ( n ) be the order statistic, the random variates ordered by magnitude, where X (1) is the smallest (minimum) and X ( n ) denotes the largest (maximum) variate. It can be shown (see Sect. A1) that the pdf of X ( i ) (1 /lessorequalslant i /lessorequalslant n ) is given by The corresponding cdf of the i -th order reads then and the distribution function of the smallest and the largest value are found to be and In the limit of very large sample sizes both F ( n )( x ) and F (1)( x ) can be described by a member of the general extreme value (GEV) distribution (Fisher & Tippett 1928; Gnedenko 1943)   where α is the location-, β the scale- and γ is the shape-parameter. Usually these parameters are obtained directly from the data or from an underlying model (see for instance Coles (2001)). Apart from the distributions of the single order statistics, it is very interesting to derive joint distribution functions for several orders. The joint pdf of the two order statistics X ( r ) , X ( s ) (1 /lessorequalslant r < s /lessorequalslant n ) is for x < y given by (see Appendix A for a more detailed discussion) The joint cumulative distribution function can e.g. be obtained by integrating the pdf above or by a direct argument and is found to be given by Analogously the above relations can be generalised to the joint pdf of Xn 1 , . . . , Xn k (1 /lessorequalslant n 1 < · · · < nk /lessorequalslant n ) for x 1 /lessorequalslant · · · /lessorequalslant xk , which is given by Further details and derivations concerning order statistics can be found in the Appendix A. In the remainder of this work we will repeatedly make use of percentiles. In statistics, a percentile is defined as the value of a variable below which a certain percentage, p , of observations fall. Percentiles can be directly obtained from the inverse of the cdf and will be hereafter denoted as Qp .", "pages": [ 2, 3 ] }, { "title": "2.2 Connection to cosmology", "content": "As outlined in the previous subsection, the only quantity that is needed for calculating the cdfs, F ( i )( x ), of the order statistics (see equation 2) is the cdf, F ( x ), of the underlying distribution from which the sample is drawn. Assuming the random variates, Xi , to be the masses of galaxy clusters, then the cdf, F ( m ), can be calculated (see e.g. Harrison & Coles (2012)) by means of where the total number of clusters, N tot, is given by Here, f sky is the fraction of the full sky that is observed, (d V / d z ) is the volume element and n ( m , z ) is the halo mass function. If needed, the corresponding pdf can always be obtained by f ( m ) = d F ( m ) / d m . Analogously, the order statistics can be calculated as well for the redshift instead of the mass. In this case the cdf reads where For the latter, the order statistics does no longer depend only on the survey area via f sky but, in addition the selection function of the survey has to be included via a limiting survey mass, m lim( z ). In this work, we do not attempt to model the possible redshift dependence of m lim and assume it to be constant throughout the remainder of this work. With the distributions F ( m ) and F ( z ) at hand, we can now easily derive the cdfs of the corresponding order statistics. Since we will focus in this work on the few largest values, we will refer to the distribution of the maximum, F ( n )( x ), as first order, to the second largest as second order and so on. We calculated the distributions of the first fifty orders from F ( n )( x ) to F ( n -49)( x ), where n = N tot, and present the results in Fig. 1 for the mass (left panel) and redshift (right panel). In both panels the color decodes the order of the distribution, ranging from the blue for F ( n )( x ) to the green for F ( n -49)( x ). For both cases we assumed f sky = 1, a redshift range of 0 /lessorequalslant z /lessorequalslant ∞ and the Tinker et al. (2008) mass function. In the case of the order statistics in redshift, we assume a limiting survey mass of m lim = 10 15 M /circledot . It can be nicely seen how, with increasing order (from blue to green), the cdfs shift in both cases to smaller values of the mass or redshift. A first important result is that, with the increasing order, the cdfs steepen, which results in an enhanced constraining power, since small shifts in the mass or redshift may yield large di ff erences in the derived probabilities. In this sense the higher orders will be more useful for falsification experiments than the extreme value distribution which, due to its shallow shape, requires extremely large values of the observable to statistically rule out the underlying assumptions. Since higher orders encode information from the n most extreme objects, deviations from the expectation are statistically more significant for n values instead of a single extreme one. In addition, we compare the distribution of the maxima F ( n )( m ) and F ( n )( z ) to those obtained from a extreme value approach (Davis et al. 2011; Waizmann et al. 2012a, Metcalf & Waizmann in prep.) based on the void probability (White 1979), using equation 5. For both cases presented in Fig. 1, the red, dashed curve of the GEV distribution, G ( x ), agrees very well with the directly calculated F ( n )( x ). In order to allow a quick estimation of the distributions of the order statistics, we provide in the Appendix B also fitting formulae for F ( x ) as a function of the survey area for the cases of mass and redshift. The fitting formulae for the distribution in mass allow an estimation of the quantiles in the range from the 2-percentile, Q 2, to the 98-percentiles, Q 98, with an accuracy better than one per cent for A s /greaterorsimilar 200 deg 2 and for the ten largest masses. In the instance of the order statistics in redshift, the quality of the fits depends on m lim as well. For m lim = 10 15 M /circledot an accuracy of better than two per cent can be achieved for A s /greaterorsimilar 2000 deg 2 and for m lim = 5 × 10 14 M /circledot the same accuracy is obtained down to A s = 100 deg 2 . A more detailed discussion of the fitting functions and their performance can be found in Appendix B. In the remaining part of this work, we will discuss how the underlying cosmological model a ff ects the order statistics and confront the theoretically derived order statistics with observations, afterwards.", "pages": [ 3, 4 ] }, { "title": "3 DEPENDENCE OF THE ORDER STATISTICS ON THE UNDERLYING COSMOLOGY", "content": "Eventually, the order statistics in mass and redshift is determined by the number of galaxy clusters in a given cosmic volume. The quantities that impact on this number can be categorised into two classes. The first one contains all e ff ects that modify structure formation itself, like the choice of the mass function or the amplitude of the mass fluctuations, σ 8, for instance. These e ff ects manifest themselves most strongly in the exponentially suppressed tail of the mass function, hence at high masses. The second class contains all the e ff ects that modify the geometric evolution of the Universe. By changing the evolution of the cosmic volume, the number of clusters in a given redshift range can be substantially di ff erent, even if both cosmologies yield the same the number density of objects of a given mass (see e.g. Pace et al. 2010).", "pages": [ 4 ] }, { "title": "3.1 Impact of cosmological parameters", "content": "In order to quantify the impact of di ff erent cosmological parameters on the order statistics in mass and redshift, we study the e ff ect on the 98-percentile, Q 98, which we use to define possible outliers from the underlying distribution. In Fig. 2, we present the relative di ff erence in Q 98 as a function of four di ff erent cosmological parameters comprising σ 8, Ω m (assuming the flatness constraint), the equation of state parameter, w 0, and the derivative wa from the relation w ( a ) = w 0 + wa (1 -a ), where a denotes the scale factor. A non-vanishing value of the latter indicates a time-varying equation of state. In each panel of Fig. 2, we show the relative di ff erences for 5 di ff erent orders, of the order statistic in mass with z ∈ [0 , ∞ ] (blue lines) and z ∈ [1 , ∞ ] (green lines), as well as in redshift (red lines) assuming m lim = 10 15 M /circledot . For all calculations we assumed the full sky and the Tinker et al. (2008) mass function. It can be seen that order statistics is very sensitive to σ 8, such that the relative di ff erences in Q 98 would amount to ∼ 7 per cent for the range allowed by WMAP7 of ( σ 8 = 0 . 811 ± 0 . 023). All three order statistics exhibit the same functional behaviour, with the mass-based ones being more sensitive than the redshift-based one. This can be understood by the fact that the mass-based order statistics probe the most massive clusters and hence the exponential tail of the mass function which is highly sensitive to σ 8. For modifications of the matter density, Ω m, assuming the flatness constraint ΩΛ = 1 -Ω m, the situation is substantially di ff erent from the previous case (see upper right panel of Fig. 2). Overall, the order statistics are less sensitive and they do not exhibit the same functional behaviour. The order statistics in mass (blue lines) performs best for larger value of Ω m because the most massive clusters will reside at rather low redshifts. At high redshifts (green and red lines), the increase in Ω m and hence, the decrease in ΩΛ , yields a smaller number of very massive clusters. Despite the increase in the matter density, the decrease in volume is dominating for the range of Ω m shown in the plot and, thus, the relative di ff erence decreases. In this sense the volume e ff ects dominate at high redshifts over the increase in matter density, whereas at low redshifts the increase in matter density dominates. The lower left panel of Fig. 2 shows the sensitivity of the order statistics to changes in the constant equation of state, w 0. Evidently, the most massive clusters at low redshifts (blue line) have no sensitivity to w 0, whereas at high redshifts (green and red line) the sensitivity is better. The volume e ff ects are, compared to modifications in Ω m, less important and the observed increase in the relative difference in Q 98 with decreasing w 0 is dominated by modifications of the exponential tail of the mass function (for a more thorough discussion, see e.g. Pace et al. 2010). When assuming a time-dependent equation of state, modelled by w ( a ) = w 0 + wa (1 -a ), as presented in the lower right panel of Fig. 2, the observed functional behaviour can be explained by identical arguments as before. The results exhibit again the high sensitivity of the high redshift order statistics on modifications of wa . It should be noted that we fixed w 0 = -1 . 0 for all cases. It can be summarised that for modifications that strongly affect the structure formation, like σ 8 for instance, the order statistics in mass for z ∈ [0 , ∞ [ is comparable in its sensitivity to the redshift based order statistics. Modifications that strongly alter the geometric evolution of the Universe a ff ect more strongly the order statistics in redshift. However, one should keep in mind that in the case of the order statistics in mass, the relative di ff erences are on the same level as the inaccuracies in cluster mass estimates. This problem does not occur for redshifts, which can be measured to a very high accuracy. Of course, in this case the observational challenge is transferred to compiling a sample with a precise mass limit. Apart from the cosmological parameters also the choice of the mass function is expected to have a strong e ff ect on the order statistics as will be discussed in the following subsection.", "pages": [ 4 ] }, { "title": "3.2 Impact of the choice of the mass function", "content": "When performing a falsification experiment of Λ CDM using the n most massive or n highest redshift clusters, then one has to specify the reference model against which the observations have to be compared with. Apart from the cosmological parameters that are usually fixed to the obvious choice of the WMAP7 values, a halo mass function has to be chosen as well. As mentioned earlier, this is particularly important for galaxy clusters since the exponentially suppressed tail of the mass function is naturally very sensitive to modifications. In order to quantify the impact of di ff erent mass functions on the order statistics in mass and redshift, we computed the cdfs, F ( n -9) , . . . , F ( n ), for the Press & Schechter (1974) (PS), the Tinker et al. (2008) and the Sheth & Tormen (1999) (ST) mass functions for f sky = 1 and present them from top to bottom in Fig. 3. Comparing the panels to each other reveals the tremendous sensitivity of the distributions to the choice of the mass function. Taking the Tinker mass function as a reference, the median, Q 50, changes for both types of order statistics by -20 per cent for the PS case and by + 15 percent for the ST case. These di ff erences can be explained by the fact that the ST mass function leads to a substantial increase in the number of haloes, particularly at the high mass end, whereas the PS mass function results in much fewer haloes in the mass and redshift range of interest. For the remainder of this paper we will use the Tinker mass function as reference because the halo masses are defined as spherical overdensities with respect to the mean background density, a definition that is closer to theory and actual observations than friend-of-friend masses. However, considering that due to statistical limitations, current fits for the mass function are still not very accurate for the highest masses ( > 3 × 10 15 M /circledot ) and that systematic uncertainties allow even smaller masses an accuracy of a few per cent at most (Bhattacharya et al. 2011), one has to be very cautious with falsification experiments that are based on extreme objects. The uncertainty in the mass function alone will allow a rather wide range of distributions.", "pages": [ 4, 5 ] }, { "title": "4 SUITABLE SAMPLES OF GALAXY CLUSTERS FOR AN ORDER STATISTICAL ANALYSIS", "content": "Having introduced the order statistics of the most massive or the highest redshift clusters, we intend now to compare observed clusters with the theoretical distributions. To do so, it is necessary to select suitable samples of galaxy clusters, which we will discuss in the following.", "pages": [ 5 ] }, { "title": "4.1 General considerations", "content": "The selection of a suitable sample of galaxy clusters for an order statistical analysis is by no means a trivial task. The necessary ordering of the quantities mass and redshift by magnitude requires that they have been derived in an identical way across the sample. Otherwise, systematics and biases, like the di ff erences between lensing and X-ray mass estimates for instance (see e.g. Mahdavi et al. 2008; Zhang et al. 2010; Planck Collaboration et al. 2012; Meneghetti et al. 2010; Rasia et al. 2012), will render the ordering meaningless. Despite an increasing amount of data from di ff erent surveys, a lack of large homogeneous samples persists. Thus, we decided to base our comparison on clusters that stem from catalogues like the SPT massive cluster sample (Williamson et al. 2011) and the MCXC cluster catalogue (Pi ff aretti et al. 2011), which will be discussed in further detail below.", "pages": [ 5 ] }, { "title": "4.2 The SPT massive cluster sample", "content": "The SPT survey (Carlstrom et al. 2011) is ideally suited for the intended purpose of an order statistical analysis. Being based on the Sunyaev Zeldovich (SZ) e ff ect (Sunyaev & Zeldovich 1972, 1980) the SPT survey is able to detect massive galaxy clusters up to order sun Tinker mass function order mass m [M sun ] ST mass function order sun Tinker mass function ST mass function order sun high redshifts. The fact that the limiting mass of SZ surveys varies weakly with redshift (Carlstrom et al. 2002) allows in principle to construct mass limited cluster catalogues. However, it should be emphasised that the assumption of an m lim independent of redshift depends critically on the sensitivity and the beam width of an actual survey. For this work, we take the catalogue of Williamson et al. (2011) which comprises the 26 most significant detections in the full survey area of A SPT s = 2500 deg 2 . Ensuring a constant mass limit of M 200m ≈ 10 15 M /circledot , clusters were selected on the basis of a signal-to-noise (S / N) threshold in the filtered SPT maps. For all 26 catalogue members, either photometric or spectroscopic redshifts were determined as well. The cluster masses given in the catalogue are defined with respect to the mean cosmic background density and need no further conversion to match the mass definition of the reference Tinker et al. (2008) mass function. To each cluster of the sample we assign the error bars that we obtained by adding the reported statistical and systematic errors in quadrature.", "pages": [ 5, 6 ] }, { "title": "4.3 The MCXC cluster catalogue", "content": "The MCXC catalogue (Pi ff aretti et al. 2011) is based on the publicly available compilation of clusters' detections from ROSAT All-Sky Survey (NORAS, REFLEX, BCS, SGP, NEP, MACS, and CIZA) and other serendipitous surveys (160SD, 400SD, SHARC, WARPS, and EMSS), and provides the physical properties of 1743 galaxy clusters systematically homogenised to an overdensity of 500 (with respect to the cosmic critical density). This metacatalogue is not complete in any sense, but it is constituted by X-ray flux-limited samples that ensure that the X-ray brightest objects in the nearby ( z /lessorsimilar 0 . 3) Universe, and therefore the most massive Xray detected clusters, are all included. We have then simply ranked the objects accordingly to their estimated M 200m, that is obtained from the tabulated M 500c as where Ω z = Ω m(1 + z ) 3 / E 2 z , Ez = ( Ω m(1 + z ) 3 + ΩΛ ) 1 / 2 , and the ratio between the radii at di ff erent overdensities has been obtained by assuming an NFW profile (Navarro et al. 1996) with c 200 = 4.", "pages": [ 6, 7 ] }, { "title": "4.4 Preparations of the ordered samples", "content": "We order the SPT and MCXC catalogues by magnitude of the observed mass and present the ten most massive systems in Table 1. For statistical comparisons the observed masses have to be corrected for the Eddington bias (Eddington 1913) in mass. As a result of the exponentially suppressed tail of the mass function and the substantial uncertainties in the mass determination of galaxy clusters, it is more likely that lower mass systems scatter up while higher mass systems scatter down, resulting in a systematic shift. Thus, before an observed mass can be compared to a theoretical distribution, this shift has to be corrected for. To do so, we follow Mortonson et al. (2011) and shift the observed masses, M obs, to the corrected masses, M corr, according to where /epsilon1 is the local slope of the mass function (d n / d ln M ∝ M /epsilon1 ) and σ ln M is the uncertainty in the mass measurement. We corrected the observed masses in both, the SPT and the MCXC catalogues, using the values of σ ln M listed in the fifth column of Table 1 which we deduced from the reported uncertainties in the nominal masses. The larger the observational errors are, the larger is the correction towards lower masses. As an exemplary exception from the SPT catalogue, we used for the mass of SPT-CL J0102-4915 the value reported by Menanteau et al. (2012), which is based on a combined SZ + Xrays + optical + infrared analysis. The multi-wavelength study shifts M obs = (1 . 89 ± 0 . 45) × 10 15 M /circledot (Williamson et al. 2011) to a larger value of M obs = (2 . 16 ± 0 . 32) × 10 15 M /circledot , changing the rank from the fifth to the third most massive. This shows that with the expected increase in the quality of cluster mass estimates, the ordering of the most massive cluster will undergo significant changes. We expect that the reshu ffl ing will a ff ect more strongly the most massive clusters due to the fact that the large error bars will cause lower ranked clusters to scatter up. We will discuss the impact of the reshu ffl ing in more detail in Sect. 5.1. In addition, we sorted the SPT catalogue by redshift and list the ten highest redshift clusters above m lim ≈ 10 15 M /circledot in Table 2.", "pages": [ 7 ] }, { "title": "5 COMPARISON OF THE INDIVIDUAL ORDER STATISTICS WITH OBSERVATIONS", "content": "In this section we will compare the individual ranked systems listed in Table 1 for the mass and in Table 2 for the redshift with the individual distributions for each rank, as e.g. shown in Fig. 3. s s s s s", "pages": [ 7, 8 ] }, { "title": "5.1 Order statistics in cluster mass", "content": "In order to demonstrate the impact of the survey area on the distributions of the order statistics in mass, we show in Fig. 4 the dependence of di ff erent quantiles ( Q 2, Q 25, Q 50, Q 75 and Q 98) on the survey area for the nine most massive clusters. In addition, the green error bars show the clusters from the SPT and MCXC catalogues listed in Table 1 for the respective survey areas of A SPT s = 2500 deg 2 and A MCXC s = 27490 deg 2 . From the individual panels in Fig. 4 it can be inferred that, as expected, a larger survey area yields a larger expected mass for the individual rank. Furthermore, with increasing rank towards higher orders, the interquantile range, like (Q2-Q98), narrows. A behaviour that can also be seen in Fig. 1 as steepening of the cdf with increasing rank. Therefore, the largest mass (first order) is expected to be realised in a much wider mass range than the higher orders. s s We will now compare the observations in more detail with the theoretical expectations in the form of box-and-whisker diagrams as shown in Fig. 5. Here, the blue-bordered, grey filled box denotes the interquartile range (IQR) which is bounded by the 25 and 75-percentiles ( Q 25, Q 75) and the median ( Q 50) is depicted as a red line. The black whiskers denote the 2 and 98-percentiles ( Q 2, Q 98) and we follow the convention that observations that fall outside are considered as outliers. As before, the nominal observed cluster masses are denoted as green error bars where for the left column the SPT catalogue and for the right column the MCXC catalogue was used. In addition we plot the Eddington bias corrected masses, M Edd 200m , from the sixth column of Table 1 as orange triangles with dashed error bars. We performed the analysis for three di ff erent mass functions, comprising from the top to the bottom panel, the PS, the Tinker and the ST mass functions. In addition to the Eddington bias in mass, we expect a shift to larger masses caused by the reshu ffl ing of orders due to the uncertainties in mass. In order to quantify this e ff ect, we Monte Carlo (MC) simulated 10 000 realisations of the 26 SPT and 123 MCXC (with M > 10 15 M /circledot ) cluster masses after their correction for the Eddington bias and order them by mass. The masses were randomly drawn from the individual error interval, assuming Gaussian distributions. We present the results as violet, empty circles with dash-dotted 1 σ error bars in Fig. 5. It can be seen that the highest ranks are more strongly a ff ected by the reshu ffl ing than the lower ones and that they are on average shifted to larger values. Of course, the amount of this e ff ect will depend on the size of the error bars. Further, the reshuffling yields mass values that fall between the nominal (green error bars) and the Eddington bias corrected ones (orange error bars). For the SPT catalogue, it can be seen from the top left panel of Fig. 5 that the outdated PS mass function seems to be disfavoured by the reshu ffl ed and the nominal masses of the ten largest objects. However, the error bars are large and do not allow an exclusion of the PS mass function. For the Tinker and the ST mass function, the boxes indicating the theoretical distributions move to larger mass values and therefore they match the observed masses better than the PS mass function. In particular, the third ranked (second ranked after Eddington bias correction) system SPT-CL J0102 with its smaller errors and, hence, giving the tightest constraints, is consistent with Λ CDM for both mass functions. All other ranks are consistent as well due to their large error bars. The reshu ffl ed sample matches perfectly the Tinker mass function consolidating the conclusion that the most massive clusters of the SPT sample are in agreement with the statistical expectations. The conclusions for the MCXC catalogue are identical, however the jump between the fourth and the fifth largest order yield to an inconsistency of the observed higher orders with the expectations based on the ST mass function. This jump is clearly caused by the incompleteness of the MCXC catalogue and, thus, the inclusion of the missing clusters would most certainly move the observed sample to higher masses in the direction of the results we obtained from the SPT sample. In this sense we do not see any indication of a substantial di ff erence between the small and wide field survey. The analysis of the SPT sample illustrates the potential of utilising the n most massive galaxy clusters to test underlying assumptions, like e.g. the mass function. For instance, a multi-wavelength study of the 26 SPT clusters would reduce the error bars to the level of SPT-CL J0102 (the nominal third ranked cluster in the left column of Fig. 5 ), which would significantly tighten the constraints on the underlying assumptions like e.g. the halo mass function. In turn, by assuming the Λ CDM reference cosmology, the comparison of the observed masses with the individual order distributions allows to check the completeness of the observed sample. In the upper panel of Fig. 6, we present the dependence of di ff erent percentiles ( Q 2, Q 25, Q 50, Q 75 and Q 98) on the order for a survey area of A s = 20 000 deg 2 . Choosing the Q 98 percentile as exclusion criterion, one would need roughly to find ten clusters with m /greaterorsimilar 2 . 5 × 10 15 M /circledot , three clusters with m /greaterorsimilar 3 . 2 × 10 15 M /circledot or one cluster with m /greaterorsimilar 5 × 10 15 M /circledot in order to report a significant deviation from the Λ CDM expectations. Of course, the observed masses might have to be corrected for the Eddington bias in mass and a possible reshu ffl ing as previously demonstrated. In general, exclusion criteria based on order statistics extend previous works (Mortonson et al. 2011; Waizmann et al. 2012a) from statements about single objects to statements about object samples which considerably improves the reliability of the entire study.", "pages": [ 8, 9, 10 ] }, { "title": "5.2 Order statistics in cluster redshift", "content": "We performed an identical analysis for the individual order statistics for the SPT massive cluster catalogue ranked by redshift listed in Table 2. For the theoretical distributions we assume a limiting mass of m lim = 10 15 M /circledot and a survey area of A SPT s = 2500 deg 2 . As before, we present in Fig. 7 the dependence of the order statistical distributions on the survey area for the first nine orders. Again, an increase in the survey area yields a shift of the theoretical distributions to higher redshifts and, as shown in the right panel of Fig. 1, the cdfs steepen for the higher ranks, resulting in a shrinking interquantile range. In Fig. 8, we present the box-and-whisker diagram in redshift, again for the PS, the Tinker and the ST mass functions (from top to bottom). The definition of boxes and whiskers remains unchanged with respect to Fig. 5. Again, the data from Table 2 is denoted by green error bars, which are negligibly small in the case of spectroscopic redshifts. Thus, we abstained from the MC simulation of the reshu ffl ing in the case of redshift. While for the order statistics in mass the results only depended on the choice of the survey area, the situation is di ff erent for the order statistics in redshift. Here, a constant survey limiting mass is assumed, which will be subject to uncertainties for a real survey and, furthermore, will also exhibit some redshift dependence. Thus, the theoretical distributions are intrinsically less accurate than the ones with respect to cluster mass. Indeed, the comparison with the data in Fig. 8 exhibits a di ff erent behaviour with respect to the one in Fig. 5. Here, first four orders seem to be fit better by the Tinker mass function while the higher orders seem to favour the PS mass function. Taking the Tinker mass function as reference it seems that a few systems with M > 10 15 M /circledot are missing at redshifts z /greaterorsimilar 0 . 7. The di ff erence with respect to the findings for the order statistics in mass for the same sample could, along the lines of Sect. 3, be interpreted as a signature of a deviation from the reference Λ CDM model. However, considering the previously mentioned simplifying assumptions in the modelling of the theoretical distributions, we do not infer any cosmological conclusions and leave a better, more realistic, modelling of m lim( z ) of the SPT survey to a future work. In the lower panel of Fig. 6, we present the dependence of di ff erent percentiles ( Q 2, Q 25, Q 50, Q 75 and Q 98) on the order for a survey area of A s = 20 000 deg 2 and a constant limiting mass of m lim = 10 15 M /circledot . Taking the Q 98 percentile as exclusion criterion, one would need to find ten clusters with z /greaterorsimilar 1, three clusters with z /greaterorsimilar 1 . 2 or one cluster with z /greaterorsimilar 1 . 55 in order to report a significant deviation from the Λ CDM expectations. Currently, SPT-CL J2106 s s is the only known cluster of such a high mass having a redshift z > 1. With an assigned survey area of As = 2800 deg 2 (ACT + SPT), it might from a statistical point of view still be possible to find ten objects that massive at z > 1 in the larger survey area. The method presented in this work allows to construct similar exclusion criteria for any kind of survey design.", "pages": [ 10, 11 ] }, { "title": "6 COMPARISON OF THE JOINT ORDER STATISTICS WITH OBSERVATIONS", "content": "Having studied the individual order statistics in mass and redshift in the previous section, we turn now to the study of the joint distributions of the order statistics as introduced in Sect. 2.1. The simplest case of a joint order distribution is twodimensional. In this case the pdf and cdf are given by equation 6 and equation 7, respectively. Starting with the joint pdf, we present in Fig. 9 the joint distributions in mass (left panel) and redshift (right panel) for several order combinations as denoted in the individual panels. All calculations assume the full sky and the Tinker mass function. In the case of the joint distributions in redshift, we assume a constant limiting survey mass of m lim = 10 15 M /circledot . Due to the condition that x < y , all distributions are limited to a triangular domain. An inspection of the di ff erent pdfs in Fig. 9 reveals that, for the combination of the first and the second largest order (upper leftmost panel), the most likely combination of the observables is very close to the diagonal. This means that it is more likely to find the two largest values close to each other, at absolute values that are smaller than the extreme value statistics would imply for the maximum alone. Then, when moving to combinations of the first with higher orders (first row), it can be seen that the peaks of the pdfs move away from the diagonal and that they extend to larger values for the larger observable. This indicates that it is more likely to find the two systems with a larger separation in the observable when the s s s s di ff erence between the considered orders is larger. Accordingly, for higher order combinations (lower rows), the peaks of the joint pdf move to smaller values of the observables. It should also be noted that the peaks steepen for higher order combinations, confining the pdfs to smaller and smaller regions in the observable plane. As an example, the first and second largest observations (upper leftmost panel) can be realised in much larger area than the sixth and eighth largest one (lower rightmost panel). Apart from the joint pdfs, it is also instructive to study the joint cdfs as presented in Fig. 10 for the observed mass (left panel) and redshift (right panel). In order to add observational data from the SPT catalogue, we assume a survey area of A SPT s = 2500 deg 2 and a m lim = 10 15 M /circledot for the joint distribution in redshift. Additionally, we added the two largest nominal observed (red error bars) and the Eddington bias corrected masses (grey error bars) from Table 1 to the left panel and the two highest redshifts of the SPT massive cluster sample from Table 2 to the left panel. In the case of the mass, we find F ( n -1)( n ) ≈ 0 . 92 for the nominal and F ( n -1)( n ) ≈ 0 . 1 for the Eddington bias corrected masses. Hence, using the central values, in ∼ (8 -90) percent of the cases a mass larger than the one of SPT-CL J0658 and a mass larger than the one of SPT-CL J2248 are observed. Thus, also the joint cdf confirms that the two largest masses do not exhibit any tension with the concordance cosmology. The same conclusion applies in the case of the joint distribution in redshift. By means of equation 8 these results can be extended to the n -dimensional case, allowing the formulation of a likelihood function of the ordered sample of the n most massive or highest redshift clusters.", "pages": [ 11, 12 ] }, { "title": "7 SUMMARY", "content": "In this work, we studied the application of order statistics to the mass and redshifts of galaxy clusters and compared the theoretically derived distributions with observed samples of galaxy clusters. Our work extends previous studies that hitherto considered only the extreme value distributions in mass or redshift. On the theoretical side, our results can be summarised as follows. (i) We introduce all relations necessary to calculate pdfs and cdfs of the individual and joint order statistics in mass and redshift. In particular, we find a steepening of the cdfs for higher order distributions with respect to the extreme value distribution of both mass and redshift. This steepening corresponds to a higher constraining power from distributions of the n -largest observations. The presented method extends previous works to include exclusion criteria based on the n most massive or n highest redshift clusters for a given survey set-up. (ii) Conceptually, we avoid the bias due to an a posteriori choice of the redshift interval in the case of the order statistics in mass by selecting the interval 0 /lessorequalslant z /lessorequalslant ∞ . Hence, we study the statistics of the hierarchy of the most massive haloes in the Universe, which mostly stem from redshifts z /lessorsimilar 0 . 5. On the contrary, when choosing the order statistics in redshift, focus is laid on haloes that stem from the highest possible redshifts. However, the calculations will require a model of the survey characteristics in the form of a limiting survey mass as a function of redshift. (iii) By putting the emphasis on either the most massive or on the highest redshift clusters above a given mass limit, the order statistics is e.g. particularly sensitive to the choice of the mass function. While the order statistics in mass is very sensitive to σ 8 and Ω m due to the domination of low redshift objects, the order statistics in redshift proves to be very sensitive to w 0 and wa . For a fixed cosmology, both order statistics are e ffi cient probes of the functional shape of the mass function at the high mass end. (iv) In addition to the individual order statistics, we study as example case also the joint two order statistics. We find that for the combination of the largest and the second largest observation, it is most likely to find them to be realised with very similar values with a relatively broadly peaked distribution. (v) In order to allow a quick estimation of the distributions of the order statistics, we provide in the Appendix B fitting formulae for F ( x ) as a function of the survey area for the cases of mass and redshift. The fitting formulae for the distribution in mass allow for a percentile estimation in the range from Q 2 to Q 98 with an accuracy better than one per cent for A s /greaterorsimilar 200 deg 2 and for the ten largest masses. In the case of the order statistics in redshift, the quality of the fits depends on the chosen m lim. However, for survey areas of A s /greaterorsimilar 2000 deg 2 accuracies better than two per cent can be achieved for large values of m lim = 10 15 M /circledot and a lowering of m lim further improves the accuracy. After introducing the theoretical framework, we compared the theoretical distributions with actually observed samples of galaxy clusters that we ranked by the magnitude of the observables mass and redshift. We decided to compile two catalogues, the main one is based on the SPT massive cluster sample (Williamson et al. 2011) and additionally we analysed the meta-catalogue of X-ray detected clusters of galaxies MCXC (Pi ff aretti et al. 2011) based on publicly available flux-limited all-sky survey and serendipitous cluster catalogues. This meta-catalogue can be considered as complete for z /lessorsimilar 0 . 3 and, hence, by no means as complete as the SPT one. The results of the comparison can be summarised as follows. (i) In the case of the order statistics in mass, we compared the theoretical expectations for the ten largest masses for the PS, the Tinker and the ST mass functions. Assuming WMAP7 parameters, we find that the nominal and the Eddington bias corrected values", "pages": [ 12, 13 ] }, { "title": "14 J.-C. Waizmann, S. Ettori and M. Bartelmann", "content": "for the observed masses favour the Tinker and the ST mass functions. When considering the possible bias due to a reshu ffl ing of the ranks caused by the large error bars (statistical + systematic errors), we find that the SPT sample matches the Tinker mass function very well. The constraints are expected to tighten considerably once the error bars of all objects are scaled down by combining several cluster observables in multi-wavelength studies. (ii) In contrast to the ranking in mass, the order statistics of the SPT clusters in redshift is less well fit by the theoretical distributions based on the Tinker mass function. It appears that a few systems with M > 10 15 M /circledot are missing at redshifts z /greaterorsimilar 0 . 7. One explanation could be found in a non-standard cosmological evolution to which the order statistics in redshift is more sensitive. However, it is more likely that a more precise modelling (including the redshift dependence) of the true limiting survey mass of SPT will account for the observed deviations. (iii) Instead of utilising order statistics to perform exclusion experiments, it can also be used for consistency checks of the completeness of the observed sample and of the modelling of the survey selection function as indicated by the analysis of the MCXC (mass) and the SPT (redshift) samples.", "pages": [ 14 ] }, { "title": "8 CONCLUSIONS", "content": "We introduced a powerful theoretical framework which allows to calculate the expected individual and joint distribution functions of the n -largest masses or the n -highest redshifts of galaxy clusters in a given survey area. This approach is more powerful than the extreme value statistics that focusses on the statistics of the single largest observation alone. As a proof of concept, we compared the theoretical distributions with observed samples of galaxy clusters. However, data of su ffi cient quantity, uniformity and completeness is still sparse such that constraints are not particularly tight. This situation will most certainly improve in the near and intermediate future. Since the emphasis of this work lies on the introduction of the theoretical framework of order statistics and its application to galaxy clusters, we contended ourselves with a study of cluster masses and redshifts. Unfortunately, the mass of a galaxy cluster is not a direct observable and subject to large scatter and observational biases. In a follow-up work, we intend to extend the formalism to direct observables, like for instance X-ray luminosities, and to include the scatter in the scaling relations into the theoretical distributions.", "pages": [ 14 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We acknowledge financial contributions from contracts ASIINAF I / 023 / 05 / 0, ASI-INAF I / 088 / 06 / 0, ASI I / 016 / 07 / 0 COFIS, ASI Euclid-DUNE I / 064 / 08 / 0, ASI-Uni Bologna-Astronomy Dept. Euclid-NIS I / 039 / 10 / 0, and PRIN MIUR 2008 Dark energy and cosmology with large galaxy surveys . M.B. is supported in part by the Transregio-Sonderforschungsbereich TR33 The Dark Universe of the German Science Foundation. J.C.W. would like to thank Lauro Moscardini and Ben Metcalf for the very helpful discussions.", "pages": [ 14 ] }, { "title": "REFERENCES", "content": "Allen S. W., Evrard A. E., Mantz A. B., 2011, ARA&A, 49, 409 Williamson R. et al., 2011, ApJ, 738, 139 Zhang Y.-Y. et al., 2010, ApJ, 711, 1033", "pages": [ 14, 15 ] }, { "title": "APPENDIX A: ORDER STATISTICS", "content": "In this appendix, we outline the derivation of the most important relations of the order statistics and some subtleties considering their implementation. For more details we refer to the excellent textbooks on the topic by Arnold et al. (1992) and by David & Nagaraja (2003) which we closely follow for the remainder of this appendix.", "pages": [ 15 ] }, { "title": "A1 Individual distributions", "content": "Let X 1 , X 2 , . . . , Xn be a random sample of a continuous population with the cumulative distribution function, F ( x ). Further, let X (1) /lessorequalslant X (2) /lessorequalslant · · · /lessorequalslant X ( n ) be the order statistic, the random variates ordered by magnitude, where X (1) is the smallest (minimum) and X ( n ) denotes the largest (maximum) variate. The event x < X ( i ) /lessorequalslant x + δ x is the same as the one depicted in panel ( a ) of Fig. A1 and, thus, we have Xk /lessorequalslant x for i -1 of the Xk , exactly one Xk in x < Xk /lessorequalslant x + δ x and the remaining n -i of the Xk in Xk > x + δ x . Now, the number of ways how n observations can be arranged in the three regimes is given by where each of them has a probability of Therefore, under the assumption that δ x is small, we find for the probability neglecting terms of O ( δ x ) 2 . Dividing by δ x and performing δ x → 0 yields the pdf as given in equation 1 The corresponding cdf of the i -th order, as given by equation 2 in Sect. 2, can now either be obtained by integrating the above equation or by the following argument for -∞ < x < ∞ . Hence, the cdf of X ( i ) is equivalent to the tail probability (starting from i ) of a binomial distribution with n trials and a success probability of F ( x ). By setting i = n or i = 1 one obtains the cdfs for the smallest and the largest order statistics as given by equation 3 and equation 4. Order statistics of galaxy clusters", "pages": [ 15 ] }, { "title": "A2 Joint distributions", "content": "The joint pdf of the two order statistics X ( r ) , X ( s ) (1 /lessorequalslant r < s /lessorequalslant n ) for x < y can be derived by similar arguments as for the single order statistics. The derivation scheme is now extended according to panel ( b ) of Fig. A1. Analogously to equation A4 we obtain where The joint cumulative distribution function can be obtained by integrating the pdf from above or again by the following direct argument This is exactly identical to the tail probability of a bivariate binominal distribution. Following the same line of reasoning as for the joint two order statistics, the above relations can be generalised to the joint pdf of Xn 1 , . . . , Xn k (1 /lessorequalslant n 1 < · · · < nk /lessorequalslant n ) for x 1 /lessorequalslant · · · /lessorequalslant xk , which is given by The right hand side of this relation can be written in a more compact form (David & Nagaraja 2003) as", "pages": [ 15 ] }, { "title": "A3 Regarding the implementation", "content": "The implementation of the order statistics for the intended application of this work, as discussed in Sect. 2.1 and Sect. 2.2, is rather straightforward. However, one important subtlety arises from the combinatoric prefactors that contain factorials of n , which due to the large number of haloes cannot be calculated directly. However, for all prefactors the factorials of n can be avoided by writing them as products and by dividing out common terms. As a simple example, we take the prefactor from equation A1. In this case the index i will, depending on the order, be given by a term like i = ( n -j ) with j = 0 for the distribution of the maximum, j = 1 for the second largest and so on. Thus, we obtain which can be calculated for rather large values of n . In a similar manner, all combinatoric prefactors can be simplified and implemented.", "pages": [ 16 ] }, { "title": "APPENDIX B: A FITTING FUNCTION FOR THE ORDER STATISTICS", "content": "In this additional section, fitting functions for the order statistics in mass and in redshift are defined. As functional form for the numerical fits, we will use equation 5 in combination with the relation equation 4, which yields     Here, x is the observable, either mass or redshift, and the GEV parameters α , β and γ as well as the number of haloes 1 , n , are functions of the survey area via the variable y = log 10 ( A s). Once the cdf, F ( x ), is known, all order statistics can be calculated by means of the relations discussed in the previous Appendix A. Inverting the cdfs of order statistics allows to obtain the percentiles which can then be utilised as Λ CDMexclusion criteria (see e.g. Fig. 6).", "pages": [ 16 ] }, { "title": "B1 Order statistics in mass", "content": "In order to determine the fitting function for the order statistics in mass, we calculate the GEV parameters according to Davis et al. (2011) and Waizmann et al. (2011) and the number of haloes, n , as a function of the survey area and fit them by the following functions where y = log 10 ( A s). The observable x in equation B1 is defined to be x = log 10 ( M 200 mh ). We present the results in Fig. A2 in the form of relative di ff erences between the fitted and directly calculated values of five selected percentiles ( Q 2, Q 25, Q 50, Q 75 and Q 98) as a function of the survey area. The di ff erent colors denote the largest order statistics, F ( n )( x ) (black lines), the fifth largest order statistics, F ( n -4)( x ) (blue lines) and the tenth largest order statistics, F ( n -9)( x ). The relative errors in the five di ff erent percentiles are for almost the complete range of survey areas on the sub-per cent level (only Q 98 for F ( n )( x ) exhibits a slightly larger error for very small survey areas).", "pages": [ 16 ] }, { "title": "B2 Order statistics in redshift", "content": "For fitting the order statistic in redshift, we proceed in a similar way as for the mass, setting x = z in equation B1. For calculating the GEV parameters as a function of the survey area, we follow the approach presented in (Metcalf & Waizmann in preparation). However, since in contrast to the order statistics in mass, the distributions depend on the choice of the limiting survey mass, we fitted the distributions for two choices of m lim. First, we set m lim = 10 15 M /circledot , identical to the setup we discussed in this paper for the SPT massive cluster sample. Secondly, we lower the threshold to m lim = 5 × 10 14 M /circledot . In the first case, we obtain and for the second choice we find where y = log 10 ( A s) for both cases. We present the results in Fig. B1 again as relative di ff erences. It can be seen that in the case of high limiting mass (upper panel), the fit performs poorly for survey areas smaller than ∼ 1000 deg 2 due to the insu ffi cient number of haloes that are expected to be found. However, above ∼ 2000 deg 2 the percentiles of the first ten orders can be fitted with an accuracy better than two per cent. s s If the limiting mass is lowered, the quality of the fit improves drastically as shown in the lower panel of Fig. B1 for m lim = 5 × 10 14 M /circledot . In this case, sub-percent-level accuracy is reached for A s /greaterorequalslant 1000 deg 2 and an accuracy better than two per cent down to 100 deg 2 .", "pages": [ 16, 17 ] } ]

[ { "title": "ABSTRACT", "content": "We present a model of γ -ray emission through neutral pion production and decay in twotemperature accretion flows around supermassive black holes. We refine previous studies of such a hadronic γ -ray emission by taking into account (1) relativistic e ff ects in the photon transfer and (2) absorption of γ -ray photons in the radiation field of the flow. We use a fully general relativistic description of both the radiative and hydrodynamic processes, which allows us to study the dependence on the black hole spin. The spin value strongly a ff ects the γ -ray emissivity within ∼ 10 gravitational radii. The central regions of flows with the total luminosities L < ∼ 10 -3 of the Eddington luminosity ( L Edd) are mostly transparent to photons with energies below 10 GeV, permitting investigation of the e ff ects of space-time metric. For such L , an observational upper limit on the γ -ray (0.1 - 10 GeV) to X-ray (2 - 10 keV) luminosity ratio of L 0 . 1 -10GeV / L 2 -10keV /lessmuch 0 . 1 can rule out rapid rotation of the black hole; on the other hand, a measurement of L 0 . 1 -10GeV / L 2 -10keV ∼ 0 . 1 cannot be regarded as the evidence of rapid rotation, as such a ratio can also result from a flat radial profile of γ -ray emissivity (which would occur for nonthermal acceleration of protons in the whole body of the flow). At L > ∼ 10 -2 L Edd, the γ -ray emission from the innermost region is strongly absorbed and the observed γ -rays do not carry information on the value of a . We note that if the X-ray emission observed in Centaurus A comes from an accretion flow, the hadronic γ -ray emission from the flow should contribute significantly to the MeV / GeV emission observed from the core of this object, unless it contains a slowly rotating black hole and protons in the flow are thermal. Key words: accretion, accretion discs - black hole physics - gamma-rays: theory", "pages": [ 1 ] }, { "title": "Andrzej Nied'zwiecki, 1 /star Fu-Guo Xie 2 , 3 /star and Agnieszka Ste¸pnik 1 /star", "content": "1 Department of Astrophysics, University of Ł'od'z, Pomorsk a 149 / 153, 90-236 Ł'od'z, Poland 2 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China 3 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China 30 July 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Early investigations of black hole accretion flows indicated that tenuous flows can develop a two-temperature structure, with proton temperature su ffi cient to produce a significant γ -ray luminosity above 10 MeV through π 0 production (e.g. Dahlbacka, Chapline & Weaver 1974). The two-temperature structure is an essential feature of the optically-thin, advection dominated accretion flow (ADAF) model, which has been extensively studied and successfully applied to a variety of black hole systems (see, e.g., reviews in Yuan 2007, Narayan & McClintock 2008, Yuan & Narayan 2013) over the past two decades, following the work of Narayan & Yi (1994). Mahadevan, Narayan & Krolik (1997; hereafter M97) pointed out that γ -ray emission resulting from proton-proton collisions in ADAFs may be a signature allowing to test their fundamental nature. The model of M97 relied on a non-relativistic ADAF model and their computations were improved by Oka & Manmoto (2003; hereafter OM03) who used a fully general relativistic (GR) model of the flow. However, both M97 and OM03 neglected the Doppler and gravitational shifts of energy as well as gravitational focusing and capturing by the black hole, which is a major deficiency because the γ -ray emission is produced very close to the black hole's horizon. Furthermore, both works neglected the internal absorption of γ -ray photons to pair creation, which e ff ect should be important in more luminous systems. ADAFs are supposed to power low-luminosity AGNs, like Fanaro ff -Riley type I (FR I) radio galaxies or low-luminosity Seyfert galaxies, and a measurement, or even upper limits on their γ -ray emission, may put interesting constraints on the properties of the source of high-energy radiation in such objects. M97 and OM03 considered only the CGRO / EGRET source in the direction of the Galactic Center for such an analysis. Significant progress in exploration of the γ -ray activity of AGNs which has been made after their works, thanks to the Fermi mission, motivates us to develop a more accurate model of the hadronic γ -ray emission from ADAFs. Detections of γ -ray emission from objects with misaligned jets (e.g. Abdo et al. 2010b) are most relevant for our study. Their γ -ray radiation is usually explained as a jet emission; we show that emission from an accretion flow may be a reasonable alternative, at least in some FR Is. We focus on modelling of radiation in 100 MeV - 10 GeV energy range, relevant for the Fermi -LAT measurements of the FR I radio galaxies (Abdo et al. 2010b) and over which the upper limits in Seyfert galaxies are derived (Ackermann et al. 2012). The dependence of the γ -ray luminosity on the black hole spin parameter makes a particularly interesting context for such an investigation. Already a rough estimate by Shapiro, Lightman & Eardley (1976) indicated a strong dependence of the γ -ray luminosity from a two-temperature flow on the spin of the black hole and, then, they suggested that this e ff ect may serve as a means to measure the spin value (see also Eilek & Kafatos 1983 and Colpi, Maraschi & Treves 1986). OM03, who made GR calculations for the modern ADAF model, found a dramatic dependence of the γ -ray luminosity on the spin value in models with thermal distribution of proton energies, however, they concluded that the dependence is weak if protons have a nonthermal distribution. In this work we extend the analysis of this issue and clarify some related properties. Wefind global solutions of the hydrodynamical ADAF model, which follows Manmoto (2000), and use them to compute the γ -ray emission. Similarly to M97 and OM03 we take into account emission resulting from thermal and nonthermal distribution of proton energies; we use similar phenomenological models, with some modifications which allow to illustrate separately e ff ects due to local distribution of proton energies and to radial profile of γ -ray emissivity. We also use our recently developed model of global Comptonization (Nied'zwiecki, Xie & Zdziarski 2012; hereafter N12, see also Xie et al. 2010) to compute the X-ray emission, which allows to investigate the internal absorption of γ -ray photons to pair creation in the flow. In our computations we assume a rather weak magnetic field, with the magnetic pressure of 1 / 10th of the total pressure, supported by results of the magnetohydrodynamic (MHD) simulations in which amplification of magnetic fields by the magneto-rotational instability typically saturates at such a ratio of the magnetic to the total pressure (e.g. Machida, Nakamura & Matsumoto 2004, Hirose et al. 2004, Hawley & Krolik 2001). We investigate the dependence on the poorly understood parameter in ADAF theory, δ , describing the fraction of the turbulent dissipation that directly heats electrons in the flow. We take into account only one value of the accretion rate, but the considered ranges of the spin and δ parameters yield a rather large range of bolometric luminosities of ∼ 10 -4 to 10 -2 of the Eddington luminosity. In our paper we present both the spectra a ff ected by γγ absorption and those neglecting the absorption effect; the latter may be easily scaled to smaller accretion rates, for which the γγ absorption becomes unimportant.", "pages": [ 1, 2 ] }, { "title": "2 HOT FLOW MODEL", "content": "We consider a black hole, characterised by its mass, M , and angular momentum, J , surrounded by a geometrically thick accretion flow with an accretion rate, ˙ M . We define the following dimensionless parameters: r = R / R g, a = J / ( cR g M ), ˙ m = ˙ M / ˙ M Edd, where ˙ M Edd = L Edd / c 2 , R g = GM / c 2 is the gravitational radius and L Edd ≡ 4 π GMm p c /σ T is the Eddington luminosity. Most results presented in this work correspond to M = 2 × 10 8 M /circledot , in Fig 5a we present also results for M = 2 × 10 6 M /circledot . We consider ˙ m = 0 . 1 and three values of the spin parameter, a = 0, 0.95 and 0.998. The inclination angle of the line of sight to the symmetry axis is given by θ obs. We assume that the density distribution is given by ρ ( R , z ) = ρ ( R , 0) exp( -z 2 / 2 H 2 ), where H is the scale height at r . We assume the viscosity parameter of α = 0 . 3 and the ratio of the gas pressure (electron and ion) to the total pressure of β B = 0 . 9. The fraction of the dissipated energy which heats directly electrons is denoted by δ . Our calculations of hadronic processes are based on global solutions of the fully GR hydrodynamical model of two-temperature ADAFs, described in N12, which follows closely the model of Manmoto (2000). Here we recall only the ion energy equation, which is most important for the present study: where Λ ie is the Coulomb rate, the compressive heating and the advection of the internal energy of ions, respectively, are given by and the viscous dissipation rate, per unit area, is given by where p = ( p i + p e) /β B, p i is the ion pressure, p e is the electron pressure, Γ i is the ion adiabatic index, Ω is the angular velocity of the flow, γφ is the Lorentz factor of the azimuthal motion and A = r 4 + r 2 a 2 + 2 ra . The form of the energy equation given in equation (1) is standard in ADAFs theory, although actually it should include an additional term describing the direct cooling of protons to pion production, Q γ . In our calculation of hadronic processes we find that Q γ is approximately equal to Λ ie at r < 10. At ˙ m = 0 . 1, considered in this work, both Q γ and Λ ie are much smaller, by over 3 orders of magnitude, than the e ff ective heating Q vis + Q compr and the heating is fully balanced by the advective term, Q int. This justifies our neglect of the direct hadronic cooling. The only di ff erence between our GR model and that of Manmoto (2000) involves the simplifying assumption of d ln( R ) / d ln( H ) = 1 adopted in the latter; we do not follow this simplification and an exact H ( R ) profile is considered in all our hydrodynamic equations. We note that the simplification has a considerable e ff ect in the central part of the flow, e.g. it results in an underestimation of the proton temperature by a factor of ∼ 1 . 5 within the innermost 10 R g. Applying the above simplifying assumption we get exactly the same flow parameters as Manmoto (2000); note, however, that Manmoto (2000) assumed an equipartition between the gas and plasma pressures, with β B = 0 . 5, which in general gives a smaller proton temperature than β B = 0 . 9 assumed here. In particular, for a = 0 and δ = 10 -3 , models with β B = 0 . 9 give the proton temperature larger by a factor of /similarequal 4, close to the horizon, than models with β B = 0 . 5. This underlies also the di ff erences in the γ -ray luminosity levels between the thermal models of Oka & Manmoto (2003) and ours, as discussed in Section 3. To obtain global transonic solutions we have to adjust the specific angular momentum per unit mass accreted by the black hole, for which the accretion flow passes smoothly through the sonic point, r s. We note that this condition permits for two kinds of solutions, below referred to as a 'standard' and a 'superhot' solution. The latter (superhot) has much larger proton temperature and density, furthermore, the sound speed is large and the sonic point located in the immediate vicinity of the event horizon, e.g. r s /similarequal 1 . 2 for a = 0 . 998. In the standard solutions the sonic point is located at larger distances, r s > 2. Taking into account rather extreme properties of the superhot solutions (specifically, a very large magnitude of Q vis illustrated in Fig. 1(a) and discussed in Section 6.1) we neglect them in this work and for all values of a we consider only the standard solutions which are consistent with solutions of the model investigated in several previous studies (e.g. Manmoto 2000, Yuan et al 2009, Li et al. 2009). Note, however, that in our previous works (N12, Nied'zwiecki, Xie & Beckmann 2012) we considered the superhot solution with a = 0 . 998, then, the results for a = 0 . 998 discussed in those works correspond to flows with larger proton temperature and density (both by a factor of ∼ 5) than these considered in the present study.", "pages": [ 2, 3 ] }, { "title": "3 HADRONIC γ -RAY EMISSION AND RELATIVISTIC TRANSFER EFFECTS", "content": "The hydrodynamical solutions set the proton density, n p, and temperature, T p, as a function of radius. In principle, this should allow to determine the γ -ray emissivity, resulting from neutral pion production in proton-proton collisions and their subsequent decay into γ -ray photons, in the rest frame of the flow. However, details of this process are subject to an uncertainty related to the distribution of proton energies, which is unlikely to be thermal in optically thin flows (see discussion in Section 6.2). Following M97 and OM03 we assume that the temperature from the global solution functions as a parameter specifying the average energy of protons in the plasma which, however, does not have to have a thermal distribution. We consider several phenomenological models which must satisfy the obvious requirements that at each radius (1) the number density of protons equals n p( r ), determined by the global ADAF solution and (2) the average energy of protons equals the average energy of the Maxwellian proton gas with temperature, T p( r ), determined by the global ADAF solution, where θ p = kT p / m p c 2 and we use the simplified (cf. Gammie & Popham 1998) relativistic form of U th( θ p). We consider models involving various combinations of thermal and power-law distributions of proton energies, where n th and n pl are the local densities of these two populations. The thermal model (model T) assuming a purely Maxwellian distribution of protons and the nonthermal model (model N, same to nonthermal models of M97 and OM03), assuming that the total energy is stored in the power-law distribution of a small fraction of protons, allow us to estimate the minimum and maximum level of γ -ray luminosity, respectively, for a given set of ( M , ˙ m , a , α , β , δ ). Mahadevan (1999) and OM03 considered the model involving the mixture of the thermal and powerlaw distributions, with the radius-independent parameter characterizing the fraction of energy that goes into the two distributions. Deviations of such a model from model N are trivial, with the γ -ray luminosity linearly proportional to the fraction of energy going to the power-law distribution. In this work we consider a di ff erent hybrid model (model H) with the radius dependent normalization between the power-law and the thermal distribution, which allows us to illustrate some additional e ff ects. The detailed assumptions on the parameters of these models are as follows ( n p( r ) and T p( r ) denote values given by the global ADAF solution): Model T assumes a purely Maxwellian distribution of protons, equation (5), with n th = n p( r ) and θ p = kT p( r ) / m p c 2 . Model N assumes that a fraction ψ of protons form the powerlaw distribution, equation (6), with the radius-independent index s and n pl = ψ ( r ) n p( r ), and the remaining protons are cold, with the Lorentz factor γ /similarequal 1 ( γ = 1 is assumed in the computations). The radius-dependent fraction ψ is determined by Model H assumes that an e ffi cient nonthermal acceleration operates only within the central ∼ 15 R g, where the average proton energies resulting from the ADAF solutions become relativistic. Specifically, we assume that at each radius at r < 15 a fraction of protons form a thermal distribution at a subrelativistic temperature of T = 4 . 3 × 10 11 K ( θ p = 0 . 04), and the remaining form a power-law distribution (equation 6) with a constant (i.e. radius-independent) index s and n pl = ψ ( r ) n p( r ). The relative normalization of these two distributions is determined by (where the factor 6 . 6 / 4 . 2 results from equation (4) with θ p = 0 . 04). At r > 15, where T p < 4 . 3 × 10 11 K, there are no non-thermal protons in this model, which then results in a negligible pion production at such distances, similar as in model T. The chosen value of T = 4 . 3 × 10 11 Kgives a smooth transition between a purely thermal and a hybrid plasma at r = 15, however, radiative properties of model H are roughly independent of the specific value of the temperature of the subrelativistic thermal component. We remark also that T = 4 . 3 × 10 11 K is close to the limiting temperature above which the pion production prevents thermalization of protons (see Stepney 1983, Dermer 1986b) The e ffi ciency of pion production by protons with the powerlaw distribution increases with the decrease of the power-law index s . On the other hand, the fraction ψ decreases with decreasing s , roughly as ψ ∝ ( s -2). These two e ff ects balance each other yielding the largest luminosity in 0.1-10 GeV range, L 0 . 1 -10GeV, for s /similarequal 2 . 5 -2 . 6. For 2 . 3 < s < 2 . 8, the dependence of L 0 . 1 -10GeV on s is weak; for s = 2 . 1, L 0 . 1 -10GeV is by a factor of ∼ 2 smaller than for s = 2 . 6. To estimate the maximum value of L 0 . 1 -10GeV that can be produced in a flow with given parameters, in our computations for models N and H we use s = 2 . 6. For all values of a , θ p > 0 . 1, and also ψ > 0 . 1 in models H and N with s = 2 . 6, within the innermost several R g. In our solutions of the flow structure we assume that protons are thermal and we use the thermal form of the gas pressure. Then, our models N and H with non-thermal proton distributions are not strictly self-consistent, as their pressure may deviate from the thermal prescription. However, this is a rather small e ff ect, e.g. the pressure of the purely non-thermal distribution (model N) di ff ers by 20-30 per cent from the pressure of a thermal gas with the same internal energy. For a given distribution of proton energies we determine the γ -ray spectra in the flow rest frame, strictly following Dermer (1986a,1986b), in a manner similar to M97 and OM03; however, we do not apply the following simplification underlying their nonthermal model. As argued in M97, the fraction of nonthermal protons should be small, ψ /lessmuch 1, and, therefore, interactions of nonthermal protons with other nonthermal protons may be neglected; hence, only interaction of nonthermal protons with cold protons are taken into account in their computations. We remark that such an approach underestimates the γ -ray luminosity, e.g. by a factor of ∼ 2 in model N with a = 0 . 95 and s = 2 . 6 (for which ψ /similarequal 0 . 4 in the innermost region). In all our models we take into account interaction of protons with all other protons. To compute the γ -ray luminosity and spectra received by distant observers we use a Monte Carlo method similar to that described in Nied'zwiecki (2005). We generate γ -ray photons isotropically in the plasma frame, make a Lorentz transformation from the flow rest frame to the locally non rotating (LNR) frame and then we compute the transfer of γ -ray photons in curved space-time; see, e.g., Bardeen et al. (1972) for the definition of LNR frames and the equations of motion in the Kerr metric. The dashed lines in Fig. 2 show the radial profiles of the vertically-integrated γ -ray emissivity, Q γ ( Q γ gives the energy emitted from the unit area per unit time) for models T, H and N with a = 0 . 95. The solid lines in Fig. 2 show the radial profiles of the vertically-integrated local luminosity (the energy per unit time reaching infinity from the unit area at a given r ). The local luminosity profiles shown by the solid lines neglect the γγ absorption, so the di ff erence between the dashed and solid lines is only due to the relativistic transfer e ff ects. Fig. 3 shows the corresponding γ -ray spectra and compares them with the spectra for a = 0. At r < 10 both models N and H are characterised by similar values of ψ and produce similar amounts of γ -rays. In both models T and H the contribution from r > 10 is very weak; in model N the radial emissivity is much flatter despite ψ being small, e.g. ψ < 5 × 10 -3 at r > 100. Comparing models T and H we can see the e ff ect of the local proton distribution function and by comparing models H and N we can see the e ff ect of the radial emissivity. For the thermal distribution of protons, the rest-frame photon spectra are symmetrical, in the logarithmic scale, around ∼ 70 MeV but in EFE units they peak around 200 MeV; the position of the maximum in the spectra observed by distant observes is slightly redshifted. Note that the di ff erence of γ -ray luminosities, L γ , between a = 0 and 0.95 in our model T is much smaller than that derived by OM03, whose thermal models with a = 0 and 0.95 give L γ di ff ering by approximately three orders of magnitude. The difference is due to di ff erent values of β B assumed here and by OM03, which result in di ff erent θ p, as discussed in Section 2. The dependence of L γ on θ p changes around θ p ≈ 0 . 1 (see, e.g., fig. 3 in Dermer 1986b). At lower temperatures, L γ is extremely sensitive to θ p, with the increase of θ p by a factor of 2 yielding the increase of L γ by over two orders of magnitude. At θ p > 0 . 1, the dependence is more modest, e.g. the increase of θ p from 0.2 to 0.4 results in the increase of L γ by only a factor of ∼ 2. For β B = 0 . 9 assumed in this work, θ p > 0 . 1 at small r for all values of a , making the γ -ray luminosity much less dependent on the black hole spin. For β B = 0 . 5, assumed by OM03, the proton temperature is small, with the maximum value of θ p ≈ 0 . 03 for a = 0, which leads to the strong dependence of L γ on a . For both model H and N, the spectrum at E > 1 GeV has the same slope as the power-law distribution of proton energies. For model H with s = 2 . 6, L γ is by a factor of 3 larger than in model T. Rather small di ff erence between L γ in our thermal and nonthermal models is again due to our assumption of a weak magnetic field. At smaller β B, resulting in smaller θ p, the presence of even a small fraction of non-thermal electrons leads to the increase of L γ by orders of magnitude, as can be seen by comparing the emissivities of our models N and T at r > 10 (see also M97). For models T and H the bulk of the γ -ray emission comes from r < 10 (Fig. 2ab) and the GR transfer e ff ects reduce the detected γ -ray flux by approximately an order of magnitude. In model N the magnitude of the GR e ff ects on the total flux is reduced due to strong contribution from r > 10 (which is weakly a ff ected by GR). Also in model N, the contribution from r > 10, which for a = 0 approximately equals the contribution from r < 10, reduces the di ff erence between the γ -ray fluxes observed for a = 0 and a = 0 . 95 to only a factor of ∼ 2, see Fig. 3(cd). The viewing-angle dependent spectra for model H, which would be observed (if unabsorbed) by distant observers, are shown by the dashed and dotted lines in Fig. 4. The flows considered in this work are quasi-spherical and optically thin and hence their appearance depends on the viewing angle primarily due to the relativistic transfer e ff ects. Most importantly, trajectories of photons emitted close to a rapidly rotating black hole are bent toward its equatorial plane. Therefore, the γ -ray radiation has a significant intrinsic anisotropy in models with large a , with edge-on directions corresponding to larger γ -ray fluxes.", "pages": [ 3, 4, 5 ] }, { "title": "4 COMPTONIZATION AND γγ ABSORPTION", "content": "The absorption of γ -rays in the radiation field of the flow has been calculated in a fully GR model by Li et al. (2009). Here we use a similar approach with the major di ff erence involving the computation of target photon density. Li et al. (2009) considered the propagation of photons with energies of 10 TeV, which are absorbed mostly in interactions with infra-red photons. Those low energy photons are produced primarily by the synchrotron emission which can be simply modelled using its local emissivity. In turn, photons with energies in 0.1-10 GeV range, considered in this work, are mostly absorbed by the UV and soft X-ray photons, which are produced by Comptonization. Then, an exact computation of the angular-, energy- and location-dependent distribution of the target photon field requires the precise modelling of the Comptonization taking into account its global nature. In our model we apply the Monte Carlo (MC) method, described in detail in N12, with seed photons for Comptonization from synchrotron and bremsstrahlung emission. We find self-consistent electron temperature distributions using the procedure described in N12; we iterate between the solutions of the electron energy equation (analogous to equations 1-3; note that here we include the direct electron heating, while N12 assumes δ = 0) and the GR MC Comptonization simulations until we find mutually consistent solutions. In Fig. 5 we show the resulting spectra. Fig. 6 shows the radial profiles of the radiative cooling of electrons (strongly dominated by Comptonization), Q Compt, for δ = 10 -3 and compares them with the γ -ray emissivity, Q γ , for model T. Note that Q γ is much steeper than Q Compt so the GR effects are more important for the γ -ray than for the X-ray emission. As we can see in Fig. 6 and also in the corresponding spectra in Fig. 5a, for δ = 10 -3 the black hole spin negligibly a ff ects the Comptonized radiation; this property results from a large magnitude of the compression work, which is roughly independent of a and dominates the heating of electrons for small values of δ (cf. N12). For δ ≥ 0 . 1 the direct heating contributes significantly to the heating of electrons and for δ = 0 . 5 it dominates over other heating processes at r < 100 for all values of a . Then, the dependence of Q vis on a results in a noticeable dependence of the Comptonized radiation on a for δ ≥ 0 . 1 (see also Xie & Yuan 2012 for a recent study of the dependence of X-ray luminosity on δ ). The radiative e ffi ciency increases from η = 0 . 004 for all values of a at δ = 10 -3 to η = 0 . 02 for a = 0, η = 0 . 08 for a = 0 . 95 and η = 0 . 1 for a = 0 . 998 at δ = 0 . 5. Despite considering only one value of accretion rate, our solutions span a range of bolometric luminosities, from L ≈ 4 × 10 -4 L Edd (for δ = 10 -3 ) to L ≈ 10 -2 L Edd (for δ = 0 . 5 and a = 0 . 998). The corresponding X-ray spectral slopes harden from Γ X /similarequal 1 . 7 to Γ X /similarequal 1 . 5 with the increase of L . Note that these values correspond to the range of parameters close to the turning point in the L -Γ correlation observed in AGNs (e.g. Gu & Cao 2009). Then, we likely consider here the range of the largest luminosities of the flows in which synchrotron emission is the dominant source of seed photons for Comptonization (see discussion and references in N12). Having found the self-consistent solutions, described above, we apply our MC model to tabulate the distribution of all photons propagating in the central region (up to r out = 1000), d n ph( R , θ, E LN , Ω LN) / d E LNd Ω LN (in photons cm -3 eV -1 sr -1 ), where R and θ are the Boyer-Lindquist coordinates, E LN is the photon energy in the LNR frame and d Ω LN is the solid angle element in the LNR frame. To compute the optical depth to pair creation, τγγ , we closely follow the method for determining an optical depth to Compton scattering in the Kerr metric, see Nied'zwiecki (2005) and Nied'zwiecki & Zdziarski (2006), however, here we calculate the probability of pair creation in the LNR frame whereas for the Compton e ff ect an additional boost to the flow rest frame is applied. While Compton scattering is most conveniently described in the plasma rest frame, pair production can be simply modelled in the LNR frame and, thus, the transformation to the flow rest frame is not necessary here. We solve equations of the photon motion in the Kerr metric and we determine the increase of the optical depth along the photon trajectory from where d l LN is the length element in the LNR frame, σγγ ( E LN , E γ LN , θ LN) is the pair production cross section (e.g., Gould & Schreder 1967), E γ LN is the energy of the γ -ray photon in the LNR frame and θ LN is the angle between the interacting photons in the LNR frame. GR a ff ects the γγ opacity through (1) bending the trajectories of both the γ -ray photon and target photons and (2) changing energies of both the γ -ray photon and target photons. As an example, the neglect of the gravitational shift of the γ -ray photon energy, by using σγγ ( E LN , E γ, θ LN) (where E γ is the energy at infinity) instead of σγγ ( E LN , E γ LN , θ LN) in equation (9), underestimates τγγ by a factor of ≈ 2-3 for photons emitted from the innermost region. In Fig. 7 we show values of the total optical depth, τγγ ( r ), integrated along the outward radial direction in the equatorial plane from the emission point at the radial coordinate r to the outer boundary at r out. As we can see, the γγ opacity is a strong function of both the γ -ray energy and the location in the flow. The dotted lines in Fig. 2 show how the γγ absorption attenuates γ -rays observed from a given r . It is apparent that around ˙ m ∼ 0 . 1 flows undergo transition from being fully transparent to mostly opaque to γ -rays. In our models with the Eddington ratio L / L Edd = 4 × 10 -4 , the flow is fully transparent to photons with energies < ∼ 100 MeV; at higher energies the absorption leads to moderate attenuation, with the increase of the photon index at E > 1 GeV by ∆Γ /similarequal 0 . 2, see Fig. 4. The size of the γ -ray photosphere (the surface of τγγ = 1) increases with increasing L and for L /similarequal 10 -2 L Edd the GeV photons cannot escape from r < 10. At such L , our model H gives spectra with τ a clear cut-o ff around 1 GeV (see the solid line in Fig. 5b) which could be measured by Fermi . In other cases absorption leads to a smooth softening of the spectra. In terms of the 2-10 keV luminosity, L 2 -10keV, flows with L 2 -10keV < 10 -5 L Edd should be fully transparent to MeV / GeV photons. Flows with L 2 -10keV > 10 -3 L Edd can emit significant amounts of unabsorbed γ -rays only if their γ -ray emissivities are strong at large r . E.g. in our model N, the luminosity of the flow at r > 50, which region would be outside the photosphere of 1 GeV photons even at much larger L 2 -10keV ∼ 10 -2 L Edd, is L 0 . 1 -10GeV /similarequal 10 40 erg / s. Then, the γ -ray luminosity exceeding 10 41 erg / s can be expected at ˙ m > 0 . 3 if the γ -ray emitting flow extends out to several tens of R g, which property is, however, unclear as objects with high luminosities often show signs of a cold disc extending to rather small radii (so the transition between the hot and cold flow may occur within the γ -ray photosphere). Note that for such a scenario, with γ -ray emission from a hot flow at large L , we expect a small luminosity ratio of L 0 . 1 -10GeV / L 2 -10keV ∼ 10 -3 regardless of the value of a .", "pages": [ 5, 6, 7 ] }, { "title": "5 X-RAY VS γ -RAY LUMINOSITY", "content": "In Fig. 8 we summarize our results regarding the relation between the X-ray and γ -ray luminosities. The range of expected L 0 . 1 -10GeV is constrained from below by values indicated in Fig. 8a for model T, and from above by values in Fig. 8b for model N. As we can see, the models give the luminosity ratios L 0 . 1 -10GeV / L 2 -10keV between ∼ 0 . 002 and 0.2. In model T, L 0 . 1 -10GeV for a = 0 and a = 0 . 998 di ff er by a factor of several; the unabsorbed luminosities di ff er by over an order of magnitude but for L close to 10 -2 L Edd the γγ absorption reduces the di ff erence to a factor of ∼ 4. In model N, L 0 . 1 -10GeV for a = 0 and a = 0 . 998 di ff er by only a factor of ∼ 2; as noted before, the di ff erence is reduced here due to contribution from large r . Model N for a = 0 gives similar L 0 . 1 -10GeV as model T with large a ; larger density and average energy for large a is approximately compensated by a larger fraction of protons above the pion production threshold for model N. Note, however, that - despite similar luminosities - the spectra for these two regimes di ff er significantly, see Fig. 3. In model H, L 0 . 1 -10GeV has a similar dependence on a as in model T, with a large di ff erence between small and high values of a . We conclude that it is the radial distribution of γ -ray emissivity, rather than the local proton distribution function, which reduces the dependence on a . This is even more hindering for attempts of assessing the spin value basing on the γ -ray luminosity level. The proton energy distribution function is reflected in the produced spectral shape and, therefore, it could be constrained observationally and taken into account in this kind of analysis. On the other hand, one cannot expect to derive information on the radial emissivity profile from observations, so an investigation of black hole spin values using the γ -ray luminosity should be subject to significant uncertainty. In models with a dominant contribution from the central ∼ 10 R g, intrinsic γ -ray luminosities of flows around submaximally ( a = 0 . 95) and maximally ( a = 0 . 998) rotating black holes di ff er by a factor of ∼ 2. For the latter, the flow extends to smaller radii and hence a larger proton temperature and density are achieved, see Fig. 1. However, a strong contribution from r > 10 as well as γγ absorption make the di ff erence between the observed luminosities insignificant. Regarding the dependence on δ , we notice a somewhat sur- rising property of models with δ = 0 . 5, which predict a larger γ -ray luminosity than models with smaller δ (the e ff ect is more pronounced for larger values of a ). The physical reason is that for δ = 0 . 5, the slight decrease of proton temperature is outweighed by the increase of density (through the decrease of both the radial velocity and the scale height with decreasing temperature), cf. Manmoto (2000), which leads to the increase of L γ ∝ n 2 p T p. At smaller values of δ , the intrinsic L γ depends negligibly on δ , in agreement with the results of OM03 for a = 0 and δ ≤ 0 . 3 in their nonthermal model. The scaling of density with ˙ m and M in our global GR solutions only weakly di ff ers from that of self-similar ADAF model, i.e. n ∝ ˙ m / M (e.g. Mahadevan 1997). Then, the intrinsic γ -ray luminosity can be estimated as L 0 . 1 -10GeV /similarequal L unabs( ˙ m / 0 . 1) 2 ( M / 2 × 10 8 M /circledot ) -1 erg / s, where L unabs is the unabsorbed luminosity shown by the open symbols in Fig. 8(ab). This gives also the observed luminosity at ˙ m < 0 . 1, when the absorption e ff ects are unimportant. Obviously, the above scaling with ˙ m is not relevant for ˙ m > ∼ 0 . 1, for which the increase of ˙ m results in the decrease of the observed L 0 . 1 -10GeV due to γγ absorption. On the other hand, the linear scaling of the γ -ray luminosity with M holds even when the γγ absorption is important. The spectral distribution of Comptonized radiation changes slightly with the black hole mass (due to the change of the synchrotron emission, which a ff ects the position of Comptonization bumps, see Fig 5a), however, the e ff ect is insignificant for the γγ opacity and we have checked that it negligibly a ff ects the observed γ -ray spectra.", "pages": [ 7, 8 ] }, { "title": "6 DISCUSSION", "content": "The form of the dissipation rate and the proton distribution function are two major uncertainties for predicting the γ -luminosity. We briefly discuss here the related e ff ects in accretion flows.", "pages": [ 8 ] }, { "title": "6.1 Viscous heating", "content": "The form of the viscous dissipation rate given by equation (1) results from the usual assumption that the viscous stress is proportional to the total pressure, with the proportionality coe ffi cient α . We use this form of the viscous stress for computational simplicity, however, as we discuss in N12, this may be an oversimplified approach (although it has support in MHD simulations, see below). Below we briefly compare the dissipation rate in our models with that predicted by the classical Novikov & Thorne (1973) model. We define Q vis integrated over the whole body of the flow as the total dissipation rate, Q vis , tot, in our solutions and compare it to the dissipation rate in a Keplerian disc for the corresponding value of a , Q NT , tot, given by the Novikov & Thorne model. Obviously, Q vis , tot does not need to match Q NT , tot closely. The latter value is calculated under the assumption that the shear stress vanishes at the radius of the innermost stable circular orbit (ISCO), which condition is not applicable to geometrically thick ADAFs and its release should lead to stronger dissipation. On the other hand, ADAFs are subKeplerian which property decreases the dissipation rate. Our Q vis , tot does not di ff er significantly from Q NT , tot, however, the e ffi ciency of dissipation in our model comparatively increases with increasing a . Specifically, Q vis , tot = 0 . 025 ˙ Mc 2 ≈ 0 . 5 Q NT , tot for a = 0, Q vis , tot = 0 . 2 ˙ Mc 2 ≈ Q NT , tot for a = 0 . 95 and Q vis , tot = 0 . 58 ˙ Mc 2 ≈ 1 . 5 Q NT , tot for a = 0 . 998. Note that in an ADAF with a large value of a most of the dissipation occurs very deep in the potential, where relativistic e ff ects strongly reduce the energy escaping to infinity, whereas in a Keplerian disc the dissipation is less centrally concentrated and its radiation is subject to less severe reduction. For example, the gravitational redshift alone (neglecting, e.g., the photon capture under the event horizon) would give similar luminosities received by distant observers ( /similarequal 0 . 3 ˙ Mc 2 ) both in our ADAF and in Novikov & Thorne models with a = 0 . 998, if all of the dissipated energy were converted into radiation. Obviously, in optically thin ADAFs only a small part of the dissipated energy is radiated away and most of it is accreted by the black hole, so the radiative e ffi ciency is much smaller than 0.3. As we mentioned in Section 2, the model formally allows for two solutions and the above values of Q vis , tot correspond to the standard solution, considered in previous sections. The superhot solution has an extreme dissipation, with Q vis , tot /similarequal 5-10 ˙ Mc 2 . This is not necessarily an unphysical property (note that Q vis , tot is defined in the rest frame of the flow), as the observed luminosity of such flows does not exceed the accretion rate of the rest mass energy. Nevertheless, the magnitude of Q vis suggests that the underlying assumption of Q vis ∝ p breaks down at very large p . The issue of the proper description of the Q vis term could be partially resolved by comparing analytic models such as our, aiming at a precise calculation of the produced radiation, with MHD simulations. The latter currently neglect radiative cooling, or use very approximate descriptions for it, in turn, they provide more accurate accounts of the dissipation physics. Such MHD simulations support some properties of our model, e.g. the Q vis ∝ p prescription of viscous heating (Ohsuga et al. 2009). Furthermore, the GR MHD simulations have shown that flows cross the ISCO without any evidence that the shear stress goes to zero, which leads to the increase of the radiative e ffi ciency, and the deviations from Novikov & Thorne model increase with increasing H / R ratio (e.g. Noble, Krolik & Hawley 2009, Penna et al. 2010). We could not, however, quantitatively compare our models with such simulations, as the published results have a much smaller aspect ratio than our solutions ( H / R > 0 . 5). Lastly, we remark that Gammie & Popham (1998) and Popham & Gammie (1998) present a model similar to ours but with a more elaborate description of the shear stress (in turn, they neglect radiative processes). We note that their results indicate a similar in magnitude stabilizing e ff ect of the rotation of black hole.", "pages": [ 8 ] }, { "title": "6.2 Proton distribution function", "content": "As pointed out by Mahadevan & Quataert (1998), Coulomb collisions are too ine ffi cient to thermalize protons in optically thin flows and the proton distribution function is determined by the viscous heating mechanism, which is poorly understood. Our solutions, with M = 2 × 10 8 M /circledot and ˙ m = 0 . 1, give the accretion time-scale, t a, much shorter than the proton relaxation time-scale, t pp; e.g. at r ≤ 20, t a < 10 hours and t pp > 10 5 hours. Clearly, the protons cannot redistribute their energy through Coulomb collisions. In solar flares, the best observationally studied example of particle acceleration / heating in a magnetised plasma, a significant fraction of the released energy is carried by non-thermal, high energy particles (e.g. Aschwanden 2002), which strongly motivates for considering the nonthermal distribution of protons, as originally proposed in M97. Applying the generic description of particle acceleration, see e.g. section 3 in Zdziarski, Malzac & Bednarek (2009), we check whether the conditions in ADAFs allow for proton acceleration to ultrarelativistic energies, as assumed in our computations for the power-law distributions. The magnetic field strength in our ADAF solutions is B /similarequal 10 G, 100 G and 1000 G at r = 100, 10 and 2, respectively. Assuming an acceleration rate d E / d t ∝ ξ eB , where e is the elementary charge and ξ is the acceleration e ffi ciency ( ξ < ∼ 1), we find that the maximum Lorentz factor limited by the synchrotron energy loss is γ max ∼ 10 6 -10 7 , depending on r . Another condition, of the Larmor radius being smaller than the acceleration site size, R acc, gives γ max ∼ 10 7 at r /greatermuch 100, and larger values of γ max at smaller r , even if we safely assume R acc = 1 R g ( = 3 × 10 13 cm). We conclude that the central region of a hot accretion flow may be a site of the acceleration of protons to energies which easily allow hadronic emission of photons even in the TeV range. This conclusion remains valid for the whole relevant range of accretion rates and masses of supermassive black holes, as the strength of the magnetic field scales as B ∝ ˙ m 1 / 2 M -1 / 2 . Considering processes which could compete with protonproton interactions, we note that proton-photon interactions are much less e ffi cient in the central region of the flow. Namely, the number density of γ -ray photons, which can e ff ectively interact with protons in photomeson production, is by a factor of ∼ 10 3 smaller than the number density of protons within the innermost few R g; the number density of hard X-ray photons, which can interact with protons in photopair production, is similar to n p. The cross-section for both channels of proton-photon interactions is by over two orders of magnitude smaller than the cross-section for proton-proton interaction, making the photo-hadronic production of secondary particles negligible.", "pages": [ 8, 9 ] }, { "title": "7 COMPARISON WITH OBSERVATIONS", "content": "We briefly compare here predictions of our model with γ -ray observations of objects which may be powered by ADAFs. We note, however, that for a more detailed comparison additional physical processes, related to charged pion production (cf. Mahadevan 1999), should be taken into account. In particular, relativistic electrons produced by the pion decay should be important in modelling the emission in the MeV range. We also tentatively discuss the origin in the accretion flow of the very high energy radiation detected from M87 and Sgr A /star , for the latter under assumption that the contributions from two separate sources dominate above 100 GeV (Sgr A /star ) and at lower energies (di ff use emission) in the radiation observed from the Galactic Center region. However, we note that the opacity at very high energies may be a ff ected by the nonthermal synchrotron emission of the relativistic electrons. The work on the model implementing the e ff ects of charged pions is currently in progress.", "pages": [ 9 ] }, { "title": "7.1 Misaligned AGNs", "content": "The misaligned AGNs detected by the Fermi -LATinclude seven FR I radio galaxies and four FR IIs (Abdo et al. 2010b). The low-power FR I galaxies are supposed to be powered by radiatively ine ffi cient accretion flows (e.g. Balmaverde, Baldi & Capetti 2008) and, thus, are more relevant for application of our results. Interestingly, Wu, Cao & Wang (2011) assess that supermassive black holes in FR Is rotate rapidly, with a > 0 . 9. The X-ray emission of more luminous FR Is is supposed to come from an accretion flow (e.g. Wu, Yuan & Cao 2007), but their γ -ray emission tends to be interpreted in terms of jet emission (e.g. Abdo et al. 2009, 2010ab). FR Is are supposed to be the parent population of BL Lac objects, however, the Lorentz factors required by a jet model are much lower than typical values found in models of BL Lac objects (see, e.g., Abdo et al. 2010b). Therefore, the radiation observed in FR Is and BL Lacs must have a di ff erent origin, which adds some complexity to the jet model for FR Is. At least two FR I galaxies reported in Abdo et al. (2010b), M87 and Centaurus A, are detected because of their proximity rather than a small inclination angle of their jets and, then, they are interesting targets for searching for the γ -ray emission from accretion flows. In Nied'zwiecki et al. (2012a) we roughly compared preliminary results of our model with the X / γ -ray observational data of Centaurus A from INTEGRAL and Fermi -LAT. We found that the ADAF model, which matches the X-ray emission in this object, predicts the γ -ray emission significantly weaker than measured by Fermi if the value of the spin parameter a is small. We should emend, however, that this conclusion is valid only for models assuming a significant γ -ray emission only from the central ∼ 10 R g, such as our models T and H. Regardless of the values of a and δ , our model N with s = 2 . 6 predicts the absorbed L 0 . 1 -10GeV approximately consistent with 1 . 3 × 10 41 erg / s measured in Cen A by Fermi (Abdo et al. 2010a). Also regardless of the value of a , our models with δ = 10 -3 predict the 2-10 keV flux as well as the X-ray spectral index consistent with that observed in Cen A; for δ > ∼ 0 . 1 the model overpredicts the flux and hardness of the X-ray radiation. In the above we assumed the black hole mass of 2 × 10 8 M /circledot (e.g. Marconi et al. 2001), which is also within the range allowed by the recent measurement of Gnerucci et al. (2011). The central core of the accretion system in M87 has been considered as the γ -ray emitting region e.g. by Neronov & Aharonian (2007) in their model with particle acceleration in the black hole magnetosphere. The accretion rate assessed from the highresolution observations of the nucleus of M87 by Chandra , and used to model the multiwavelength spectrum of the nucleus by emission from an ADAF, by Di Matteo et al. (2003; note that their definition of ˙ m di ff ers from ours by a factor of 10), /similarequal 0 . 1 M /circledot / year, corresponds to ˙ m /similarequal 0 . 01 for a black hole with M = 3 × 10 9 M /circledot . At such ˙ m the central region should be transparent to γ -ray photons. Then, we compare the unabsorbed luminosities from our nonthermal models with s = 2 . 2 (approximately consistent with the slope of the Fermi data above 200 MeV; Abdo et al. 2009) and M = 3 × 10 9 M /circledot with the γ -ray measurements of M87. We find that the luminosity derived in the 0.2-10 GeV range by Fermi (Abdo et al. 2009), and above 100 GeV by ground-based telescopes (e.g. Aleksi'c et al. 2012) in the low state of M87, can be reproduced by our model with a = 0 . 998 and /similarequal 0 . 14 M /circledot / year (for θ obs = 40 · ). The required accretion rate, larger by 40 per cent than the face value of the estimate in Di Matteo et al. (2003), is allowed by the precision of the estimation of the accretion rate using Bondi accretion theory. We conclude that hadronic processes in ADAF can contribute significantly to the γ -ray emission observed in M87 in the low state, or even explain these observations entirely. However, such a model requires that all the available material forms the innermost flow, i.e. it does not allow a strong reduction the accretion rate in the central region by outflows (assumed in models of Sgr A /star , see below).", "pages": [ 9 ] }, { "title": "7.2 Sgr A /star", "content": "The value of ˙ m in the central region is the major issue for applications of ADAF models to the supermassive black hole in the Galactic Center. The Bondi accretion rate corresponds to ˙ m B /similarequal 10 -3 for M /similarequal 3 × 10 6 M /circledot and early models used such ˙ m to explain the broadband spectra of Sgr A /star (e.g. Narayan et al. 1998). The measurement of the millimetre / submillimetre polarization of Sgr A /star is often assumed to limit the accretion rate to ˙ m ≤ 3 × 10 -5 (e.g. Marrone et al. 2007), but counterarguments are presented, e.g., by Mo'scibrodzka, Das & Czerny (2006) and Ballantyne, Ozel & Psaltis (2007). An updated ADAF model to Sgr A /star , with ˙ m /similarequal 10 -5 and strong electron heating, is presented in Yuan, Quataert & Narayan (2003). An alternative to ADAF model, proposed by Mo'scibrodzka et al. (2006) and recently applied by Okuda & Molteni (2012), with the low angular momentum flow, assumes ˙ m /similarequal 6 × 10 -4 . The low angular momentum flow has a similar density in the innermost region as an ADAF with the same ˙ m , but much smaller proton temperature < 4 × 10 11 K. Then, the magnitude of hadronic processes could be used to distinguish the two classes of models; however, the low T may be an artificial e ff ect resulting from the neglect of viscosity in the former (low angular momentum) class. The CGRO / EGRET source, 3EG J1746-2851, was initially considered as a possible γ -ray counterpart of Sgr A /star ; Narayan et al. (1998) and OM03 found that their ADAF models underpredicted the γ -ray luminosity implied by the EGRET measurement. However, improved analyses (e.g. Pohl 2005) subsequently indicated that 3EG J1746-2851 is displaced from the exact Galactic Center and excluded Sgr A /star , as well as the TeV source observed with HESS which may be directly related with Sgr A /star , as its possible counterparts. Recently, Chernyakova et al. (2011) analysed the Fermi -LAT observations of the Galactic Center and combined them with the HESS observational data of the point-like source (Aharonian et al. 2009). The spectrum of the source seen in the MeV / GeV band by Fermi is consistent with a π 0 -decay spectrum. The HESS data indicate flattening of the spectrum above 100 GeV, suggesting that at the highest energies a di ff erent spectral component dominates, with rather hard spectrum, Γ /similarequal 2 . 2. In the EFE plot, the normalization of the HESS source is an order of magnitude larger than the quiescent X-ray emission of Sgr A /star . To explain these observations, Chernyakova et al. (2011) discuss a model, following previous works (e.g. Atoyan & Dermer 2004), with protons accelerated in the accretion flow and then di ff using outwards to interact with dense gas at distances of ∼ 1 pc. The emission at energies below 100 GeV may be explained by interactions of protons injected into the interstellar medium during a strong flare of Sgr A /star that occurred 300 years ago, as protons generating photons with such energies are still di ff usively trapped in the γ -ray production region. On the other hand, most of the higher energy protons have already escaped and hence an additional, persistent injection of high-energy protons has to be assumed to account for the HESS observations, with the required rate of 2 × 10 39 erg / s implying a very high e ffi ciency of the conversion of the accreting rest mass energy into the proton energy. We remark that alternatively the spectral component above 100 GeV can be explained by proton-proton interactions in an accretion flow around a rapidly rotating black hole. Our model with a = 0 . 998, s = 2 . 2 and edge-on viewing direction predict the flux consistent with the HESS detection for ˙ m = 3 × 10 -4 . The scenario with the γ -ray emission produced in the accretion flow may be tested over the following years, if the fall of the gas cloud into the accretion zone of Sgr A /star (Gillessen et al. 2012) results in the increase of the mass accretion rate.", "pages": [ 9, 10 ] }, { "title": "7.3 Seyfert galaxies", "content": "Spectral properties of the Seyfert galaxy, NGC 4151, are consistent with the model of an inner hot flow surrounded by an outer cold disc (e.g. Lubi'nski et al. 2010), however, its rather large luminosity, L > 0 . 01 L Edd indicates that a potential γ -ray emission from innermost region would be strongly absorbed. The constraint of L 0 . 1 -10GeV / L 14 -195keV < 0 . 0025 derived for this object in Ackermann et al. (2012) can still give interesting information if the spectral model constrains the parameters of the inner hot / outer cold accretion system (in particular, the distance of transition between the two modes of accretion). The constraint of L 0 . 1 -10GeV / L 14 -195keV < 0 . 1, or even < 0 . 01 for some objects, found for other Seyfert galaxies by Ackermann et al. (2012) is not strongly constraining for the spin value, as can be seen in Fig. 8 (note that L 14 -195keV is by a factor of a few, depending on Γ X , larger than L 2 -10keV), especially for large X-ray luminosities (and hence strong γγ absorption) characterising most of objects analysed in that paper.", "pages": [ 10 ] }, { "title": "8 SUMMARY", "content": "We have studied the γ -ray emission resulting from proton-proton interactions in two-temperature ADAFs. Our model relies on the global solutions of the GR hydrodynamical model, same as OM03, but we improve their computations by taking into account the relativistic transfer e ff ects as well as the γγ absorption and by properly describing the global Comptonization process. We have found that the spin value is reflected in the properties of γ -ray emission, but the e ff ect is not thrilling. The speed of the black hole rotation strongly a ff ects the γ -ray emission produced within the innermost 10 R g. If emission from that region dominates, the observed γ -ray radiation depends on the spin parameter noticeably; the intrinsic γ -ray luminosities of flows around rapidlyrotating and non-rotating black holes di ff er by over a factor of ∼ 10. However, if the γ -ray emitting region extends to larger distances, the dependence is reduced. In the most extreme case with protons e ffi ciently accelerated to relativistic energies in the whole body of the flow, the regions within and beyond 10 R g give comparable contributions to the total emission, reducing the di ff erence of γ -ray luminosities between high and low values of a to only a factor of ∼ 2. The radial emissivity profile of γ -rays is very uncertain (as the acceleration e ffi ciency may change with radius), then, the level of the γ -ray luminosity cannot be regarded as a very sensitive probe of the spin value. Still, it may be possible to assess a slow rotation in a low luminosity object by putting an upper limit on the γ -ray luminosity at the level of < ∼ 0 . 01 of the X-ray luminosity. The presence of nonthermal protons may be easily assessed from the γ -ray spectrum as for a purely thermal plasma the total γ -ray emission would be observed only at energies lower than 1 GeV. We have considered the accretion rate of ˙ m = 0 . 1, for which our model gives the bolometric luminosities between /similarequal 4 × 10 -4 L Edd and 10 -2 L Edd. Such ˙ m , with the corresponding range of L / L Edd, seems to be favoured for investigation of the hadronic γ -ray emission, with the related e ff ects of the space-time metric, because the internal γ -ray emission is large and its attenuation by γγ absorption is weak. Flows with L > 10 -2 L Edd can produce observable γ -ray radiation only if the emitting region extends out to a rather large distance of several tens of R g. We have found the X-ray to γ -ray luminosities ratio as a function of the black hole spin and the e ffi ciency of the direct heating of electrons. The L 0 . 1 -10GeV / L 2 -10keV ratios reaching ∼ 0 . 1, with the corresponding levels of the γ -ray luminosities which may be probed in nearby AGNs at the current sensitivity of Fermi -LAT surveys, encourage to consider contribution from an accretion flow to the γ -ray emission observed in low-luminosity objects. We point out that such a contribution should be strong at least in Cen A. The γ -ray luminosity decreases rapidly ( ∝ ˙ m 2 ) with decreasing ˙ m . M87 and Sgr A /star are the obvious, however possibly also unique, objects for which the γ -ray emission from the flow can be searched at ˙ m /lessmuch 0 . 1. Obviously, contribution from other γ -ray emitting sites should be properly subtracted to establish the luminosity of an accretion flow, which may be particularly di ffi cult in Sgr A /star . The ADAF model with a hard (acceleration index s /similarequal 2 . 2) nonthermal proton distribution can explain the γ -ray detections of both Sgr A /star (above 100 GeV) and M87, however, it does not allow for strong reduction of the accretion rate by outflows. Nevertheless, it seems intriguing that in both nearby, low accretion-rate objects, in which the γ -ray radiation is not suppressed by γγ absorption, observations reveal very high energy components, consistent with predictions of such a model. In both objects the model requires a large black hole spin. The luminosity ratio should decrease rather slowly ( L γ/ L X ∝ ˙ m 0 . 3 , as the radiative e ffi ciency of hot flows varies as ˙ m 0 . 7 at small ˙ m ; cf. Xie & Yuan 2012) with decreasing ˙ m . The presence of the γ -ray signal in low-luminosity AGNs is often considered as the evidence for the origin of their radiation in a jet (see, e.g., Takami 2011). We note that such a diagnostic is not valid because accretion flows may produce a similarly strong γ -ray emission as jets in such objects.", "pages": [ 10, 11 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are grateful to Włodek Bednarek for helpful discussions and to the referee for a very careful review and comments which helped us to improve the presentation of the results. This research has been supported in part by the Polish NCN grant N N203 582240. FGX has been supported in part by the NSFC (grants 11103059, 11121062, 11133005, and 11203057), the NBRPC (973 Program 2009CB824800), and SHAO key project No. ZDB201204.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "Abdo A. A., et al., 2009, ApJ, 707, 55 Abdo A. A., et al., 2010a, ApJ, 719, 1433 Abdo A. A., et al., 2010b, ApJ, 720, 912 Ackermann M., et al., 2012, ApJ, 747, 104 Aharonian F., et al., 2008, A&A, 492, L25 Aharonian F., et al., 2009, A&A, 503, 817 Aleksi´c, J., et al., 2012, A&A, 544, 96 Aschwanden M. 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[ { "title": "ABSTRACT", "content": "Structures in Warm Dark Matter (WDM) models are exponentially suppressed below a certain scale, characterized by the dark matter particle mass, m x . Since structures form hierarchically, the presence of collapsed objects at high-redshifts can set strong lower limits on m x . We place robust constraints on m x using recent results from the Swift database of high-redshift gamma-ray bursts (GRBs). We parameterize the redshift evolution of the ratio between the cosmic GRB rate and star formation rate (SFR) as ∝ (1+ z ) α , thereby allowing astrophysical uncertainties to partially mimic the cosmological suppression of structures in WDM models. Using a maximum likelihood estimator on two different z > 4 GRB subsamples (including two bursts at z > 8), we constrain m x /greaterorsimilar 1 . 6-1.8 keV at 95% CL, when marginalized over a flat prior in α . We further estimate that 5 years of a SVOM-like mission would tighten these constraints to m x /greaterorsimilar 2 . 3 keV. Our results show that GRBs are a powerful probe of high-redshift structures, providing robust and competitive constraints on m x . Key words: methods: statistical -gamma-ray burst: general -cosmology: dark matter", "pages": [ 1 ] }, { "title": "Constraints on Warm Dark Matter models from high-redshift long gamma-ray bursts", "content": "R. S. de Souza 1 /star , A. Mesinger 2 , A. Ferrara 2 , Z. Haiman 3 , R. Perna 4 , N. Yoshida 5 1 Korea Astronomy & Space Science Institute, Daejeon 305-348, Korea 2 Accepted - Received -", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The current concordance cosmology, in which structure formation proceeds in a hierarchal manner driven by pressureless cold dark matter (CDM), has been remarkably successful in explaining the observed properties of largescale structures in the Universe (e.g., Tegmark et al. 2006; Benson 2010) and the cosmic microwave background (CMB) (e.g., Komatsu et al. 2011). Such observables probe scales in the range ∼ 1 Gpc down to ∼ 10 Mpc. On smaller scales, /lessorsimilar 1 Mpc, there are still some discrepancies between standard ΛCDM and observations (e.g., Menci et al. 2012). For instance, N-body simulations predict more satellite galaxies than are observed both around our galaxy (the so-called 'missing satellite problem'; Moore et al. e.g., 1999; Klypin et al. e.g., 1999), and in the field as recently noted by the ALFALFA survey (e.g., Papastergis et al. 2011; Ferrero et al. 2012). Furthermore, simulations of the most massive Galactic CDM subhaloes are too centrally condensed to be consistent with the kinematic data of the bright Milky Way satellites (e.g., Boylan-Kolchin et al. 2011). Moreover, observations of small galaxies show that their central density profile is shallower than predicted by CDM N-body simulations (e.g., Moore 1994; de Blok et al. 2001; Donato et al. 2009; Macci'o et al. 2012; Governato et al. 2012). Baryonic feedback is a popular prescription for resolving such discrepancies. Feedback caused by supernovae (SNe) explosions and heating due to the UV background may suppress the baryonic content of low-mass haloes (e.g., Governato et al. 2007; Mashchenko et al. 2008; Busha et al. 2010; Sobacchi & Mesinger 2013b), and make their inner density profile shallower (e.g., de Souza & Ishida 2010; de Souza et al. 2011). However, accurately matching observations is still difficult even when tuning feedback recipes (e.g. Boylan-Kolchin et al. 2012). An alternative explanation might be found if dark matter (DM) consisted of lower mass ( ∼ keV) particles and thus was 'warm' (WDM; e.g., Bode et al. 2001; Khlopov & Kouvaris 2008; de Vega & Sanchez 2012; de Vega et al. 2012; Kang et al. 2013; Destri et al. 2013; Kamada et al. 2013). The resulting effective pressure and free-streaming would decrease structure on small-scales, though again fine-tuning might be required to fully match all the observations (e.g. Boylan-Kolchin et al. 2011; Macci'o et al. 2012; Borriello et al. 2012). The most powerful testbed for these scenarios is the high-redshift Universe. Structure formation in WDM models (or in any cosmological model with an equivalent powerspectrum cut-off) is exponentially suppressed on smallscales (e.g., Schneider et al. 2012, 2013). Since structures form hierarchically, these small halos are expected to host the first galaxies. If indeed dark matter were sufficiently 'warm', the high-redshift Universe would be empty. Therefore, the mere presence of a galaxy at high-redshift can set strong lower limits on the WDM particle mass. Due to their high luminosity, gamma-ray bursts (GRBs) constitute a remarkable tool to probe the highz Universe and small-scale structures. They provide a glimpse of the first generations of stars (e.g., de Souza et al. 2011, 2012), as well as provide constraints on primordial non-Gaussianity (Maio et al. 2012). As pointed out by Mesinger et al. (2005), the detection of a single GRB at z > 10 would provide very strong constraints on WDM models. Here we extend the work of Mesinger et al. (2005) by presenting robust lower limits on WDM particle masses, using the latest Swift GRB data. The current data, including many redshift measurements, allows us to perform an improved statistical analysis by directly comparing the distribution of bursts in various models as a function of redshift. Furthermore, we make more conservative 1 assumptions throughout the analysis, such as normalizing the SFRGRB ratio at high redshifts (thereby using a shorter, more accurate lever arm which minimizes modeling uncertainty), using an unbiased luminosity function and allowing the SFRGRB ratio to evolve with redshift. Finally, we study the effectiveness of future observations in improving the current constraints. Current limits on dark matter masses, m x , are motivated by several observations. The Lymanα forest implies m x /greaterorsimilar 1 keV (e.g., Viel et al. 2008) and m νs > 8 keV for sterile neutrinos (Seljak et al. 2006; Boyarsky et al. 2009). Likewise, WDM models with a too warm candidate ( m x < 0 . 75 keV) cannot simultaneously reproduce the stellar mass function and the Tully-Fisher relation (Kang et al. 2013). Also, the fact that reionization occurred at z /greaterorsimilar 6 implies m x /greaterorsimilar 0 . 75 keV (Barkana et al. 2001). However, all of these limits are strongly affected by a degeneracy between astrophysical (i.e. baryonic) processes and the dark matter mass. Our approach in this work is more robust, driven only by the shape of the redshift evolution of the z > 4 SFR. Furthermore, it is important to note that the SFR is exponentially attenuated at high-redshifts in WDM models. Since astrophysical uncertainties are unable to mimic such a rapid suppression, probes at high-redshifts (such as GRBs and reionization) are powerful in constraining WDM cosmologies. The outline of this paper is as follows. In § 2 we discuss how we derive the dark matter halo mass function and SFR in WDM and CDM models. In § 3 we derive the correspond- g GRB redshift distribution. In § 4 we discuss the adopted observed GRB sample. In § 5 we present our analysis and main results. In § 6, we discuss possible future constraints using a theoretical mock sample. Finally, in § 7, we present our conclusions. 2", "pages": [ 1, 2 ] }, { "title": "2 STRUCTURE FORMATION IN A WDM DOMINATED UNIVERSE", "content": "Massive neutrinos from the standard model (SM) of particle physics were one of the first dark matter candidates. However, structures formed in this paradigm are incompatible with observations. Other alternative dark matter candidates usually imply an extension of the SM. The DM particle candidates span several order of magnitude in mass (Boyarsky et al. 2009): axions with a mass of ∼ 10 -6 eV, first introduced to solve the problem of CP violation in particle physics, supersymmetric (SUSY) particles (gravitinos, neutralinos, axinos) with mass in the range ∼ eVGeV, superheavy dark matter, also called Wimpzillas , (also considered as a possible solution to the problem of cosmic rays observed above the GZK cutoff), Q-balls, and sterile neutrinos with mass ∼ keV range, just to cite a few. For a review about dark matter candidates see Bertone et al. (2005). Two promising candidates for warm dark matter are the sterile neutrino (Dodelson & Widrow 1994; Shaposhnikov & Tkachev 2006) and gravitino (Ellis et al. 1984; Moroi et al. 1993; Kawasaki et al. 1997; Primack 2003; Gorbunov et al. 2008). In WDM models the growth of density perturbations is suppressed on scales smaller than the free streaming length. The lighter the WDM particle, the larger the scale below which the power spectrum is suppressed. In addition to this power-spectrum cutoff, one must also consider the residual particle velocities. As described in Barkana et al. (2001), these act as an effective pressure, slowing the early growth of perturbations. Bellow we describe how we include these two effects in our analysis (for more information, please see Barkana et al. 2001; Mesinger et al. 2005).", "pages": [ 2 ] }, { "title": "2.1 Power spectrum cutoff", "content": "The free-streaming scale, below which the linear perturbation amplitude is suppressed (e.g., Colombi et al. 1996; Bode et al. 2001; Viel et al. 2005), is given by the comoving scale where Ω x is the total energy density contained in WDM particles relative to the critical density, h is the Hubble constant in units of 100 kms -1 Mpc -1 , and m x is the WDM particle mass. The resulting modification of the matter power spectrum can be computed by multiplying the CDM power spectrum P CDM ( k ) by an additional transfer function (Bode et al. 2001): where µ = 1 . 12 and", "pages": [ 2, 3 ] }, { "title": "2.2 Effective pressure", "content": "Structure formation in WDM models will be further suppressed by the residual velocity dispersion of the WDM particles, which delays the growth of perturbations. As shown in Barkana et al. (2001) and Mesinger et al. (2005), this 'effective pressure' is of comparable importance to the power spectrum cutoff in determining the WDM mass functions. The pressure can be incorporated in the halo mass function by raising the critical linear extrapolated overdensity threshold at collapse, δ c ( M,z ). Using spherically symmetric hydrodynamics simulations, and exploiting the analogy between the WDM effective pressure and gas pressure, Barkana et al. (2001) computed the modified δ c ( M,z ). They showed that one can define an effective WDM Jeans mass, i.e., the mass scale below which collapse is significantly delayed by the pressure. We will denote this scale as: where we make the standard assumption of a fermionic spin 1/2 particle. Note that equation (4) differs from the original one proposed by Barkana et al. (2001) by a factor of 60. With this adjustment, we find that the collapsed fractions, F coll ( z ), simulated assuming a sharp mass cutoff at M WDM are in good agreement with the full random-walk procedure (assuming a more gradual rise in δ c ( M )) of Barkana et al. (2001) (see Fig. 2). The former approach has the advantage of being considerably faster and simpler.", "pages": [ 3 ] }, { "title": "2.3 Collapsed fraction of haloes and cosmic star formation", "content": "We use the Sheth-Tormen mass function, f ST , (Sheth & Tormen 1999) to estimate the number density of dark matter haloes, n ST ( M,z ), with mass greater than M : where A = 0 . 3222 , a 1 = 0 . 707 , p = 0 . 3 and δ c = 1 . 686. The mass function f ST can be related to n ST ( M,z ) by where ρ m is the total mass density of the background Universe. The variance of the linearly extrapolated density field σ ( M,z ) is given by where b ( z ) is the growth factor of linear perturbations normalized to b = 1 at the present epoch, and W ( k, M ) is a real space top-hat filter. In order to calculate the CDM power spectrum P CDM ( k ), we use the CAMB code 3 . For WDM models, we compute the power spectrum, P WDM ( k ), with equation (2). The fraction of mass inside collapsed halos, F coll ( > M min , z ), is then given by: and the minimum mass is estimated as where M gal corresponds to the minimum halo mass capable of hosting star forming galaxies. We use M gal ≡ ¯ M min from equation (11) in Sobacchi & Mesinger (2013a), who present a physically-motivated expression for the evolution of M gal which takes into account a gas cooling criterion as well as radiative feedback from an ionizing UV background (UVB) during inhomogeneous reionization 4 . In other words, M gal is set by astrophysics, whereas M WDM ( m x ) is set by the particle properties of dark matter. In Fig. 1, we show these two limits as a function of redshift. As we shall see, our conclusions are derived from the regime where M WDM > M gal , and hence they are not sensitive to the exact value of M gal . Another point worth highlighting is that models with m x /greaterorsimilar 3 . 5 keV have M WDM ( m x ) /greaterorequalslant M gal up to z ≈ 11. Thus, any constraint beyond ∼ 3 . 5keV will no longer be robust, since galaxy formation can be suppressed even in CDM, bellow halo masses comparable to M WDM . Therefore, observations at much higher redshifts are required to obtain tighter constraints. In figure 2, we plot the fraction of the total mass collapsed into haloes of mass > M min , F coll ( > M min , z ). The shaded region shows the collapsed fraction in CDM, with a range of low-mass cutoffs corresponding to virial temperatures 300 K < T vir < 10 4 K. The other curves correspond to WDM particle masses of m x = 3.0, 2.5, 2.0, 1.5, and 1.0 keV (top to bottom). This figure is analogous to Fig. 2 in Mesinger et al. (2005), serving to motivate equation (4). The fractions F coll ( > M min , z ), computed according to the full random walk procedure used in Mesinger et al. (2005), are shown as solid black curves, while the approximation of a sharp cutoff at M WDM (eq.4) corresponds to the red dashed curves. The comoving star formation rate as a function of redshift is assumed to be proportional to dF coll /dt : The proportionality constant is irrelevant for our analysis, as it is subsumed in our normalization procedure below.", "pages": [ 3 ] }, { "title": "3 http://camb.info/", "content": "4 Using M gal ( z ) from Sobacchi & Mesinger (2013a) for WDM models is not entirely self-consistent, since UVB feedback effects should be smaller in WDM models. However, this effect is smaller than other astrophysical uncertainties. Most importantly, our conclusions are driven by models which at high redshift have M WDM > M gal , and are therefore insensitive to the actual choice of M gal (see Fig. 2). l ( > M , z) 16 10 placements - 8 - 6 - 4 - 2 0 0 2 4 6 8 9 10 z", "pages": [ 3, 4 ] }, { "title": "3 THEORETICAL REDSHIFT DISTRIBUTION OF GRBS", "content": "Under the hypothesis that the formation rate of long GRBs (LGRBs; duration longer than 2 sec) follows the SFR (e.g., Totani 1997; Campisi et al. 2010; Bromm & Loeb 2006; de Souza et al. 2011), the comoving rate of GRBs, Ψ GRB , can be expressed as Here ζ 0 is a constant that includes the absolute conversion from the SFR to the GRB rate. The evolutionary trend described by α may arise from several mechanisms (e.g., Kistler et al. 2009), with a possible explanation provided by the GRB preference for low-metallicity environments 5 . A metallicity threshold seems to provide a natural explanation for the observed value of α ≈ 0 . 5 -1 at low redshifts ( z /lessorequalslant 4; e.g. Robertson & Ellis 2012). Such a metallicity threshold increases with redshift, whereas the characteristic halo mass decreases with redshift. In such a scenario, a value of α = 0 would be appropriate for our high-redshift ( z > 4) analysis. Nevertheless, we conservatively keep α as a free parameter. The observed number of GRBs in the range z 1 /lessorequalslant z /lessorequalslant z 2 , N ( z 1 , z 2 ), can be expressed by with where the parameter K accounts for the efficiency of finding GRBs and measuring their redshift (e.g., area coverage, the survey flux limit, beaming factor of GRBs, etc) 6 , dV/dz is the comoving volume element per unit redshift, ∆ t is the 5 Since host galaxies of long duration GRBs are often observed to be metal poor, several studies have tried to connect the origin of long GRBs with the metallicity of their progenitors (e.g., M'esz'aros 2006; Woosley & Bloom 2006; Salvaterra & Chincarini 2007; Salvaterra et al. 2009; Campisi et al. 2011). Such a connection is physically-motivated since core-collapse models could not generate a long-GRB without the progenitor system having low metallicity (e.g., Hirschi et al. 2005; Yoon & Langer 2005; Woosley & Bloom 2006). On the other hand, several authors report observations of GRBs in high metallicity environments (e.g., Levesque et al. 2010; Kruhler et al. 2012), suggesting that GRB hosts are not necessarily metal poor. Despite the apparent preference of GRBs towards metal-poor hosts, there is no clear cutoff in metallicity, above which GRB formation should be suppressed. 6 There are several selection effects known to mask the true GRB redshift distribution, e.g.: (i) the host galaxy dust extinction; (ii) the redshift desert (a redshift span, 1 . 4 < z < 2 . 5, in which it is difficult to measure absorption and emission spectra); (iii) Malmquist bias; and (iv) the difference between redshift measurements techniques (e.g., Coward et al. 2012). We therefore expect K to be redshift dependent, and this evolution is subsumed in our parameter α above. Most of the above effects (e.g. obtaining GRB redshifts, dust extinction, Malmquist bias) result in biasing the observed sample towards low redshifts. Since we calibrate the proportionality between the GRB rates and SFRs at z ∼ 3 -4, we likely overestimate the efficiency in the redshift determination of z > 4 GRBs. Therefore, we expect that even a non-evolving K (i.e. our results for α = 0) would be a conservative assumption. time interval in the observer rest frame 7 , and I ( z ) is the integral over the GRB luminosity function (LF), p ( L ), To remove the dependence on proportionality constants, we construct cumulative distribution functions (CDFs) of GRBs over the redshift range z i < z < z max : PSfrag replacements The expected number of observed GRBs in a given redshift interval, for each combination θθθ ≡ { m x , α } , can be written as where ζ 0; θθθ is normalized so that each model recovers the observed GRB rate, N obs ( z 1 , z 2 ) , at z ≈ 4, N obs (3 , 4) is equal to 24 (7) for S1 (S2) samples respectively. Note that normalizing at a relatively highz ( z =3-4) is indeed a conservative choice. If we had normalized at lower redshifts, say z ∼ 1 -2, the absolute number of GRBs predicted by m x = 0 . 5 keV and CDM, between z ∼ 3 -4, would already diverge by a factor of ∼ 2 for α values in the range 0 -2. In addition to being conservative, normalizing at high-redshifts allows us to have a relatively short lever arm over which our simple scaling, SFR ∝ dF coll /dt , is presumed to be accurate. At lower redshifts, mergers and AGN feedback (missing from our model) are expected to be important in determining the SFR. Hence to normalize our CDFs, we chose the largest redshift at which our sample is reasonably large (see below). It is important to note however that our main results are based on the z > 4 CDFs, which unlike the predictions for the absolute numbers of bursts, do not depend on our choice of normalization.", "pages": [ 4, 5 ] }, { "title": "4 GRB SAMPLE", "content": "Our LGRB data is taken from Robertson & Ellis (2012), corresponding to a compilation from the samples presented in Butler et al. (2007); Perley et al. (2009); Butler et al. (2010); Sakamoto et al. (2011); Greiner et al. (2011), and Kruhler et al. (2011). It includes only GRBs before the end of the Second Swift BAT GRB Catalog, and is comprised of 152 long GRBs with redshift measurements. It is important to consider the completeness of the sample. Several efforts have been made to construct a redshift-complete GRB sample (e.g., Greiner et al. 2011; Salvaterra et al. 2012). However, to do so, many GRBs with measured redshifts are excluded. Such requirements are even more severe for highz bursts, which makes them of little use for our purposes. To explore the dependence of our results on a possible bias in © c 56 the GRB redshift distribution, we construct two samples: (S1) we use a luminosity function based on the z < 4 subsample (consisting of 136 GRBs); and (S2) we use a subsample with isotropic-equivalent luminosities bright enough to be observable up to high redshifts (comprised of 38 bursts). The two samples are summarized in table 1. Since there is a degeneracy between a biased SFR-GRB relation and a redshift-dependent LF, we implicitly assume that any unknown bias will be subsumed in the value of the α parameter.", "pages": [ 5 ] }, { "title": "4.1 Luminosity function sample (S1)", "content": "The number of GRBs detectable by any given instrument depends on the specific flux sensitivity threshold and the intrinsic isotropic LF of the GRBs. In figure 3, we show the distribution of log-luminosities for z < 4 GRBs, which can be well described by a normal distribution The values L = log L iso / erg s -1 ; L ∗ = log 10 51 . 16 , σ L = 1 . 06 and p ∗ = 1 . 26 are estimated by maximum likelihood optimization. The luminosity threshold is given by where d L is the luminosity distance. Consistent with previous works (e.g. Li 2008), we set a bolometric energy flux limit F lim = 1 . 15 × 10 -8 erg cm -2 s -1 for Swift by using the smallest luminosity of the sample. Due to Malmquist bias, our fitted LF is likely biased towards high luminosities. To minimize this problem and increase the sample completeness, we fit our LF using only z /lessorequalslant 4 GRBs, which we show in Fig. 3. We reiterate that the Malmquist bias only serves to make our results even more conservative by predicting a flatter redshift distribution of GRBs. placements", "pages": [ 5, 6 ] }, { "title": "4.2 Luminosity-limited sample (S2)", "content": "Another approach, less dependent on the LF parametrization and the Malmquist bias, is to construct a luminositylimited subsample of the observed bursts bright enough to be seen at the highest redshift of interest. Assuming that the LF does not evolve with redshift, this subset would be proportional to the total number of bursts at any given redshift. In figure 4, we show the redshifts and isotropic luminosities of our entire sample. The dot-dashed blue line corresponds to the effective Swift detection threshold. For our luminosity-limited sample, we only use GRBs with isotropicequivalent luminosities L iso /greaterorequalslant 1 . 34 × 10 52 ergs s -1 , which comprise all GRBs observable up to z ∼ 9 . 4. Hereafter, all calculations will correspond to either the complete (LF derived) sample (S1), or the luminosity-limited sample (S2).", "pages": [ 6 ] }, { "title": "5 OBSERVATIONAL CONSTRAINTS", "content": "In this section, we test the WDM models by comparing the predicted absolute detection rates of bursts as well as the CDFs with the observed samples. We consider 3 different ranges of α : (i) a constant SFR-GRB relation, α = 0 (case 0, C0); (ii) -1 < α < 2 (case 1, C1); and (iii) a flat prior over -∞ < α < ∞ (case 2, C2) 8 . All cases are summarized in table 1.", "pages": [ 6 ] }, { "title": "5.1 Absolute detection rate of bursts", "content": "In tables 2-3, we present the absolute number of GRBs at high redshifts in CDM and WDM models with particle masses of 0.5-3.5 keV, as well as the actual number in our sample observed with Swift . All models are normalized to yield the observed number of bursts at 3 < z < 4, as described in equation (17) and the associated discussion. As expected, models with small WDM particle masses predict a rapidly decreasing GRB rate towards high redshifts. This exponential suppression can in some cases be partially compensated by an increasing GRB-to-SFR rate (i.e. α > 0). For the S1C1 case, models with m x /greaterorequalslant 2 . 5 keV show a good agreement with Swift observations for 0 < α < 1, though values of α ∼ 2 are a better fit to the observations at z > 8. For the case S2C1, all models with m x /greaterorequalslant 2 . 5 keV seem to be consistent with data for α ∼ 1 -2. In both cases, the two observed bursts in the interval 8 < z < 10, are already at odds with 1.5 keV < m x < 2.5 keV models. Finally, we see that models with m x /lessorequalslant 1 keV predict a dearth of GRBs at z > 6, which is inconsistent with current observations. Extreme models with m x ∼ 0 . 5 keV already fail at intermediate redshifts (4 < z < 6), even for values of α as high as two.", "pages": [ 6 ] }, { "title": "5.2 The redshift distribution of z > 4 bursts", "content": "Although the absolute rate of bursts is the simplest prediction, it is dependent on the normalization factor between the SFR and GRB rate at 3 < z < 4. Hence, for the remainder of the paper, we focus on comparing the theoretical and observed z > 4 CDFs. The CDFs are not dependent on normalization factors and are therefore more conservative and robust predictions. In figure 5 we plot the CDFs for CDM and WDM (under the assumption of α = 0), as well as the observed Swift distribution. The lighter the WDM particle, the sharper the CDF rise at lowz . There is a clear separation between CDM and WDM models with m x /lessorsimilar 1 . 5 keV. Both the S1 and S2 samples (top and bottom panels respectively) show the same qualitative trends. As we saw above, the highz suppression of structures in WDMmodels can be compensated for by allowing the GRB rate/SFR to increase towards higher redshifts. How degenerate are these cosmological vs astrophysical effects? In figure 6, we show the CDF for m x = 0 . 5 keV for several values of α for S2 sample. The exponential suppression of DM halo abundances in this model is so strong, that an unrealistically high value of α ∼ 15 is required to be roughly consistent with observations. Such a high value is ruled out by low-redshift observations, which imply α /lessorsimilar 1 (e.g. Robertson & Ellis 2012; Kistler et al. 2009; Trenti et al. 2012).", "pages": [ 6 ] }, { "title": "5.3 Constraints from the redshift distribution of z > 4 bursts", "content": "To quantify how consistent are these CDFs with the observed distribution from Swift , we make use of two statistics: (i) the one-sample Kolmogorov-Smirnov (K-S) test; and (ii) a maximum likelihood estimation (MLE). Both tests are described in detail in appendix A. The K-S test provides a simple estimate of the probability the observed distribution was drawn from the underlying theoretical one. We compute this probability, for fixed α first, for our models S1C0, S1C1, S2C0 and S2C1. Consistent with the more qualitative analysis from the previous section, models with m x /lessorsimilar 1 . 0 keV are ruled out at 90% CL assuming -1 /lessorequalslant α /lessorequalslant 1. For α = 0 (S1C0 and S2C0), the limits are even more restrictive and models with m x /lessorsimilar 1 . 5 keV are ruled out at 90% CL for both samples. So far, we have analyzed each model individually in order to quantify a lower limit on m x , given a single value of α . Using a χ 2 MLE (see appendix A) allows us to compute posterior probabilities given conservative priors on α . Thus we are able to construct confidence limits in the twodimensional, ( m x , α ) parameter space. The results for cases S1C2, S2C2 are shown in figure 7 at 68%, 95%, 99% CL. Both samples show the same qualitative trends, with the data preferring higher values of m x and CDM. Marginalizing the likelihood over -3 /lessorequalslant α /lessorequalslant 12, with a flat prior, shows that models with m x /lessorequalslant 1 . 6 -1 . 8 keV are ruled out at 95% CL for S1C2 and S2C2 respectively.", "pages": [ 7 ] }, { "title": "6 FUTURE CONSTRAINTS", "content": "In the previous section, we have quantified the constraints on WDM particle masses using current Swift GRB obser- vations. We obtain constraints of m x /greaterorsimilar 1 . 6-1.8 keV. We now ask how much could these constraints could improve with a larger GRB sample, available from future missions? As a reference, we use the Sino-French space-based multiband astronomical variable objects monitor (SVOM) 9 mission. The SVOM has been designed to optimize the synergy between space and ground instruments. It is forecast to observe ∼ 70 -90 GRBs yr -1 and ∼ 2 -6 GRB yr -1 at z /greaterorequalslant 6 (see e.g., Salvaterra et al. 2008). We first construct a mock GRB dataset of 450 bursts with redshifts obtained by sampling the CDM, α = 0 PDF given by equation (13). This sample size represents an optimistic prediction for 5 yrs of SVOM observations 10 (see e.g., Salvaterra et al. 2008). We then perform the MLE analysis detailed above on this mock dataset at z > 4. The resulting confidence limits are presented in Fig. 8. This figure shows that ∼ 5 yrs of SVOM observations would be sufficient to rule out m x /lessorequalslant 2 . 3 keV models (from our fiducial CDM, α = 0 model) at 95% CL, when marginalized over α . This is a modest improvement over our current constraints using Swift observations. As already foreshadowed by figures 1 and 2, as well as the associated discussion, it is increasingly difficult to push constraints beyond m x > 2 keV. On the other hand, the α constraint improves dramatically due to having enough highz bursts to beat the Poisson errors. We caution that the relative narrowness around α = 0 of the contours in Fig. 8 is also partially due to our choice of (CDM, α = 0) as the template for the mock observation. placements placements ass (keV) ass (keV) z", "pages": [ 7, 8 ] }, { "title": "7 CONCLUSION", "content": "Small-scale structures are strongly suppressed in WDM cosmologies. WDM particle masses of m x ∼ keV have been invoked in order to interpret observations of local dwarf galaxies and galactic cores. The high-redshift Universe is a powerful testbed for these cosmologies, since the mere presence of collapsed structures can set strong lower limits on m x . GRBs, being extremely bright and observable to well within the first billion years, are a promising tool for such studies. Here we model the collapsed fraction and cosmic SFR in CDM and WDM cosmologies, taking into account the 1 . 5 2 . 0 2 . 5 3 . 0 effects of both free-streaming and effective pressure due to the residual velocity dispersion of WDM particles. Assuming that the GRB rate is proportional to the SFR, we interpret 5 years of Swift observations in order to place constraints on m x . We conservatively account for astrophysical uncertainty by allowing the GRB rate/SFR to evolve with redshift as ∝ (1 + z ) α . In order to fold completeness limits into our analysis, we used a low-z sample to estimate the intrinsic LF, or else restricted our analysis to a luminosity-limited subsample detectable at all redshifts. For each model ( m x , α ), we compute both the absolute detection rates and CDFs, at z > 4. A K-S test between the model and observed CDFs rules out m x < 1 . 5 (1.0) keV, assuming α = 0 ( < 2), at 90% CL. Using a maximum likelihood estimator, we are able to marginalize over α . Assuming a flat prior in α , we constrain m x > 1 . 6-1.8 keV at 95% CL. A future SVOM-like mission would tighten these constraints to m x /greaterorsimilar 2 . 3 keV. The strong and robust constraints we derive show that GRBs are a powerful probe of the early Universe. Their utility would be further enhanced with insights into their formation environments and their relation to the cosmic SFR.", "pages": [ 8 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank Emille Ishida for the careful and fruitful revision of the draft of this work and Andressa Jendreieck for useful comments. RSS thanks the Max Planck Institute for Astrophysics (Garching, Germany) for its hospitality during his visit. Mass (keV) placements 5 10 15 20 placements 5 10 15 20 PSfrag replacements 5 10 15 20 Boyarsky A., Lesgourgues J., Ruchayskiy O., Viel M., 2009, JCAP, 5, 12 Boyarsky A., Ruchayskiy O., Shaposhnikov M., 2009, Annual Review of Nuclear and Particle Science, 59, 191 Boylan-Kolchin M., Bullock J. 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The null hypothesis that the observed GRB redshifts are consistent with a model distribution can be evaluated by estimating a p-value , which corresponds to one minus the probability that the null hypothesis can be rejected. The K-S test consists in comparing the statistical parameter where F ( z ) and G ( z ) are the CDF for the theoretical and observed sample and sup is the the supremum of a totally or partially ordered set. We estimate the p-value via nonparametric bootstrap, which consists of running Monte Carlo realizations of the observed CDF using a random-selectionwith-replacement procedure estimated from the data. This provides a histogram of the statistic D, from which a valid goodness-of-fit probability can be evaluated. 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[ { "title": "ABSTRACT", "content": "We have analysed archival data taken with the Australia Telescope Compact Array (ATCA) during 2001-2003 and detected nine new interstellar and circumstellar H 2 O masers in the LMC. This takes the total number of star formation H 2 O masers in the LMCto 23, spread over 14 different star forming regions and three evolved stars. Three H 2 O maser sources (N105a/MC23, N113/MC24, N157a/MC74) have been detected in all the previous observations that targeted these sites, although all show significant variability on timescales of decades. The total number of independent H 2 O maser sources now known in the LMC means that through very long baseline interferometry astrometric measurements it will be possible to construct a more precise model of the galactic rotation of the LMC and its orbital motion around the Milky Way Galaxy. Key words: masers - stars:formation, mass-loss - Magellanic Clouds.", "pages": [ 1 ] }, { "title": "H. Imai 1 , 2 /star , Y. Katayama 1 , S. P. Ellingsen 3 and Y. Hagiwara 4 , 5", "content": "1 Department of Physics and Astronomy, Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima 890-0065, Japan International Centre for Radio Astronomy Research, M468, The University of Western Australia, 35 Stirling Hwy, Crawley, Western Australia, 6009, Australia 2-21-1, Osawa Mitaka, 181-8588 Tokyo, Japan Accepted 2013 February 20. Received 2013 February 19; in original form 2013 February 13", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "There are more than a thousand H 2 O maser sources currently known in the Milky Way (MW) galaxy (e.g., Valdettaro et al. 2001; Walsh et al. 2011). In contrast, the total number of maser sources detected in all species in the Large and Small Magellanic Clouds (LMC, SMC) is around 20 (see Ellingen et al. 2010, hereafter EBCQF10, and references therein). The maser sources in the LMC and the SMC are important objects for addressing a range of astrophysical questions. H 2 O masers are typically associated with outflow activity from young stellar objects (YSOs) and the final copious stellar mass loss phase of red giant and supergiant stars. All of the H 2 O masers found in the Magellanic Clouds (MCs) to date are associated with either massive YSOs or red supergiants, both of which trace the present sites of star formation in these galaxies. Taking into account the relatively low metallicity of the MCs compared to the MW, some differences are expected in the dominant physical mechanisms and the timescales that govern the process of star formation and stellar mass loss. Very long baseline interferometric (VLBI) observations can resolve each maser source into a cluster (or clusters) of compact maser features, whose three-dimensional motions (line-of-sight velocities and proper motions) can be measured. With current instruments with which astrometric accuracy of 10 micro-arcseconds ( µ as) is achievable, such studies are possible for star forming regions in nearby galaxies at distances of up to 1 Mpc (e.g. Brunthaler et al. 2005). Measuring the motions of the maser sources with respect to nearby (in terms of angular separation) quasars, it is possible to determine both a space motion and a trigonometric parallax distance (given sufficient astrometric accuracy) of the maser source. Within the MW on kilo-parsec scales, H 2 O maser sources are confined to the MW thin disk and have been used to derive fundamental Galactic structure parameters (Reid et al. 2009; Honma et al. 2012). The LMC and SMC are at distances of approximately 50 and 60 kpc respectively (e.g., Cioni et al. 2000), so the amplitude of the trigonometric parallax is approximately 20 µ as. This is too small to enable accurate parallax distance determinations with current instruments and techniques (e.g. Reid et al. 2009), but may be feasible through statistical approaches, or with future space-based VLBI missions. mine the space motion of maser sources within the MCs by measuring and correcting for the intrinsic internal motions of maser features in the source. The locations and past orbits of the MCs are important and controversial topics in studies of the star formation history and process of interactions of the galaxies in the MW-MC system (e.g., Diaz & Bekki 2012). Astrometric observations using data from groundbased optical telescopes and the Hubble Space Telescope have been used to measure the proper motions of the MCs (Piatek et al. 2008; Vieira et al. 2010; Kallivayalil et al. 2013). However, there exist non-negligible discrepancies among the results. The optical astrometry results are affected by the inclusion of stars from a variety of populations with different dynamical characteristics located along the same lines of sight. They are also affected by the galactic rotation model adopted (e.g., Haschke, Grebel & Duffau 2012), which is required to extract the centre-of-mass space motions of the galaxies from the observed proper motions. In order to analyse the 3-D kinematics of the individual H 2 O maser sources in the MCs and to derive the dynamical parameters of these galaxies, efforts to increase the number of identified H 2 O masers in these galaxies are important. Statistical analysis of the 3-D kinematics of the maser sources will enable us to compare the kinematic properties of the MCs with those of the MW. In order to construct dynamical models of the MCs, each galactic model requires the present centre-of-mass space motion, the velocity field of the galactic rotation and the distance to the galaxy as free parameters in the model fitting. A large number of maser sources ( /greaterorsimilar 20) are required to make such an analysis feasible. In this paper, we present H 2 O masers in the LMC detected with the Australia Telescope Compact Array (ATCA), including nine newly discovered since the work of EBCQF10.", "pages": [ 1, 2 ] }, { "title": "2 ANALYSIS OF ATCA ARCHIVAL DATA", "content": "We analysed archival observations of H 2 O maser sources in the MCs taken in ATCA observations made on 2001 January 7 (program C901), and in the period of 2002 February-2003 May (C973) 1 . Table 1 summarises the observing strategy and correlator setup used for the observations. In each of the observing sessions all six 22-m antennas were used in the array, but for some observations only three baselines were 1 The C973 observations also targeted sources in the SMC, which we also reduced. However, the SMC data have been independently analysed and published by Breen et al. (2013) and in this paper we focus only on the sources in the LMC. useable. Data reduction was undertaken using the MIRIAD package, following the standard procedures for ATCA data. The visibility amplitude and phases were calibrated by referencing them to observations of flux density and phase calibrator sources. Phase calibration solutions were obtained from the scans on the phase calibrators, J045005.4 -810102 and J050644.0 -610941, one of which was observed every 1020 min. Although there was a difference in the channel resolution for the correlator outputs in the different sessions, all data were smoothed to the achieve the same velocity resolution of ∼ 0.5 km s -1 . We then constructed synthesis images for each source to search for H 2 O masers and to check the image fidelity. For sources where a maser detection was confirmed and there was sufficient coverage of the uv-plane (mainly in the C973 2003-June session), we attempted to obtain the spectrum and the coordinates of the maser emission from the image cube. Even for the sources with sufficient uv-coverage, there existed significant side lobes, making the astrometry difficult. The angular resolution of the ATCA observations was typically ∼ 0 '' .5 (corresponding to a linear resolution of ∼ 0.1 pc at the distance of the LMC), within which most velocity components of a single maser source will be contained. Because we are mainly interested in the number of independent maser sources rather than the internal structures of the individual maser sources in this paper, we have only determined the coordinates of the brightest velocity components of the maser sources. The uncertainty in the measured maser positions is 1-2 '' in the cases where we could not uniquely determine the brightest point of the maser source due to the high side lobe levels. Columns 2 and 3 of Table 2 give the coordinates of the detected maser sources where we were able to determine them.", "pages": [ 2 ] }, { "title": "3 RESULTS", "content": "A total of twelve 22-GHz H 2 O masers were detected towards the LMC in the C901 and C973 ATCA observations, eight and one of which are newly discovered interstellar and circumstellar sources, respectively. Table 2 lists the sixteen interstellar H 2 O masers in the LMC known to date (see also the latest review of maser source surveys towards the MCs in van Loon 2012). Table 3 gives the parameters of the circumstellar maser newly detected. Figure 1 shows the ATCA cross-power spectra of the H 2 O masers in the LMC. The stellar source O-AGB 815 (2MASS J05351409 -6743558) corresponds to an M4-type variable star (HV 1001) with a K -magnitude of 8.14. Its H 2 O maser is apparently weak ( ∼ 0.2 Jy) and the maser luminosity is comparable to or lower than that seen in supergiants in the MW (e.g., VY CMa with an H 2 O maser flux density of up to 1000 Jy at a distance of 1.1 kpc, Choi et al. 2008). The eight newly detected interstellar H 2 O masers (those where Column 7 of Table 2 contains only the reference 4) are IRASF04521 -6928, HII 1107, HII 1186, N113a (in N113/MC24), IRAS 05202 -6655, NGC 1984/OH 279.6 -32.6, N214b and N180. In some regions, H 2 O maser sources were spread over an area of order 1 ' . Taking into account the position uncertainty, we regarded maser emission located within a radius of 1 '' -2 '' as a single source and they are shown in a single line in Table 2. There are fifteen independent H 2 O maser sources that are separated from each other by more than 1 ' (15 pc at the distance of the LMC). Some bright H 2 Omaser sources were first detected more than 25 years ago (e.g., Scalise Jr. & Braz 1981). However, even for regions where there has been persistent maser emission over a long period, the sources have exhibited large flux density variability and sometimes drop below the detection threshold. For example, in the brightest maser source N113/MC24, the peak velocity of the brightest maser component has dramatically changed and the strongest spectral feature observed in this paper (observation in 2003 May) was not detected in the observations of EBCQF10 (2008 August). As a result, apparent position shifts ( > 1 '' or > 0.25 pc) may occur due to the maser sources with such flux variability in the different locations (a Christmas tree effect). Clustering of H 2 O maser sources on parsec scales is observed in some Galactic star formation regions (e.g., W51 North and Main; W3(OH) and W3 IRS5). Therefore, the observed position shifts may be due to variation in the relative flux density between maser sources within a cluster, although we cannot rule out the possibility that these shifts occur just due to the relative positional errors of the observations made at different epochs. N113/MC24 and N113a present one example of a cluster of H 2 O maser sources (Lazendic et al. 2002; Oliveira et al. 2006; EBCQF10). We looked for corresponding objects within 10 '' of the maser sources and these are listed in Column 9 of Table 2. Only the nearest sources from the maser position are listed. The corresponding 2MASS sources are located within 3 '' of the masers. Note that IRAS 05202 -6655 seems to be associated with an extended infrared source (2MASX J05201653 -6652544) and its population is unclear. Proximity to sources listed by Gruendl & Chu (2009) may be a good indication of association with a YSO.", "pages": [ 2, 3, 4 ] }, { "title": "4 DISCUSSION", "content": "Here we estimate possible detections of H 2 O masers in the LMC using an H 2 O maser luminosity function (H 2 O-LF). Greenhill et al. (1990) derived the H 2 O-LF for the MW, which will be improved by further unbiased H 2 O maser surveys in the MW (e.g., Walsh et al. 2011). Similar to the approach used by Darling (2011) to estimate the expected number of H 2 O masers in M31, we need to rescale the H 2 OLF using the relative star formation rates of the MW and the LMC ( ∼ 4 and ∼ 0 . 4 M /circledot yr -1 , respectively, Diehl et al. 2006; Skibba et al. 2012). The shortest on-source integration time for the present data was ∼ 3 min. With a velocity resolution of 0.5 km s -1 , the corresponding detection limit for the peak of any H 2 O maser emission is estimated to be 1 Jy. Adopting a distance to the LMC of 51 kpc (corresponding to a distance modulus of 18.55, e.g., Cioni et al. 2000; Haschke, Grebel & Duffau 2012) our detection limit corresponds to an isotropic H 2 O maser luminosity of 2 . 4 × 10 -5 L /circledot . We calculate the expected number of H 2 O masers with the luminosities higher than this value in the LMC to be ∼ 15. This is in good agreement with the actual number of the identified sources over 1 Jy (16 sources in Table 2). This supports the argument that the number of H 2 O maser detections in the LMC is consistent with MW expectations when the difference in the star formation rates between the LMC and the MW are taken into account (Green et al. 2008; EBCQF10, and references therein). Although the same approach when applied to M31 results in an underestimation of the expected number of detections (prediction of ∼ 3 compared to the 5 detected, Darling 2011), the difference is less than a factor of 2 and within the estimated uncertainty of this approach. EBCQF10 used the LMC YSO catalogue of Gruendl & Chu (2009) to investigate the infrared properties and spectral energy distribution (SED) of star formation regions with an associated H 2 O maser and compared them to the entire sample. They showed that H 2 O-associated YSOs have higher central mass, larger envelope radius, higher ambient density and higher total luminosity than the general YSO population. EBCQF10 classified YSO within 2 '' of the H 2 O maser position as being associated. Applying the same criterion to the 8 new LMC H 2 O masers listed in Table 2, we find three (HII 1186, N214b & N180) have a definite YSO each within 2 '' (table 9 of Gruendl & Chu 2009). In contrast, for the 16 LMC interstellar masers investigated by EBCQF10, 11 were within 2 '' of a definite YSO identified by Gruendl & Chu (2009). The cause of this discrepancy is not clear, it may indicate that the newly detected LMC H 2 O maser sources are primarily associated with more evolved star formation regions than the sample of EBCQF10, which targeted all the CH 3 OH, OH and H 2 O masers known at that time. The three associations from the newly detected masers have similar SED properties to the maser-associated sources of EBCQF10. However, none of them are sources identified by EBCQF10 as a high-probability maser candidate, so future searches towards these sources may provide additional H 2 O masers in the LMC. Including the three circumstellar H 2 O masers associated with red supergiants (IRAS 04553 -6825, 05280 -6910, van Loon et al. 2001; HV 1001) as members of the young stellar population, there are 19 independent maser sources in the LMC. Even limited to 16 interstellar masers brighter than 0.5 Jy, through measurement of the proper motions of these masers (better than 100 µ as yr -1 in a time baseline of one year) it is possible to obtain 48 observables (the threedimensional velocity vector components for each source). A basic dynamical model of the LMC requires nine free parameters in the fitting: a source distance, a three-dimensional centre-of-mass space motion vector, the galactic rotation axis inclination and position angle and a third (or higher order) polynomial galactic rotation curve. Fig. 2 shows that the H 2 O masers are distributed throughout the LMC, mitigating the likelihood of bias in the model fitting which may occur if only a small number of independent locations were available. If the distance to the LMC and the axis of the galactic rotation are fixed in the model fitting (using previous accurate determinations), it will be possible to extract the velocity field of the LMC and more precisely determine the centre-of-mass space motion of the LMC from maser proper motion observations.", "pages": [ 4, 5 ] }, { "title": "5 CONCLUSION", "content": "With the current sample of H 2 O masers in LMC (19 in total) and the existence of position-reference quasars within a few degrees of arc of these masers (Imai et al. in preparation), VLBI astrometric observations to estimate the centreof-mass space motion vector for the LMC are feasible. They may provide an opportunity for independently checking the results of optical astrometry measurements. Furthermore, measurement of the trigonometric parallax of the LMC ( π ≈ 20 µ as ) may be possible. With a position measurement accuracy achieved by the state-of-the-art VLBI astrometry mentioned previously, an individual measurement of the parallax will yield only a marginal detection. However, with independent measurements from ∼ 10 regions, each with potentially a few detections of the maser spot parallax, it will be possible to statistically improve the reliability and precision of the parallax detection. In the same observations it will also be possible to investigate the spatio-kinematical structure for each H 2 O maser source and to study the physical properties of the star formation regions and the final stellar mass loss in the MCs.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Breen S. L., Lovell J. E. J., Ellingsen S. P., et al. , MNRAS, in preparation Brunthaler, A., Reid, M. J., Falcke, H., Greenhill, L. J., Henkel, C., 2005, Science, 307, 1440 Choi Y.-K., et al. , 2008, PASJ, 60, 1007 Cioni M.-R., van der Marel R. 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[ { "title": "ABSTRACT", "content": "The recently published GALFA-HI Compact Cloud Catalogue lists 20 neutral hydrogen clouds that might pinpoint previously undiscovered high-latitude dwarf galaxies. Detection of an associated gamma-ray dark matter signal could provide a route to distinguish unambiguously between truly dark matter dominated systems that have accumulated neutral hydrogen but have not successfully ignited star formation and pure gaseous structures devoid of dark matter. We use 4.3 years of Fermi observations to derive gamma-ray flux upper limits in the 1-300 GeV energy range for the sample. Limits on gamma rays from pair annihilation of dark matter are also presented depending on the yet unknown astrophysical factors. Key words: (cosmology:) dark matter - gamma-rays: observations - Galaxy: halo Galaxy: structure", "pages": [ 1 ] }, { "title": "N. Mirabal 1 , 2 /star", "content": "1 Ram'on y Cajal Fellow 2 Dpto. de F'ısica At'omica, Molecular y Nuclear, Universidad Complutense de Madrid, Spain", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The number of dark matter subhalos surrounding the Milky Way has puzzled dark matter aficionados for over more than a decade. While cosmic large-scale structures are well described by the Lambda Cold Dark Matter (ΛCDM) cosmological model, a number of discrepancies exist between the standard theory of galaxy formation and observations of substructures at smaller scales (Frenk & White 2012). In particular, numerical ΛCDM simulations consistently predict that galaxies must be surrounded by a huge population of subhalos (Klypin et al. 1999; Moore et al. 1999). Intuitively, subhalos would encompass anything from the largest satellites ( e.g. dwarf galaxies) to substructures with masses around 10 -4 M /circledot (Loeb & Zaldarriaga 2005). Unfortunately, completing a survey of the smallest gravitationally bound systems is not straightforward (Ando et al. 2008). At faint flux levels, it becomes ever more difficult to recognise sparsely populated stellar systems. Important progress has been made recently resulting from newly discovered ultrafaint dwarf galaxies, which have nearly doubled the number of known dwarfs (Willman et al. 2005; Belokurov et al. 2007). Yet the dwarf counts in the Milky Way remains far to small compared to the number of subhalos predicted by simulations. Interestingly, the discovery of compact hydrogen clouds potentially located in the Galactic halo has revived the possibility of a larger population of galaxy candidates that could be traced by neutral hydrogen (Lockman 2002; Ryan-Weber et al. 2008). This point was made even earlier by Klypin et al. (1999) who recognised that the much broader and physically distinct set of enigmatic HI structures commonly referred to as high-velocity clouds (HVCs) could represent the missing dark matter subhalo population (Oort 1970; Bregman 1980; Braun & Burton 1999; Blitz et al. 1999). Although intriguing, to date, a firm HI cloud-subhalo link has not been established (Quilis & Moore 2008). Prompted by the recent release of the GALFA HI Compact Cloud Catalogue (Saul et al. 2012), we revisit the possibility that some compact hydrogen clouds can be used as a proxy for missing dark matter subhalos. There are inherent problems when trying to test this possibility, especially the lack of distance and mass information for such systems. Another major hurdle is the absence of associated stellar populations in most compact hydrogen clouds. In fact, it is even possible that many of these systems never formed stars (Ricotti 2009). Lewis et al. (2000) proposed exploiting 'pixel gravitational lensing' as a way to map the dark matter content in hydrogen clouds. However, the shortage of properly aligned nearby background galaxies prevents a generalised application to a large sample. For this reason, we explore a different approach to determine their dark matter content. If compact hydrogen clouds are dark matter dominated systems, nearby dense objects could potentially produce a detectable dark matter annihilation signal in gamma rays (Bergstrom et al. 1999; Baltz et al. 2000). In that vein, Flix et al. (2005) looked for spatial coincidences between unidentified EGRET sources and HVCs. With the vast improvements in gammaray sensitivity and angular resolution afforded by Fermi (Atwood et al. 2009), we can search for annihilating dark matter in GALFA-HI compact hydrogen clouds directly. This study should serve not only to investigate a possible tracer of gas-bearing dark matter seeds that never formed stars, but also to set a general upper bound on the annihilation cross section 〈 σv 〉 χ of a hypothesised weakly interacting massive particle (WIMP). Searches for dark matter signals using Fermi have been conducted unsuccessfully in a wide variety of astrophysical systems (Buckley & Hooper 2010; Ackermann et al. 2011; Ando & Nagai 2012; Mirabal et al. 2012). This work simply extends the hunt to a new sample. The paper is structured as follows. In Section 2 we explain the selection of high-latitude galaxy candidates from the GALFA-HI Compact Cloud Catalogue, as well as the Fermi LAT analysis. In Section 3 we derive gamma-ray flux upper limits for Segue 1 and set dark matter annihilation constraints for the galaxy candidates. Finally, we briefly summarise our interpretation and possible future directions in Section 4.", "pages": [ 1, 2 ] }, { "title": "2 GALAXY CANDIDATES AND FERMI LAT ANALYSIS", "content": "The GALFA-HI Compact Cloud Catalogue (Saul et al. 2012) is generated from the Galactic Arecibo L -Band Feed Array HI (GALFA-HI) Survey Data Release One (Peek et al. 2011). At completion, GALFA-HI will cover 13,000 deg 2 of the sky in the 1420 MHz hyperfine transition of hydrogen between V LSR = ± 650 km s -1 . Using a novel cloud detection algorithm, Saul et al. (2012) identified a total of 1964 compact ( < 20 ' ) hydrogen clouds in the initial 7520 deg 2 Data Release One. The catalogue breaks down the clouds into a scheme that includes high-velocity clouds (HVCs), galaxy candidates, cold low-velocity clouds (CLVC), warm low-velocity clouds, and warm positive lowvelocity clouds in the third Galactic quadrant. For our purposes, we are only concerned with the 27 possible galaxy candidates that might form the core sample of potentially undiscovered dark matter subhalos. In order to guard against possible gaseous disk interlopers that may have been pushed into the halo by stellar feedback (Ford et al. 2008), we also prune galaxy candidates at | b | /lessorequalslant 10 · . Finally, we are left with a subset of 20 high-latitude galaxy candidates. In Fig. 1, we show the distribution of these systems on the sky. In order to explore the gamma-ray emission, we use the publicly available dataset acquired by the Large Area Telescope (LAT) instrument on board the Fermi Gammaray Space Telescope (Atwood et al. 2009). The LAT is a pair-conversion gamma-ray detector sensitive to photon energies from 20 MeV to 300 GeV. We retrieve all photons of 'source' class ( evclass=2 ) within a 10 · circular region centred at the position of each galaxy candidate. The data analysed here were collected between 2008 August 4 and 2012 November 20 (approximately 4.3 years of data). Good time intervals were processed using the available v9r27p1 Fermi Science Tools with the standard P7SOURCE V6 instru- nt response function. Throughout, we apply a maximum zenith angle cut of 100 · . We further filter the data using the gtmktime filter expression recommended by the LAT team, namely '(DATA QUAL==1) && (LAT CONFIG==1) && ABS(ROCK ANGLE) < 52' The final analysis for each region includes all the point sources listed in the 2FGL catalogue (Nolan et al. 2012), the current Galactic diffuse emission model gal 2yearp7v6 v0.fits , and the extragalactic isotropic model iso p7v6source.txt . The resulting dataset is analysed with a binned likelihood method using the gtlike tool in the standard Fermi Science Tools 1 . For each position, we create a count map made up of 30 logarithmically uniform energy bins using gtbin . We next construct a binned exposure map with gtexpcube2 , and a model source and diffuse count map with gtsrcMaps . Fig. 2 shows a typical Fermi count map from the sample. In the absence of emission, flux upper limits are then derived using the implementation of LATAnalysisScripts 2 . This set of Python libraries unifies the pyLikelihood module included in the standard Fermi Science tools. Upper limits are computed with calcUpper assuming a power law spectrum of high-energy emission E -2 within a radius of 10 · . We restrict our analysis to photons in the 1-300 GeV energy range. The Fermi LAT point spread function (PSF) is typically 0 . 8 ( E/ 1GeV) -0 . 8 deg, which in our selected energy range restricts the photons to less than 1 · around each location (Geringer-Sameth & Koushiappas 2012). Table 1 summarises the 95% confidence level upper limits. Since the Fermi exposure is rather uniform over the sky the upper flux limits are fairly similar, except for a handful of locations with higher diffuse background emission or neighbouring bright gamma-ray sources.", "pages": [ 2, 3 ] }, { "title": "3 DARK MATTER CONSTRAINTS", "content": "If compact hydrogen clouds are highly dark matter dominated objects their gamma-ray emission should be well characterised by a differential spectrum that can be written as where 〈 σv 〉 χ is the thermally averaged annihilation cross section, m χ is the dark matter particle mass, and dN γ dE is the photon spectrum of annihilation products (Abdo et al. 2010). The second term, or the so-called astrophysical factor J (∆Ω), corresponds to the integration of the dark matter density squared ρ 2 ( l, Ω) along the line of sight l of the compact cloud, over a solid angle ∆Ω, so that In principle, we expect compact clouds to be virtually free of gamma-ray emission from embedded diffuse and individual point sources. As a result, it is reasonable to expect that the differential spectrum from dark matter annihilation should dominate any gamma-ray signal in the direction of observation to these systems. Formally, a robust computation of an upper limit on the annihilation cross section 〈 σv 〉 χ requires some knowledge of the astrophysical factor J (∆Ω) of the system under consideration. Given a lack of direct observational constraints on the astrophysical factors of these galaxy candidates J gc , we are only able to estimate 〈 σv 〉 χ bounds from our sample by tying them to potentially similar systems in the Milky Way. Under the Ansatz that they are strongly dark matter dominated, the newly discovered ultra-faint dwarf galaxies would appear to be the closest relatives to compact hydrogen clouds (Strigari et al. 2008). As shown by Simon et al. (2011), Segue 1 represents the darkest of these systems with a very high mass-to-light ratio ( ∼ 3400 M /circledot /L /circledot ) and dark matter density 2 . 5 +4 . 1 -1 . 9 M /circledot pc -3 . It also boasts the largest astrophysical factor for known dwarfs J Segue 1 = 10 19 ± 0 . 6 GeV 2 cm -5 (Essig et al. 2010). Hereafter, we derive upper limits on 〈 σv 〉 χ relative to the bounds already imposed for Segue 1 using Fermi measurements (Abdo et al. 2010; Essig et al. 2010; Scott et al. 2010; Geringer-Sameth & Koushiappas 2012). Accordingly, we repeat the previous Fermi LAT analysis now centred on Segue 1 ( /lscript, b ) = (220 . 5 · , 50 . 4 · ). The corresponding gammaray upper limit for Segue 1 is F lim Segue1 (1-300 GeV) = 3.5 × 10 -11 ph cm -2 s -1 . In order to turn our flux upper limits into a bound on 〈 σv 〉 χ , we need to assume a specific annihilation channel. We adopt the strictest possible limit on dark matter annihilation into bottom quarks b ¯ b for Segue 1 based on 3 years of Fermi data, which translates into 〈 σv 〉 χ /lessorsimilar 3 × 10 -26 cm 3 s -1 for particle masses m χ /lessorsimilar 40 GeV (Geringer-Sameth & Koushiappas 2012). Assuming the annihilation to be purely into b ¯ b and an average flux for the galaxy candidates 〈 F lim ( gc ) 〉 = 7 . 7 × 10 -11 ph cm -2 s -1 , we can approximate 〈 σv 〉 χ from these systems as for m χ /lessorsimilar 40 GeV WIMP masses annihilating to b ¯ b , depending on the yet unknown J gc .", "pages": [ 3, 4 ] }, { "title": "4 DISCUSSION AND FUTURE PROSPECTS", "content": "We have reported gamma-ray flux upper limits for a subset of galaxy candidates from the GALFA-HI Compact Cloud Catalogue. It is difficult to imagine any of these systems beating the astrophysical factors of known dwarf galaxies that spans the gamut from 4 × 10 17 GeV 2 cm -5 for Carina to 1 . 3 × 10 19 GeV 2 cm -5 for the ultra-faint dwarf galaxy Segue 1 (Essig et al. 2010). Without mass and distance constraints, it will be difficult to place compact clouds in the context of other measurements. It is generally assumed that in order to be self gravitating, compact clouds could pack masses as low as a few M /circledot to greater than ∼ 10 6 M /circledot for outer systems (Giovanelli et al. 2010; Saul et al. 2012). Our naive expectation is that J Segue 1 /J gc /greatermuch 1 should hold in most cases. As an illustration, the average compact cloud properties indicate a mass of ∼ 2 × 10 4 M /circledot and a physical size of ∼ 100 pc at distance of 100 kpc (Saul et al. 2012). Since the astrophysical factor is proportional to the density squared, we can roughly approximate J Segue 1 /J gc ≈ 10 3 for M Segue 1 ∼ 6 × 10 5 M /circledot (Simon et al. 2011). To be sure, arguments consistent with an interpretation that Segue 1 is a tidally disrupting star cluster contaminated by the Sagittarius stream should be definitively ruled out (Niederste-Ostholt et al. 2009). For the entire family of ultra-faint galaxies, it is also critical to derive more reliable estimates of the J -factor (Walker et al. 2011). This null result joins the ranks of past dark matter annihilation searches, which have failed to detect a gammaray signal in systems suspected of high dark matter content (Bringmann & Weniger 2012). Unfortunately, as with the rest of dark matter pursuits, this is an everything or nothing undertaking. Here, we are further hampered by the fact that we are seeking a detection with nearly no information about distances and masses of the objects involved. Nonetheless, Fermi will continue to collect data and stricter gamma-ray limits for these systems can be reached. Above E /greaterorsimilar 100 GeV, the Cherenkov Telescope Array (CTA) will be crucial to escalate the dark matter search to unprecedented bounds (CTA Consortium 2011; Doro et al. 2013). There is also sufficient motivation to explore signatures for other reasonable dark matter candidates at other wavelengths (Feng 2010). From our measurements, it is still unclear where GALFA-HI galaxy candidates fit in the larger dark matter subhalo picture. Searches for stellar counterparts associated with these gas-bearing systems are underway and should intensify (Saul et al. 2012). If this goal is accomplished, member stars could be used to produce a dark matter density profile. In an optimistic scenario, compact clouds could validate a sort of 'hiding in plain sight' model whereby dark matter subhalos at small scales would be traced directly through neutral hydrogen. Alternatively, dedicated observational studies could finally establish that all compact clouds are purely baryonic and hence devoid of dark matter (Plockinger & Hensler 2012). The disagreement between observations and ΛCDM simulations might then have to be invoke more inventive so-l ions (Boylan-Kolchin, Bullock & Kaplinghat 2012). Regardless, the potential dark matter fingerprint discussed here stands a possible diagnostic of suspected nearby dark matter dominated galactic candidates. With the advent of Skymapper (Keller et al. 2007) and the Large Synoptic Survey Telescope (Ivezi'c et al. 2008), it might be possible to trace the subhalo population directly using faint stars. A cross-match between stellar concentrations and compact hydrogen cloud positions should be conducted as soon as said surveys are completed.", "pages": [ 4 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank the referee for useful suggestions that improved the paper. N.M. acknowledges support from the Spanish government through a Ram'on y Cajal fellowship, the Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064, and the Spanish MINECO under project code FPA2010-22056-C06-06. We also acknowledge the use of public data from the Fermi data archives. Lastly, we thank Jeremy S. Perkins for contributing a wonderful set of Python libraries for routine Fermi analysis.", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "We revisit an analytical model to describe the halo-matter cross-power spectrum and the halo auto-power spectrum in the weakly nonlinear regime, by combining the perturbation theory (PT) for matter clustering, the local bias model, and the halo bias. Nonlinearities in the power spectra arise from the nonlinear clustering of matter as well as the nonlinear relation between the matter and halo density fields. By using the 'renormalization' approach, we express the nonlinear power spectra by a sum of the two contributions: the nonlinear matter power spectrum with the e ff ective linear bias parameter, and the higher-order PT spectra having the halo bias parameters as the coe ffi cients. The halo auto-power spectrum includes the residual shot noise contamination that needs to be treated as additional free parameter. The higher-order PT spectra and the residual shot noise cause a scale-dependent bias function relative to the nonlinear matter power spectrum in the weakly nonlinear regime. We show that the model predictions are in good agreement with the spectra measured from a suit of high-resolution N -body simulations up to k /similarequal 0 . 2 h Mpc -1 at z = 0 . 35, for di ff erent halo mass bins. Key words: galaxies: clusters: general - cosmology: theory - dark energy - large-scale structure of Universe", "pages": [ 1 ] }, { "title": "Atsushi J. Nishizawa /star , Masahiro Takada and Takahiro Nishimichi", "content": "Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo, Chiba 277-8582, Japan 12 June 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Clustering statistics of galaxies such as the two-point correlation function of galaxies or the Fourier-transformed-counterpart power spectrum are powerful tools to constrain cosmology. In particular, the baryon acoustic oscillation (BAO) experiment with wide-area galaxy redshift survey is recognized as a robust probe of cosmological distances. There are various on-going and planned galaxy surveys aimed at achieving high-precision BAO measurements over a wider range of redshifts: the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) 1 , Subaru Prime Focus Spectrograph (PFS) Survey 2 , and the ESA Euclid satellite experiment 3 . The BAO scale is one particular length scale measured from the pattern of galaxy clustering. Much more significant signal-tonoise ratios are inherent in the broad-band shape and amplitude information of the galaxy power spectrum at BAO scales. However, to reliably use the amplitude information, we need to resolve various systematic uncertainties in the weakly nonlinear regime: the nonlinear clustering e ff ect and galaxy bias uncertainty. There are promising developments towards a more accurate modeling of the nonlinear clustering of matter based on N -body simulations (Springel et al. 2005; Angulo et al. 2008; Takahashi et al. 2009; Nishimichi et al. 2009; Valageas & Nishimichi 2011) as well as perturbation theory (PT) of structure formation (Juszkiewicz 1981; Vishniac 1983; Makino et al. 1992; Jain & Bertschinger 1994; Jeong & Komatsu 2006; Crocce & Scoccimarro 2006; Matsubara 2008a; Taruya & Hiramatsu 2008; Saito et al. 2008; Nishimichi et al. 2007). The galaxy bias uncertainty is a harder problem, because physical processes involved in galaxy formation / evolution are highly nonlinear and still very challenging to model from the first principles. Hence a practical approach often used assumes an empirical parametrization of galaxy bias; the peak bias model (Kaiser 1984; Mo&White 1996; Sheth & Tormen 1999; Schmidt et al. 2012) and the local bias model assuming a 'local' mapping relation between galaxy and matter distributions at each spatial position (Coles 1993; Fry & Gaztanaga 1993; Scherrer & Weinberg 1998; Schmidt et al. 2012). In the peak bias model, a galaxy or more precisely halo is assumed to form at or around the peak of the initial matter density field, where a typical scale of the peaks corresponds to scales of the halo that host galaxies at low redshifts (although one halo can contain several galaxies inside). Thus the distribution of halos are by nature biased relative to the underlying matter distribution (Kaiser 1984), because only the density peaks can be places to have halos today, while the under-density regions or the density minima are very di ffi cult (or impossible) to form halos. Furthermore, the longwavelength perturbation mode in the initial density field causes modulations in the heights of the small-scale peaks, and in turn alter subsequent formation of halos at low redshifts. The amount of halo bias can depend on the long-wavelength modes as a result of mode coupling in the weakly nonlinear regime relevant for BAO scales. The halo bias or peak bias models have been studied in the literature (Mo & White 1996; Sheth & Tormen 1999; Desjacques et al. 2011; Scoccimarro et al. 2012; Schmidt et al. 2012). The nonlinear e ff ect on the galaxy power spectrum arises from two e ff ects: the nonlinear clustering of matter (mostly dark matter) and the nonlinear bias relation between the galaxy and dark matter distributions. At BAO scales in the weakly nonlinear regime, we expect that the nonlinear clustering of galaxies can be accurately modeled by incorporating the PT of structure formation, the local bias model and / or the peak-background split bias model (halo bias). Such an attempt was first made in (Heavens et al. 1998), and followed by various works (McDonald 2006; Matsubara 2008a; Jeong &Komatsu 2009; Saito et al. 2009; McDonald & Roy 2009; Manera et al. 2010; Baldauf et al. 2010a; Pollack et al. 2012; Sato & Matsubara 2011; Chan et al. 2012; Baldauf et al. 2012), which continuously show an improved understanding of the nonlinear galaxy power spectrum. In this paper, we revisit the method of modeling the nonlinear power spectra of halos, more explicitly halo-matter and halo-halo power spectra, by incorporating the PT, the local bias model and the halo bias. In doing this, we employ 'renormalization approach' developed in McDonald (2006) in order to re-sum contributions of the nonlinear matter clustering up to the higher-order loop corrections. This yields the term expressed by the product of the 'full' nonlinear matter power spectrum and the renormalized linear bias parameter. The remaining terms, for which we keep the one-loop correction order based on the standard PT, give the e ff ect of scaledependent halo bias in the halo power spectrum. Thus our approach fully utilizes the recent improvement in modelling the nonlinear matter power spectrum (in this paper we will use the refined PT prediction developed in Taruya et al. (2009)). We test the accuracy of our model predictions by comparing with the halo spectra measured from high-resolution N -body simulations done in Nishimichi &Taruya (2011). This paper is organized as follows. In Sec. 2, we develop a method of modeling the nonlinear halo-matter and halo-halo power spectra by incorporating the PT, the local bias model and the halo bias. In this section, we also show the detailed comparison of the model predictions with the simulation results for the halo catalogs of di ff erent mass bins. Throughout this paper we assume the Λ dominated, cold dark matter model ( Λ CDM) as for our fiducial model which is consistent with Komatsu et al. (2011): the density parameters of matter, baryon and the cosmological constant are Ω m0 = 0 . 279, Ω b0 / Ω m0 = 0 . 165, and ΩΛ = 0 . 721 (i.e. flat geometry), the Hubble parameter h = 0 . 701, the tilt of the primordial power spectrum ns = 0 . 96 and the power spectrum normalization σ 8 = 0 . 817.", "pages": [ 1, 2 ] }, { "title": "2 PRELIMINARIES: PERTURBATION THEORY AND HALO BIAS", "content": "Our model is based on three ingredients; the perturbation theory (PT) of structure formation (Juszkiewicz 1981; Vishniac 1983; Fry 1984; Goro ff et al. 1986; Suto & Sasaki 1991; Jain & Bertschinger 1994), the local bias model (Coles 1993; Fry & Gaztanaga 1993; Scherrer & Weinberg 1998) and the halo bias model (Mo & White 1996; Sheth & Tormen 1999) (also see Cooray & Sheth 2002; Bernardeau et al. 2002, for thorough reviews). In this section, we briefly review the PT and the halo bias we will employ in the following sections.", "pages": [ 2 ] }, { "title": "2.1 Standard Perturbation Theory", "content": "Throughout this paper, we consider a pressure-less, irrotational fluid system and assume cold dark matter as the dominant fluid component to drive gravitational instability of structure formation. The nonlinear dynamics in an expanding universe is fully characterized by the density fluctuation field, δ m , and the peculiar velocity field θ m (Bernardeau et al. 2002). Given the initial conditions, the time evolutions of the fields are governed by the continuity equation, the equation of motion and the Poisson equation. By using the standard PT, we can solve the nonlinear dynamics. The density fluctuation field at a given redshift z is expanded as The PT solution for the n -th order density fluctuation field can be found to be   where δ m (1) is the linear density field today, D + ( z ) is the linear growth rate normalized as D + ( z = 0) = 1, and δ 3 D ( k ) is the Dirac delta function. The n -th order density fluctuation field has the amplitude of the order O [( δ m (1)) n ]. The growth rate can be computed, e.g. by solving Eq. (7) in Oguri & Takada (2011). The Fourier kernel Fn ( q 1 , · · · , q n ) describes a coupling between di ff erent Fourier modes due to nonlinear clustering, and we will use the expression in Eq. (10a) of Jain & Bertschinger (1994). Note that, although the form of the Fourier kernel is exact only for an Einstein de-Sitter model with Ω m = 1, it was shown to be a good approximation of the exact solution for a Λ CDMmodel.", "pages": [ 2 ] }, { "title": "2.2 Halo Mass Function and Halo Bias", "content": "Dark matter halos that host galaxies and / or galaxy clusters are useful tracers of large-scale structure, and can be used to infer the underlying dark matter distribution. However, the halo and dark matter distributions are not the same, which leaves an uncertainty, the so-called bias uncertainty. In this paper we employ the halo bias model developed in Mo & White (1996); Sheth & Tormen (1999) (also see Cooray & Sheth 2002). Let us start with defining the halo mass function n ( M , z ) dM , which gives the comoving number density of halos in the mass range [ M , M + dM ] and at redshift z . We employ the mass function given in Sheth & Tormen (1999): where ¯ ρ m 0 is the mean mass density today; ν ≡ [ δ c / D + ( z ) σ m ( M )] 2 ; δ c is the threshold over-density for spherical collapse model; σ m ( M ) is the present-day rms fluctuations in the mass density tophat smoothed over scale R = (3 M / 4 π ¯ ρ m 0) 1 / 3 . We will throughout this paper employ the coe ffi cients a = 0 . 75 and p = 0 . 3, which are obtained by comparing the fitting formula to N -body simulations. Note that the normalization coe ffi cient A is determined so as to satisfy the normalization condition ∞ 0 d ν f ( ν ) = 1. ∫ The mass function above holds only in an ensemble average sense, i.e. the average of the halo distribution over a su ffi ciently large volume. In other words, the number density of halos in a finite volume is modulated according to fluctuations of the underlying matter distribution within the volume, δ m . Employing the local bias model for halos, we assume that the halo distribution at a given spatial position x is locally related to the underlying matter distribution at the position x as where F is the functional to govern the local mapping relation. In the second line of the r.h.s., we have Taylor-expanded the relation in terms of δ m ( x ), and F ( n ) denotes the n -th derivatives of F with respect to δ m . Exactly speaking, as stressed in Schmidt et al. (2012), the local bias relation would hold to a good approximation in a 'peak-background split' picture (also see Mo & White 1996; Sheth & Tormen 1999, for the pioneer work). In the peakbackground model, the matter density field is divided into long- and short-wavelength modes, which correspond to 'background' and 'peak' density modes, respectively. The short-wavelength modes are at the scales responsible for formation of halos corresponding to about 10Mpc at maximum in the initial density fields and therefore are well below BAO scales (up to k ∼ a few 0.1 h / Mpc -1 in wavenumber). The long-wavelength modes are a 'coarse-grained' field responsible for a modulation of the peak heights of shortwavelengths, and in this paper we assume that the long-wavelength modes are at BAO scales. Hence we assume that δ m ( x ) in Eq. (4) is the coarse-grained field, even though we did not explicitly denote a notation to express the smoothing nature of δ m ( x ). The term 〈 [ δ m ( x )] n 〉 in the above equation is introduced to enforce 〈 δ h 〉 = 0. As shown in Schmidt et al. (2012), the expansion coe ffi cients in Eq. (4), F ( n ) , can be related to the peak-background split bias parameters or halo bias parameters in an ensemble average sense. Since we focus on the halo correlation functions in this paper, we empirically assume that the halo density field in Fourier space is given as where bn is the halo bias parameters and we have set F ( n ) = bn when converting Eq. (4) to the above equation. The 1st and 2nd-order bias parameters, which are relevant for the results in the following sections, are given in terms of the derivatives of halo mass function (Eq. 3): where In Eq. (5), to keep more generality, we included the additional term /epsilon1 ( k ) to model the noise field that is uncorrelated to the matter density field, i.e. 〈 /epsilon1 δ m 〉 = 0 (see McDonald 2006). The term 〈 [ δ m ( x )] n 〉 in Eq. (4) contributes only to the monopole mode of k = 0, so we ignored the contribution as it is not relevant for the halo power spectra. We again notice that Eq. (5) is not exact, and rather ansatz we employ in this paper. We will test how well our empirical, analytical model can describe the halo power spectra in the weakly nonlinear regime by comparing the model predictions with the simulation results.", "pages": [ 2, 3 ] }, { "title": "3 RENORMALIZEDPERTURBATION THEORY FOR NONLINEAR HALO POWER SPECTRA", "content": "In this section, we model nonlinear cross-power spectrum of matter and halos and nonlinear auto-power spectrum of halos by combining the 'renormalized' PT approach (McDonald 2006; Saito et al. 2009; Jeong & Komatsu 2009; Saito et al. 2011) with the local bias model, the halo bias and the perturbation theory described in the preceding section.", "pages": [ 3 ] }, { "title": "3.1 Halo-matter cross-power spectra", "content": "First, let's consider the matter power spectrum defined as Using Eq. (1), we can find that the power spectrum including up to the one-loop corrections are given as where P L m ( k ; z ) is the linear power spectrum, and Pm (13) and Pm (22) are the one-loop corrections arising from the ensemble averages of the higher-order matter density fluctuation fields; 〈 δ m (1) δ m (3) 〉 and 〈 δ m (2) δ m (2) 〉 , respectively. The one-loop corrections at a given redshift z can be computed once the linear power spectrum at the redshift is specified: Similarly, using the standard PT and halo bias prescription, we can compute the cross-power spectrum between matter and halos of mass M , which is the quantity that halo-shear cross-correlation can directly probe. By inserting Eq. (1) into Eq. (5), we can find the formal expression of the halo-matter cross-spectrum in a selfconsistent manner by including up to the one-loop correction terms of O ( δ 4 m (1) ): where σ 2 ≡ ∫ d 3 q / (2 π ) 3 P L m ( q ) and we employed notational simplification in the halo bias parameters; bi = bi ( M ). Thus, a formal implementation of the standard perturbation theory (SPT)based halo-matter spectrum yields the divergence term, i.e σ 2 ∼ ∫ ∞ q 3 P L m ( q )d ln q → ∞ , for a CDM-type power spectrum. In practice, since halo formation involves a coarse-grained smoothing of the underlying matter distribution corresponding to halo scales (also see discussion below Eq. 4), the divergence does not arise in the power spectrum we actually observe. Also note that the prefactor coe ffi cient of the linear power spectrum P L m is independent of k . From the first two terms of the r.h.s. of Eq. (11), we might re-write the two terms as where we have defined the notations δ b 1 ≡ σ 2 ( b 3 + 68 b 2 / 21) / 2 and δ Pm ≡ Pm (13) + Pm (22), which are the higher-order contributions to the linear bias and the linear matter power spectrum by the order of O ( δ 2 m (1) ) with respect to leading order in the PT formalism. Motivated by the equation above as well as the similar idea proposed by McDonald (2006), we propose the ' renormalized ' power spectrum as The first term is given by the nonlinear matter power spectrum, P NL m , multiplied by the 'e ff ective' or 'renormalized' linear bias parameter, b e ff 1 , while the second term includes the bare halo bias, b 2( M ), in Eq. (5). Thus the renormalized term can include the nonlinear corrections of matter clustering up to the higher orders. From the PT viewpoint, this is not self-consistent in a sense that the term b e ff 1 P NL m includes the higher-order contributions than the one-loop order. There are several nice features in our renormalization prescription: We will test an accuracy of the renormalized cross-power spectrum (Eq. 13) by comparing the predictions with simulation results for halos of various mass ranges. We will below show how the use of the 'full' nonlinear matter power spectrum in Eq. (13) can give a better fit to the simulation.", "pages": [ 3, 4 ] }, { "title": "3.2 Halo auto-power spectrum", "content": "Similarly, we propose the renormalized auto-power spectrum of halos with masses M and M ' (for generality of discussion, we consider the case that halos are in di ff erent mass ranges): where we have introduced the e ff ective linear bias b e ff 1 and b e ff ' 1 for halos of masses M and M ' , respectively, and used the collapsed notations such as b 2 = b 2( M ) and b ' 2 = b 2( M ' ) and similarly those for b 1 and b ' 1 . The last term δ N ' denotes the residual shot noise term arising from 〈 /epsilon1 ( k ; M ) /epsilon1 ∗ ( k ' ; M ' ) 〉 in Eq. (5) as follows. As in the literature, we refer to the term as the residual noise component after subtracting the Poisson shot noise, 1 / n h, without addressing the origin. Assuming that the residual shot noise is constant over the scale, but may vary with halo masses, /epsilon1 ( k , M ) → /epsilon1 ( M ), we will below study whether the model including the residual shot noise can give a better fit to the N -body simulation results for halos of di ff erent mass ranges. In the second line on the r.h.s. of the above equation, we redefined the shot noise term as δ N ' = δ N + b 2 b ' 2 ∫ (d 3 q / (2 π ) 3 ) P L m ( q ) 2 so that the following term (the second term in the second line), which contributes to the scale-dependent bias, becomes finite for the limit k →∞ : We would like to notice features of the renormalized halo power spectrum: Since ∫ d 3 q P L m ( q ) 2 is constant, the residual shot noise term modified in this way is still constant in space, but can vary with halo masses both through the dependence on b 2( M ) b 2( M ' ) and /epsilon1 . The constant δ N ' needs to be treated as an additional free parameter for predicting P hh( k ). In the following, we refer to the residual shot noise term as δ N , instead of δ N ' , for notational simplicity. The e ff ective bias parameter b e ff 1 in Eq. (14) is the same to that in Eq. (13) up to the order O ( δ 2 m (1) ). It is worth mentioning di ff erence between our approach and McDonald (2006). In McDonald (2006), all the bias coe ffi cients are replaced with the renormalized bias parameters; b 1 → b e ff 1 and b 2 → b e ff 2 . Hence, the bias parameters need to be treated as free parameters, and their relations with the halo bias parameters were not discussed. We will below test the validity of Eq. (14) using simulations. The lower panel of Fig. 1 shows the halo model predictions for the e ff ective bias functions, which are defined in terms of the halo and matter power spectra: b cross ( k ) ≡ P h m ( k ) / Pm ( k ) or b auto ( k ) ≡ P hh( k ) / P h m ( k ) (see Eqs. 13 and 14). Note that we did not include the residual shot noise contribution in this plot (set δ N = 0). Our model predicts a scale dependent bias in the weakly nonlinear regime, and the degree of scale-dependence changes with halo masses. The top panel of Fig. 1 shows that b 2( M ) is negative for low mass halos, goes to zero around M /similarequal 2 × 10 13 M /circledot / h and then becomes positive for more massive halos. Hence, Eqs. (13) and (14) tell that the nonlinear bias due to b 2( M ) suppresses the power spectrum amplitudes for low mass halos, while it enhances for high mass halos. Also the model shows that, unlike the linear theory prediction, the scale-dependent halo bias generally di ff ers in the estimators, P h m / Pm and P hh / P h m . Hence r /nequal 1 in the weakly nonlinear regime, where r is the correlation coe ffi cient of halo bias, defined as r ≡ P hh / √ P h mPm .", "pages": [ 4, 5 ] }, { "title": "3.3 HOD model: Relating halos to galaxies", "content": "Although we have focused on the halo power spectra, halos are not a direct observable from a galaxy redshift survey and need to be inferred from the distribution of galaxies. A practically useful approach to relate galaxies to halos is using the halo occupation distribution (HOD) (Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001). In this paper, for simplicity we assume that we can implement the method developed in Reid & Spergel (2009) (also see Reid et al. 2009, 2010; Hikage et al. 2012b,a) for reconstructing the halo distribution from the measured galaxy distribution. In the halo catalog, each halo hosts only one galaxy. In this setting, there is only one galaxy per halo. The crosspower spectrum of matter and galaxies and the galaxy auto-power spectrum are given in terms of the halo spectra as where the spectra P h m and P hh ' are given by Eqs. (13) and (14), respectively. Here N HOD( M ) is the halo occupation distribution (HOD), and ¯ n g is the mean number density of galaxies, defined as ¯ n g ≡ ∫ d M (d n / d M ) N HOD( M ). Exactly speaking, the equations above are valid only if each galaxy is at the center of halo. When the galaxies have an o ff set from the halo center, we need to include a convolution of the average o ff set distribution of galaxies, ˜ p o ff ( k ; M ), with the HOD distribution (see Hikage et al. (2012b,a) for details). However, for the real-space power spectrum we focus on in this paper, the e ff ect is negligible on scales of interest; ˜ p o ff ( k ; M ) ≈ 1 at the scales of interest. For the redshift-space power spectrum, the o ff -centered galaxies cause a significant Fingers-ofGod e ff ect. The main focus of this paper is the scale-dependent galaxy bias due to nonlinearities of structure formation, so we focus on the real-space power spectrum. Recently Hamaus et al. (2011) proposed a method, more directly based on the halo model approach, to model the nonlinear power spectrum of halos. In this method, the nonlinear halo-matter and halo-halo power spectra are given by a sum of the linear power spectrum, multiplied with linear bias parameter, and the 1-halo term. The 1-halo term of Phm is calculated by the mass weighted integral of the mass function, because the Fourier-transformed halo profile ˜ ρ ( k , M ) / M /similarequal 1 in the weakly nonlinear regime of interest. We will below compare the performance of our method and theirs by comparing the model predictions with our own high-resolution N -body simulations.", "pages": [ 5 ] }, { "title": "4.1 N -body simulations and the halo catalogs", "content": "To test the accuracy of our method for modeling the halo-matter and halo-halo power spectra in the weakly nonlinear regime, we use N -body simulations for Λ CDM model done in Nishimichi & Taruya (2011). In brief, we adopted 1280 3 N -body particles and the box size of volume 1 . 5 (Gpc h -1 ) 3 , and used the simulation outputs at z = 0 . 35. We defined halos using the Friends-of-Friends (FoF) finder algorithm with linking length 0.2 times the mean particle separation. For each halo, we use the total mass of member N -body particles as the halo mass, i.e. the FoF mass, and the center-of-mass positions of the particles as the halo position. Then we computed the halo power spectrum from the discrete distribution of halos in each simulation realization using the cloud-in-cells interpolation method and Fourier transformation. To reduce the statistical scatters, we use the mean spectra from 15 realizations. To explore the validity of our model for di ff erent ranges of halo masses, we divided the halos into di ff erent mass bins. Table 1 shows the parameters of the halo catalogs; we consider the halo catalogs divided into 9 mass bins, with mean masses ranging from 2 . 96 × 10 12 to 7 . 03 × 10 13 M /circledot / h . Note that, for the following results, we use the shot-noise-subtracted halo spectra, where we subtracted the theoryexpectation 1 / ¯ n h from the halo spectra measured from the simulations (¯ nh is the mean number density of halos in a given simulation). The halo bias and the halo mass function we use are given as a function of the virial mass, rather than the FoF mass. We use the conversion relation M vir = 0 . 88 M FoF at z = 0 . 35 to estimate the average virial mass for each mass bin using the method in Hu & Kravtsov (2003), where we assume that the FoF halo mass is close to the enclosed mass M 180 b (White 2002) inside which the mean density is 180 times the mean mass density and also assume that each halo follows an Navarro-Frenk-White profile (Navarro et al. 1997) with concentration parameter c vir = 4. We should note that this mass conversion only slightly changes the model predictions for the halo spectra, by up to a few % in the amplitudes. To compute the power spectrum for halos in the finite mass range used for the halo catalogs in Table 1, we use the following HOD in on our model (Eqs. 16 and 17):", "pages": [ 5, 6 ] }, { "title": "4.2 Halo-matter cross-power spectrum", "content": "First, we study the halo-matter cross-power spectrum and define the bias function, for convenience of the following discussion, as where P h im is the cross-spectra between dark matter ( N -body) particles and halos of the i -th mass bin, and Pm is the matter power spectrum computed from the original N -body simulations. The shot noise is negligible for the cross-spectrum P h im . The data points in Fig. 2 show the bias function b cross h i ( k ) mea- sured from the simulations. The spectra P h im and Pm in each simulation share the same large-scale structure, and therefore the scatters due to the sampling variance mostly cancel in the ratio, yielding relatively smoothly-varying data points over k . The figure clearly shows that the halo bias has greater amplitudes for more massive halos. At su ffi ciently large scales or small k , k < ∼ 0 . 08 h Mpc -1 , the halo bias appears to be constant, implying that the linear bias model is valid at the large scales. On the other hand, at the larger k , the simulation results manifest a scale-dependent halo bias for all the halo mass bins. The dot-dashed curves show our model predictions computed using Eq. (13). To compute the model predictions, we need to fix one free parameter, the renormalized linear bias b e ff 1 , for which we determined b e ff 1 by fitting the model-predicted b cross ( k ) to the simulation b cross ( k ) in the linear regime, at k /lessorequalslant 0 . 05 h Mpc -1 . Table 1 shows that the best-fit b e ff 1 di ff ers from the linear halo bias by about 10%, which is also found by Manera et al. (2010). In our method, we interpret that the discrepancy between b 1 and b e ff 1 arises due to the renormalization; b e ff 1 has a contribution of the higher-order moments (see around Eq. 12) in addition to b 1. However, the discrepancy might also be ascribed partly to the inaccuracy of the analytical halo mass function (Eq. 3), compared with our N -body simula- tions, as well as to violation of the universality of the mass function. (e.g. Tinker et al. 2008). However, exploring these issues is beyond the scope of this paper, so we leave these for future work. Besides this free parameter, we used the input Λ CDM model parameters and halo mass range to compute the model prediction. Once the parameter b e ff 1 is determined, our model can be in remarkably good agreement with the simulation results, including the k -dependence and the halo mass-dependence. For the largest mass bin, our model shows a sizable disagreement, possibly due to an inaccuracy of the halo mass function used for the model calculation or the breakdown of perturbation theory to describe too strong nonlinear bias. At large k , the perturbation theory ceases to be accurate, and is indeed not accurate enough up to k ∼ 0 . 2 h Mpc -1 . There have been many e ff orts to improve the PT prediction of nonlinear matter power spectrum by including the higher-order loop corrections, e.g. the renormalized PT (Crocce & Scoccimarro 2006; Taruya et al. 2009, also see references therein). The solid curve shows the results if we use the improved PT prediction for the nonlinear matter power spectrum P NL m that is taken from Nishimichi & Taruya (2011) (also see Taruya et al. 2009) (the closure theory; hereafter CPT), instead of the standard PT including up to the one-loop correction. The improved P NL m has smaller amplitudes in the weakly nonlinear regime k > ∼ 0 . 1 h Mpc -1 at the redshift z = 0 . 35 than the standard PT predicts, yielding a slightly stronger scale-dependence of b ( k )( = P h m / P NL m ) as evident from Eq. (13). The improved P NL m shows a similar-level agreement with the simulation results. In order to see the accuracy of our model for P h m ( k ), we compare the simulation result for P h m with the di ff erent model predictions in Fig. 3. Here we considered the intermediate halo mass bin (bin 4), and we use the parameter b e ff 1 determined in Fig. 1. Encouragingly, the figure clearly shows that our model prediction well agrees with the simulation P h m ( k ) up to k /similarequal 0 . 2 h Mpc -1 , if we use the improved PT model (CPT) for the nonlinear matter power spectrum. If we use the standard PT theory instead, the agreement is only up to k /similarequal 0 . 12 h Mpc -1 , and the standard PT breaks down at the larger k , in the weakly nonlinear regime. Hence we conclude that the apparent agreement of the standard PT up to the higher k bins in Fig. 1 is due to a cancellation of inaccuracies in the two spectra P h m and Pm , in the numerator and denominator of the bias function b cross ( k ). Comparing the solid and short-dashed curves shows the e ff ect of nonlinear scale-dependent bias that arises from the term proportional to b 2( M ) in Eq. (13). The scale-dependent bias becomes important at k > ∼ 0 . 1 h Mpc -1 for this redshift z = 0 . 35, and our model can reproduce the simulation result. The dashed curve shows the model prediction recently proposed in Hamaus et al. (2011), where the nonlinear power spectrum is computed based on the halo model in combination with the halo bias parameters. The figure shows that the agreement is not as good as our model prediction.", "pages": [ 6, 7 ] }, { "title": "4.3 Halo auto-power spectrum", "content": "The model parameters for the halo power spectrum P hh( k ) are b e ff 1 and the residual shot noise parameter δ N . For convenience of our discussion, we use the following parametrization of the residual shot noise term relative to the standard shot noise term: Fig. 4 compares the PT predictions and the simulation results for P hh( k ), as in Fig. 3. To compute the PT predictions, we determined the free parameters b e ff 1 and C for each halo catalog as follows. For b e ff 1 , we used the same values as those used for the halo-matter cross-power spectra in Fig. 2. For C , we determined the value by fitting the model prediction to the simulation result up to k max = 0 . 15 or 0 . 20 h Mpc -1 for the SPT or the improved PT (CPT), respectively. The maximum wavenumber k max is chosen because each of the PT models for nonlinear matter power spectrum is accurate enough up to the k max-wavenumber to within about 3% accuracy (Nishimichi & Taruya 2011). In doing this fitting, we accounted for the statistical uncertainties in estimating the power spectrum from the 15 simulation realizations; i.e., we used the weighting in each k -bin, given as ( ∆ P hh) 2 ∝ 1 / (2 π k 2 i ∆ k )[ P hh( ki ) + 1 / ¯ n h] 2 ( ki is the central value of the i -th k -bin and ∆ k is the width). The best-fit residual shot noise parameter C is about 30% compared to the standard shot noise for this halo mass bin (bin 4). Table 1 shows a strong anti-correlation between C and halo mass. The anti-correlation might be ascribed to a mass dependence of the stochastic halo bias (Matsubara 1999; Taruya & Suto 2000). As can be found from Fig. 4, our model prediction is in good agreement with the simulation result, apparently up to k /similarequal 0 . 25 h Mpc -1 , if we use the improved PT prediction (solid curve) 4 . The nice agreement is found by including the residual shot noise contribution, which can account for a part of the nonlinear bias effect. The standard perturbation theory cannot achieve the similar level agreement, even if varying the residual shot noise parameter. The figure also shows other model predictions, and the comparison of di ff erent model predictions manifests the importance of nonlinear clustering e ff ect and scale-dependent bias in the weakly nonlinear regime. Combining the results in Figs. 3 and 4 implies that the halo bias parameters b 1( M ) and b 2( M ), in combination with the nonlinear matter power spectrum and the residual shot noise, can well reproduce the halo spectra P h m ( k ) and P hh( k ). Now we study another bias function defined in terms of the halo power spectrum as This bias function is di ff erent from the bias function b cross ( k ) we studied in Fig. 1, as can be explicitly found from Eqs. (13) and (14) (also Eqs. 16 and 17). Fig. 5 compares the PT predictions and the simulations results for the bias function of each halo catalog b auto hi ( k ), where the PT predictions are computed by using the bestfit power spectra P h m and P hh as estimated in Figs. 2 and 4. Note that the model in Fig. 5 appears to show a larger disagreement with the simulation result at k > 0 . 2 h Mpc -1 , compared to Fig. 4, especially when using the CPT for P NL m . This is mainly because of the unphysical damping of the CPT model for the P NL m at that scales (Taruya et al. 2009), and partly because the plotting range of y -axis in Fig. 5 is narrower than in Fig. 4. We should notice that the discrepancy of the model from the simulation is less than 1% at k < 0 . 2 h Mpc -1 for both the plots. The best-fit value of the residual shot noise pa- rameter, C , for each halo mass bin is given in Table 1. The amount of the residual shot noise varies with halo masses, ranging from a few % to 70% compared to the standard shot noise term, and changes from negative to positive values from less to more massive halos. The bias function b auto ( k ) shows a scale-dependence over the range of k we consider. The scale-dependence of b auto indeed di ff ers from that of b cross ( k ) in Fig. 2 as the PT model predicts. The model predictions can fairly well reproduce the simulation results up to k /similarequal 0 . 2 h Mpc -1 , but then show a larger disagreement at the larger k than in Fig. 2 due to inaccuracies in both the model predictions for P h m and P hh.", "pages": [ 7, 8, 9 ] }, { "title": "4.4 Comparison of halo bias model with other models", "content": "The main di ff erence between our method and the previous works (McDonald 2006; Jeong & Komatsu 2009; Saito et al. 2009; Baldauf et al. 2010a; Saito et al. 2011), is whether or not to incorporate the renormalized bias approach and the halo bias into the PT approach for computing the nonlinear power spectra. In this section, wecompare our model predictions with other models that have similar forms. The nonlinear power spectra in these models are expressed by the following general forms (see Eqs. 13 and 14 for our model): where In terms of these forms, we can categorize the di ff erent models as Note that, to have a fair comparison, we use CPT to compute P NL m ( k ) in the first term of Eq. (22) for all the above models. Thus the different models have di ff erent ranges of their variations in the power spectra as a function of k with varying free parameters for a given cosmological model. Fig. 6 shows the di ff erent model predictions, described above, for P h m and P hh for di ff erent halo mass bins, compared to the simulation results. To determine the free parameters in each model, we minimize the following χ 2 by comparing the model prediction to the simulation result: where P ( ki ) = [ P h m ( ki ) , P hh( ki )], the power spectra with superscripts 'sim' or 'model' are the simulated or model power spectra, respectively, C is the covariance matrix of the power spectrum computed from 15 realizations of the simulated power spectra, and C -1 is the inverse matrix. We use the power spectrum information up to k max = 0 . 2 h Mpc -1 . We included correlation between P h m and P hh at the same k -bin in the covariance matrix, but ignored correlations between di ff erent k bins for simplicity 5 . Our choice of k max = 0 . 2 h Mpc -1 is based on the fact that the CPT prediction for P NL m ( k ) is accurate to within a 3% level up to k = 0 . 2 h Mpc -1 compared to the simulated spectrum at z = 0 . 35 as shown in Nishimichi &Taruya (2011). Fig. 6 shows that the di ff erent models well reproduce the simulated P h m or P hh to an equal-level accuracy up to k /similarequal 0 . 15 or 0 . 2 h Mpc -1 in some cases. Again note that our model has least free parameters among these models, because our model uses the halo bias parameters b 1( M ) and b 2( M ) to model the scale-dependent bias of the nonlinear halo power spectra (Eqs. 22), and therefore restricts a range of the model variations compared to other models. Nevertheless, our model appears to be reasonably accurate compared to other models. For comparison, the shaded region around the curves in Fig. 6 shows 1 σ statistical errors of the power spectrum measurements expected for a survey with volume coverage of about 3 . 4 (Gpc h -1 ) 3 , which roughly corresponds to the volume of a BOSS-like survey with redshift range 0 . 5 /lessorequalslant z /lessorequalslant 0 . 7 and area coverage 10 , 000 square degrees. We estimated the error bars by scaling the scatters at each k bin from the 15 simulation realizations assuming that the scatters scale with a survey volume as σ ( P ) ∝ 1 / √ Vs . It can be found that our model is accurate at least enough up to k /similarequal 0 . 20 h Mpc -1 within the 1 σ errors for a BOSS-like survey, for a di ff erent range of halos masses. The linear theory is not accurate at BAO scales, at k > ∼ 0 . 1 h Mpc -1 , although the PT based model also ceases to be accurate at k > 0 . 2 h Mpc -1 . ∼ Fig. 7 and Table 2 give a more quantitative comparison of our model with other models, where we consider the BOSS-like survey to compute the χ 2 given the expected measurement accuracies as in Fig. 6. Fig. 7 compares the best-fit coe ffi cients ( α 1 , . . . , α 4), which can be read as e ff ective bias parameters (see below Eq. 22), with the results of our model; the halo bias parameters ( b 1( M ) , b 2( M )) and the best-fit parameters b e ff 1 and δ N . The figure shows that the results for 'Saito et al. (2009)' or 'McDonald 2006' reproduce the similar results to our model. The model '4 free parameters' shows a sizable di ff erence from our result, especially for α 3 and δ N , implying that even small changes in the coe ffi cients give the similar nonlinear power spectra for P NL h m and P hh at the scales. This also means a strong degeneracy between α 3 and δ N parameters. Table 2 gives the reduced χ 2 of the best-fit model power spectra for each model and for each halo mass bin. Here we consider k max = 0 . 15 or 0 . 2 h Mpc -1 for the maximum wavenumber to use for the model fitting. The degrees of freedom are defined by the number data point of the simulated spectra (58 or 78 in total for P h m and P hh when employing k max = 0 . 15 or 0 . 2 h Mpc -1 , respectively) minus the number of free parameters (either 2, 3 or 4). Again our model and the models 'Saito et al. (2009)' or 'McDonald (2006)' are in a similar-level accuracy, and the model '4 free parameters' is slightly better due to more free parameters. Note that the reduced χ 2 value is smaller than unity, partly because the central values of the simulated spectra are taken from the simulations of 22.5 (Gpc h -1 ) 3 volume and therefore the central value have smaller scatters than expected from a BOSS-like survey.", "pages": [ 9, 10 ] }, { "title": "5 SUMMARYAND DISCUSSION", "content": "In this paper, we have studied a method of modeling the nonlinear halo power spectra, P hm( k ) and P hh( k ), by combining the PT approach of structure formation, the local bias ansatz and the halo bias. The nonlinearities of halo power spectra, which are deviations form the linear theory prediction, arise from the two e ff ects: the nonlinear matter clustering and the nonlinear relation between the matter and halo density fields. In deriving the nonlinear halo power spectra, we employed the renormalization approach (McDonald 2006) to re-sum the higher-order terms so that the terms are replaced with the nonlinear matter power spectrum, multiplied by the 'renormalized' linear bias parameter. The remaining terms in the nonlinear halo spectra are given as a function of the linear matter power spectrum and the halo bias, where the terms at the one-loop correction order are included. As a result, we expressed the halo-matter cross-power spectrum (Eq. 13) in terms of the nonlinear and linear matter power spectra, the halo bias ( b 2( M )), and one free parameter, the renormalized linear bias parameter ( b e ff 1 ), which needs to be determined in the linear regime. Similarly, the halo auto-power spectrum (Eq. 14) is given as a function of the nonlinear and linear matter power spectra, the halo bias functions ( b 1( M ) , b 2( M )), and the two free parameters, b e ff 1 and the residual shot noise parameter δ N . Thus our method utilizes the recent development in an accurate model of the nonlinear matter power spectrum based on the refined perturbation theory and / or N -body simulations. In our model, the halo power spectra are specified by cosmological parameters, halo mass, redshift, and a less number of free parameters. We showed that our model predictions for P h m ( k ) and P hh( k ) are in nice agreement with the simulation results, up to k /similarequal 0 . 2 h Mpc -1 , at simulation output z = 0 . 35 (see Figs. 3 and 4), if using the improved PT theory prediction for the nonlinear matter power spectrum in the model calculation. The linear power spectrum breaks down at k /similarequal 0 . 1 h Mpc -1 . Thus our model might allow a factor 2 gain in the maximum wavenumber k max up to which to include the power spectrum information when constraining cosmological parameters. In the sampling variance limited regime, the factor 2 gain is equivalent to a factor 8 larger volume, yielding greater statistical power of the power spectrum measurement. In addition, a wider coverage of wavenumbers in the power spectrum amplitudes gives a higher sensitivity to some of intriguing cosmological parameters such as the total neutrino mass (Saito et al. 2008, 2009, 2011) and the running index of the primordial power spectrum. Thus developing a su ffi ciently accurate model of the nonlinear halo power spectrum is very important in order for us to have improved cosmological constraints, yet without having any significant biases in the derived parameters. Our model naturally predicts that, in the weakly nonlinear regime, the halo power spectra show a scale-dependent bias relative to the nonlinear matter power spectrum (see Figs. 2 and 5). The PT based model naturally predicts that the two bias functions, defined as b cross ( k ) = P h m ( k ) / Pm ( k ) and b auto ( k ) = √ P hh( k ) / Pm ( k ), di ff er in the weak nonlinear regime. Furthermore, by incorporating the halo occupation distribution (HOD) model, we can predict the nonlinear power spectra of galaxy-matter and galaxy-galaxy in the weakly nonlinear regime. We hope that our model can give a better description of the nonlinear galaxy power spectra, and then allows for improved cosmological constraints via the measured power spectra. Our model can be further refined by including the higher-order loop corrections to the nonlinear bias functions. We showed that our model using the halo bias can give a good fit to the simulation results for the halo spectra. This o ff ers a promising synergy between imaging and spectroscopic galaxy surveys, because a cross-correlation of the spectroscopic galaxies with images of background galaxies, the so-called galaxy-galaxy weak lensing, can directly probe the mean mass of host halos and in turn constrain the halo bias. Here the halo mass is constrained from the small-scale weak lensing signals arising from the mass distribution within one halo, which is complementary to the large-scale information of galaxy clustering at k < ∼ 0 . 2 h Mpc -1 we focus on in this paper. This synergy is available if the spectroscopic and imaging surveys see the same region of the sky. This is the case for upcoming surveys: the BOSS and the Subaru HSC Survey, the Subaru HSC and PFS surveys, the Euclid, the WFIRST and a combination of the LSST survey with spectroscopic surveys. However, our method rests on simplified assumptions one of which is the local bias model. Our model can be further improved by including the non-locality of halo bias such as the dependence of halo bias on the curvature of the initial density peaks (Desjacques et al. 2010) and / or the tidal field around the density peaks (Chan et al. 2012; Baldauf et al. 2012). This is an interesting possibility, and will be explored in our future work. In this paper, we have focused on the real-space power spectra of halos or galaxies. Actual observable for galaxy redshift survey is the redshift-space power spectrum of galaxies, which is affected by the redshift-space distortion e ff ect due to peculiar motions of galaxies. Towards a more accurate modeling of the nonlinear galaxy power spectrum in redshift space, we need to further include the nonlinear coupling between the redshift-space distortion e ff ect and the nonlinear galaxy bias. There are encouraging developments in modeling the redshift-space matter power spectrum in redshift-space, based on the refined perturbation theory and N -body simulations (Matsubara 2008b; Taruya et al. 2009, 2010; Tang et al. 2011; Matsubara 2011; Sato & Matsubara 2011). The redshift-space distortion e ff ect due to virial motions of galaxies within halos, the so-called Fingers-of-God (FoG) e ff ect, is harder to model, but Hikage et al. (2012b,a) recently developed an empir- A. J. Nishizawa et al. ical method to model the FoG e ff ect based on the halo model and proposed a method to remove the FoG contamination by combining with galaxy-galaxy weak lensing measurement. It seems straightforward to incorporate these methods in the method developed in this paper, in order to include all the e ff ects, nonlinear clustering, nonlinear bias, nonlinear redshift-space distortion and FoG e ff ect. This is our future work and will be presented elsewhere. These efforts are very important in order to attain the full potential of future high-precision galaxy surveys as well as to obtain unbiased, robust cosmological constraints from the surveys.", "pages": [ 10, 11, 12 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "Wethank Issha Kayo, Ravi Sheth and Atsushi Taruya for useful discussion. In particular, we thank Atsushi Taruya for making the code to compute the nonlinear matter power spectrum publicly available to us. This work is supported in part by the Grant-in-Aid for the Scientific Research Fund (No. 23340061), by JSPS Core-to-Core Program 'International Research Network for Dark Energy', by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the FIRST program 'Subaru Measurements of Images and Redshifts (SuMIRe)', CSTP, Japan. T. N. is supported by a Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows (PD: 22-181).", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "We study scaling behaviour of statistics of voids in the context of the halo model of nonlinear large-scale structure. The halo model allows us to understand why the observed galaxy void probability obeys hierarchal scaling, even though the premise from which the scaling is derived is not satisfied. We argue that the commonly observed negative binomial scaling is not fundamental, but merely the result of the specific values of bias and number density for typical galaxies. The model implies quantitative relations between void statistics measured for two populations of galaxies, such as SDSS red and blue galaxies, and their number density and bias. Key words: large-scale structure of Universe - methods: numerical - methods: statistical - galaxies: statistics", "pages": [ 1 ] }, { "title": "J. N. Fry 1 , 2 /star and S. Colombi 2 /star", "content": "1 Department of Physics, University of Florida, Gainesville FL 32611-8440, USA 2 Institut d'Astrophysique de Paris, CNRS UMR 7095 and UPMC, 98bis bd Arago, F-75014 Paris, France 6 June 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Understanding the behaviour of voids in the galaxy distribution is one of the remaining unsolved problems of largescale structure. Voids are a powerful probe of nonlinear large scale structure. They probe high order statistical properties, but do so on scales that should be accessible in perturbation theory. One interesting property of voids is a scaling behaviour implied in the hierarchical model of higher order clustering. The hierarchical scaling has been verified many times, in a variety of samples, including the CfA redshift survey (Maurogordato & Lachi'eze-Rey 1987; Vogeley et al. 1994), Perseus-Pisces Fry et al. (1989), the Southern Sky Redshift Survey (Maurogordato et al. 1992), and IRAS 1.2Jy (Bouchet et al. 1993). Particularly intriguing are recent results from 2dFGRS (Croton et al. 2004a, 2007) and from DEEP2 and SDSS (Conroy et al. 2005; Tinker et al. 2008), in which the scaling continues to hold with improved precision over larger scales, for both magnitude selected subsamples and random dilutions. However, in data (Bouchet et al. 1993; Gazta˜naga 1994; Croton et al. 2004b; Ross et al. 2006, 2007), and in numerical simulations (Fry et al. 2011), the hierarchical clustering on which the scaling is based is not obeyed. The hierarchical normalization removes much of the variation, but the hierarchical amplitudes still depend on scale, and the premise of the scaling does not hold in detail. A recent alternative to purely hierarchical behaviour is provided by the halo model (Ma & Fry 2000a,b; Scoccimarro, Sheth, Hui, & Jain 2001). In this paper we show that void scaling can be understood in the halo model. In Section 2, we review statistics and display the connection between voids and correlation functions, and we apply the halo model. Many common models are realizations of the halo model, and we present several of these in Section 3. In Section 4 we test the model against numerical results and present several halo model scaling relations. Section 5 contains a final discussion.", "pages": [ 1 ] }, { "title": "2.1 Generating Functions", "content": "The probability that a volume be empty of galaxies, or void, is an intriguing statistical measure, accessible to perturbation theory on large scales and yet an inherently nonlinear statistic on all scales. We study the properties of voids in the context of the halo model, the essence of which is that galaxies come in groups or clusters embedded in haloes; the number of galaxies is then the sum over haloes of the number within each halo. The generating function formulation of the halo model (Fry et al. 2011) is useful for studying combinatorics, and particularly voids. Let the probability and moment generating functions be where P n is the probability that a randomly placed volume contains n galaxies, and m k is the order k factorial moment of the distribution, m k = ¯ N k ¯ µ k = 〈 n [ k ] 〉 , where n [ k ] = n ( n -1) · · · ( n -k + 1) = n ! / ( n -k )!. Since both probabilities and moments can be obtained as derivatives of G , probabilities as ∣ we see that the generator M ( t ) of moments, so that m k = d k M ( t ) / d t k | t =0 is thus M ( t ) = G ( t + 1) (cf. Szapudi & Szalay 1993). Connected, irreducible, discreteness corrected moments k n = ¯ N n ¯ ξ n are similarly obtained from K ( t ) = ln M ( t ) = ln G ( t +1). In terms of the irreducible ¯ ξ n , the probability that a volume V be empty of galaxies, or void, is then a sum over all orders, a result also obtained by considering Venn diagrams and contour integrals in the complex plane (Fall et al. 1976; White 1979; Fry 1986). From the void probability, we can write the statistic In observations, in perturbation theory, and in stable clustering, we often take the volume-averaged correlations to follow the so-called hierarchical pattern, ( S 2 = 1). In the hierarchical case, the void probability becomes a power series in the variable ¯ N ¯ ξ . This is the hierarchical scaling relation: the void statistic -log P 0 / ¯ N is a function of the scaling variable ¯ N ¯ ξ , where the void probability P 0 , the mean count ¯ N = 〈 N 〉 , and the scaling variable ¯ N ¯ ξ = ( 〈 N 2 〉 -〈 N 〉 2 -〈 N 〉 ) / 〈 N 〉 are all observationally measurable quantities. When ¯ N ¯ ξ is small, χ → 1, the Poisson result P 0 = e -¯ N , with clustering appearing only as a small correction. When ¯ N ¯ ξ is large, the void scaling behaviour is a strong test of hierarchical clustering to high orders. A similar scaling behaviour has been found for gaps in the rapidity distribution resulting from proton-antiproton, proton-nucleus, and relativistic heavy ion collisions (Hegyi 1992; Malik 1996; Ghosh et al. 2001). Several models have been presented with specific analytic forms for the void scaling function χ ( ¯ N ¯ ξ ), useful against which to compare observational and numerical results. Details are contained in Appendix A. Void scaling has been tested and found to hold in observational data from the CfA redshift survey (Maurogordato & Lachi'eze-Rey 1987; Vogeley et al. 1994), Perseus-Pisces (Fry et al. 1989), the Southern Sky Redshift Survey (Maurogordato et al. 1992), the IRAS 1.2-Jy redshift catalog (Bouchet et al. 1993), and more recently in the 2dFGRS (Croton et al. 2004a, 2007), and DEEP2 and SDSS (Conroy et al. 2005). However, it is not clear that the scaling should be obeyed: although the normalization to S k = ¯ ξ k / ¯ ξ k -1 removes much of the dependence of ¯ ξ k on scale, the S k are not in fact constant (Bouchet et al. 1993; Gazta˜naga 1994; Croton et al. 2004b, 2007; Ross et al. 2006, 2007), and the galaxy distribution does not obey equation (7). To understand this, is interesting to look at implications for voids in the halo model.", "pages": [ 1, 2 ] }, { "title": "2.2 The Halo Model", "content": "Reduced to its most basic terms, in the halo model total galaxy count is the sum over clusters of the number of objects in a cluster. On large scales, boundary effects are unimportant and clusters can be considered as point objects that are either entirely inside or entirely outside, the point cluster limit of the halo model (Fry et al. 2011). In the limit that clusters are unresolved (the point cluster limit, each cluster is either entirely within V or entirely outside of V ), and all clusters have identical occupation distribution (each cluster has the same mean count ¯ N i and higher order moments ¯ µ n,i ), the generating function total count probabilities is the composition of the halo number and halo occupancy generating functions, and galaxy count moments are simply related to correlations ¯ ξ k,h of halo number and moments ¯ µ k of halo occupation, with ¯ N g = ¯ N h ¯ N i , and The moments ¯ ξ k are in general the sum of many terms, with different dependences on scale, and galaxies do not in general have the constant amplitudes of the hierarchical scaling pattern. However, if only the underlying cluster correlations obey ¯ ξ k,h = S k ¯ ξ k -1 h , these relations contain a more subtle scaling. The combinatorics implied by the composition of generating functions in equation (9) and the general pattern of equations (10)-(13) remain true in the full halo model, in which occupation statistics depend on halo mass, with a distribution described by the halo mass function d n/ d m and in which haloes can span the boundaries of V , with two modifications (Fry et al. 2011). First, when haloes are not identical but range over a distribution of masses, every term in G ( z ) = 〈 z N 〉 , and in particular the occupation moments ¯ µ n,i , are further averaged over the halo mass function. After averaging over haloes of different mass, with mass-dependent occupation distribution and correlation strength, the net effect is to replace the occupation moment ¯ µ k with and halo correlations with where the mean halo bias ¯ b h is b ( m ) as given in Mo et al. (1997), averaged over the occupied halo mass function, the mean galaxy bias ¯ b is weighted by occupation number, and ¯ b k is weighted by the occupation number factorial moment 〈 N [ k ] 〉 , ( ¯ b = ¯ b 1 ). The weighted bias factors ¯ b k are generally of order unity (but since higher order bias factors are weighted by higher powers of mass and bias is typically an increasing function of mass, ¯ b k is rising with k ). If halo correlations are hierarchical, ¯ ξ k,h = S k,h ¯ ξ k -1 2 ,h , galaxy correlations are then found to be polynomial functions of the combination ¯ N h ¯ ξ h , or ¯ b ¯ N h ¯ ξ h / ¯ b h , etc. For small R the quantities ¯ µ k rise monotonically with scale, but on large scales ¯ µ k and b k become constant (Fry et al. 2011). In the halo model, the galaxy correlations are not simply hierarchical, but every term in equation (6), though no longer a simple power of ¯ N ¯ ξ , is a (polynomial) function of ¯ N h ¯ ξ h , If we assume that the halo distribution follows the hierarchical pattern ¯ ξ k,h = S k,h ¯ ξ k -1 h , then χ is a function of the variable ¯ N h ¯ ξ h . But, by equation (19), ¯ N g ¯ ξ g is a (linear) function of ¯ N h ¯ ξ h . Thus, in the halo model, although the galaxy amplitudes S k are not constant, χ remains a function of ¯ N g ¯ ξ g : the hierarchical scaling for voids holds, even though ¯ ξ k,g no longer follows the simple hierarchical pattern. The pattern is seen in general in the generating function formulation. The probability generating function is additionally averaged over halo mass m , leading to the replacements of eqs. (14) and (15); and so this pattern continues to all orders. With no empty haloes, the halo occupancy generating function has g i (0) | m = p 0 | m = 0 for every halo mass m , and so we have the very useful result This is not a surprise: even when averaged over a distribution of haloes of different mass, no haloes means no galaxies, no galaxies means no haloes.", "pages": [ 2, 3 ] }, { "title": "3 NUMERICAL RESULTS", "content": "We present results for statistics of voids in the distribution of dark matter, galaxies, and haloes for the numerical simulation studied in Fry et al. (2011). The simulation is performed with the adaptive mesh refinement (AMR) code RAMSES (Teyssier 2002) for a ΛCDM cosmology with Ω m = 0 . 3, Ω Λ = 0 . 7, H 0 = 100 h kms -1 Mpc -1 with h = 0 . 7, and normalization σ 8 = 0 . 93, where σ 8 is the root mean square initial density fluctuation in a sphere of radius 8 h -1 Mpc extrapolated linearly to the present time. The simulation contains 512 3 dark matter particles on the AMRgrid, initially regular of size 512 3 , in a periodic cube of size L box = 200 h -1 Mpc; The hierarchical amplitudes S k for mass, galaxies, and haloes in this simulation are presented by Fry et al. (2011), and further details can be found in Colombi, Chodorowski & Teyssier (2007). From the simulation data, we compute for spheres of radius R = 0 . 5-25 h -1 Mpc the probability P 0 that the volume be empty and the moments and The binomial uncertainty in the void probability is (Maurogordato & Lachi'eze-Rey 1987; Hamilton 1985) where N tot is the total number of independent volumes sampled, with is an additional cosmic variance contribution proportional to ¯ ξ ( L ), the variance on the scale of the sample (Colombi, Bouchet, & Schaeffer 1995), which is often insignificant. In computing the uncertainty in χ = -ln P 0 / ¯ N , the numerator and denominator are far from independent, but in fact are almost exactly anticorrelated, so that ∣ ∣ (Colombi, Bouchet, & Schaeffer 1995). We adopt this as our error. We see that the numerical results follow the predictions of hierarchical scaling, but this is not necessarily what is expected. Within the uncertainties of sampling a small number of objects, the amplitudes for haloes may be consistent with constant values, but those for mass, and especially for galaxies, are not. The normalization from ¯ ξ k to S k = ¯ ξ k / ¯ ξ k -1 2 removes much of the variation with scale (the unnormalized five-point function for mass covers more than ten decades), but the resulting S k for galaxies are not constant, as shown in figs. 5 and 6 of Fry et al. (2011), where the residual variation is still a factor of up to 10. Thus, we are faced with the fact that hierarchical void scaling is observed, but its premise does not hold. The halo model provides an explanation: in eqs. (19)-(22), ¯ ξ 2 and the higher order ¯ ξ k are all functions of ¯ N h ¯ ξ h , and ¯ N h ¯ ξ h is linearly related to ¯ N g ¯ ξ g . Figures 3 and 19 illustrate this in the simulation results. Figure 3 shows ¯ N g ¯ ξ g and ¯ N h ¯ ξ h as a function of scale R , for the full samples and for the two-, four-, an eight-fold dilutions. Measured values are widely distributed; however, from equation (19), on large scales ¯ N g ¯ ξ g is related to ¯ N h ¯ ξ h , as illustrated in Fig. 4. On large scales, where halo size is negligible, we expect to have no galaxies only if we have no haloes, a result also implied in the composition of generating functions. Figure 5 shows P 0 ,g vs. P 0 ,h for the same volumes. The halo model contains the requirement P 0 ,g = P 0 ,h , no galaxies means no haloes, no haloes means no galaxies, from equation (24) or from the fundamental sum over the occupancy of each halo, N g = ∑ N i . From this, it is possible to obtain relations between void scaling curves for galaxies and their haloes, or between two different galaxy populations. Figure 6 illustrates a mapping suggested by the halo model. from the halo curve to the galaxy curve. The figure shows the scaling statistic for galaxies -ln P 0 ,g / ¯ N g (filled squares), for haloes -ln P 0 ,h / ¯ N h (filled circles), and for haloes mapped vertically by the ratio of number and horizontally by the ratio of the factors in ¯ Nb 2 , (open circles). The mapping is indicated by arrows for a selected sample of points, but every open circle originates from a filled circle. The mapped halo curve and the galaxy curve are different for ¯ N ¯ ξ /lessorsimilar 1, where resolved halo form factors affect the statistics (Fry et al. 2011), but they merge for ¯ N ¯ ξ /greaterorsimilar 1. See also Tinker & Conroy (2009). We obtain some inequalities comparing galaxies with their parent haloes. Write the scaling variable as x = ¯ N ¯ ξ . All halo scaling curves lie above the minimal scaling curve, and so χ h ( x h ) > χ min ( x h ) > 1 /x h . With χ g /χ h = ¯ N g / ¯ N h and with x g /x h = b 2 g ¯ N g /b 2 h ¯ N h , this becomes a limit on χ g , Since b ( m ) is an increasing function of mass and b g (equation 17) is weighted to larger m than b h (equation 16), we thus expect χ g > 1 /x g . As a horizontal scaling, we expect χ g = χ h at a value Since both b g /b h > 1 and ¯ N g / ¯ N h > 1, we expect that in general, galaxy scaling curve will be to the right of the halo scaling curve. To the extent that the scaling curve can be represented as a power law, χ = Ax -ω (Balian & Schaeffer 1988), we can write quantitative relations. Power-law behaviour implies a vertical mapping between two scaling curves at the same value of x , or a horizontal mapping, between two curves at the same value of χ . Since ω is often near 1, the horizontal mapping is typically much more dependent on relative bias and only weakly on relative number. This horizontal mapping is illustrated in Fig. 6, with ω = 0 . 79. We can compare two galaxy populations, say i and j . The simpler case is when both derive from essentially the same halo population; then we have the mappings With a power-law halo scaling function χ h = Ax -ω , we find the horizontal mapping χ i = χ j at If halo scaling follows the minimal model, with ω ≈ 1, number density does not enter at all. For the negative binomial model, on scales of interest ω ≈ 0 . 8 and 1 -1 /ω ≈ -0 . 25, still a very weak dependence. Such a weak dependence on number means that we can expect scaling curves for populations with higher bias to be shifted to the right, by a factor of approximately b 2 rel , or shifted upwards, by a slightly larger factor. We can also compare galaxy populations that derive from distinct halo populations that have different number density and correlation strength, as long as they follow the same scaling curves, as in Figure 2. Here, it is the relative bias b g,i /b h,i and relative number density ¯ n g,i / ¯ n h,i , etc., that appear in the scaling relation. For power law χ ∼ x -ω , this reduces to the single horizontal scaling Finally, we present results for voids in the mass or dark matter distribution. The number of dark matter particles is so large that unless diluted substantially, only very small volumes are empty. We begin with a random sample of one out of eight, or 256 3 particles, for which we compute the full P N for volumes with R = 0 . 2 h -1 Mpc to R = 25 h -1 Mpc by factors of √ 2. We then take advantage of the generating function to plot results for dilutions by a factor of λ , for λ = 2 k/ 2 , k = 0 to 20, or effective number of points from 256 3 = 16777 216 down to 16 284. Figure 7 shows the scaling function χ = -ln P 0 / ¯ N plotted against the scaling variable ¯ N ¯ ξ for the full sample and the twenty dilutions. We note that Croton et al. (2004a) do not test scaling, but present results for only one density, which from their simulation parameters should be equivalent to the second curve below the median in Fig. 7. The dark matter results do not follow the scaling implied by gravitational instability, but this is because most of the volumes sampled are not in the large-scale, perturbative regime. To compare with perturbative gravitational instability, Fig. 8 shows results restricted to large volumes, R = 6 . 3, 8 . 8, 12.5, and R > 17 h -1 Mpc. For sufficiently large R the measurements seem to approach the curve predicted for gravitational instability for the appropriate value of n ≈ -2, where d(ln ¯ ξ ) / d(ln R ) = -(3 + n ). The smallR behaviour of the halo model also has its own, modified scaling behaviour, with power-law correlations ¯ ξ k ∝ R -( k -1) γ + δ , where γ = (9 + 3 n ) / (5 + n ) and δ = (3 + n ) p ' / (5 + n ); p ' characterizes the small-mass behaviour of the halo mass function, d n/ d m ∼ ν p ' /m 2 (Ma & Fry 2000a; Scoccimarro, Sheth, Hui, & Jain 2001). In terms of ¯ ξ 2 , this is with ∆ = δ/γ = p ' / (3 -p ' ), independent of spectral index n . This dependence implies a modified scaling, expected to hold at small R . This behaviour was anticipated numerically by Colombi, Bouchet, & Hernquist (1996), who also point out that this modified scaling cannot persist on all scales. Figure 9 shows the success of this scaling for p ' = -0 . 2, ∆ = -0 . 0625. This value is different, even in sign, from the scaling exponent inferred from low order hierarchical amplitudes, although both are numerically small, and may indicate a change in the mass dependence of halo mass function at smaller masses. In evaluating results for dark matter particles, we must also keep in mind that it is possible that some effect remains of the initial grid. Colombi, Bouchet, & Schaeffer (1995) suggest that void results should be reliable only P 0 /greaterorsimilar 1 /e , but we see no change of behaviour on different sides of this boundary.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "4 DISCUSSION", "content": "The implication of the halo model for void probability on large scales is simple: to have no galaxies means no halos, no halos means no galaxies. This point cluster limit of the halo model provides a natural answer to the otherwise puzzling question, Why do voids obey the hierarchical scaling when the correlation functions do not satisfy the hierarchical premise, constant S n . In the limit that the volume considered is large compared to halo sizes, void of galaxies means void of haloes, the void probability P 0 is the same, and scaling curves are related by number and by clustering strength or bias. Since ¯ N g > ¯ N h (a halo contains one or more galaxies) and ¯ ξ g > ¯ ξ h (bias is an increasing function of mass), we anticipate points on the scaling plot move down and to the left. In the simulations as well as in observations, the negative binomial scaling curve is a good approximation to that for galaxies; the weak clustering lognormal curve is much less favored, and the more strongly clustered lognormal even less so. We expect the void scaling relation to provide different scaling curves for different galaxy populations; that galaxy results are often in agreement with the negative binomial curve can be attributed to the number density and clustering strength of typical galaxies. With different results for different galaxy populations, in a direction consistent with relative bias and number density, we conclude that there is no fundamental reason that galaxies follow the negative binomial scaling curve, but that this follows from typical galaxy parameters. The success of void scaling for galaxies requires that the underlying halo distribution follows the hierarchical pattern of higher order clustering. In our simulations (Fry et al. 2011), the halo S k,h are approximately constant, roughly S 3 ,h ≈ 1, S 4 ,h ≈ 2, and S 5 ,h ≈ 3, over a limited range of scales squeezed between the finite size of the simulation on the large end and the ability to separate extended objects on the small end. These values of the S k,h do not change much for different mass ranges. More important, as seen in Fig. 2, different halo samples have remarkably similar scaling curves: for mass thresholds ranging over a factor of 20 and number densities different by a factor of more than 4, the scaling curves are indistinguishable and seem to follow well the geometric halo mode curve of equation (A16). The scaling is important, because there is essentially no direct observational information for S k,h . What results do exist are only for much higher mass thresholds: Jing (1990) measures the void scaling function for ACO clusters and Cappi et al. (1991) for samples defined by Postman et al. (1986) and Tully (1987), but their results only reach ¯ N ¯ ξ /lessorsimilar 2, for which all model scaling curves are much the same. Jing & Zhang (1989) find that Abell clusters have a hierarchical three- ction with amplitude independent of richness class, and Cappi & Maurogordato (1995) also find, to a degree, constant S k amplitudes for Abell and ACO clusters (but with a systematic difference between northern and southern galactic hemispheres), with numerical values S 3 ≈ 3, S 4 ≈ 15, S 5 ≈ 100, appropriate to the high threshold, rare halo limit S k = k k -2 of Bernardeau & Schaeffer (1999). These do not apply directly to statistics of halos that host galaxies, including single galaxies and so extend down to galaxy masses; a theory that predicts the halo amplitudes S k,h or the halo scaling curve for mass thresholds of 10 11 or 10 12 M /circledot has yet to be found. Dark matter behaves differently. For dark matter the behaviour of voids depends strongly on the density of particles. In the quasilinear regime on large scales, the behavior seems to follow the predictions of gravitational instability. For mass, there is no smallest object, no smallest cluster, and for any scale there are always clusters smaller and larger than that size. The halo model has implications for high order functions, Small scales follow a modified scaling predicted by the halo model, as in equation (41). Halo model mappings have been derived in order to apply to observational data. Tinker et al. (2008) present in their fig. 7( d ) void scaling curves for SDSS blue and red galaxies in which the locus for red galaxies is shifted substantially to larger values of ¯ N ¯ ξ ; similar results are found for red and blue 2dFGRS galaxies by Croton et al. (2007). Figure 10 shows the SDSS data (filled circles) and the results of halo model scalings applied to red and blue galaxies for three different assumptions about the underlying haloes: assuming red and blue galaxies reside in the same haloes (open triangles), assuming a power-law halo scaling curve with ω = 1 (open circles), and assuming their parent haloes trace same halo scaling curve (open squares), using the values b red = 1 . 02, b blue = 0 . 85, n red = 0 . 00328, n blue = 0 . 00433 h 3 Mpc -3 , and ratios ( b g /b h ) red = 1 . 53, ( n g /n h ) red = 1 . 93, ( b g /b h ) blue = 1 . 18, ( n g /n h ) blue = 1 . 21. computed from analytic halo occupation distributions for red and blue samples given by Tinker et al. (2008). Blue squares and red triangles begin to show departures from simple scalings, which should apply only in the large-scale limit. The last, relative scaling is perhaps the most realistic, but the halo assumptions overlap and all of the scalings behave similarly. This is confirmation that the ideas of the halo model apply to observations, as well as to simulations. The void scaling results illustrate yet another success of the halo model, in describing nonlinear phenomena that it was not designed and not optimised to explain. Applied to dark matter, the model may still be at best an approximation, but for galaxies, on scales where details of the structure of haloes are irrelevant, it is almost necessarily true: the total number of galaxies is the sum over haloes of the number of galaxies in each halo, and the combinatoric results of the halo model are independent of whether there is such a thing as a universal profile shape or not.", "pages": [ 7, 8 ] }, { "title": "Acknowledgments", "content": "The question of why is it that voids in the galaxy distribution obey hierarchical scaling when their correlations do not follow the hierarchical pattern was raised by Darren Croton at the Summer 2007 workshop on Modelling Galaxy Clustering at the Aspen Center for Physics. We thank David Weinberg for providing the SDSS results. JNF acknowledges support from the City of Paris, Research in Paris program and thanks the Pauli Institute for Theoretical Physics, University of Zurich, and the Institut d'Astrophysique de Paris for hospitality during this work. This research has made use of NASA's Astrophysics Data System.", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "Alimi J.-M., Blanchard A., & Schaeffer R., 1990, ApJ, 349, L5 Balian R., Schaeffer R., 1988, ApJ, 335, L43 Bernardeau F., 1992, ApJ, 392 , 1 Bernardeau F., 1994, A&A, 291, 697 Bernardeau F., Schaeffer R., 1999, A&A, 349, 697 Bouchet F. R., Strauss M. A., Davis M, Fisher K. B., Yahil A., Huchra J. P., 1993, ApJ, 417, 36 Cappi A., Maurogordato S., 1995, ApJ, 438, 507 Cappi A., Maurogordato S., Lachieze-Rey M., 1991, A&A, 243, 28 Carruthers P., Minh D.-V., 1983, Phys. Lett. 131B, 116 Carruthers P., Shih C. C., 1983, Phys. Lett 127B, 242 Carruthers P., Shih C. C., 1987, Internat. J. Mod. Phys. 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S., Geller M. J., Park C., Huchra J. P., 1994, AJ, 108, 745 White S. D. M., 1979, MNRAS, 186, 145 Yang W., Xuan R., Li L., 1983, Geochemistry, 1, 52", "pages": [ 8, 9 ] }, { "title": "APPENDIX A: MODELS", "content": "In this Appendix we present several models with specific analytic forms for the void scaling function χ ( ¯ N ¯ ξ ), useful against which to compare observational and numerical results. Many of these models, introduced previously in a variety of different contexts (see Fry 1986; Mekjian 2007), can be realised as halo models with a Poisson halo distribution; several were discussed by Sheth (1996). With mean µ , probabilities p n = µ n e -µ /n !, the Poisson generating function is in particular, the void probability is P 0 = e -¯ N h = e -¯ N/ ¯ N i . For an unclustered halo distribution, correlations of galaxy number are given by the last term in eqs. (10)-(13), and a Poisson halo distribution is always hierarchical of a sort, with scaling function χ = -ln P 0 / ¯ N = 1 / ¯ N i , scaling variable ¯ N ¯ ξ = ¯ N i ¯ µ 2 ,i , and amplitudes S k = ¯ µ k,i / ¯ µ k -1 2 ,i all determined by the occupation distribution (although not every occupation distribution has constant S k ). Fig. A1 compares models detailed in the following.", "pages": [ 9 ] }, { "title": "A1 Minimal Poisson Model", "content": "A Poisson sum of clusters with mean µ with Poisson occupancy distribution with mean ν has", "pages": [ 9 ] }, { "title": "10 J. N. Fry and S. Colombi", "content": "From derivatives of G we have moments 〈 N [ k ] 〉 (equation 4), from which we obtain ν = ¯ N ¯ ξ , µ = 1 / ¯ ξ , void probability P 0 = G (0) = exp [ -(1 -e -¯ N ¯ ξ ) / ¯ ξ ] , and thus This minimal hierarchical model, with S k = 1 for all k , saturates the Schwarz inequality requirement that the hierarchical amplitudes obey S 2 m S 2 n /greaterorequalslant S 2 m + n (Fry 1986). A Poisson occupation distribution formally includes possibility empty haloes. The same result can be achieved by excising empty haloes and rescaling (Fry et al. 2011), so that the occupation generating function becomes Equation (A5) is the limit a →∞ of the hypergeometric model of Mekjian (2007), which has The remainder of the calculation is straightforward; although the relation between ν and ¯ N i changes, again ν = ¯ N ¯ ξ and χ = (1 -e -¯ N ¯ ξ ) / ¯ N ¯ ξ . The minimal model scaling curve is plotted as the dotted (black) line in Fig. A1.", "pages": [ 10 ] }, { "title": "A2 Negative Binomial Model", "content": "The negative binomial distribution with mean ¯ N and parameter K (also called Pascal, if K is an integer, or P'olya distribution if K is real), has count probabilities For K = 1 this reduces to the Bose-Einstein distribution, and is sometimes also referred to as modified Bose-Einstein. This distribution appears in the frequency of industrial accidents (Greenwood & Yale 1920), the distribution of ancient meteorites found in China (Yang, Xuan, & Li 1983), in quantum optics (Klauder &. Sudarshan 1968), and in the multiplicity of charged particles produced in high energy collisions (Carruthers & Shih 1983, 1987; Carruthers 1991) and cosmic ray showers (Teich, Campos, & Saleh 1987), as well as in large-scale structure (Neyman et al. 1953; Carruthers & Shih 1983; Carruthers & Minh 1983; Carruthers 1991; Gazta˜naga 1992; Elizalde & E. Gazta˜naga 1992; Gazta˜naga & Yokohama 1993), where it is often found to be a good approximation to the observed scaling curve of galaxies. The negative binomial can be realised as a Poisson sum of clusters with logarithmic occupation distribution (Sheth 1995). The halo and occupation generating functions are The probability of a void is G (0) = e -µ , and χ = µ/ ¯ N ; it is only necessary to relate these to moments ¯ N , ¯ ξ obtained from G ' (0) and G '' (0) as in eqs. (A3), (A4), Then, It has been suggested that convergence of the logarithmic series defined by equation (8) with S k = ( k -1)! limits ¯ N ¯ ξ < 1; but the probability generating function formulation has no restriction. Equation (A14) is the limit a → 1 of the hypergeometric model of Mekjian (2007). The negative binomial model scaling curve is plotted as the solid (blue) line in Fig. A1.", "pages": [ 10 ] }, { "title": "A3 Geometric Hierarchical Model", "content": "An occupancy distribution with probability p n ∝ p n for n /greaterorequalslant 1 has occupancy generating function g i ( z ) = z (1 -p ) / (1 -pz ), and From the first and second moments, ¯ N = µp/ (1 -p ) and ¯ N 2 ¯ ξ = 2 µp/ (1 -p ) 2 , we find p = 1 2 ¯ N ¯ ξ/ (1 + 1 2 ¯ N ¯ ξ ), and The geometric halo model is the case a = 2 of the hypergeometric model of Mekjian (2007), the model of Hamilton (1988) with Q = 1 2 , and also the ω = 1 instance of the form χ = 1 / (1 + ¯ N ¯ ξ/ 2 ω ) ω cited in Alimi et al. (1990). Although not plotted, the geometric model falls between the minimal and negative binomial curves in Fig. A1.", "pages": [ 10 ] }, { "title": "A4 Quasi-Equilibrium Model", "content": "Saslaw & Hamilton (1984) apply thermodynamics to obtain a gravitational quasi-equilibrium distribution function. The resulting distribution is once again a halo model, a Poisson sum of haloes, with Borel occupation distribution (Sheth & Saslaw 1994; Sheth 1996), and with total count probabilities The void probability is P 0 = e -¯ N (1 -b ) , and as the second moment gives the scaling function is Saslaw and Hamilton assume the functional form b (¯ nT -3 ) = b 0 ¯ nT -3 / (1 + b 0 ¯ nT -3 ) to interpolate between ideal gas ( b → 0) and virialized ( b → 1) limits. Sheth (1995) shows that invoking instead the form b = 1 -ln(1 + b 0 ¯ nT -3 ) /b 0 ¯ nT -3 , (which has the same limits), the negative binomial also arises as a quasi-equilibrium model. The quasi-equilibrium model scaling curve is plotted as the long dash/short dash (green) line in Fig. A1. This model is also the ω = 1 2 instance of the form χ = 1 / (1 + ¯ N ¯ ξ/ 2 ω ) ω cited in Alimi et al. (1990).", "pages": [ 10, 11 ] }, { "title": "A5 Lognormal Model", "content": "It has been found that in the limit of very high threshold, a clipped Gaussian field produces a distribution with Q k = 1, S k = k k -2 for all k (Politzer & Wise 1984; Szalay 1988), the ν = 0 model of (Schaeffer 1984) and a result that holds in the rare halo limit under some very general condition (Bernardeau & Schaeffer 1999). For this set of amplitudes the scaling function is written parametrically (Schaeffer 1984) as This also constitutes the lower envelope of the lognormal distribution, suggested by Hubble (1934) and more recently considered by Coles & Jones (1991); although the full lognormal distribution does not in general scale, lognormal voids approach this curve for ¯ ξ /lessmuch 1 (numerically found to hold for ¯ ξ /lessorsimilar 1). The Schaeffer model, or lower bound of the lognormal distribution, is plotted as the long dash/short dash (green) line in Fig. A1.", "pages": [ 11 ] }, { "title": "A6 Gravitational Instability", "content": "The gravitational instability amplitudes S k can be computed in perturbation theory, which gives S 3 = 34 / 7 (Peebles 1980), S 4 = 60 712 / 1 312 (Fry 1984), etc. The complete set of amplitudes can be obtained from a generating function (Bernardeau 1992). In particular, the function is obtained as a transform of the vertex generating function G ( τ ) by with χ ( y ) = 1 + ϕ/y . The function G ( τ ) is found parametrically, the same hypercycloid functions that describe the time evolution of spherical underdensities (Peebles 1980). A useful analytic approximation to this function has been found to be The gravitational instability scaling curve is plotted as the long-dashed (cyan) line in Fig. A1. The smoothing in computing volume-averaged moments modifies the values of the S k and so also the scaling curve. For a power-law power spectrum, P = Ak n , Bernardeau (1994) shows that the windowed vertex generating function becomes G s = G [ τ (1+ G s ) -(3+ n ) / 6 ]. With the approximation of equation (A26), the effect of smoothing on a scale where ¯ ξ ( R ) has effective power index d(ln ¯ ξ ) / d(ln R ) = -(3 + n ) then follows from which can in some cases be solved analytically and in all cases can be used to obtain G , ϕ , and χ numerically. Dashed (cyan) curves in Fig. A1 show the windowed gravitational instability result for n = -3, -2, -1, 0, and +1 (top to bottom). The n = +1 windowing of the gravitational instability scaling function is remarkably similar to the minimal model, and the n = 0 mapping of the gravitational instability function is remarkably similar to the negative binomial model.", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "A covariant scattering kernel is a core component in any self-consistent general relativistic radiative transfer formulation in scattering media. An explicit closed-form expression for a covariant Compton scattering kernel with a good dynamical energy range has unfortunately not been available thus far. Such an expression is essential to obtain numerical solutions to the general relativistic radiative transfer equations in complicated astrophysical settings where strong scattering effects are coupled with highly relativistic flows and steep gravitational gradients. Moreover, this must be performed in an efficient manner. With a self-consistent covariant approach, we have derived a closed-form expression for the Compton scattering kernel for arbitrary energy range. The scattering kernel and its angular moments are expressed in terms of hypergeometric functions, and their derivations are shown explicitly in this paper. We also evaluate the kernel and its moments numerically, assessing various techniques for their calculation. Finally, we demonstrate that our closed-form expression produces the same results as previous calculations, which employ fully numerical computation methods and are applicable only in more restrictive settings. Key words: radiative transfer - scattering - relativity.", "pages": [ 1 ] }, { "title": "Ziri Younsi 1 /star and Kinwah Wu 1", "content": "1 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey, RH5 6NT, UK Accepted ***. Received *** in original form ****", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Compton scattering of photons by relativistic electrons is an efficient process to produce high-energy cosmic X-rays and γ -rays. It plays an important role in determining spectral formation and in regulating energy transport in a variety of astrophysical systems, e.g. accretion disks of black hole systems (Sunyaev & Titarchuk 1985; Dermer & Liang 1989; Haardt 1993; Poutanen & Vilhu 1993; Titarchuk 1994; Hua & Titarchuk 1995; Stern et al. 1995), relativistic AGN jets (Begelman & Sikora 1987; McNamara, Kuncic & Wu 2009; Krawczynski 2012), neutron-star X-ray bursts (Titarchuk 1988; Madej 1991; Titarchuk 1994; Madej, Joss & R'o˙za'nska 2004), and in some accreting white dwarfs (Kylafis & Lamb 1982; Matt 2004; McNamara et al. 2008; McNamara, Kuncic & Wu 2008; Titarchuk, Laurent & Shaposhnikov 2009). Compton scattering of cosmic microwave background photons by hot gases trapped inside the potential wells of large gravitating systems, such as galaxy clusters, also leads to Sunyaev-Zel'dovich effects (Sunyaev & Zeldovich 1980; Rephaeli 1995; Dolgov et al. 2001; Colafrancesco, Marchegiani & Palladino 2003), through which various aspects of cosmology and the evolution of large-scale structures in the Universe may be investigated. Compton scattering in astrophysical plasmas is often investigated using Monte-Carlo simulations (e.g. Pozdnyakov, Sobol & Syunyaev 1983; Hua & Titarchuk 1995). The Monte-Carlo approach is an approximation scheme to proper radiative transfer calculations, where the radiative transfer equation is derived from the laws of conservation (see Rybicki & Lightman 1979; Chandrasekhar 1960; Peraiah 2001). It has the advantage of being able to handle complicated system geometries, as well as the flexibility to incorporate relevant additional physics, such as absorption and pair production, into the system. However, it is not straightforward to implement the usual Monte-Carlo method in certain extreme astrophysical environments, such as systems with steep density gradients or fractal-like inhomogeneities, and ultra-relativistic flows near the event-horizon of a black hole. In the latter, relativistic and space-time curvature effects are important, and radiative transfer in these systems requires a covariant formulation (Lindquist 1966; Baschek et al. 1997; Fuerst & Wu 2004; Younsi, Wu & Fuerst 2012). In the absence of scattering, the covariant radiative transfer can be solved along the null geodesic (see Viergutz 1993;", "pages": [ 1 ] }, { "title": "2 Z. Younsi and K. Wu", "content": "Reynolds et al. 1999; Dexter & Agol 2009) using a ray-tracing technique (e.g. Fuerst & Wu 2004; Vincent et al. 2011). In the presence of scattering, the covariant transfer equation is much more complicated, and the transfer equation is no longer a differential equation but an integro-differential equation. A key ingredient in the radiative transfer formulation is the scattering kernel, which describes how photons interact with electrons. The moment expansion (Thorne 1981; Turolla & Nobili 1988; Rezzolla & Miller 1994; Challinor 2000; Fuerst 2006; Wu et al. 2008; Shibata et al. 2011) of this kernel is essential in deriving a practical (numerical) scheme to solve the integro-differential radiative transfer equation (see Fuerst 2006; Farris et al. 2008; Zanotti et al. 2011). This article shows explicitly the derivation of the invariant scattering kernel for Compton scattering in a general relativistic setting and finds a closed-form expression in terms of hypergeometric functions. The method is not limited by energy range and is valid both for Compton and inverse Compton scattering. The article is organised as follows. § 2 introduces the covariant radiative transfer equation in the presence of scattering and discusses methods for its solution. § 3 derives the covariant KleinNishina cross-section for relativistic Compton scattering. § 4 derives from first principles the relativistic electron distribution function, which must later be convolved with the Klein-Nishina cross-section. § 5 presents an outline of the derivation of the integral from of the covariant Compton scattering Kernel. § 6 outlines a method to simplify the calculation of successive angular moments of the scattering kernel through changing the order of integration. § 7 derives algebraic expressions for the first three angular moments of the scattering kernel. § 8 outlines a method for deriving the angular moments of the scattering kernel, through employing recursion identities. § 9 demonstrates how these moment integrals may be expressed in closed-form, for arbitrary order n , in terms of Gauss hypergeometric functions. This yields an analytic result for moments of the Klein-Nishina cross-section, M n . § 10 performs the integration of the convolution of the moments of the Klein-Nishina cross-section with the relativistic electron distribution function, using the methods outlined in the previous chapters. § 11 is devoted to the discussion and § 12 the summary.", "pages": [ 2 ] }, { "title": "2 RADIATIVE TRANSFER WITH SCATTERING", "content": "In Newtonian space-time the radiative transfer equation in a medium reads (see Mihalas & Mihalas 1984; Peraiah 2001) where I ν ( ˆ Ω ) is the intensity of the radiation at a frequency ν propagating in the ˆ Ω -direction, j ν and κ ν are the emission and absorption coefficient respectively, and σ ( ν, ˆ Ω ; ν ' , ˆ Ω ' ) is the scattering kernel which determines the amount of radiation intensity at a frequency ν ' in a direction ˆ Ω ' being scattered into the intensity I ν ' ( ˆ Ω ' ). For instance, in the photon-electron scattering process, the scattering kernel is determined by the momentum distribution of the electrons and the differential scattering cross-section, the Klein-Nishina (Klein & Nishina 1929) differential cross-section where e is the electron charge, m e is the electron mass, σ T is the Thomson cross-section (Thomson 1906; Compton 1923), k i and k f are the wave numbers of the photon before and after the scattering respectively, and ˆ /epsilon1 i and ˆ /epsilon1 f are the corresponding polarisation vectors of the photon. The function f ( k f , ˆ /epsilon1 f ; k i , ˆ /epsilon1 i ) is given by In the absence of scattering, the covariant form of the radiative transfer equation may be written as (Baschek et al. 1997; Fuerst & Wu 2004; Wu et al. 2008; Younsi, Wu & Fuerst 2012), where I is the invariant intensity of the radiation, x α is a position 4-vector, η 0 and χ 0 are the invariant emission and absorption coefficients respectively (evaluated in a local inertial frame), ξ is the affine parameter, k α is the propagation (wave number) 4-vector of the radiation, and u α is the 4-velocity of the medium interacting with the radiation. Equation (4) is similar in form to equation (1) without the scattering term. The term k α u α | ξ is a correction factor for the aberration and energy shift in the transformation between reference frames. For covariant transfer of radiation in the presence of scattering, the radiative transfer equation is of the form analogous to equation (1). Several methods have been proposed to solve the above equation or to obtain an approximate solution. For instance, one could transform the integro-differential radiative transfer equation into a set of differential equations using a moment expansion (Thorne 1980, 1981; Fuerst 2006; Wu et al. 2008; Shibata et al. 2011). Nevertheless, one needs to specify the properties of the medium spanning the space-time. In addition to the global flow dynamics, one also needs to know how the radiation interacts with the medium (via the emission coefficient, absorption coefficient and the scattering λ ≡ - kernel), at least in the local inertial frame. The invariant emission and absorption coefficients can be easily derived from the conventional emission and absorption coefficients (see Fuerst & Wu 2004, 2007). The derivation of the scattering kernel is more complicated. Some attempts have been made (e.g. Shestakov, Kershaw & Prasad 1988), but only numerical results were obtained due to the complexity of the underlying mathematics. To date a closed-form expression for the corresponding scattering kernel is not available. The lack of a closed-form scattering kernel hinders the development of fast and accurate numerical algorithms to solve the covariant radiative transfer equation, which itself can be numerically intensive.", "pages": [ 2, 3 ] }, { "title": "3 COVARIANT COMPTON SCATTERING", "content": "Here and hereafter this article adopts the geometrical unit convention (with G = c = h = 1) and employs the ( -, + , + , +) metric signature. Energy-momentum conservation implies that in a photon-electron scattering process. Here unprimed and primed variables denote, respectively, variables evaluated before and after scattering. The 4-momentum of a photon k α and the 4-momentum of an electron p α satisfy k α k α = k ' α k ' α = 0 and p α p α = p ' α p ' α = -m 2 e , respectively. Energy-momentum conservation also leads to the invariance relation and a covariant generalised energy-shift formula for the scattered photon, As the scattering process occurs in a relativistic fluid, the derivation of the scattering opacity due to ensembles of photons and electrons requires expressing the scattering variables of the particles in the local reference rest-frame (co-moving with the fluid 4-velocity), as well as specifying the transformation between the fluid rest-frame and the observer's frame. The fluid 4-velocity, in the fluid rest frame, is denoted as u α . The electron 4-velocity is v α . Clearly u α u α = -1 and v α v α ≡ v < 1. The directional unit 4-vector of the photon in the fluid rest frame may be specified as n α , which is given by where the tensor P αβ = g αβ + u α u β projects onto the 3-surface orthogonal to k β . A variable may be constructed, from which n α may be expressed as Hence, it follows the photon 4-momentum may be expressed as Similarly, for the electrons, p α u m α e Clearly λ = 1 / √ 1 -v 2 , which is simply the Lorentz factor of the electron. The directional 4-velocity of the electron in the fluid frame is therefore It therefore follows that and Note that the photon 4-momentum after the scattering event is . (13) Thus, the following expressions are obtained: where ζ = n α n ' α is the direction cosine of the angle between the incident and scattered photon. Hence energy-momentum conservation, equation (8), may be expressed as: The cross-section for scattering of a photon by an electron is given in Kershaw, Prasad & Beason (1986) as: where D ≡ 1 -ˆ Ω · v /c = 1 -v α n α , and similarly for D ' . Using equations (18)-(20), the photon-electron scattering cross-section, equation (22), may be expressed in the following covariant form: where δ denotes the Dirac delta function, and T , P are defined respectively as T ( p ( k α k α α k ' α )( p 2 ) β p ' β ) It follows that P represents energy and momentum conservation of the scattering process. The delta function enforces the conservation of energy and momentum in the scattering process, by weighting the scattering cross-section such that it is zero if energy and momentum are not conserved. Integrating this cross-section, equation (23), over a relativistic electron distribution function yields the kernel for Compton scattering.", "pages": [ 3, 4 ] }, { "title": "4 ELECTRON DISTRIBUTION FUNCTION", "content": "In order to calculate the Compton scattering kernel the relativistic electron distribution function, f ( λ ), must be determined. This may be derived as follows. The energy of an electron is E = λ m e c 2 , and its linear momentum is given by p = λ m e v , from which it follows that As an example, consider an ensemble of relativistic electrons with isotropic momenta for which the distribution function is given by the pseudo-Maxwellian where E is the electron energy, T e the electron temperature, k B the Boltzmann constant and C is a normalisation constant. Note that the distributions of electrons in momentum space and in velocity space are related via which may be expressed as It immediately follows that f ( v ) = C ' λ ( v ) 5 e - λ ( v ) /τ , (30) where C ' = m 3 e C is a constant and τ = k B T e /m e . The normalisation of the distribution function f ( v ) to unity, i.e. yields the familiar relativistic Maxwellian form, where K 2 denotes the modified Bessel function of the second kind. = , (24) A 2 4", "pages": [ 4, 5 ] }, { "title": "5 COMPTON SCATTERING KERNEL", "content": "The Compton scattering kernel, as seen in equation (5), is essential in solving the radiative transfer equation. It is determined by the convolution of the photon-electron scattering cross-section with the electron velocity distribution, i.e. where ρ is the electron density. To evaluate the above integral, first consider the argument of the delta function Rewriting (34) in terms of a linear combination of a scalar and an inner product of two unit vectors is a succinct way of expressing the energy-momentum conservation. More importantly, aside from the more compact notation, the inner product of two unit vectors (the magnitude of which never exceeds unity) provides constraints on the electron energy. This makes the subsequent integrals easier to solve, and is the most natural way of proceeding with the problem. Substituting equations (18)-(20) into (34) yields where and hence (35) is split into a scalar and vector component. It immediately follows that y may be rewritten as where and q , akin to the resultant photon energy along the direction of photon momentum transfer, is defined as Therefore ˆ w α represents a unit vector along the direction of photon momentum transfer and ˆ v α ˆ w α is simply the projection of the electron velocity onto this preferred direction. Under integration, the delta function can be rewritten as δ (Γ /w +ˆ v α ˆ w α ) /w , and the energy-momentum conservation may be rewritten as From this it immediately follows || -Γ /w || /lessorequalslant 1 and therefore which is akin to solving the quadratic equation Aλ 2 -Bλ -C = 0, with coefficients A , B and C given by: , (44) C = q + . (46) Taking the positive solution to (43) yields, upon employing the identity q 2 = ( γ ' -γ ) 2 -A , which is essentially the minimum electron energy in the Compton scattering process. The form of λ as a function of ζ is crucial in later calculations involving integrations over λ and ζ . The integral in equation (33) may now be rewritten as 2", "pages": [ 5 ] }, { "title": "6 Z. Younsi and K. Wu", "content": "Hence it follows that the delta function fixes this preferred direction naturally (Prasad, Kershaw & Beason 1986; Beason, Kershaw & Prasad 1991), and this is clearly the most straightforward approach. Note, as in Kershaw, Prasad & Beason (1986), the angular addition formula: where ˆ m α is equal to ˆ w α or ˆ w ' α , the unit vector of the photon velocity before or after collision respectively. It is easily verified that As such, in equation (48) only the φ integral need be evaluated explicitly. The square-bracketed term in the kernel may be rewritten (e.g. Kershaw, Prasad & Beason 1986) as which must be integrated term-by-term over φ . The integrals to solve have the forms: where α = 1 - v ( ˆ w α α ˆ v ) β = -v √ 1 -( ˆ w α n α √ Clearly, the two integrals are related, via I 2 = -d I 1 d α and therefore only I 1 need be evaluated, yielding 2 ) , (56) 1 - ( ˆ w α ˆ v α ) 2 . (57) where the coefficients α ≡ α ( x ), β and α 2 -β 2 are given by wherein the notation x ≡ γ + λ prior to collision and x ≡ γ ' -λ after collision is adopted. Additionally, ω 2 = (1 + ζ ) / (1 -ζ ). The φ -integrals immediately follow, yielding n α )( ˆ w α 1 , 2", "pages": [ 6, 7 ] }, { "title": "Covariant Compton Scattering Kernel 7", "content": "The Compton scattering kernel in equation (33) may now be rewritten as where the function R ( x ) is defined as where w ≡ w ( x ), with w ( x ) = [ x ( γ -1 + γ '-1 ) -1 ] . The scattering kernel, as it is written in equation (70), is highly symmetric and essentially the sum of three terms: the resultant photon energy along the direction of momentum transfer, a pre-collisional photon-electron interaction term, and less a post-collisional photon-electron interaction term, with the interaction term defined in equation (71).", "pages": [ 7 ] }, { "title": "6 ANGULAR MOMENTS OF THE COMPTON CROSS SECTION", "content": "In solving the full radiative transfer equation with Compton scattering, a generalised Eddington approximation (Eddington 1926; Rybicki & Lightman 1979) to compute successive angular moment integrals of σ s may be employed (Thorne 1981; Fuerst 2006; Wu et al. 2008). In this section, angular moments of the form ζ n (e.g. Shestakov, Kershaw & Prasad 1988, and references therein) are used to define the moment expansion of the Compton scattering kernel. This requires solving integrals of the form However, as equation (72) stands, integrating over f ( λ ) is analytically impossible. Rather than perform the λ integration first, it is more straightforward to switch the order of integration. Not only does this enable the derivation of analytic results, performing the λ integration after the ζ integration affords the method greater generality, since the ζ integral is independent of the assumed electron distribution function (in the isotropic case). To change the order of integration, first consider λ + ( ζ ) (which must be inverted, i.e. ζ ( λ + ) found), with the left boundary λ + ( -1) found as whereas is the right boundary. The minimum value of λ + , i.e. the value of ζ such that λ + is minimised, is found as ζ = 1 ± and hence ( ) 2 . (77) ∣ ∣ γ -1 -γ '-1 ∣ ∣ /greaterorequalslant Rearranging λ + to find ζ as a function of λ yields It immediately follows that the order of integration may be reversed as at the expense of evaluating two different integrals. However, if λ L /lessorequalslant λ min then λ min = λ L and the second term in equation (79) vanishes, necessitating evaluation of the first double integral only (see Fig. 1). γ - 1 - γ '- 1 , (75)", "pages": [ 7 ] }, { "title": "7 PERFORMING THE ANGULAR MOMENT INTEGRALS", "content": "In evaluating equation (72) with equation (79), three different types of moment integral arise, namely Note the identity d S n, 2 d x ≡ -xR n . With the aforementioned definitions the angular moment function of order n , M n , may be written as where A n = 2 γγ ' Q n , (85) In equations (81) and (82), the integrals have an x -dependence which is crucial to their evaluation. As noted earlier, x ≡ ( γ + λ ) or x ≡ ( γ ' -λ ), depending on whether the integral is pre-collisional or post-collisional. The evaluation of these integrals yields different results depending on whether x 2 < 1, x 2 = 1 or x 2 > 1. The moment integrals may be integrated analytically, although the resultant expressions are algebraically cumbersome. The n = 0 , 1 , 2 moments for A 0 and B 0 are as follows: (87)", "pages": [ 8 ] }, { "title": "For x 2 = 1:", "content": "/negationslash where and the function C ( x ) is defined as For x 2 = 1: In principle equations (80)-(82) may be integrated for arbitrary n , but, as seen in equations (87)-(97), the resultant algebraic expressions become extremely cumbersome. Moreover, the expressions for A n must be evaluated either two or four times per scattering event, and B n either four or eight times per scattering event. Given the inherent algebraic complexity, and the number of calls required per scattering event, this will lead to significant loss of precision, in particular between cancellations of terms of similar value or of particular smallness (e.g. Poutanen & Vurm 2010). Using equations (84)-(86) the Compton scattering kernel may be written more compactly as where, as noted before, the second term in square brackets in the above equation vanishes when λ L /lessorequalslant λ min , saving significant computational expense. In the case of x 2 = 1, the moment integrals simplify significantly. This is as far as it proves possible to proceed analytically. Integrations over λ would have to be performed with an appropriate numerical scheme. Naturally, the question arises as to whether the integrals in equation (98) can be performed analytically. As it stands, the method presented thus far would require arbitrary precision arithmetic to evaluate, and therefore be computationally expensive and time consuming. In the following section, the evaluation of integrals (80)-(82) is demonstrated analytically and in closed form, for arbitrary moment order.", "pages": [ 8, 9 ] }, { "title": "8 EVALUATING THE MOMENT INTEGRALS FOR ARBITRARY ORDER", "content": "The previous section derived analytic expressions for the first three moments of the Compton scattering kernel. As the order of the moments increases, the algebraic complexity of the resultant expression grows rapidly. Clearly the method, as it stands, does not lend itself readily to the evaluation of higher-order moments. These are necessary for more accurate evaluation of radiation transport problems. A much faster method is to evaluate equations (76)-(78) recursively. Firstly, consider equation (80) for Q n . By employing the identity upon integrating Q n by parts, the following recurrence relation immediately follows With the seed Q 0 = ( √ γ 2 + γ ' 2 -q ) /γγ ' , Q n may be evaluated for arbitrary n . Next consider equation (81) in the form where the substitution u = 1 -ζ has been employed, and c ≡ (1 -x 2 ) / 2. By expanding in series the term (1 -u ) n , equation (101) may be written as Defining the integral a recursion relation for equation (103) may be found by integrating by parts The value I R (0) immediately follows, but to perform recursively the seed value I R (1) is also needed Therefore R n may now be defined as which can be solved for arbitrary n . Similarly, for S n,m where After some working, the recursion relation for I S ( k, m ) is obtained as This identity requires four different seed values for the cases m = 0, 1 and 2: The numerical evaluation of these recursion relations in Fortran95 is shown in Fig. 2 for Q n and S n, 2 . For Q n it is clear the method is inaccurate for n > 20, regardless of the cosine of the scattering angle, ζ . For R n the method is numerically unstable for n > 30 for ζ = -1, as well as slowly convergent, regardless of the value of x . However, for ζ > -1 the method appears both numerically stable and rapidly convergent, even for n = 50. Similar results are obtained for S n,m as for R n , with the exception that for lower energies, S n, 0 is numerically unstable both for extreme backward scattering and extreme forward scattering beyond n = 30. More accurate evaluation would require the implementation of arithmetic precision beyond that of standard double precision. Thus equations (100), (106) and (107) enable (84) to be solved iteratively. In computing angular moments of the KleinNishina cross-section this will greatly reduce the computational time and resources required. Each moment integral can be computed recursively using the stored numerical value of the previous moment. Unfortunately, as the order increases, there will inevitably be loss of precision through differences of terms in the recursion relations. Further, it is impossible to perform the final integral over the electron distribution function without either an algebraic expression for each moment, or an appropriate closed-form expression for each moment in terms of more generalised functions. The following sections detail such a method based on the latter.", "pages": [ 9, 10 ] }, { "title": "9 EVALUATING MOMENT INTEGRALS - HYPERGEOMETRIC FUNCTION METHOD", "content": "In this section the moment integrals in equations (80)-(82) are evaluated in terms of ordinary hypergeometric functions (Bateman 1955). In terms of this function, the problem of relativistic Compton scattering is greatly simplified (Aharonian & Atoyan 1981). Hypergeometric functions are a very general class of functions which contain many of the known mathematical functions as special or limiting cases (Luke 1969; Abramowitz & Stegun 1972). The ordinary hypergeometric function of one variable, or Gauss hypergeometric function (Gauss et al. 1866), is defined by the series is the rising factorial or Pochhammer symbol (Bateman 1955). The series is absolutely convergent for | z | < 1, and terminates after a finite number of terms if either a or b is a negative integer. The case | z | /greaterorequalslant 1 may be solved by analytic continuation (Zhang & Jin 1996). Although z may take complex values, in this paper z is always real. With this definition the integrals Q n , R n and S n,m may be solved. Having written R n and S n,m in summation form in equations (106) and (107) simplifies things considerably. Using the series expansion (1 -u ) n = ∑ n k =0 ( -1) k ( n k ) u k , the following expressions for (80)-(82) are found Here are a few notes about the continuity of expressions (116)-(118). Q n is always within the convergence region, and only lies on the boundary in the case of a perfectly elastic collision i.e. Thomson scattering (Thomson 1906). Equations (117) and (118) can be divided into two cases: those which lie within the convergence region ( | z | < 1) and those that lie on the boundary or outside it ( z /lessorequalslant -1). The case z /lessorequalslant -1, i.e. ζ /greaterorequalslant ( x 2 +1) / ( x 2 -1), may be solved by analytic extension with the following expression which brings R n and S n,m into the convergence region. The Gauss hypergeometric function is well documented in the literature and there exist several codes in Fortran which can evaluate it both accurately and rapidly (e.g. Forrey 1997; Zhang & Jin 1996), in addition to handling all cases of differences of parameters and values which can give rise to numerical problems (e.g. Zhang & Jin 1996). In the special case x 2 = 1 the expressions for R n and S n,m reduce to which are detailed in Appendix A. Thus the moment integrals for all values of x have been defined in closed-form. Results of the direct numerical evaluation of the moment integrals Q n and S n, 2 are presented in Fig. 3. For Q n the direct hypergeometric function method is a significant improvement. This is obvious since, in closed-form, Q n only ever requires one function evaluation, irrespective of the moment order. However, for R n and S n,m this method fares no better, and is in fact worse for larger scattering angles than the recursive method. This is due to oscillating sums in the corresponding expressions. However, the closed-form nature of these expressions is necessary to define the scattering kernel analytically. Plots of the numerical evaluation of the moment integral M n as a function of n , evaluated in Python to high numerical precision, are shown in Fig. 4. For very low scattering angles the angular moments are oscillatory, as can be seen in the ζ = -1 case. However, this is not a numerical issue, but rather an intrinsic physical issue with the form of the Compton scattering kernel itself. Recall equation (41), which was derived in taking the direction of photon momentum transfer as the z -axis of integration. In doing this, q is uniquely defined by equation (41) and so the method is inherently somewhat oscillatory for ζ close to -1, i.e. scattering angles close to 0. The remainder of the paper proceeds with the hypergeometric function method, with the aforementioned numerical considerations in mind. The final step in computing the Compton scattering cross-section is integrating over the relativistic electron distribution function, which is detailed in the next section. In Fig. 5, M n is plotted as a function of ζ for low order and high order, odd and even moments n . Odd and even moments are plotted separately to emphasise the change in shape and decrease in size of M n as the order increases. Odd and even moments have a distinct shape which flattens and decreases in magnitude as the order increases. Clearly as the moment order increases, M n becomes less sensitive to moderate scattering angles and remains unchanged over an increasingly large range of ζ . The effect of increasing electron velocity is to shift the maximum of M n towards ζ = 1, i.e. back scattering, as well as reducing the absolute magnitude of M n .", "pages": [ 10, 11 ] }, { "title": "10 INTEGRATING OVER THE ELECTRON DISTRIBUTION FUNCTION", "content": "In the general case, in all of the literature at present, only integration over ζ or λ has been performed analytically generally a choice must be made between performing integrals of the angular moments or integrating over the electron distribution function. The sixth and final integration over photon energy can be performed numerically during the radiative transfer calculations at each point along a ray. Regardless, with the methods at present, one is left with at best two further sets of integrals to evaluate. Further, the problem as formulated in the current literature (Prasad, Kershaw & Beason 1986; Nagirner & Poutanen 1993; Poutanen & Vurm 2010) is algebraically cumbersome. It is common to resort to Monte-Carlo methods to solve the multi-dimensional integrals. To have a closed-form solution to the first five integrals, including the electron distribution function, would eliminate the need for evaluating multi-dimensional integrals and entail solving only the photon frequency integral along the ray, as is common in ray-tracing (see e.g. Vincent et al. 2011; Younsi, Wu & Fuerst 2012).", "pages": [ 11, 12 ] }, { "title": "10.1 Integrating over the electron distribution function for constant ζ", "content": "Convolving the moment integrals with the electron distribution function necessitates solving integrals of the form where the change of variable for pre-collision (post-collision) as ˜ x = γ + λ (˜ x = λ -γ ' ), followed by y = 1 -˜ x 2 /τ 2 has been introduced. The ± sign indicates pre/post-collision and α = u (1 -τ 2 ) / 2, β = uτ 2 / 2. Consider the Taylor expansion where the recursion s p is defined as Next consider the integral The integral (127) may be easily solved recursively: where Hence equation (122) may now be written as For fixed values of ζ this method works well. However, in the case of evaluating the full Compton scattering kernel, from the limits of integration in equation (79) it is clear that ζ ± is a function of λ . Consequently, the integral in equation (127) is no longer trivial and cannot be expressed in closed-form. The resultant integrations over λ must the be performed numerically. An algorithm to perform the integration is presented in the following subsection.", "pages": [ 12 ] }, { "title": "10.2 Integrating over the electron distribution function in general", "content": "To evaluate the full Compton scattering kernel, in full generality, there are two expressions of importance, namely A n = 2 γγ ' Q n , (131) Computing the Compton scattering kernel involves evaluating the following expression: I 7 ( ζ, x, λ 1 , λ 2 ) = The second term in equation (133), T 2 , vanishes if the condition given by equation (77) is satisfied, as noted previously. Solving equation (133) necessitates the definition of the following seven integrals I 2 λ ∫ 1 ) = d λ e - R n ( ζ, x ) 2 - 2 γγ ' d λ e S n, 2 ( ζ, x ) ∫ λ λ 1 where z ≡ ( 1 -x 2 ) (1 -ζ ) / 2 and the dependence of ζ and x on λ has been neglected, i.e. ζ ≡ ζ ( λ ) and x ≡ x ( λ ). Consider the functions where the dependence of I on ζ , x , λ 1 and λ 2 has been suppressed for the sake of brevity. The Compton scattering kernel may then be expressed as the composition of the following ten terms -∞ (149) (150) (151) (152) (148) λ/τ ( ζ, x, λ 1 , λ 2", "pages": [ 12, 13 ] }, { "title": "14 Z. Younsi and K. Wu", "content": "where Q n ( -1) is equivalent to Q n evaluated at ζ = -1. Recall ζ ± ≡ ζ ± ( λ ), as given in equation (78). Terms t 1 (pre-collisional) and t 2 (post-collisional) are independent of x . With the above ten terms T 1 and T 2 may now be written as where, as noted before, T 2 vanishes if condition (77) is satisfied. With T 1 and T 2 expressed, one may now evaluate equation (133) numerically. It is easily shown that the number of numerical integrals scales linearly with the moment order n and is given by 48 n + 51 or 24 n + 25, depending on whether T 2 need be evaluated. However, this is assuming the independent evaluation of each moment. In reality, in evaluating a moment n , all lower-order moments must also have been evaluated, and so the order of the method at each order n is given by ( n +1)(24 n +51) or ( n +1)(12 n +25). The angular moments of the full Klein-Nishina Compton scattering kernel may now finally be written as: where T ( γ, γ ' , τ ) ≡ T , as given in equation (133) and C = 3 ρσ T / 32 πm e .", "pages": [ 14 ] }, { "title": "10.3 Numerical implementation", "content": "In implementing the formulation in the previous subsection numerically, several considerations and modifications of the formulae need to be considered. A prominent problem is the magnitude of the 1 /τ K 2 (1 /τ ) term in the expression for the scattering kernel at electron temperatures below 10 keV. For an electron temperature of 10 keV its value is 4 . 378 × 10 24 , at 1 keV its value is 7 . 717 × 10 225 and moving down to temperatures of 1 meV, the lower-end of temperatures we will investigate numerically, the corresponding value is 1 . 649 × 10 221924493 . On this basis alone, any numerical computation of the scattering kernel would immediately require very high numerical precision indeed, particularly at temperatures below 1 keV. Accordingly, all of the numerical integrals in Equations (134)-(140), particularly in the case of nearly elastic collisions, will be of corresponding numerical smallness so as to cancel such large terms, since the value of the scattering kernel in this case is generally of the order of unity. Consequently, these numerical integrals will also require substantial numerical precision in memory storage alone. Another issue is the need to define an efficient algorithm which computes the integrals and sums in Equations (134)-(140) with the minimum of computational overhead. Some integrals are repeated and consequently we introduce a new notation to make the formulation and its numerical implementation more transparent. Consider the following integral definition: We may rewrite Equations (134)-(140) as follows where, Note that the integrals in Equations (158) and (160) are identical, thus only the integral F n +1 (3 / 2 , n +3 / 2 , 0) need be computed in I 4 . With this, the scattering kernel may be written as which is far less expensive to compute numerically. Now, the modified term e -1 /τ /τ K 2 (1 /τ ), at an electron temperature of 10 keV has the value of 2 . 811 × 10 2 , at 1 keV its value is 9 . 183 × 10 3 and at 1 meV its value is now 9 . 217 × 10 12 . This method is readily parallelised, with each integral, or group of integrals, performed per CPU. Additionally, if the array D ( n, k, l ) is populated prior to runtime, and care is taken to handle positive and negative terms, performing one final subtraction at the end, then the method can be made very accurate. In the following subsection we detail a numerical investigation of a basic code we have written in Python to evaluate angular moments of the Compton scattering kernel.", "pages": [ 14, 15 ] }, { "title": "10.4 Numerical tests", "content": "The computation of the angular moments of the Compton scattering kernel is based on the solution of many integrals of the form given in equations (156) and (157). We have written a code using the arbitrary-precision mathematics package mpmath in Python 2.7.3 from the Enthought Python Distribution 7.3-1 (64 bit). All calculations were performed on a Mid-2009 MacBook Pro with a 3.06GHz Intel Core 2 Duo CPU with 8GB of 1067 MHz DDR3 RAM - no computer-specific optimisations were performed. The code was designed and tested on Mac OSX 10.8.2, compatible with any OS with Python and mpmath installed. To illustrate the functionality of the method, we show the relative errors, ε , for the first six angular moments of the Compton scattering kernel, for a broad range of photon energies, from γ = 1 meV to γ = 1 GeV. The relative error is defined with respect to an arbitrary precision code written in Mathematica with no less than 100 digits of accuracy. Values of σ KN of magnitude less than 10 -100 are neglected. The code is evaluated first with 53 bits of numerical precision (double precision - D). If the relative error is not less than 10 -12 we then evaluate σ KN with 106 bit precision (double-double - DD), 159 bit precision (triple-double - DD) and, if necessary, with 212 bits of precision (quad-double - QD). We have chosen two electron temperatures for numerical testing, 1 meV and 1 keV. We have chosen to iterate γ ' as γ ' = (1 + δ ) γ , with δ taking the values 10 -6 , 10 -4 , 10 -2 and 1 (for δ > 1, σ KN is always negligibly small and so we omit those results). To our knowledge there are no freely available codes in the literature which can compute successive angular moments of the Compton scattering kernel. Consequently, we have written a code in Mathematica 8 which computes the angular moments to arbitrary order. We then compare the results from Mathematica with those obtained from our Python code, evaluating the relative error ε between the two. In Table 1 the relative errors are computed for an electron temperature of 1 meV. It is clear that at low incident photon energies, namely 1 meV and 1 eV, double precision arithmetic is insufficient. Further, at γ = 1 meV, even the errors at double-double precision are not sufficiently small, and so we display the result for triple-double precision. By photon energies of γ = 1 keV double precision results become no worse than a few parts in 1000. In Table 2 the relative errors are computed for an electron temperature of 1 keV. Again, at very low photon energies we have to resort to double-double, and even triple-double arithmetic precision. However, by photon energies of 1 keV double precision arithmetic is again sufficient. In those regions where γ is large and the relative error at double precision is of the order of 10 -3 or greater, the value of the scattering kernel is significantly less than unity, generally of the order of 10 -50 or less. As the electron temperature increases still higher the results become even more accurate at double precision, and follow the same underlying trends, so we neglect them for the sake of brevity. Clearly the method presented does not fare so well at low photon energies ( γ, γ ' /lessmuch 1), as well as regions where | γ -γ ' | /lessmuch 1 and τ /lessmuch 1 and so we must resort to numerical precision greater than that of standard double precision. Regarding computation time, at double precision the numerical results can take from a few tenths of seconds to a few tens of seconds. Computation time increases drastically with increased numerical precision. We stress the system architecture these calculations were performed on was simply a laptop, and there is tremendous scope to improve the implementation of the underlying method. Since the method centrally revolves around solving specific definite integrals, it is easily parallelised and can be made significantly faster on that basis alone. Further, by careful consideration of positive and negative terms, only one subtraction need be performed per moment evaluation, greatly reducing round-off error (since the integral in equation (156) is always positive). The terms D ( n, k, l ) may be tabulated prior to runtime and all values of F k can be stored in an appropriate array. In addition, the integrals themselves could be pre-calculated on a standard grid of cases, with interpolation performed on this grid at run-time. In the regions where double precision accuracy is insufficient, asymptotic series expansions can be employed, particularly where τ → 0, γ, γ ' → 0, γ/γ ' → 1 and γ fixed with γ ' → 0 (and vice-versa). However, in most regions of astrophysical interest, the electron and photon energies are of the order of keV energies or greater. The aforementioned refinements would make the Compton scattering code very robust across a much broader energy range, particularly at lower energies.", "pages": [ 15 ] }, { "title": "11 RESULTS AND DISCUSSION", "content": "We remark that this method can easily be generalised to include evaluation of moments of the cross-section in terms of more general functions of ζ , such as Legendre polynomials. This is shown in Appendix B. We show in Figures 6-10 the computed moments of the Compton scattering kernel (in arbitrary units, i.e. C = 1) obtained from the closed-form expression that we have derived as a function of scattered photon energy. Figures 6 and 7 illustrate the dependence of the zeroth moment of the scattering kernel on the electron temperature, for various incident photon energies. Fig. 8 shows the dependence of the 1st, 2nd, 3rd, 4th and 5th moments of the scattering kernel for an incident photon energy of 40 keV and an electron temperature of 1 keV (top) and 20 keV (bottom). Fig. 9 is similar to Fig. 8, except that the incident photon energy is 100 keV. Fig. 10 is as in Fig. 9, except the incident photon energy is now 300 keV. The parameters for the plots in these figures were chosen to enable comparison with previous numerical calculations by Pomraning (1972, 1973) in which the angular moments are expanded in terms of Legendre polynomials P n ( ζ ). Without a closedform expression for the scattering kernel, Pomraning (1972, 1973) employed a fully numerical approach in his calculations. Although Pomraning (1972, 1973) employed a Legendre polynomial moment expansion and we have considered different functions for the moment expansions, in the classical limit, the zeroth order terms in both calculations are identical. Figures 6-7 indeed show that the zeroth order moments obtained by our derived closed-form expression are the same as those obtained by the Legendre polynomial expansion of Pomraning (1972). We also note that the zeroth order moments of the kernel that we computed for various electron temperatures are consistent with Monte-Carlo simulations of Compton scattering of monochromic emission lines shown in Pozdniakov, Sobol & Sunyaev (1979) and Pozdnyakov, Sobol & Syunyaev (1983). In practical radiative transfer calculations, the full radiative transfer equation with scattering may in principle be decoupled, in a truncated moment expansion, into a series of coupled ordinary differential equations (Thorne 1981; Fuerst 2006; Wu et al. 2008). In solving the full radiative transfer equation in curved space times, a covariant generalisation of the Eddington approximation (Fuerst 2006; Wu et al. 2008; Shibata et al. 2011) may be employed, which, coupled with the aforementioned closed-form expressions for the angular moments, yields a semi-analytic approach, necessitating the evaluation of two numerical integrals, namely over λ and γ (or γ ' , by detailed balance). The detailed procedures for such a decomposition are beyond the scope of this study, and we leave this to a future article.", "pages": [ 16 ] }, { "title": "12 SUMMARY", "content": "We have derived a covariant expression for the relativistic Compton scattering kernel self-consistently. By specialising the z -axis of integration along the direction of photon momentum transfer, and re-arranging the order of integration, the problem of computing angular moments of the Klein-Nishina cross-section has been reduced to one of solving three types of moment integral. Further, in re-arranging the order of integration, our method is not restricted to the particular assumed electron distribution function, although for this work we assumed a relativistic Maxwellian distribution for the electrons. The analytical representation of these moment integrals in terms of hypergeometric functions enabled us to express the Klein-Nishina scattering kernel in the particularly elegant form given in equation (155). The problem of evaluating moments of the Klein-Nishina cross-section has been reduced to simply computing a series of one-dimensional integrals over the electron energy, λ , which are easily evaluated by quadrature methods. This is a significant improvement over current approaches. We investigated the numerical stability of the evaluation of the angular moment integrals in Fortran95 , both by recursive and direct evaluation of the hypergeometric functions. It was found that for n > 30, numerical stability becomes an issue and double-precision arithmetic is no longer adequate. Further, as already described, the case of very low scattering angle ( ζ →-1) is oscillatory, and slowly convergent, owing to the geometry of the problem. We also investigated the convergence of the angular moments of the Klein-Nishina scattering kernel ( M n ) and found the case of inverse-Compton scattering to be more slowly convergent than conventional Compton scattering, but also that the rate of convergence is strongly dependent on the electron velocity. We found that as the electron velocity increases, M n converges much more rapidly as the moment order increases. We carried out demonstrative calculations of the first six moments of the Klein-Nishina scattering kernel, convolved with a relativistic Maxwellian distribution for electrons, for various incident photon energies and electron temperatures. The results we obtained were consistent with those obtained by fully numerical calculations in which the moment expansion is performed in terms of Legendre polynomials (Pomraning 1972, 1973) and by Monte-Carlo simulations of emission line broadening (Pozdniakov, Sobol & Sunyaev 1979; Pozdnyakov, Sobol & Syunyaev 1983). We note that our closed-form expression enables us to perform covariant radiative transfer calculations efficiently in astrophysical settings where general relativistic effects are important, with the moment truncation carried out via an Eddington approximation scheme (see Fuerst 2006; Wu et al. 2008).", "pages": [ 16, 18 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank Curtis Saxton for comments, helpful suggestions and carefully proof reading the manuscript, as well as assistance with the presentation of some figures. We also thank the referee for pointing us towards some of the more recent works on this subject.", "pages": [ 18 ] }, { "title": "REFERENCES", "content": "Abramowitz M., Stegun I. A., 1972, Handbook of Mathematical Functions, Dover: New York Aharonian F. A., Atoyan A. M., 1981, Ap&SS, 79, 321 Appell P., 1880, Ann. Sci. Ecole Norm. 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Transf., 36, 273 Klein O., Nishina T., 1929, Zeitschrift fr Physik A Hadrons and Nuclei, 52, 853, 10.1007/BF01366453", "pages": [ 18 ] }, { "title": "APPENDIX A: APPELL HYPERGEOMETRIC FUNCTION METHOD", "content": "The Appell F 1 hypergeometric function is one of a set of four hypergeometric series of two variables (Appell 1880; Appell & Kamp'e de F'eriet 1926). It is a very general class of special function, containing many other special functions as particular or limiting cases, including hypergeometric functions of one variable like the Gauss 2 F 1 . The Appell F 1 function is defined by the series expansion This series is absolutely convergent for | z 1 | < 1, | z 2 | < 1. Cases outside of the unit disc of convergence can be calculated through analytic extension (Olsson 1964), hence an algorithm can be constructed to evaluate the function numerically (e.g. Colavecchia, Gasaneo & Miraglia 2001; Colavecchia & Gasaneo 2004). Consider R n and S n,m . R n , after an appropriate substitution, may be expanded into a doubly-infinite series as where α ≡ 1+ x 2 , β ≡ 1 -x 2 and u ≡ 1 -ζ . Performing the integral over u and using the identity (2 k +2 l +1) = (3 / 2) k + l / (1 / 2) k + l the following closed-form expression for the moment integral R n is obtained: By the same process, a closed-form expression for the moment integral S n,m also follows In the case x 2 = 1 these expressions simplify to Gauss hypergeometric functions of one variable through the following identity: As expected from the integral expressions for the moment integrals in equations (81) and (82), equations (A3) and (A4) are identical in argument and differ only in their parameters ( a, b 2 , c ). For both of these expressions the parameter b 1 = -n , and are hence absolutely convergent, since ( -n ) k = 0 for k /greaterorequalslant n . That is to say, by writing the Appell hypergeometric function as a single sum over Gauss hypergeometric functions (Srivastava & Karlsson 1985) the series always converges in n +1 terms. Although it may appear profitable to compute the scattering kernel in terms of Appell hypergeometric functions, since these simplify to finite sums of Gauss hypergeometric functions, it is not computationally cheaper and so the results in Section 9 are expressed in terms of the latter.", "pages": [ 28 ] }, { "title": "APPENDIX B: MOMENT EXPANSION IN TERMS OF MORE GENERAL POLYNOMIALS", "content": "As has already been observed (see Fig. 4), a moment expansion in terms of ζ n , although convenient, is not strongly convergent for very small scattering angles. The expansion is inherently oscillatory in this instance, since even moments will always yield strictly positive results for the Compton scattering kernel, and odd moments are both positive and negative. The question naturally arises as to how the behaviour changes if a different moment formalism is chosen. This method can also be applied if the electron distribution is no longer isotropic, introducing a ζ -dependence in the electron distribution function. Consider a generalised function of ζ , F ( ζ ), which can be represented as a Taylor series: Defining tilde variables as those which represent a moment expansion in terms of F n ( ζ ), it is readily shown that the generalised moment integrals may be written in terms of the usual Q n , R n and S n,m as", "pages": [ 28 ] }, { "title": "Covariant Compton Scattering Kernel 29", "content": "from whence it follows Therefore", "pages": [ 29 ] } ]

[ { "title": "ABSTRACT", "content": "We present the results of fitting simulations of an unbiased sample of SDSS galaxies utilizing the fitting routine GALFIT and analysis pipeline PyMorph. These simulations are used to test the two-dimensional decompositions of SDSS galaxies. The simulations show that single S'ersic models of SDSS data are recovered with σ mag ≈ 0 . 025 mag and σ radius ≈ 5%. The global values (half-light radius and magnitude) are equally well constrained when a two-component model is used. Sub-components of two-component models present more scatter. SDSS resolution is the primary source of error in the recovery of models. We use a simple statistical correction of the biases in fitted parameters, providing an example using the S'ersic index. Fitting a two-component S'ersic + Exponential model to a single S'ersic galaxy results in a noisier, but unbiased, recovery of the input parameters ( σ totalmag ≈ 0 . 075 mag and σ radius ≈ 10%); fitting a single S'ersic profile to a two-component system results in biases of total magnitude and halflight radius of ≈ 0 . 05 -0 . 10 mag and 5%-10% in radius. Using an F-test to select the best fit model from among the single- and two-component models is sufficient to remove this bias. We recommend fitting a two-component model to all galaxies when attempting to measure global parameters such as total magnitude and halflight radius. Key words: galaxies: structural parameters - galaxies: fundamental parameters galaxies: catalogs - methods: numerical - galaxies: evolution", "pages": [ 1 ] }, { "title": "Alan Meert, 1 ∗ Vinu Vikram, 1 † and Mariangela Bernardi 1 ‡", "content": "1 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA Accepted 2013 May 7. Received 2013 April 29; in original form 2012 November 26", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Measurement of fundamental galaxy properties is an essential step in analyzing galaxy structure, formation, and evolution. At the most basic level, luminosity, size, and morphology are important properties, useful in evaluating dynamical and evolutionary models (e. g., Shankar et al. 2013). Nonparametric methods exist to estimate luminosity, size, and structure without imposing a functional form on a galaxy (e. g., Petrosian 1976; Abraham et al. 1996; Blanton et al. 2001). However, non-parametric methods are sensitive to the depth of the image and to the PSF. This can result in underestimating the luminosity and size of an object due to missing flux in faint regions of the galaxy or when the true size of the galaxy becomes small relative to the size of the angular PSF (Blanton et al. 2001, 2003). Parametric methods offer a reasonable way to extrapolate galaxy light profiles into fainter regions at the expense of introducing a potentially incorrect functional form for the galaxy. Common functional forms used in parametric fitting include the r 1 / 4 and the r 1 /n models originally presented by de Vaucouleurs (1948) and S'ersic (1963). Empirical study suggests that bulges and elliptical galaxies are better described by de Vacouleurs profiles or S'ersic profiles with S'ersic index n ≈ 4. Disks and late-type spirals are best described by exponential profiles or S'ersic profiles with S'ersic index n ≈ 1 (Freeman 1970). More recent work has shown that the relationship between S'ersic index and the photometric or kinematic components of a galaxy is more complicated. Following Kent (1985), many studies simultaneously fit a second component in order to better accommodate the qualitative differences of bulges and disks in galaxies. Additionally, Caon, Capaccioli & D'Onofrio (1993) showed that the S'ersic profile is a better fit to many early-type galaxies than the traditional de Vacouleurs profile. There have been several catalogs of photometric galaxy decompositions presented recently (Simard et al. 2011; Kelvin et al. 2012; Lackner & Gunn 2012) with particular interest on the applicability of large sets of image decompositions to evolutionary models. However, systematic effects continue to be of concern, and the reliability of two-component decompositions in cases of low to moderate signal-to-noise are often viewed with some skepticism. In order to quantify the systematics and robustness of the ∼ 7 × 10 5 fits of g, r, and i band SDSS spectroscopic galaxies to be presented in Meert, Vikram & Bernardi (2013), hereafter referred to as M2013, we generate simulations of single and two-component galaxies, referred to as 'mocks,' and fit them using the same PyMorph pipeline (Vikram et al. 2010) used for the photometric decompositions presented in M2013. The M2013 catalog has already been used in Bernardi et al. (2013) to study systematics in the sizeluminosity relation, in Shankar et al. (2013) to study sizeevolution of spheroids, and in Huertas-Company et al. (2012) to study the environmental dependence of the masssize relation of early-type galaxies. Following several detailed studies which have used simulations to test the robustness of different fitting algorithms (e. g., Simard et al. 2002; Haussler et al. 2007; Lackner & Gunn 2012), the main goal of this paper is to test the robustness of PyMorph pipeline software on SDSS galaxies. We use these simulations to test the effects of increased signal-to-noise as well as increased resolution, PSF errors, and sky determination. Our simulations are specifically applicable to SDSS galaxies and are useful for evaluating the decompositions presented in M2013. We use unbiased samples to estimate and correct the systematic error on recovered parameters as well as estimate reasonable uncertainties on fit parameters. A description of the simulation process is presented in Section 2. This includes constructing a catalog of realistic galaxy parameters (Section 2.1); generating galaxy surface brightness profiles based on these parameters (Section 2.2); generating sky and noise (Sections 2.3 and 2.4); and including seeing effects in the final image. The completed simulations are run through the fitting pipeline, and the fits are analyzed in Section 3. We examine the recovery of structural parameters in noise-free images (Section 3.1) and parameter recovery in realistic observing conditions including both neighboring sources and the effects of incorrect PSF estimation (Section 3.2). Recovery of mock galaxies is unbiased for single S'ersic models. However, two-component mocks are biased when fitted with single S'ersic profiles. This bias consists of an overestimate of the size and luminosity of the galaxy. PyMorph is further tested by inserting mocks into real SDSS images to test the dependence on local density (Section 3.3). We examine dependence of the fits on resolution and signal-to-noise (Section 3.4). The effect of changing the fitted cutout size (Section 3.5) and the effect of incorrect background estimation (Section 3.6) are also examined. In Section 4 we discuss the overall scatter in our fits and the implications of the simulations. Finally, in Section 5 we provide concluding remarks. We generate single-component S'ersic galaxy models (hereafter referred to as Ser ) and two forms of twocomponent galaxy models: one is a linear combination of de Vacouleurs and an exponential profile ( DevExp ) and the other replaces the de Vacouleurs with a S'ersic profile ( SerExp ). A good overview of the S'ersic profile used throughout this paper is presented in Graham & Driver (2005). Throughout the paper, a ΛCDM cosmology is assumed with ( h ,Ω m ,Ω Λ ) = (0.7,0.28,0.72) when necessary.", "pages": [ 1, 2 ] }, { "title": "2.1 Selecting the simulation catalog", "content": "We create a set of mocks using fits from the photometric decompositions presented in M2013. These galaxy parameters represent the r-band image decompositions of a complete sample of the SDSS spectroscopic catalog containing all galaxies with spectroscopic information in SDSS DR7 (Abazajian et al. 2009). The sample contains galaxies with extinction-corrected r-band Petrosian magnitudes between 14 and 17.77. The lower limit of 17.77 mag in the r-band is the lower limit for completeness of the SDSS Spectroscopic Survey (Strauss et al. 2002). The galaxies are also required to be identified by the SDSS Photo pipeline (Lupton et al. 2001) as a galaxy ( Type = 3 ), and the spectrum must also be identified as a galaxy ( SpecClass = 2 ). Additional cuts on the data following Shen et al. (2003) and Simard et al. (2011) are applied. Any galaxies with redshift < 0.005 are removed to prevent redshift contamination by peculiar velocity. Galaxies with saturation, deblended as a PSF as indicated by the Photo flags, or not included in the Legacy survey 1 are also removed from the sample. In addition, following Strauss et al. (2002) and Simard et al. (2011), we apply a surface-brightness cut of µ 50, r < 23 . 0 mag/arcsec 2 because there is incomplete spectroscopic target selection beyond this threshold. After applying all data cuts, a sample of 670,722 galaxies remains. We select an unbiased sample of galaxies from the DR7 sample and use the fitted models from PyMorph to generate the mocks used in this paper. For each model ( Ser , DevExp , and SerExp ), we select a representative sub-sample physically meaningful photometric decompositions. In order to ensure that the galaxies are representative of the full catalog, we examined the distributions of basic observational parameters of SDSS galaxies (surface brightness, redshift, apparent Petrosian magnitude, Petrosian half-light radius, and absolute magnitude). Some restrictions on fit parameters are necessary to ensure that outliers are removed from the parameter space used to generate the simulations. Galaxies that do not satisfy these basic cuts are removed to ensure that the parameters used to generate the images are physically motivated. The cuts do not significantly bias our galaxy distribution as is shown in Figure 1. The cuts are: 1 A list of fields in the Legacy survey is provided at http://www.sdss.org/dr7/coverage/allrunsdr7db.par Conditions (i) and (ii) are used to prevent selection of Ser models with extended profiles that are likely the result of incorrect sky estimation during the fitting process. Condition (iii) ensures that any disk dominated galaxies have a bulge component that is smaller than the disk. After enforcing the cuts on the sample, 10,000 fitted galaxy profiles for each of the Ser , DevExp , and SerExp models are selected at random without regard to the morphological classification of the original galaxy. The fitted parameters of these sample galaxies are used to generate the mocks used in testing the pipeline. Selecting galaxy samples independent of galaxy morphology allows the DevExp and SerExp samples to contain some galaxies that do not truly possess a second component. Additionally, there will be some truly two-component galaxies (i. e., both bulge and disk components are present) that are misrepresented by a single S'ersic fit. However, this sampling method will not invalidate the results of our tests. Since we seek to test the ability to recover simulated galaxy parameters, we only require a realistic sample of galaxy profiles. Our samples satisfy this requirement. Single S'ersic galaxies in the original sample, simulated as mock Ser galaxies and fit with Ser models, test the ability to recover S'ersic parameters. Similarly, Ser mocks with SerExp models, show bias resulting from over-fitting a galaxy. Fitting the SerExp mocks with a Ser model shows the bias due to under-fitting. Fitting a single-component model regardless of galaxy structure or morphology is a common practice (e.g., Blanton et al. 2005; Haussler et al. 2007; Simard et al. 2011). In Figure 8b we show that bias of 0.05 mags and 5% of the halflight radius result from fitting a two-component galaxy with a single component and that this bias increases to 0.1 mags and 10% of the halflight radius for brighter galaxies. These biases are important in analyzing the results of a single-component fitting catalog. For example, Bernardi et al. (2013) shows that intermediate B/T galaxies can often be fit by S'ersic models with large S'ersic indicies, which can lead to misclassification if cuts similar to Shen et al. (2003) are used. Figure 1 shows the distributions of surface brightness, redshift, extinction-corrected r-band Petrosian magnitude, r-band Petrosian halflight radius, and absolute magnitude of all SDSS spectroscopic galaxies (in black) and our simulation samples: Ser (red), DevExp (green), and SerExp (blue). The distribution of mock galaxies reproduces the observed distribution for all three samples for each observational parameter as verified by a KS 2-sample test. Figure 1 also presents the signal-to-noise (S/N) of the mock samples as compared to the parent distribution. The S/N of the images is a limiting factor in the fitting process, so care must be taken to ensure that the S/N is not artificially increased in the simulations when compared to true SDSS galaxies. This S/N is calculated using the r-band Petrosian magnitude and r-band Petrosian halflight radius. Petrosian quantities are used to make a fairer comparison among all the samples. Because the Petrosian quantities are non-parametric, they avoid the complications that arise in assessing the possible biases introduced during fitting. Any differences in S/N are not large enough to significantly bias the distributions as verified by the KS 2-sample test. Therefore, we conclude that our samples are fair representations of the underlying distribution of SDSS spectroscopic galaxies. The S/N is discussed further in Section 2.4. Testing the accuracy of the PyMorph fitting routine does not necessarily require an unbiased parameter distribution. In reality, all that is required is a sample with sufficient coverage of the parameter space represented by the data. The simulations use smooth profiles, simplifications of the true galaxies that are observed in SDSS. Examination of the results of fitting these simplified models and comparison to fits of true observed galaxies can potentially yield useful information regarding galaxy structure. In Bernardi et al. (2013), the simulations are used together with the decompositions of the SDSS spectroscopic sample to characterize the scatter in the size-luminosity relation as well as examine possible biases. In order to make these comparisons, an unbiased sample is required. The distributions shown in Figure 1 show that the simulations are appropriate to use for this purpose.", "pages": [ 2, 3 ] }, { "title": "2.2 Generating the images", "content": "We generate the two-dimensional normalized photon distributions from the one-dimensional S'ersic profiles and the onedimensional exponential profiles of each bulge and disk component. Disk components are only simulated where required, as is the case for two-component fits. When multiple components are to be simulated, each component's normalized photon distribution is generated separately and combined prior to generating the simulated exposure. Two-dimensional galaxy profiles are treated as azimuthally symmetric one-dimensional galaxy light profiles that are deformed according to an observed ellipticity. The galaxy profiles are generated using the structural parameters generated from photometric decompositions as described in the previous section. Single-component galaxy profiles and the bulges of two-component galaxies are generated according to the S'ersic profile where S'ersic index (n), half-light radius ( R e ), and surface brightness at R e ( I e ) are selected simultaneously from the catalog described in the previous section. For the DevExp and SerExp cases, an exponential disk (Equation 1 with n = 1) is added to the S'ersic component to model the disk component of the galaxies. The disk is modeled using a slightly modified version of Equation 1. This model requires input parameters scale radius ( R d ) and central surface brightness ( I d ). After generating the two-dimensional profile, the image is pixelated by integrating over each pixel area. The details of this integration are largely unimportant. However, the simulation must take careful account of the integration in the central pixels, where the profile can vary greatly over a single pixel. Various oversampling methods have been devised to properly correct this common problem (e. g., Peng et al. 2002; Haussler et al. 2007). The simulations in this paper have been tested to ensure that the pixelby-pixel integration is accurate to ≈ 3% of the corresponding Poisson noise in a given pixel. Therefore, we treat the simulations as exact calculations of the galaxy photon distributions since any noise from the integration contributes only a small amount to the total noise budget. The pixelated galaxy is numerically convolved with a PSF extracted from SDSS DR7 data using read PSF program distributed by SDSS 2 . The choice of this PSF is discussed in Section 3.2.", "pages": [ 3, 4 ] }, { "title": "2.3 Creating the background", "content": "Two hundred background images, each equal in size to an SDSS fpC image, are also simulated for testing purposes. These images contain constant background and a randomly selected field of galaxies taken from an SDSS fpC image. The SDSS catalog provides rudimentary photometric decompositions of each star and galaxy. Galaxies are fit with an exponential disk and a de Vacouleurs ( n = 4) bulge independently. The best fit is reported as a linear combination of the two fits using the fracdev parameter to express the ratio of the de Vacouleurs model to the total light in the galaxy. For the simulated background used in this paper, each 2 read PSF is part of the readAtlasImages-v5 4 11 package available at http://www.sdss.org/dr7/products/images/read_psf.html galaxy is generated using the combined profile of the two fits. The de Vacouleurs bulge and exponential disk component are separately simulated according to the magnitude, radius, ellipticity, and position angle reported in SDSS. Each component is simulated using the method described in Section 2.2. The background galaxy is constructed by adding the two components using the fracdev parameter. The galaxy is then inserted into the fpC image. Any foreground stars are also simulated as point sources and inserted into the image. For the background sky counts in the image, we use the mean sky of all SDSS observations as given in the SDSS photoobj table by the sky r parameter. The distribution of the sky flux is plotted in Figure 2 in units of counts (or DN) per pixel per exposure. The median and mean values for a 54 second SDSS exposure are ≈ 125 and ≈ 130 counts per pixel, respectively. We use the mean value of 130 counts per pixel as the background in our simulations. This sky background is applied to the entire chip as a constant background; no gradient is simulated across the image. Background gradients should be approximately constant across a single galaxy. This assumption is verified by inserting the simulated galaxies into real SDSS fpC images near known clusters, where the sky contribution should be higher and gradients are more likely. In Section 3.3 we show that there is little change in the behavior of the fits in these types of environments. Previous work has improved the measurements of sky background (see Blanton et al. 2011). However, these cor- r rections tend to focus on areas of large, bright galaxies or on making the sky subtraction stable for purposes of tiling fpC images together. Since we are only focused on maintaining the proper S/N for our simulations, the sky levels provided in the SDSS database are sufficient, provided that they maintain the correct S/N. We discuss the S/N distribution of our simulations and the original SDSS galaxy sample in Section 2.4 below. Diffraction spikes and other image artifacts are not directly simulated. However, the SDSS photo pipeline often misidentifies additional phantom sources along an observed diffraction spike. These phantom sources are modeled in our background, and so these effects are approximately modeled. It is reasonable to expect that the diffraction effects should not have a large effect on the fitting process, as their elongated straight structure does not mimic galaxy structure. The dominant effect produced by the bright stars in the field is bias in the background estimation in the nearby neighborhood of a star. After simulation of the background images, and prior to adding noise, each background image is convolved with a random SDSS PSF selected from original fpC image upon which the individual image is based. Selecting PSFs from original SDSS images introduces a variation in PSF size between mock galaxies inserted into images and the background galaxies. However, this variation is not of concern for us in the fitting process because the vast majority of galaxies (over 90% of all galaxies) do not have neighbors near enough to the target galaxy to require simultaneous fitting. For these galaxies, the PSF applied to neighboring galaxies is of no interest in the fitting process because the sources are masked out. The details of this masking are not discussed in the remainder of the paper. Modifying the masking conditions produce no noticable difference in the fitted values for our simulations. For the remaining 10% of galaxies, there may be some variation in the fit due to differing PSFs. PSF sizes can differ between target and neighboring galaxies by up to a factor of 2. However in practice, this happens for less than 1% of galaxies of the galaxies with neighbors. Furthermore, incorrect PSF tends to only cause effects at the centers of galaxies. So although using a PSF that is different from the background PSF will affect the recovered parameters of the neighbor, it will not affect the target galaxy.", "pages": [ 4, 5 ] }, { "title": "2.4 Noise", "content": "After generating a target galaxy and inserting it into a background, Poisson noise is added using the average inverse gain of an SDSS image (4.7 e -/DN) and the average contribution of the dark current and read noise, referred to as the 'dark variance,' (1.17 DN 2 ), to determine the standard deviation for each pixel. Specifically, is the total flux in pixel ( i, j ) (i. e., the sum of the source and background fluxes in the pixel), and so for a single pixel. Since the fitting pipeline is dependent on the S/N, it is essential that the simulated S/N is comparable to SDSS. The distribution of the average S/N per pixel within the halflight radius for the simulations and the original galaxies is plotted in Figure 1f. The S/N distribution of simulations and the SDSS spectroscopic galaxies agree as verified by a KS 2-sample test, therefore the simulations appropriately approximate the S/N of SDSS galaxies contained in M2013. An unbiased selection in the previously mentioned parameters is not sufficient to guarantee fair sampling of the S/N with respect to magnitude, nor does it prevent fictitious correlations among multiple fit parameters. In fact, correlations among fit parameters are to be expected if the PyMorph pipeline is robustly measuring properties of the target galaxies (many correlations exist among physical parameters). It is difficult, and largely unnecessary, to examine every possible relationship for correlations introduced by biases in the sample selection process. Examining the S/N and the halflight radius versus apparent magnitude help to ensure the appropriateness of the simulation. Systematic differences in radius will lead to systematic variation in the S/N of the sample. We also examine the scatter in recovered fitting parameters as a function of magnitude. Therefore, the S/N as a function of apparent magnitude should appropriately reflect that of the parent sample from SDSS. shown here, tends to have smaller radii and higher S/N at brighter magnitude. The results of tests using the DevExp model are not discussed in the remainder of this paper. They can be found in M2013.", "pages": [ 5, 6 ] }, { "title": "2.5 Final processing for fitting", "content": "For each mock galaxy, we also generate a weight image of the σ i,j values according to Equation 4. This image is supplied along with the input image to the pipeline in order to calculate the χ 2 value for the fit. Figure 5 shows some examples of mock galaxies throughout the simulation process. This includes the noiseless mock galaxy, the noiseless simulated background, the composite image of galaxy and background, and the composite image after adding Poisson noise with σ i,j defined in Equation 4. The final image size used for fitting is 20 times the Petrosian r-band halflight radius. A discussion of this choice of stamp size is presented in Section 3.5.", "pages": [ 6 ] }, { "title": "3 TESTING PYMORPH IMAGE DECOMPOSITIONS", "content": "In order to test the parameter recovery of the PyMorph pipeline on SDSS spectroscopic galaxies, we apply the PyMorph pipeline to the mocks described in Section 2. The PyMorph pipeline uses GALFIT to fit smooth profiles to the the mock galaxies. We apply the pipeline to several different realizations of our mock galaxies. These realizations increase in complexity from a noiseless image to an image with real noise and (possibly clustered) neighboring sources. We show that the ability of PyMorph to reliably recover model parameters is limited by both the S/N and the resolution of the mock galaxy. Understanding the systematic effects of S/N and resolution is useful in interpreting the data presented in M2013. It may also be used to correct biases in the data as described later in Section 4.", "pages": [ 6 ] }, { "title": "3.1 Noiseless images", "content": "As an initial test, the pipeline is applied to simulations prior to adding noise, background counts, or neighboring sources. This produces the minimum scatter in the data, serves to verify that our simulations are correct, and shows that PyMorph is properly functioning. The total apparent magnitude, halflight radius, and additional fit parameters recovered by fitting the noiseless images of the Ser and SerExp models are presented in Figures 6a, 7a, 8a, 9a, and 10a. The plots show the difference in simulated and fitted values (fitted value - input value). The difference is shown versus the input magnitude as well as the input value of the respective fit parameter. The gray-scale shows the density of points in each plane with red points showing the median value. Error bars on the median value are the 95% confidence interval on the median obtained from bootstrapping. Blue dashed lines show the regions which contain 68% of the objects. Figures 6a and 9a show the corresponding fit is well constrained ( Ser fit with Ser , and SerExp with SerExp ). The total magnitude and halflight radius are both constrained well within 1% error on the flux or radius ( σ total mag ≈ 0 . 01 mag and σ radius ≈ 1%). However, the scatter increases somewhat for the sub-components of the SerExp fit (see Figure 10a). As the components of the SerExp model become dim (bulge/disk magnitude approaches 18.5), the component contribution to the total light becomes small. The origin of the magnitude limit is merely an artifact of our selection criteria requiring that all galaxies have total magnitude brighter than 17.77. This implies that components with magnitude of ≈ 18.5 or dimmer are necessarily sub-dominant components and contribute at most ≈ 50% of the light to the total profile. On average, components dimmer than 18.5 magnitudes contribute about 25% of the total light to a typical galaxy in this sample, and this contribution drops rapidly to about 10% by 19 magnitudes. In these cases, the sub-dominant component will be much less apparent in the image and, therefore, less important to the overall χ 2 of the fit, allowing for greater error in the parameters of that component. In addition, once Poisson noise is considered, these dimmer components suffer from much lower S/N. Later tests (Section 3.4) show substantial error on these components due to the low flux and resulting low S/N. Additionally, sub-dominant components (in particular, bulges) may be much smaller than the overall size of the galaxy. This makes bulge parameter recovery susceptible to resolution effects. These effects are also explored in Section 3.4. The magnitude and halflight radius are also well constrained when a Ser galaxy is fit with a SerExp profile (Figure 7a). However, a SerExp galaxy fit with a Ser profile produces large biases in the magnitude and halflight radius (Figure 8a). As already mentioned, the total magnitude and halflight radius are well constrained ( σ total mag ≈ 0 . 01 mag and σ radius ≈ 1%) in cases where the correct model is applied to the mocks (i. e., Ser mock fit with a Ser model). This is not always the case when the wrong model is applied (i. e., SerExp mock fit with a Ser model). When attempting to fit the simulated SerExp mocks with a Ser model, we find measurable bias of order .01 magnitudes in total magnitude. We also find the scatter of both the size and magnitude to be increased by an order of magnitude. This bias and increased scatter becomes even larger in later tests. It is obvious that a single-component galaxy cannot properly model a two-component galaxy in general, and therefore, we would expect significant problems in attempting to fit a single-component profile to a two-component galaxy. Nevertheless, this type of fit is often performed on real data at low to moderate resolution and S/N where it is unlikely to recover a robust two-component fit. An important observation is that the SerExp fit provides the most stable estimate of the halflight radius and total magnitude regardless of the true simulated galaxy model ( Ser , DevExp , or SerExp ). The additional freedom in the SerExp model and the fact that the Ser and DevExp models are special cases of the SerExp model would lead us to expect this result. Therefore, it is advisable to always use a SerExp fit in the course of fitting SDSS galaxies unless there is specific evidence to the contrary. One systematic effect in the pipeline that has been noted by other groups (e. g., Blanton et al. 2005; Guo et al. 2009), is the underestimate of S'ersic index at larger S'ersic indexes. At S'ersic indexes of n ≈ 4, we underestimate the S'ersic index by less than 1%. However, this underestimate increases in the later tests. The data suggest that a substantial component of this error is due to the resolution limits of the SDSS sample. At larger S'ersic index, a high sampling rate at the center of the galaxy is useful in distinguishing the preferred value of the S'ersic index. We further explore the effect of image resolution in Section 3.4. Since no Poisson noise is added to these images, the scatter apparent in these fits is a combination of the limitations of the SDSS data (in particular resolution), systematics inherent in the PyMorph routine (as well as the GALFIT routine used by PyMorph), and any parameter degeneracies inherent in the models. GALFIT uses the Levenberg-Marquardt minimization method (Press et al. 1992) to find the minimum of the χ 2 distribution of the fit. The Levenberg-Marquardt method is not a global search algorithm but rather follows the steepest decent to a local minimum. As the parameter space becomes more complicated, GALFIT has more trouble accurately recovering parameters. Adding components to the fit (i. e., going from a one-component to two-component fit or going from a fixed S'ersic index component to one with a free S'ersic index) will not only complicate the χ 2 surface, making convergence less likely, but may introduce true degeneracies in the parameter space. For instance, the SerExp fit of a galaxy of very late type often suffers from over-fitting. The bulge component will tend to fit the disk of the galaxy as a second disk component with n bulge ≈ 1. This is obviously an unintended solution to the fitting but one that is equally valid from an χ 2 perspective. In practice, it is difficult to prevent this type of convergence without artificially constraining the fitting routine. Such constraints are generally discouraged and can lead to other negative effects including convergence to a non-optimal solution. The best solution to the parameter degeneracy is close examination of any two-component fits in cases where n bulge ≈ 1, or B/T ≈ 0 or 1. In addition, PyMorph reports statistical error estimates on the fitted parameters as returned from GALFIT. These errors are found to be an underestimate of the true error in the fits by as much as an order of magnitude. This gross underestimation of the error is also reported by Haussler et al. (2007) as well as being discussed in the GALFIT user notes 3 . Following Haussler et al. (2007), we examine the ratio of the uncertainty reported by GALFIT to the deviation of the measured parameters (referred to as σ/ ∆). σ/ ∆ should be greater than 1 for approximately 68% of the data if the estimated uncertainty is appropriate. However, this is not the case for any of the parameters in the fits. We discuss a simple method for correcting the systematic bias and estimating the uncertainty in Section 4.", "pages": [ 6, 7, 8 ] }, { "title": "3.2 The effects of background, neighbor sources, and incorrect PSF extraction", "content": "When analyzing real data, it is not possible to extract the PSF at the target galaxy to arbitrary accuracy. Interpolation is required and generally performed on a network of the nearest stars to the target galaxy. We test this effect through extraction of a neighboring PSF to be used during fitting in place of the PSF used to generate the image. The neighbor PSF used in fitting is randomly selected from a location within a 200 pixel box surrounding the source. This provides approximately even sampling of distances from nearly 0 to about 170 pixels in separation from the source which corresponds to a separation of ≈ 0 to ≈ 67 . 32 arcseconds between the target galaxy and the location used for PSF extraction. This inserts some PSF error into the process of fitting as would be expected in the case of real data. However, it also retains the similarity between the PSF used for simulation and the PSF used for fitting. A strong similarity between the two would be expected since the PSF generally will not vary greatly over the area of a single fpC image. Target galaxies are randomly inserted into the simulated fpC images described in Section 2.3. The simulated fpC images contain sky as well as neighboring sources. The PSF of the neighboring sources will have a different PSF than the target galaxy. This effect is not of concern in this work. Prior to fitting, a new cutout is extracted from the total image (containing the target galaxy and background) ensuring that the target galaxy is at the center of the stamp image. By constructing new postage stamp images in this manner, we ensure that there is sufficient variation in the background while preventing us from fitting the incorrect galaxy. These fits (containing error in PSF reconstruction, neighboring sources, and noise) are the closest simulation to actual observing conditions that we have analyzed. Therefore the fits and the resulting measures of scatter and bias are adopted as our fiducial estimates of scatter and bias when using the pipeline. Figures 6b, 7b, 8b, 9b, and 10b show that we recover the input values with marginal scatter. The total magnitude and halflight radius remain well constrained ( σ totalmag ≈ 0 . 05 mag and σ radius ≈ 5%) in cases where the correct model is fit to the mock galaxy. However, this scatter becomes larger when the wrong model is fit. The underestimate of the S'ersic index, particularly at large values, persists. Further examination of the two-component fits show that the pipeline has difficulty extracting dim components (bulge or disk magnitude dimmer than ≈ 18 . 5). In these ranges, the components are observed at lower S/N and the pipeline looses sensitivity to the model parameters. The SerExp fit shows an underestimate of S'ersic index, which is even stronger than in the single-component case, and an underestimate of bulge radius. However, the disk parameters remain unbiased with an increase in scatter of the model parameters. The increased stability of the disk parameters relative to the bulge parameters was also noted in Simard et al. (2011). In their paper, the authors comment that this may be due to the fixed profile shape (due to the fixed S'ersic index, n = 1) or to the fact that on average bulges are more compact than disks leading to a resolution effect. This stability is the result of the increased resolution as disk sizes in our sample are roughly 3 times the FWHM of the PSF while bulges are smaller, on average approximately equal to the FWHM of the PSF in size. We discuss this further in Section 3.4. In general, the SerExp fits are problematic and require much care when analyzing individual components. However, as we have already shown, total magnitude and halflight radius are still tightly constrained. Table 1 summarizes the bias and scatter in the fits; they 3 See the technical FAQs at http://users.obs.carnegiescience.edu/peng/work/galfit/TFAQ.html exhibit trends with both the input value of the parameter and the input magnitude of the galaxy. This behavior is not properly encapsulated in the overall measure of bias, so these values are useful only as an example of the relative scale of bias and scatter for each parameter. Errors can be correlated across many fit parameters, so we also calculate a correlation matrix for the parameter errors. Figure 11 shows an example of the correlation matrix for the SerExp mocks fit with a SerExp model. We see the expected strong correlations between bulge-to-light ratio and the bulge and disk magnitudes as well as the correlation among the radii of the bulge component with the S'ersic index. While the correlation matrix suggests that there is little correlation between sky estimation error and the fitted parameters, we will show later that there is indeed a strong correlation in model errors with sky estimation error. The apparent lack of correlation of sky error with the other fitting parameters is somewhat surprising. However, Figures 14 and 15 suggest a possible explanation for the apparent lack of correlation. Correlation of parameter errors with sky errors is non-linear and asymmetric with respect to over- or underestimating the sky. The fits discussed in this section are shown on Figures 14 and 15 in red. These points lie in a region where sky error does not significantly bias most parameters. In addition, the scatter of the sky values is quite small. This small scatter prevents us from sampling the broader covariance of the sky. If, for example, the recovered sky value was an underestimate of 0.5%, then there would be a measurable covariance of fitting parameters with sky due to the steepness of the parameter bias with respect to sky level. We discuss the sky estimation further in Section 3.6.", "pages": [ 8, 9 ] }, { "title": "3.3 Testing with real images", "content": "To verify the validity of the simulated background and to test the fitting pipeline in clustered environments, we insert the mock galaxies into real SDSS fpC images. The fpC images are selected from SDSS DR7 images containing spectroscopic galaxy targets. We omit plots of the fitted values here, because the scatter and the bias in the fits remain unchanged, sug- gesting that we have properly modeled the sky background and neighboring sources common to an SDSS spectroscopic galaxy. Dense environments provide an additional test for our pipeline. To select fpC images that contain dense environments, we use the GMBCG catalog (Hao et al. 2011). We match brightest cluster galaxies (BCGs) with galaxies in our original catalog to select fpC images with cluster members including the BCG. Our mock galaxies are then inserted into the image which is run through the pipeline. In our previous simulations, intracluster light and gradients in the sky were not modeled. These tests allow us to see what the effects may be. Once again, the errors remain unchanged, showing that no environmental correction is necessary when using the fits from the pipeline. Placing mock galaxies near cluster members allows us to test for systematic effects in crowded fields. However, further examination of BCG galaxies is necessary before we are able to properly model them for this purpose. For example, the curvature at the bright end observed in the size-luminosity relation of early-type galaxies (see Bernardi et al. 2013) appears to be due to an increasing incidence of BCGs, which define steeper relations than the bulk of the early-type population (e. g., Bernardi et al. 2007, 2013). However, the curvature could also be due to intracluster light (e. g., Bernardi 2009). Our ability to test the systematic effects associated with BCGs using the method outlined above is severely limited due to the existence of a BCG at the location we would prefer to place our test galaxy (i. e., the center of the cluster). Therefore, the stability of recovered fit parameters with respect to environment cannot be assumed to extend to BCGs based on the tests presented here alone. Further tests for the largest, brightest galaxies are needed to explore this possibility. We have not presented these tests in this text.", "pages": [ 9, 13 ] }, { "title": "3.4 Varying the S/N and pixel size", "content": "In addition to the previous tests, we isolate the effects of the S/N and image resolution to quantify the contributions to the bias and scatter in our fits. Figures 6c, 7c, 8c, 9c, and 10c show the effect of increasing the S/N by a factor of 4 while holding all other parameters fixed. Similarly, Figures 6d, 7d, 8d, 9d, and 10d show the effect of increasing resolution by a factor of 2 while holding S/N constant. Corresponding decrements in these parameters were performed, although they are not presented in this paper. Improving the resolution by a factor of two substantially improves the ability to recover the radius and S'ersic index with reduced bias. For instance, the S'ersic index bias is reduced to ≈ 0 . 1 at the larger values. Additionally, the bulge parameters of the SerExp fit improve substantially with improved resolution. Corresponding changes in the S/N reduce the scatter, but by a small amount relative to the effect of the resolution change. In addition, changing the S/N does not remove the observed bias in S'ersic index or bulge size. This leads us to conclude that the limitations of the resolution of SDSS are the leading factor in causing systematic offsets in the halflight radius, S'ersic index, and other fitting parameters (including the bulges of the SerExp fits). While increasing the S/N will reduce the scatter in the fits, increased resolution is necessary to properly recover unbiased values. Lackner & Gunn (2012) also examined the effects of changing S/N and resolution on SDSS galaxies (see Figures 5-11 of their paper). The authors found that decreased resolution and S/N increases the relative error in the S'ersic index and radius. They recommended that Ser galaxies (and the bulge and disk sub-components of two-component galaxies) have radii, R hl /greaterorsimilar 0 . 5 × FWHM. This cut removes ≈ 1% of the Ser mocks and ≈ 22% of the SerExp mocks from our simulated samples with a preference toward galaxies above z = 0 . 05. While this condition is sufficient to keep the relative error in the halflight radius and S'ersic index comparable to the error in the magnitude, we find that this condition fails to remove the bias in our galaxy samples. Figure 6b shows that the underestimate of S'ersic index occurs at larger values. These galaxies tend to exhibit radii larger than the PSF. Given that the average FWHM of PSFs in our sample is ≈ 1 . 3 '' , if we apply the suggested cut in radius, we are unable to remove the bias in S'ersic index. Clearly, reliable measurements are dependent on both the S'ersic index of the object and its radius relative to the resolution. Both parameters must be accounted for when deciding on an appropriate resolution cut. If we extend the Lackner & Gunn (2012) recommendation to include a S'ersic index dependent term, this is sufficient to provide for recovery of S'ersic index > 4 with bias ≈ 0 . 1 or ≈ 1%. Galaxies should have circularized halflight radii R hl /greaterorsimilar 0.5*FWHM × n . This removes nearly 75% of the sample. While such large cuts are sufficient to remove the bias in radius and S'ersic index for the Ser fits, the data are certainly biased relative to our original catalog after the cuts. Rather than remove these galaxies, we correct for the bias following a simple statistical argument presented in Section 4.", "pages": [ 13 ] }, { "title": "3.5 Effect of cutout size", "content": "We select postage stamp cutouts for use in fitting. It is important to select a cutout size that does not significantly bias the fits produced by PyMorph. The most important consideration is to provide enough sky pixels to allow the PyMorph program to properly determine the sky level in the images. Reducing cutout size may cause overestimation of background and corresponding errors in the other fit parameters. However, we use the PyMorph pipeline and GALFIT to fit a constant background to the galaxy image. Since a larger image could make sky gradients more significant, this could bias the fits when a larger cutout is used. We seek to minimize error when estimating the sky level without introducing a gradient term and further complicating the fitting process. To test for optimal cutout size, we fit mocks with cutout sizes between 10 and 25 Petrosian half-light radii. We plot the average difference between simulated and measured fit parameters below. In Figure 12 we present the error and 1σ scatter in the error on the total magnitude, halflight radius and sky (showing SExtractor sky in blue and our estimates in red) as a function of cutout size. Smaller sizes clearly bias sky estimates made by SExtractor, but only minor improvement in the scatter of any parameters is achieved by using cutout sizes above 20 halflight radii. Since we use SExtractor sky as a starting point for our fitting, we choose a size of 20 halflight radii for our images. The sky estimates of SExtractor improve substantially. However, GALFIT sky estimation is stable over these sizes. Because GALFIT sky estimation is largely independent of the initial starting SExtractor value (which we would expect if we are truly finding the best fit to the galaxy), it is likely the case that cutout sizes smaller than even 10 halflight radii could be used for analysis. Additional plots of other parameters are omitted in this section. The other fitted parameters show little or no sensitivity to cutout size in the range of cutout sizes used. However, as previously discussed, the bias and scatter may not be equally affected across all model parameters. The effects may be concentrated in a small part of the parameter space.", "pages": [ 14 ] }, { "title": "3.6 The effect of incorrect sky estimation", "content": "Estimation of the sky in the vicinity of the target galaxy has a high level of uncertainty. Indeed, accurate sky determination is likely not even a solvable problem as discussed briefly in Blanton et al. (2011). To determine the bias introduced by our sky estimation, we have tested our fitting pipeline in cases of both underestimation and overestimation of the sky. We fix the sky at the simulated sky level, as well as at simulated sky level ± 0 . 5% and ± 1 . 0%. These ranges were chosen to represent the range of differences between our sky estimations and those provided in the CASJOBS database for the SDSS spectroscopic sample. Figure 13 shows a comparison of sky estimates using PyMorph to those provided from the SDSS photometric data pipeline. This comparison is performed on data from the catalog presented in M2013. The Figure shows the normalized distribution of differences in sky estimation in bins of 0.1%. A negative difference indicates that the sky measured by PyMorph is lower than that reported by SDSS. The vertical red solid line indicates the median of the distribution. The red dashed, dot-dashed, and dotted lines indicate the 68-95-99% ranges of the data, respectively. The 95% range of sky values is approximately between ± 1% difference. For the test, we adopt this range as the range to test for sky variation. The results of incorrectly estimating the sky are shown in Figures 14 and 15. In red, we show the results of fitting galaxies using the standard PyMorph pipeline, treating sky level as a free parameter in the fit. PyMorph systematically underestimates the sky at the 0.1% level. However, the scatter is very tight as indicated by the vertical dashed red lines. In black we have plotted the fitting results at fixed sky levels of the correct value and ± 0 . 5% and ± 1 . 0%. Errors approaching 0.5% clearly introduce a large bias in the fits. The 0.5% level is an important level because it is the approximate level of overestimation shown in the preceding section (Section 3.5) found by SExtractor. Note the asymmetry of the effects of incorrect sky estimation on fitting parameters. In particular, an underestimate of sky is much more detrimental to the fit than the corresponding overestimate. The reason for this asymmetry is due to changes in the perceived 'flatness' of the profile at large radii. When the sky is overestimated, the galaxy profile tends to 0 flux too early. This causes a decrease in the S'ersic index and a decrease in the radius. However, when the sky is underestimated, there will be an extended, approximately constant brightness profile at larger radii. The only way to model such a profile is for S'ersic index to diverge to larger values which produce flat, extended profiles at large radii. Guo et al. (2009) examined the effects of sky uncertainties in regards to the covariance between magnitude and both S'ersic index and halflight radius. They randomly sampled sky estimates from a distribution contained mostly within ± 1%. They found similar variation of S'ersic index (varying by 2 or more in some cases of underestimating the sky and varying by less than 1 in the case of overestimation). The asymmetry in bias due to incorrect sky estimation is apparent in Figure 5 of Guo et al. (2009), but not explicitly commented upon. Figure 13 shows that PyMorph consistently estimates the sky ≈ 0.25% lower than that of the SDSS pipeline. Figures 12, 14, and 15 show that PyMorph has a systematic underestimate of the true sky at the ≈ 0.1% level. This bias is much smaller than the bias associated with using the SExtractor sky estimate as shown in Figure 13 (especially for smaller cutout size), which suggests that the sky values in SDSS are slightly overestimated. SerExp disk components are remarkably robust to the errors in sky estimation, while bulge parameters suffer greatly, especially when the sky is underestimated. Upon further examination of Figure 15, the bulge parameters of the model are more accurately estimated when the sky is treated as a free parameter in the fit rather than when the sky is fixed at the correct value. However, this improvement does not suggest that underestimate of the sky is the preferred fitting outcome. It merely reflects the fact that the systematic effects due to underestimation of the sky are opposite to the underlying biases in halflight radius and S'ersic index. If we were to apply the PyMorph pipeline to an image with higher S/N and increased resolution, we would prefer the correct estimate of the sky to prevent systematic overestimate of these parameters. r", "pages": [ 14, 15 ] }, { "title": "4 DISCUSSION", "content": "In the preceding sections we have shown the covariance, bias, and scatter in our parameter estimation for the Ser and SerExp models. In reality, the effects above will combine to yield a total scatter, covariance, and bias that should approach those shown in 3.2. Our simulations give us an idea of the behavior of the PyMorph pipeline when fitting SDSS galaxies as presented in M2013. The simulations show that the recovery of global fitting parameters (total magnitude and halflight radius) in the case of SDSS galaxies is remarkably robust, even in the case of the SerExp fits. Two-component fits present a more difficult test for the pipeline. Both the bulge and disk components exhibit increased scatter relative to the scatter of the global parameters. In addition, the bulge component exhibits a systematic underestimation of the radius, S'ersic index, and magnitude, particularly for bulges with larger radii or higher S'ersic index. The galaxies fit in M2013 have a median size roughly equivalent to the average PSF of SDSS. For most galax- ies, the resolution necessary to accurately resolve bulge substructure is not present. As shown in Section 3.4, the ability to recover small bulges is improved by a factor of 2 increase in resolution. Finer resolution in central regions of the galaxy is also necessary to fully recover larger S'ersic indexes without bias. Even with these systematics, the two-component fits are still necessary to recover unbiased global parameters and can provide insight into the structure of galaxies. The use of two-component models is potentially ill-advised for many SDSS galaxies as the respective sub-components may be too small to be well-resolved. This is suggested by Simard et al. (2011) as well as Lackner & Gunn (2012) (if we use the suggested resolution cut based on the PSF FWHM). However, our data show that this recommendation should be conditional on the galaxy parameters of interest. While it may be true that bulge parameters of the SerExp fit become unreliable at small radii, we show that using only the Ser fit radius will bias a sample of SDSS galaxies containing both single and two-component profiles (see Figure 8b). However, there are no cases where the SerExp fit introduces bias. It is advisable to use the SerExp halflight radius and magnitude as the total magnitude of the galaxy when examining a sample such as this. The F-test offers a potentially powerful way to distinguish when it is necessary to use a more complicated twocomponent model. The F-test can compare the χ 2 values among nested linear models with Gaussian errors (Lupton 1993). Although our models are not linear and our error distribution is not strictly Gaussian, we apply the F-test to our fits. Following Simard et al. (2011), we adopt an F-test probability of 0.32 as the cutoff indicating a more complicated model is required. When we find a low F-test probability, P correct < 0 . 32, the more complicated model (i. e., going from a one-component to two-component fit, or allowing the S'ersic index of the bulge to vary) provides a better fit to the observed profile. In cases where a Ser fit is used rather than a SerExp fit, the improvement in fitting is large enough to justify using a model with more free parameters. The improved fit is not merely the result of using a more flexible model. A similar test was performed by Lackner & Gunn (2012) to select among a pure disk or disk+bulge model. If the selection based on the F-test is correct, then the resulting measurements of total magnitude and halflight radius will be unbiased. Using the SerExp mocks fit with each of the Ser and SerExp models, we select the fitted model by performing the F-test comparing the Ser and SerExp fits. The preferred fit (either Ser or SerExp ) of the SerExp mocks is then used to assess the bias in the halflight radius and magnitude. By examining the subset of SerExp mocks for which the F-test determines the Ser model to be the appropriate fit, we test the ability of the F-test to select galaxies that are correctly represented by Ser models. In Figure 16 we show the resulting distribution of total magnitude and halflight radius of this subset of SerExp mocks fit with Ser models. The bias originally observed in Figure 8b is not evident. However, the scatter in the recovered values are approximately twice as wide as in Figure 9b, indicating that while the fits are unbiased, some sensitivity is lost by using the simpler (and ultimately incorrect) model. The remaining SerExp mocks, for which the SerExp fit is determined by F-test to be most appropriate, are also unbiased in total magnitude and halflight radius. From this test, we conclude that using the F-test to determine the most appropriate fitted model allows for unbiased measurement of the halflight radius and total magnitude. Using the Ser mocks, the false positive rate ( Ser mocks classified as needing a SerExp fit according to the F-test) for the F-test with a significance level of 0.32 is 5%, suggesting that there is a low level of contamination in a twocomponent sample selected using the F-test. Using SerExp mocks with 0 . 2 < B/T < 0 . 8 and n bulge > 2, which we consider true two-component galaxies, the false negative rate ( SerExp mocks classified as needing only a Ser fit according to the F-test) is 34%, missing a substantial fraction of the galaxies with two components. While selection using the F- test is sufficient to remove the measured bias in global fitting parameters and is able to select a relatively pure sample of two-component galaxies, it does not select a complete sample of two-component galaxies. Clearly caution is necessary when using the F-test to select two-component galaxies from fitting routines. However, the F-test can indicate when the global parameters of a Ser model are likely unbiased regardless of the underlying galaxy type. Following Simard et al. (2011), we can also select the fitted model based on a tiered approach, first performing the F-test on the Ser and DevExp fits. Galaxies for which the DevExp fit gives a statistically significant improvement are then tested again to determine whether the SerExp fit is preferable to the DevExp fit. The preferred fit (either Ser , DevExp , or SerExp ) of the SerExp mocks is then used to assess the bias in the halflight radius and magnitude. We tested this approach and found that it did not significantly alter the results. Many galaxies exhibit more complex structure than a single- or two-component structure. Even the case of a two-component model often oversimplifies galaxy structure. Bars, rings, central sources, clumpyness, or asymmetry cannot be effectively modeled in our simulations. Because of this, we can only determine a lower-bound on the uncertainty in our parameter estimates. However, correcting fits using this lower bound improves the fit of the observed galaxy. We can apply a simple example of bias correction following the procedure outlined in Simard et al. (2002). Given the simulated and fitted values of the S'ersic index for the Ser model, we plot the bias as a function of the fitted value output by PyMorph. In this case, the output value represents the measured value in real data. The simulated value represents the true underlying value of the galaxy S'ersic index. We can determine an average bias and uncertainty in the bias, labeled as Bias and ∆ Bias , as a function of output S'ersic index. Additionally, we can measure the random error in the fits from the width of the bias distribution as a function of S'ersic index, labeled as ∆ Random . Then the corrected S'ersic index and uncertainty on the corrected index is Applying this correction allows us to correct bias as a function of both simulated and fitted S'ersic index for the sample of galaxies used in M2013. We show the results of this process in Figure 17. We are able to statistically correct for the bias in our sample in both the simulated and fitted bases for most values of the S'ersic index. However, there is an under-correction at high simulated S'ersic value. This effect appears to be due to the boundaries of the parameter space that PyMorph is allowed to search for the best fit model. By restricting PyMorph to values of n < 8, galaxies simulated with S'ersic index of 8 will be preferentially underestimated. However, the highest bins of fitted S'ersic index contain many more galaxies with over-estimated S'ersic index. Therefore the net correction will be negative and not appropriate for the highest bins. We could improve the error correction at higher bins by allowing GALFIT to explore larger values of the S'ersic index. However, this is beyond the scope of this paper. Additional corrections may also be considered (i. e., divide in both magnitude and S'ersic index prior to computing the bias correction) depending on the specifics of a given study. For properties of the global population, the corrections measured in this paper are applicable to the sample presented in M2013. Our tests were performed on r-band data from SDSS. The performance of the pipeline can change when observing in different bands. This change is primarily dependent on the change in the S/N and resolution between bands (due to the changing brightness of the sky, color of the galaxy, and size relative to the PSF) and on the different galactic structures to which neighboring SDSS filter bands are sensitive. In principle, these effects could be measured from the simulations presented in this paper by adjusting the S/N and background level. Additionally, one may have to adjust the size of the galaxies or redraw the sample to match the size distributions in the different band. In M2013, we fit the SDSS g, r, and i band data. It is unlikely that the images change drastically enough over the wavelength and redshift range observed to require additional testing in the i band. However, these simulations become an increasingly poor estimate of error in bluer bands where the photometry becomes more sensitive to star forming regions. These regions tend to be clumpier and, therefore, less well represented by a smooth profile. Therefore, g band fits may present more scatter than the r or i band data. These clumpy regions are difficult to model with the smooth models presented here. One could attempt a hybrid approach to generating simulated data whereby one isolates clumpyness in nearby galaxies and use this as a template to add clumpyness to smaller SDSS galaxies. However, the details of this process are beyond the scope of this paper. It is also potentially useful to use information about the r-band to inform the fits of neighboring bands. Indeed Simard et al. (2011) attempted this by requiring many parameters (i. e., S'ersic index, radius, ellipticity) of the fitting model to be identical across the g and r bands, essentially using the two bands as a form of coadded data to increase the S/N. This increase of S/N comes at the expense of dis-allowing variation in the matched parameters, which may or may not be an appropriate assumption (i. e., in a two-component fit, we might expect the bulge size to change across bands, which is dis-allowed). Additionally, Haußler et al. (2013) enforced simple polynomial relationships in parameters across bands, using the neighboring bands to further constrain the acceptable parameter space to be searched by the fitting algorithm. The most flexible method is to fit each band independently and examine the systematic effects of each band as necessary, making additional cross-band comparisons including color (for example, see Lackner & Gunn 2012). This is our preferred method for the data presented here and in M2013.", "pages": [ 15, 16, 17, 18 ] }, { "title": "5 CONCLUSION", "content": "We presented the simulations used to test fitting of SDSS galaxies using PyMorph. Simulations of the Ser and SerExp models were presented and examined in many different cases. The simulations were generated using the results of the fits presented in M2013. We showed that our simulations are recoverable in the case of no noise, which demonstrates that our simulations are correct. We then showed that we can recover the parameters in the case of a simulated background and noise representative of the average SDSS image (see Figures 6, 7,8, 9, and 10). Several individual effects on the fitting were examined. Weshowed that our choice of 20 halflight radii for cutout size does not significantly bias our fitting results (see Figure 12). In addition, we examined the effect of incorrect background estimation, which can significantly affect fitting results (Figure 14 and 15). Effects of increasing the S/N are somewhat limited for this sample. However, an increase in the resolution of the sample would greatly improve parameter measurements, removing many biases in the two-component fits and improving the estimation radius and S'ersic index for Ser galaxies as shown in Figures 6d and 10d. We also examined the bias created when fitting incorrect models to galaxies. Fitting a two-component S'ersic + Exponential model to what is really just a single S'ersic results in a noisier recovery of the input parameters, but these are not biased (see Figure 7b); fitting a single S'ersic to what is truly a two-component system results in an overestimate of 0.05 magnitudes in total magnitude and 5% halflight radius for dim galaxies, increasing to 0.1 magnitudes and 10% for galaxies at the brighter end of the apparent magnitude distribution (see Figure 8b). These biases are used to correct the systematics of our fitted SDSS sample and suggest that magnitude and radius values of a SerExp fit are the least likely to be biased across many galaxy types. Therefore it is advisable to use SerExp values when examining global parameters for galaxies. These simulations can be analyzed together with the fits presented in M2013 to give a more detailed understanding of galaxy structure and formation as presented in Bernardi et al. (2013).", "pages": [ 18, 19 ] }, { "title": "ACKNOWLEDGMENTS", "content": "The authors would like to thank the anonymous referee for many useful comments that helped to greatly improve the paper. AM and VV would also like to thank Mike Jarvis and Joseph Clampitt for many helpful discussions. This work was supported in part by NASA grant ADP/NNX09AD02G and NSF/0908242. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.", "pages": [ 19 ] } ]

[ { "title": "ABSTRACT", "content": "By combining high resolution, radiative transfer cosmological simulations of z ≈ 6 galaxies with a sub-grid multi-phase model of their interstellar medium we derive the expected intensity of several far infrared (FIR) emission lines ([C II ] 158 µm , [O I ] 63 µm , and [N II ] 122 µm ) for different values of the gas metallicity, Z . For Z = Z /circledot the [C II ] spectrum is very complex due to the presence of several emitting clumps of individual size < ∼ 3 kpc; the peak is displaced from the galaxy center by ≈ 100 kms -1 . While the [O I ] spectrum is also similarly displaced, the [N II ] line comes predominantly from the central ionized regions of the galaxy. When integrated over ∼ 500 km s -1 , the [C II ] line flux is 185 mJy km s -1 ; 95% of such flux originates from the cold ( T ≈ 250 K) H I phase, and only 5% from the warm ( T ≈ 5000 K) neutral medium. The [O I ] and [N II ] fluxes are ∼ 6 and ∼ 90 times lower than the [C II ] one, respectively. By comparing our results with observations of Himiko , the most extended and luminous Lyman Alpha Emitter (LAE) at z = 6 . 6, we find that the gas metallicity in this source must be sub-solar. We conclude that the [C II ] line from z ≈ 6 galaxies is detectable by the ALMA full array in 1 . 9 < t ON < 7 . 7 hr observing time, depending on Z . Key words: cosmology - ISM -", "pages": [ 1 ] }, { "title": "Livia Vallini 1 /star , Simona Gallerani 1 , Andrea Ferrara 1 & Sunghye Baek 1 1", "content": "Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy 21 August 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Highz galaxies are mainly discovered by means of their Lymanα emission line (Lyman Alpha Emitters; LAEs, e. g. Malhotra et al. 2005; Shimasaku et al. 2006; Hu et al. 2010; Ouchi et al. 2010) or through drop-out techniques (Lyman Break Galaxies; LBGs, e.g. Steidel et al. 1996; Castellano et al. 2010; Bouwens et al. 2011; McLure et al. 2011). Both methods are plagued with intrinsic limitations: the Ly α detection is hampered by the increasingly neutral InterGalactic Medium (IGM), while the source redshift cannot be precisely determined with drop-out techniques; in addition the restframe optical/UV radiation is strongly affected by presence of dust. It is then important to assess whether other probes, as the far infrared (FIR) metal lines ([C II ],[O I ], [N II ]) originating from the interstellar medium (ISM) of galaxies, could be used to detect new distant sources or better determine the properties of those already discovered. These lines are not affected by H I or dust attenuation, can deliver the precise redshift of the emitter, and open a window to investigate the structure of the galactic ISM. Among FIR lines, the 2 P 3 / 2 → 2 P 1 / 2 fine-structure transition of ionized carbon [C II ], a major coolant of the ISM, is by far the most widely used to trace the diffuse neutral medium (e.g. Dalgarno & McCray 1972; Stacey et al. 1991; Wolfire et al. 1995; Lehner et al. 2004). Up to now, high redshift ( z > 4) detections of [C II ] lines have been obtained mainly in sources with high star formation rates (SFRs) (e.g. Cox et al. 2011; De Breuck et al. 2011) or in those hosting Active Galactic Nuclei (AGN) (e.g. Maiolino et al. 2005; Gallerani et al. 2012). Recently, Walter et al. (2012) put upper limits on the [C II ] luminosity arising from a Gamma Ray Burst (GRB) host galaxy and two LAEs with moderate SFR. Other interesting finestructure lines are [O I ] 63 µ m, tracing neutral (higher density) gas, and [N II ] 122 µ m probing the ionized ISM phase. [O I ] detections have been reported in two lensed Ultra-Luminous Infrared Galaxies at z = 1 . 3 and z = 2 . 3 (Sturm et al. 2010); z > 4 nitrogen lines (including the [N II ] 205 µ m) have been detected in quasars and submillimeter galaxies (Ferkinhoff et al. 2011; Nagao et al. 2012; Decarli et al. 2012; Combes et al. 2012). The unprecedented sensitivity of ALMA will revolutionize the field allowing the detection of FIR lines from the known 'normal' population of highz galaxies (e.g. Carilli & Walter 2013, and references therein) as in the case of [C II ] detections in two z = 4 . 7 LAEs presented by Carilli et al. (2013). Therefore, developing models to predict FIR line luminosities and relate them to other physical features such as metallicity, Z , and SFR is fundamental to design and interpret future experiments. In this work, we present the first detailed predictions for the intensity of several FIR emission lines ([CII] 158 µm , [OI] 63 µm , and [NII] 122 µm ) arising from the ISM in high- tar forming galaxies. Our work is similar in spirit to that of Nagamine et al. (2006), who computed the [C II ] galaxy luminosity function based on a SPH simulation coupled with a sub-grid multi-phase model of the ISM. We improve upon Nagamine et al. (2006) work in at least two ways: (a) we concentrate on a single prototypical highz galaxy, a z = 6 . 6 LAE, hence reaching a sufficiently high resolution to properly describe the ISM small-scale density structure; (b) we implement radiative transfer which is crucial to model the intensity of the galactic UV field and the gas ionization structure.", "pages": [ 1, 2 ] }, { "title": "2 NUMERICAL SIMULATIONS", "content": "We run cosmological SPH hydrodynamic simulations using GADGET-2 (Springel 2005). We use the recent WMAP7+BAO+ H 0 cosmological parameters: Ω m = 0 . 272, Ω Λ = 0 . 728, Ω b = 0 . 0455, h = 0 . 704, σ 8 = 0 . 807 (Komatsu et al. 2011). We simulate a (10 h -1 Mpc) 3 comoving volume with 2 × 512 3 baryonic+dark matter particles, giving a mass resolution of 1.32 (6.68) × 10 5 M /circledot for baryons (dark matter) and gravitational softening /epsilon1 = 2 h -1 kpc. We select a snapshot at redshift z = 6 . 6, and we identify the most massive halo (total mass M h = 1 . 17 × 10 11 M /circledot , r vir ≈ 20 kpc) by using a Friend-of-Friend algorithm. We select a (0 . 625 h -1 Mpc) 3 comoving volume around the center of the halo, and post-processed UV radiative transfer (RT) using LICORICE (Baek et al. 2009). LICORICE uses a Monte Carlo ray-tracing scheme on an adaptive grid. We set the adaptive grid parameter to have a minimum RT size of 0 . 61 h -1 kpc. Starting from the density field provided by GADGET, we recompute gas temperature including atomic cooling from the initial temperature T 0 = 10 4 K. The initial ionization fraction is set to x HII = 0. To define the position of the ionizing sources we assume that stars form in those cells characterized by a gas density ρ /greaterorequalslant ρ th . We choose ρ th = 1cm -3 in order to reproduce the typical size ( ∼ 1 -2 kpc) of star forming regions at z ≈ 6 (Bouwens et al. 2004; Ouchi et al. 2009), as inferred by UV continuum emitting images. The projected position of stellar sources is shown in white in the upper left panel of Fig. 1. A central large stellar cluster is clearly visible, along with other 3 minor stellar clumps displaced from the center. We use the population synthesis code STARBURST99 (Leitherer et al. 1999) to obtain the ionizing spectrum of the galaxy. Theoretical works suggest that highz galaxies might be relatively enriched ( Z > ∼ 0 . 1 Z /circledot ) galaxies (Dayal et al. 2009; Salvaterra et al. 2011). We adopt Z = Z /circledot as a fiducial value for our study but we also consider a lower metallicity case, i.e. Z = 0 . 02 Z /circledot . We assume a Salpeter initial mass function with a slope of α = 2 . 35 in the mass range 1-100 M /circledot , a continuous star formation rate of 10 M /circledot yr -1 , obtained from the SFRM h relation at z = 6 . 6 (Baek et al. 2009, 2012). Ionizing UV luminosity is about L UV ≈ 7 × 10 43 erg s -1 . RT calculations are performed until equilibrium between photoionizations and recombinations is achieved; this occurs within ≈ 10 Myr. The public version of GADGET-2 used in this work does not include the star formation process, neither the radiative cooling, nor supernova feedback. The inclusion of radiative cooling may affect the baryon density profile, enhancing the density towards the center of the galaxy, whereas supernova feedback tends to smooth out density inhomogeneities. We have checked that the baryon density profile resulting from the simulations used in this work fits well with our previous low resolution simulations which include all these processes (Baek et al. 2009). Finally, we note that the large gravitational potential of massive galaxies reduces the effects of SN feedback on star formation, as exemplified by Fig. 1 of Vallini et al. (2012) and related discussion. We interpolate all gas physical properties around the halo center on a fixed 512 3 grid using the SPH kernel and smoothing length, within a (0 . 156 h -1 Mpc) 3 comoving volume. We achieve a higher resolution by interpolating on a finer grid as shown in Fig. 6 of (Baek et al. 2012). This method also allows us to have continuous density PDF at low and high dense region thus increases the maximum density about 50% from 64 3 grid to 512 3 grid. The resulting hydrogen column density map is shown in the upper right panel of Fig. 1.", "pages": [ 2 ] }, { "title": "3 MULTIPHASE ISM MODEL", "content": "With current computational resources, it is not possibile to self-consistently include sub-kpc scale physics in the above RT simulations. To this aim we adopt a sub-grid scheme based on the model by Wolfire et al. (1995, 2003, hereafter W95, W03), in which ISM thermal equilibrium is set by the balance between heating (cosmic rays, X-rays, and photoelectric effect on dust grains) and cooling (H, He, collisional excitation of metal lines, recombination on dust grains) processes (see Table 1 in W95): where n Γ ( n 2 Λ) is the heating (cooling) rate per unit volume [erg s -1 cm -3 ], and n is the total gas density. The ISM can be described as a two-phase gas 1 in which the cold (CNM) and the warm neutral medium (WNM) are in pressure equilibrium. Each cell of the simulated volume is characterized by a pressure p = (1+ x e ) nk b T , where x e is the ionized fraction, taken from the RT simulation output. We determine the density and the temperature of the CNM and WNM by solving Eq. 1, coupled with the ionization equilibrium equation. As metal cooling is not implemented neither in GADGET-2, nor in LICORICE the gas cannot cool below T min ≈ 7700 K. We apply the sub-grid ISM multi-phase model only to cells with T min /lessorequalslant T /lessorequalslant 10 4 K where the upper limit is determined by the fact that metals dominate the cooling for temperatures below 10 4 K. The rates of photoelectric effect and recombination on dust grains strongly depend on the FUV incident radiation. The incident radiation field ( G ) in the Habing band (6 -13 . 6 eV) is computed 1 Our multi-phase model does not include molecular clouds and therefore emission from dense photodissociation regions (PDRs) which we plan to address in future work. at each pixel position /vectorr = ( x, y, z ), summing contributions from all sources as following, where L ν,i is the monochromatic luminosity per source, n ∗ is the number of sources, and /vector r i represents their positions. By scaling the flux with the Habing value (1 . 6 × 10 -3 erg cm -2 s -1 ) (Habing 1968) we obtain the standard flux G 0 . Within our simulated galaxy we obtain 0 . 5 < log G 0 < 5. We find that the mean CNM (density, temperature) is ( 〈 n CNM 〉 = 50 cm -3 , 〈 T CNM 〉 = 250 K, while for the WNM we obtain instead ( 〈 n WNM 〉 = 1 . 0 cm -3 , 〈 T WNM 〉 = 5000 K). In the lower panels of Fig. 1 we show the WNM and CNM column densities. The WNM distribution closely traces regions of high ( N H ≈ 10 22 cm -2 ) total hydrogen column density that are sufficiently far from the central star forming region in order not to become ionized; cold gas lies instead only in small ( D /lessorequalslant 2 kpc) overdense clumps at the periphery of the galaxy. The maps show that cold gas clumps are surrounded by diffuse halos of warm neutral medium.", "pages": [ 2, 3 ] }, { "title": "3.1 FIR emission lines", "content": "For each simulated cell we estimate the line luminosities L i = /epsilon1 i V cell , where the emissivity, /epsilon1 i , is given by: where n and T are the density and temperature of the WNM/CNM, Λ H i (Λ e -i ) is the specific cooling rate due to collision with H atoms (free electrons) taken from Dalgarno & McCray (1972), and χ i is the abundance of the i-th species. The [N II ] line traces the ionized medium since its ionization potential (14 . 5 eV) exceeds 1 Ryd. Therefore, it provides a complementary view of the ISM with respect to the [C II ] and [O I ] lines. The [N II ] cooling rate due to collisions with free electrons is: where A = 7 . 5 × 10 -6 s -1 is the Einstein coefficient, ν is the frequency for the 3 P 2 → 3 P 1 transition, h is the Planck constant, g u /g l is the ratio of the statistical weights in the upper and lower levels, and n c = 300 cm -3 is the [N II ] critical density for T = 10 4 K. We finally compute the observed flux by integrating along the line-of-sight also accounting for the gas peculiar velocity field obtained from the simulation.", "pages": [ 3 ] }, { "title": "4 RESULTS", "content": "In Fig. 2 we show the predicted [C II ] 158 µm , [O I ] 63 µm and [N II ] 122 µm emission for the spectral resolution of our simulations (1 . 0 kms -1 ), a beam resolution of 0 . 1 arcsec and Z = Z /circledot , along with the maps obtained by integrating the spectra over the full velocity range -200 < v < 300 kms -1 . The [C II ] spectrum contains considerable structure due to the presence of several emitting CNM clumps distributed over the entire galaxy's body ( ∼ 20 kpc). The individual sizes of the clumps are however much smaller ( < ∼ 3 kpc). The peak of the spectrum reaches ∼ 2 . 5 mJy and it is displaced from the center of the galaxy by about 100 km s -1 . This is due to the fact that the gas within the central kpc of our galaxy is highly ionized by the massive stars that form there. We find that 95% of the total [C II ] flux originates from the CNM, and only 5% from the WNM. For the [C II ] emission line we obain a flux of 185 mJy km s -1 , integrating over ∼ 500 km s -1 . In Fig. 2 we plot in grey the spectrum extracted by integrating over a circular area of ∼ 2 kpc radius, centered on the component labeled A in the map. It dominates the peak of the [C II ] spectrum (30% contribution to the total emission), with the remaining ∼ 70% coming from less luminous substructures. This is an important point as with high spatial resolution observations a substantial fraction of the [C II ] emission may remain undetected. The FWHM of the main peak is ∼ 50 km s -1 , consistent with the marginal detection of [C II ] in highz LAEs (Carilli & Walter 2013). We have computed FIR line intensities also for a metallicity Z = 0 . 02 Z /circledot . In this case, the [C II ] and [O I ] intensities drop by a factor of ∼ 1000 and ∼ 300, respectively, whereas the [N II ] flux is reduced by a factor of 50. While the WNM emission is ∝ Z , at very low Z CNM is practically absent, since the lower metal content makes the CNM phase thermodynamically unfavorable. A thorough analysis of the relative fraction of the emission arising from CNM and WNM as a function of Z will be adressed in a forthcoming paper. The [O I ] spectrum has a shape similar to that of [C II ] since for both emission lines we are taking into account the emission arising from the neutral phase of the ISM. In the case of [O I ], 75% of the total flux arises from the CNM and 25% from the WNM. The maximum value of the [O I ] flux is ∼ 0 . 35 mJy. The [N II ] emission line reaches a maximum flux of 0 . 022 mJy at v = 0. This line traces the ionized phase of the ISM, and the bulk of its emission arises from the center of the galaxy where the ionizing field intensity is higher. In conclusion, the [O I ] and [N II ] fluxes are ∼ 6 and ∼ 90 times lower than the [C II ] one.", "pages": [ 4, 5 ] }, { "title": "5.1 LAE observations", "content": "As pointed out in the introduction, FIR line observations in highz sources have been carried out mainly in quasars and sub-millimeter galaxies. Recently, Walter et al. (2012) have tried to detect the [CII] emission in Himiko , one of the the most luminous LAEs at z = 6 . 6 (Ouchi et al. 2009). However, they end up only with a 1 σ upper limit of 0 . 7 mJy kms -1 . The large size of the Himiko Ly α emitting nebula ( /greaterorequalslant 17 kpc) makes this object one the most massive galaxies discovered at such high redshifts (Ouchi et al. 2009; Wagg & Kanekar 2012). From this point of view, Himiko 's properties closely resemble those of the prototypical galaxy selected from our simulation. Moreover, the radius of the region within which we distributed the stars ( ∼ 1 -2 kpc) is consistent with the Himiko half-light radius (1.6 kpc) observed by Ouchi et al. (2009). Other properties of Himiko are poorly constrained. The SFR is highly uncertain and its value strongly depends on the diagnostics used to infer it: SED fitting gives > ∼ 34 M /circledot yr -1 , UV luminosities yields = 25 +24 -12 M /circledot yr -1 ; the Ly α line implies 36 ± 2 M /circledot yr -1 . As for the metallicity, Ouchi et al. (2009) suggest Z = [1 -0 . 02] Z /circledot as a plausible range, i.e. consistent with the one we have chosen for our analysis. For a fair comparison with the Plateau de Bure Interferometer data by Walter et al. (2012), we smooth our [C II ] simulations to a beam resolution of 2 . 27 '' × 1 . 73 '' , and we produce channel maps of 200 km s -1 width. In Fig. 3 we show the map with the largest signal achieved. We find that, for Z = Z /circledot the maximum intensity is ∼ 0 . 72 mJy kms -1 , slightly exceeding the observed upper limit by Walter et al. (2012); thus, we can put a solid upper limit on Himiko 's metallicity Z < Z /circledot . This shows the potential of FIR lines in obtaining reliable metallicity measures in highz galaxies.", "pages": [ 5 ] }, { "title": "5.2 Low redshift observations", "content": "Haro 11 (H11), a nearby ( z ∼ 0 . 02) dwarf galaxy (Cormier et al. 2012), is considered a suitable local highz galaxy analog. Through PACS observations of the [C II ], [O I ] and [N II ] lines, Cormier et al. (2012) measured a size of ∼ 3 . 9 kpc for the H11 star forming region, a value which is comparable to the size of the clump A shown in the uppermost right panel in Fig.2. These authors also estimate the relative contribution to the observed FIR lines from the diffuse (neutral/ionized) medium and PDRs. They found that ∼ 80% of the [C II ] and [N II ] emissions come from the diffuse medium, while the [O I ] mostly originates from PDRs. We scale the luminosities of the predicted FIR emission lines to the H11 luminosity distance ( D L ∼ 88) and metal abundances (Cormier et al. 2012). For a fair comparison with the data, taken from Tab. 2 of (Cormier et al. 2012), we compute [C II ], [O I ], and [N II ] spectra by integrating over a region of ∼ 12 kpc in diameter, which corresponds to an angular size of 30 '' at the H11 redshift. For [C II ] and [N II ] lines our model predicts a flux corresponding to 20% of the observed one. For what concerns [O I ], we recover only 3% of the observed flux. However, we recall that the contribution of PDRs, not included in our model, might be non-negligible.", "pages": [ 5 ] }, { "title": "6 ALMA PREDICTIONS", "content": "In Table 1, we plot the expected total fluxes for the FIR emission lines considered, varying the metallicity between Z /circledot and 0 . 02 Z /circledot . In the solar metallicity case a [C II ] ∼ 5 σ detection over four 25 km s -1 channels requires a sensitivity of 0.2 mJy, which translates into an observing time of t ON = 1 . 9 h with the ALMA full array. We note that the predicted fluxes are sensitive to the actual value of Z , implying that a [C II ] line detection can strongly constrain LAE metallicities. On the other hand, this implies that LAEs characterized by metallicities Z < 0 . 5 Z /circledot would require a long observing time ( t ON > 7 . 7 h) to be detected even with the ALMA full array.", "pages": [ 5, 6 ] }, { "title": "7 SUMMARY AND CONCLUSIONS", "content": "We have presented the first attempt to predict the intensity of several FIR emission lines ([C II ] 158 µm , [O I ] 63 µm , and [N II ] 122 µm ) arising from the ISM of highz star forming galaxies. We combined RT simulations of a z = 6 . 6 galaxy with a sub-grid multi-phase model to predict the density and temperature of the cold and warm neutral phase of the diffuse ISM. We find that warm neutral medium lies in overdense regions located sufficiently far from the central star forming clump where the strong ionizing UV field does not allow the presence of neutral gas. Cold gas resides instead in more dense clumps. The physical properties of the cold and warm neutral medium deduced here are in agreement with previous studies (e.g. Wolfire et al. 1995, 2003): the mean density (temperature) of the CNM (WNM) gas are 〈 n CNM 〉 = 50 cm -3 , 〈 T CNM 〉 = 250 K, and 〈 n WNM 〉 = 1 . 0 cm -3 , 〈 T WNM 〉 = 5000 K, respectively. Assuming Z = Z /circledot , our model predicts for the [C II ] emission line a flux of 185 mJy km s -1 , integrating over ∼ 500 km s -1 . The [O I ] and [N II ] fluxes are ∼ 6 and ∼ 90 times lower than the [C II ] one, respectively. We have investigated also the case of Z = 0 . 02 Z /circledot . At this metallicity, the [C II ] and [O I ] intensities drop by a factor of ∼ 1000 and ∼ 300, respectively, while the [N II ] flux is reduced by a factor of 50. In the case of Z = Z /circledot , we have found that 95% (75%) of the [C II ] ([O I ]) emission arises from the cold neutral medium (CNM) of the ISM, and the remaining 5% (25%) from the warm neutral phase. In the lower metallicity case, the fluxes of the [C II ] and [O I ] emission lines drop abruptly since the lower metal content does not allow the presence of CNM phase. As a caveat we note that the [O I ] 63 m µ line could be optically thick (e.g. Vasta et al. 2010). The intensity of the [N II ] line, instead, scales linearly with the metallicity, since it arises from the ionized medium. Interestingly, the [C II ] and [O I ] lines are shifted with respect to the [N II ] line, as a consequence of the fact that they originate from different regions: while the ionized medium, which is traced by the [N II ] line, is located close to the center of the galaxy, the neutral gas, from which the [C II ] and [O I ] lines originate, is predominantly located at large galactocentric radii. This result can explain the shift between the [C II ] and [N II ] lines observed in some highz galaxies (e.g. Nagao et al. 2012). We have compared our predictions with observations of FIR emission lines in highz and local star forming galaxies. At Z = Z /circledot , our model slightly exceeds the 1 σ = 0 . 7 mJy kms -1 upper limit on the [C II ] intensity found in Himiko through PdBI observations (Walter et al. 2012). This result suggests that the gas metallicity in this source must be sub-solar. Our results are also marginally consistent with [C II ], [O I ], and [N II ] observations of Haro 11 (Cormier et al. 2012), a suitable highz galaxy analog in the Local Universe. In this case, our model predicts a flux which is ∼ 20% ( ∼ 3%) of the observed one in the case of [C II ] and [N II ] ([O I ]) emissions. We underestimate the observed flux in Haro11 as a non-negligible fraction of their flux may be provided by dense PDRs not included yet in our study. In particular the [O I ] line is expected to originate primarily from PDRs (Cormier et al. 2012). We defer the inclusion of PDRs in a forthcoming paper. According to our findings, the [C II ] emission line is detectable with the ALMA full array in 1 . 9 < t ON < 7 . 7 hr in star forming, highz galaxies with Z /circledot > Z > 0 . 5 Z /circledot . We emphasize again that our predictions provide a solid lower limit to the expected FIR emission lines flux. Finally, the results presented in this work might be very useful to FIR line intensity mapping studies. In fact, our model represents a valid tool to calibrate the intensity of these lines depending on the different properties of the first galaxies, such as the metallicity and the SFR. Since the mass of the CNM increases in weaker FUV radiation field environments, is it is likely that the specific emission from FIR emission lines as the [C II ] and [O I ] could increase towards fainter galaxies. We leave a dedicated study of this effect to future work.", "pages": [ 6 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank F. Combes, D. Cormier, S. Madden, and T. Nagao for useful discussions and comments.", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Baek S., Di Matteo P., Semelin B., Combes F., Revaz Y., 2009, A&A, 495, 389 Baek S., Ferrara A., Semelin B., 2012, MNRAS, 423, 774 Bouwens R. J. et al., 2011, ApJ, 737, 90 Bouwens R. J. et al., 2004, ApJ, 616, L79 Carilli C., Walter A., 2013, arxiv Carilli C. L., Riechers D., Walter F., Maiolino R., Wagg J., Lentati L., McMahon R., Wolfe A., 2013, ApJ, 763, 120 Castellano M. et al., 2010, A&A, 524, A28 Combes F. et al., 2012, A&A, 538, L4 Cormier D. et al., 2012, A&A, 548, A20 Cox P. et al., 2011, ApJ, 740, 63 Dalgarno A., McCray R. A., 1972, ARA&A, 10, 375 Dayal P., Ferrara A., Saro A., Salvaterra R., Borgani S., Tornatore L., 2009, MNRAS, 400, 2000 De Breuck C., Maiolino R., Caselli P., Coppin K., HaileyDunsheath S., Nagao T., 2011, A&A, 530, L8 Decarli R. et al., 2012, ApJ, 752, 2", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "We present maps of total magnetic field using 'equipartition' assumptions for five nearby normal galaxies at sub-kpc spatial resolution. The mean magnetic field is found to be ∼ 11 µ G. The field is strongest near the central regions where mean values are ∼ 20 -25 µ G and falls to ∼ 15 µ G in disk and ∼ 10 µ G in the outer parts. There is little variation in the field strength between arm and interarm regions, such that, in the interarms, the field is /lessorsimilar 20 percent weaker than in the arms. There is no indication of variation in magnetic field as one moves along arm or interarm after correcting for the radial variation of magnetic field. We also studied the energy densities in gaseous and ionized phases of the interstellar medium and compared to the energy density in the magnetic field. The energy density in the magnetic field was found to be similar to that of the gas within a factor of /lessorsimilar 2 at sub-kpc scales in the arms, and thus magnetic field plays an important role in pressure balance of the interstellar medium. Magnetic field energy density is seen to dominate over the kinetic energy density of gas in the interarm regions and outer parts of the galaxies and thereby helps in maintaining the large scale ordered fields seen in those regions. Key words: galaxies: ISM - galaxies: magnetic fields - galaxies: spiral - (ISM:) cosmic rays - ISM: general - radio continuum: ISM.", "pages": [ 1 ] }, { "title": "Magnetic fields in nearby normal galaxies: Energy equipartition", "content": "Aritra Basu 1 /star , Subhashis Roy 1 /star 1 National Center for Radio Astrophysics, TIFR, Pune University Campus, Ganeshkhind Road, Pune - 411007. 9 November 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Magnetic field strength plays an important role in determining the dynamics and energetics in a galaxy. It is believed that the magnetic pressure plays a role in determining the scale height of the galactic interstellar medium (ISM). Also, the magnetic field plays an important role in collapse of a gas cloud to help the star formation activity (Elmegreen 1981; Crutcher 1999). The density and distribution of cosmic rays depend on magnetic fields. It is thought that the seed field, before formation of galaxies, was amplified by compression during collapse and shearing by a differentially rotating disk (Beck 2006). Dynamo action within the galaxy amplifies and maintains field strength over galactic life-times of ∼ 10 9 year (see e.g, Moffatt 1978; Parker 1979; Moss & Shukurov 1996; Shukurov et al. 2006). Though the dynamo effect can amplify the large scale mean magnetic field, magnetohydrodynamic (MHD) turbulence can amplify the local magnetic field through field line stretching (Batchelor 1950; Groves et al. 2003) up to energy equipartition levels. In steady state, the energy density of magnetic field is close to energy density of the gas. Gas density is known to fall as a function of galactocentric distance (see e.g., Leroy et al. 2008). Therefore, it is expected that the field strength will fall as a function of galactocentric distance. Observationally, the magnetic field ( B ) can be traced by po- larization studies at various wavebands, e.g., Faraday rotation and synchrotron radiation polarization in radio, polarization of starlight in optical and polarized dust emission in infrared. Zeeman splitting of spectral lines can be used to estimate the local magnetic field. Intensity of the synchrotron emission at radio wavelengths can provide estimates of B though assumptions of 'equipartition' of energy between cosmic ray particles and magnetic field. Faraday rotation can probe the line-of-sight averaged magnetic field ( B ‖ ). However, this method uses polarized radio emission that may not be seen from a large fraction of a galaxy due to Faraday and/or beam depolarization (Sokoloff et al. 1998). Zeeman splitting can directly measure B ‖ , but it is highly susceptible to high localized magnetic field. Moreover, its detection is difficult in external galaxies. Estimation of B in the sky-plane using polarization of starlight or dust emission depends highly on several geometrical and physical parameters (Zweibel & Heiles 1997). Synchrotron emission is seen from large fraction of a galaxy and under the condition of 'equipartition', it provides a measure of total magnetic field. At low frequencies (0.33 GHz), more than 95 percent of the emission is synchrotron in origin (Basu et al. 2012a) therefore low radio frequency total intensity images can be used to determine B in galaxies. This method has been used to determine B in some of the nearby star forming galaxies. In M51, Fletcher et al. (2011) found generally stronger fields of ∼ 20 -25 µ G in the spiral arms and ∼ 15 -20 µ G in the interarm regions. In this case, B was determined using total intensity map at λ 6 cm assuming a constant spec- In column 3 D 25 refers to the optical diameter measured at the 25 magnitude arcsec -2 contour from de Vaucouleurs et al. (1991). Column 4 gives the inclination angle ( i ) defined such that 0 · is face-on. Distances in column 5 are taken from: 1 Karachentsev et al. (2003), 2 Karachentsev et al. (2002), 3 Karachentsev, Sharina & Huchtmeier (2000) and the NED † . Columns 6 and 7 lists the data used to trace the molecular and atomic gas respectively which were used to estimate the gas density. Column 9 lists the sources of H α maps used to estimate the energy density of ionized gas in Section 4.2. Column 10 lists the sources of archival data at λ 20 cm waveband: 4 VLA archival data using CD array configuration (project code: AW237), 5 Braun et al. (2007), 6 VLA archival data using CD array (project code: AS325), 7 VLA archival map by combining data from C and D array (Beck 2007). tral index for thermal and synchrotron emission. This might introduce errors in the results as the nonthermal spectral index steepens from center to edge (Basu et al. 2012a). In NGC 253, the field was found to be ∼ 20 µ Gtowards the center and fell to ∼ 8 µ Gtowards the edge (Heesen et al. 2009). In M82, the total field was found to be ∼ 80 µ G in the center and ∼ 20 -30 µ G in the synchrotron emitting halo (Adebahr et al. 2012). However, these galaxies are known starbursts, the magnetic field in the disk could be significantly affected by mixing of magnetic field from other parts of the galaxy through galactic fountain (Shapiro & Field 1976; Bregman 1980; Norman & Ikeuchi 1989; Heald 2012). To measure magnetic field and to compare its energy density with that in gas at high spatial resolution of /lessorsimilar 1 kpc we have observed five nearly face-on normal galaxies, namely, NGC 1097, NGC 4736, NGC 5055, NGC 5236 (M83) and NGC 6946. In Section 2, magnetic field strengths in these galaxies are determined using total intensity synchrotron emission at 0.33 GHz. In Section 3, we present the magnetic field maps and results. We discuss our results and compare the magnetic field energy with kinetic energies in turbulent gas in various phases of the ISM. Our results are summarized in Section 5.", "pages": [ 1, 2 ] }, { "title": "2 DATA ANALYSIS", "content": "The galaxies studied here were chosen from Basu et al. (2012a) where a thorough separation of thermal emission from the total emission was done at 0.33 GHz ( λ 90 cm) and near 1.4 GHz ( λ 20 cm) using H α as the tracer after correcting for dust absorption (Tabatabaei et al. 2007). Our sample comprises of NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946. The nonthermal spectral index ( α nt ) used to compute the equipartition magnetic field, was estimated between the above mentioned frequencies. The data sources for the sample galaxies are listed in Table 1. The 0.33 GHz observations were made using the Giant Meterwave Radio Telescope (GMRT). We broadly classified our studies between arm and interarm regions, i.e, regions of high and low gas density, identified from the H α images for each galaxy. Due to poorer resolution of the far infrared maps used for determining absorption correction of the H α emission, the overall resolution of the nonthermal emission maps was only 40 arcsec. To improve the resolution of nonthermal emission we used λ 24 -µ m emission from dust as a tracer of thermal emission (Murphy et al. 2008). The Spitzer MIPS λ 24 -µ m maps have a resolution of 6 arcsec, better than the resolution of the radio maps. The resolution of the nonthermal maps are determined by the lowest resolution radio maps and subsequently the λ 24 -µ m maps were convolved to it. However, λ 24 -µ m emission from dust is not a direct tracer of thermal emission, and in certain cases show differences with measurements made from using a direct tracer like H α (P 'e rez-Gonz ' a lvez et al. 2006; Calzetti et al. 2005). Moreover, the λ 24 -µ m emission arises not only from dust grains heated by ultra violet (UV) photons, but also from heating of diffuse cirrus clouds by the interstellar radiation field and also from old stars, mostly from the central regions. This could lead to overestimation of thermal emission in such regions. To avoid this shortcoming, and to ensure that both the methods give identical results at the resolution of the absorption corrected H α emission, we corrected the thermal fraction 1 determined from λ 24 -µ m to the thermal fraction from H α in the method described below. All the maps were brought to the same pixel size (3 arcsec) and aligned to a common coordinate system. All the pixels with signal-to-noise ratio more than 4 were considered for this analysis. In step (i), the convolved λ 24 -µ m emission was used to estimate the thermal emission using Equation 10 in Murphy et al. (2008) at a resolution given in Table 2. We then estimated the ther- Number 50 Number mal fraction at each pixel of the map. In step (ii), the maps made in step (i) are convolved to a resolution of 40 arcsec. At this resolution, the thermal fraction maps made from λ 24 µ m ( f th , 24 µ m ) must match the corresponding thermal fraction maps made from H α ( f th , H α ). Therefore, in step (iii) we divided the thermal fraction maps made from H α by the maps made in step (ii). The ratio is expected to be ∼ 1 . However, note that, Equation 10 in Murphy et al. (2008) uses the calibration for the galaxy M51 to scale dust emission at λ 24 µ m to trace thermal emission. This is known to vary between galaxies and may have systematic offsets between f th , H α and f th , 24 µ m . Table 3 shows the thermal fraction determined using the H α - and λ 24 µ m-method (columns 2 and 3 respectively). The two methods match well within ∼ 30 percent of each other. In step (iv), the correction factor to scale the f th , 24 µ m for each pixel was determined within beam of 40 arcsec from the ratio map determined in step (iii). This correction factor for each pixel was multiplied with map (i) to obtain the corrected thermal fraction map ( f corr th , 24 µ m ). The mean correction factor for each of the galaxies are listed in column 4 of Table 3. The correction factor would take care of the systematic calibration-offsets between galaxies. The resultant maps provide us with thermal fraction of the galaxies with a resolution better than 40 arcsec. Fig. 1, top panel, shows the histogram plot of the ratio f th , H α /f th , 24 µ m determined within regions of 40 arcsec for all the galaxies at λ 20 cm. The ratio has been normalized by the mean values of each galaxy to account for the systematic offset between galaxies. The grey and unfilled histograms are for arm and interarm regions respectively. For ∼ 65 percent of the regions, the ratio is seen to be smaller than unity suggesting λ 24 -µ m emission to be higher than the star formation rate. Fig. 1, bottom panel, compares the thermal fraction determined within 40 arcsec regions using H α - and corrected λ 24 µ m-method at λ 20 cm. Although, after correction, f corr th , 24 µ m agrees with f th , H α within ∼ 10 percent, there is significant spread. However, to the first order, when compared to f th , H α , f corr th , 24 µ m has significantly less spread and systematic offset than f th , 24 µ m . Thermal emission was estimated using f corr th , 24 µ m and was subtracted from the total emission to obtain the nonthermal emission. The resolution of the nonthermal maps thus obtained using this method are given in Table 2. For the spatially resolved study of the energy densities in the ISM, we used data from THINGS (Walter et al. 2008) to trace HI surface mass density, and from HERACLES (Leroy et al. 2009) and NRAO 12-m telescope (Crosthwaite et al. 2002) to trace H 2 surface mass density (see Appendix A for details). All the maps for a galaxy was convolved to a common resolution of the nonthermal maps (see Table 2) and re-gridded to common pixel size of 3 arcsec. They were then aligned to the same coordinate system for further analysis.", "pages": [ 2, 3 ] }, { "title": "2.1 Total magnetic field", "content": "From basic synchrotron theory, and assuming energy equipartition between cosmic ray particles and the magnetic field, the total field strength could be estimated (see e.g., Pacholczyk 1970; Miley 1980; Longair 2011). However, the limits of integration ( ν min to ν max ) over the synchrotron spectrum to estimate the total energy in cosmic ray electrons (CRe) depends on the magnetic field (Beck & Krause 2005). This was ignored while minimizing the total energy in magnetic field and cosmic ray particles to derive the 'classical' equipartition formula (Equation 2 of Miley 1980). This gives rise to overestimation of the field in regions of steep nonthermal spectral index ( α nt > 0 . 7 , defined as S ν ∝ ν -α nt ). We used the 'revised' equipartition formula given in Equation 3 of Beck & Krause (2005) to produce total magnetic field maps, where the equipartition field strength ( B eq ) is given as,", "pages": [ 3 ] }, { "title": "NGC 1097", "content": "Contour levels = 400 x (3, 4.243, 6, 8.485, 12, 16.97, 24, 120, 300, 430) microJy/beam", "pages": [ 4 ] }, { "title": "NGC 4736", "content": "DECLINATION (J2000) Contour levels = 400 x (3, 4.243, 6, 8.485, 12, 16.97, 40, 100) microJy/beam", "pages": [ 4 ] }, { "title": "NGC 5236", "content": "DECLINATION (J2000) NGC 6946 G) ✝ ( B ✠ ✞ ✟ ✡ Here, E p is the rest mass energy of protons, I ν is the nonthermal intensity at frequency ν , l is the path-length through the synchrotron emitting region. K 0 is the ratio of number density of relativistic protons and electrons, c 4 ( i ) is a constant depending on the inclination angle of the magnetic field. f ( α nt ) is a function of α nt such that, f ( α nt ) = (2 α nt +1) / [2(2 α nt -1)c 2 ( α nt )c α nt 1 ] , and c 1 , c 2 are constants defined in Appendix of Beck & Krause (2005). We assume K 0 , the ratio of number densities of relativistic protons ( n CRp ) and electrons ( n CRe ), such that K 0 = n CRp /n CRe /similarequal 100 . The path-length travelled through the source ( l ) is taken to be 2 kpc and corrected for the inclination. This could in principle be a function of galactocentric distance ( r , i.e, l ≡ l ( r ) ) depending on the shape of synchrotron emitting halo perpendicular to plane of galaxy disk. The scale height of the synchrotron emitting halo depends on the synchrotron lifetime ( τ syn ), and is expected to be uniform along the extent of the disk except perhaps near the central parts of the galaxies ( ∼ 1 kpc) or in high density regions. Also, B ( r ) ∝ l ( r ) -1 / ( α nt +3) (see Eq. 1) shows weak dependence of magnetic field on l ( r ) . Therefore, we assume that the path-length through the source to be constant ( l ≡ l ( r ) = l 0 /similarequal 2 kpc). The magnetic field thus estimated by us can be scaled by [2 × 10 -2 ( K 0 +1) /l ] 1 / ( α nt +3) due to the assumption of K 0 = 100 and l 0 = 2 kpc. The 'revised' equipartition formula in Eq. 1 diverges for α nt /lessorequalslant 0 . 5 . Thus for regions where α nt was found to be less than 0.55, mostly in the center and inner arms of NGC 5236 and some parts in the ring of NGC 4736, we used a spectral index of 0.55 to avoid sudden rise in the total field strength. Such regions have high gas densities and perhaps dominated by ionization or bremsstrahlung losses giving rise to flatter α nt (see Longair 2011). As a result, the magnetic field strength is overestimated in such regions (Lacki & Beck 2013). The regions of steep spectral index ( α nt > 1 ) towards the outer parts of the galaxies arises due to dominant energy losses of CRe. Thus the energy spectral index between Note: The mean magnetic field strength for the galaxies were computed including the low surface brightness diffuse emission and is therefore less than the mean values in arm and interarm regions. CRe and cosmic ray protons changes, which is assumed to be constant and the same between protons and electron in the equipartition formula. We have therefore set α nt as 1 for such regions. This gives ∼ 6-10 percent lower field strength as compared to steeper α nt .", "pages": [ 4, 5 ] }, { "title": "3 RESULTS", "content": "The estimated 'equipartition' magnetic field strength for the five galaxies, using Eq. 1, are shown in Figure 2. The resolution of the maps for each galaxy are tabulated in Table 2 and is shown in the lower left corner of each image. Overlaid are the 0.33 GHz contour maps of the galaxies from Basu et al. (2012a). The galaxy integrated mean values of magnetic field, 〈 B eq 〉 , are found to be 9 . 0 ± 2 . 0 µ G, 9 . 3 ± 2 . 1 µ G, 9 . 5 ± 1 . 1 µ G, 12 . 2 ± 3 . 0 µ G and 10 . 7 ± 1 . 8 µ G for NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946 respectively (see Table 4). Figure 3 shows B eq as a function of galactocentric distance ( r ) estimated by azimuthal averaging over annuli of one beam width. The field strength are found to be strongest near the central regions with 〈 B eq 〉 ∼ 20 -25 µ G. In the disk, 〈 B eq 〉 falls to ∼ 15 µ G and ∼ 10 µ G in the outer parts of the galaxy. That is, in most of the cases it is seen that the magnetic field fall by ∼ 40-50% from the center to the edge, similar to what is seen for the Milky Way (Beck et al. 1996). The errors in the magnetic field strength was estimated using Monte-Carlo method, wherein ∼ 10 4 random flux density samples were generated assuming Gaussian distribution of error in source flux densities at each frequency. These were used to determine the distribution of B eq . For high signal-to-noise regions ( > ∼ 10 σ , i.e towards the inner parts of the galaxies ) the distribution of B eq can be modelled as Gaussian. However, for regions with lower signalto-noise ( /lessorsimilar 5 σ , i.e, in the outer parts of the galaxies) the distribution has a tail. The error in the total field strength was found to be ∼ 2% towards the central regions (corresponding to red regions in Figure 2), ∼ 5 -10% in the disk (green regions in Figure 2) and ∼ 15 -20% in the outer parts (blue regions in Figure 2). We compared the magnetic field determined using the revised and the classical formula. In the central regions and inner disk where the α nt lies in the range 0.6 to 1, the fields match within ∼ 10%. However, in the outer parts of these galaxies where α nt is steeper ( > 1 . 2 ), the classical equipartition values are overestimated by > 20 %and increases with steepening of the spectral index to up to 50-60% towards the edge. Such a deviation between magnetic fields estimated by the two methods was shown in Beck & Krause (2005). Thus, the magnetic field determined using the classical formula is found to be constant or increasing as a function of galactocentric distance.", "pages": [ 5, 6 ] }, { "title": "3.1 Comparison with existing studies", "content": "NGC 1097 was studied in polarization at high resolution that revealed magnetic field in the bar to be aligned with the gas streamlines and thus a good tracer of gas flow (Beck et al. 1999). Strong radio emission is detected from the bar at λ 90 cm (Basu et al. 2012a), however, due to poor resolution of λ 20 cm maps ( ∼ 40 arcsec), the enhancement of magnetic field in the bar is only about 10-15 percent higher than the disk. In this study, the field at the center is found to be ∼ 18 µ G and decreases to ∼ 10 µ G towards the edge. The field in the northern bar is found to be lower than that in the southern bar with 〈 B eq 〉 ∼ 9 . 8 µ G and ∼ 12 . 2 µ G respectively. Our estimated field is lower than what was estimated by Beck et al. (2005) perhaps due to their assumption of 500 pc of synchrotron emitting region. NGC 4736 is a ringed galaxy with no prominent spiral structure from radio through infrared to optical. Polarized radio emission revealed ordered magnetic field in spiral shape possibly amplified by large-scale MHD dynamo (Chy ˙ z y & Buta 2008). They report mean total magnetic field of 17 µ G, slightly higher than our estimate of 14 µ G within a region of ∼ 3 . 5 arcmin (corresponds to inner 2.3 kpc radius). From our map (see Fig. 2), the total magnetic field in the center is found to be ∼ 18 µ G while in the ring, the field strength is 15-25 µ G with an average total field strength of ∼ 16 . 5 µ G close to Chy ˙ z y & Buta (2008) . Beyond the ring the magnetic field falls off to about 8-10 µ G at a distance of ∼ 3 kpc. NGC 5055 is a flocculent spiral galaxy and lacks organized spiral structure when seen in optical. Polarization observations revealed regular spiral magnetic fields believed to have been generated from turbulent dynamo action (Knapik et al. 2000). They estimated a mean equipartition magnetic field of 9.2 µ G close to our value of 9.5 µ G. Of all the galaxies in the sample, NGC 5055 has the weakest total magnetic field of ∼ 14 . 5 µ Ginthe center and falls off to about 10 µ G in the disk and ∼ 8 µ G towards the edge. No distinct spiral structure has been seen in the map of total magnetic field. NGC 5236 (M83) is a starburst galaxy with uniform magnetic field seen in the outer parts of the galaxy and lower degree of uniformity towards the inner regions hosting star formation (Sukumar & Allen 1989; Neininger et al. 1991, 1993). Towards the center and inner spiral arms which harbors the starburst (Calzetti et al. 1999), the α nt was found to be flatter and lies in the range 0.4-0.55. For those regions we have assumed the value of α nt as 0.55 to avoid any sudden discontinuities. The magnetic field strengths are overestimated in such regions. We found a mean total field strength of 24 µ G in the central 1 kpc regions. The magnetic field is found to be strong in the arms with strength ∼ 15-20 µ G and falls to ∼ 10 µ G in the interarms and towards the edge. The mean total field in the galaxy is found to be 12.2 ± 2.5 µ G, close to what was estimated by Neininger et al. (1991) within measurement errors. Magnetic field in NGC 6946 was studied in detail by Beck (2007). The total magnetic field strength was found to be ∼ 20 µ G in the spiral gas arms, close to what is estimated by us. In the arms turbulent fields dominate, while in the interarms large scale regular field was seen with high degree of polarization (30-60%) referred as the 'magnetic arms' by Beck (2007). From our maps (see Fig. 2), in the northern (roughly centered at RA = 20 h 34 m 52 s , DEC = +60 · 11 ' 59 '' J2000) and southern (roughly centered at RA = 20 h 34 m 33 s , DEC = +60 · 06 ' 46 . 75 '' J2000) magnetic arms, the average field is found to be ∼ 11 µ G, which is just 10-15% stronger than other interarm regions, and this result is at /greaterorsimilar 3 σ significance. The 'circular hole' of ∼ 1 kpc diameter seen in the galaxy with low radio emission at RA = 20 h 34 m 20 s and Dec. = +60 · 09 ' 40 '' (J2000) is seen to have low magnetic field ( ∼ 8 . 5 µ G) as compared to other parts and is ∼ 30% lower than the surrounding regions.", "pages": [ 6 ] }, { "title": "3.2 Magnetic fields in arms and interarms", "content": "We studied the variation of magnetic field along arm and interarm regions for the galaxies NGC 1097, NGC 5055, NGC 5236 and NGC 6946 after correcting for the radial variations. This was not possible for the ringed galaxy NGC 4736, which do not have any prominent arm. Arms and interarm regions were chosen using the Spitzer λ 24 µ m images after smoothing to the resolution of nonthermal maps. For each of the galaxies, the beginning of arm or interarm were chosen leaving the central ∼ 1 kpc region. We determined magnetic field within an area of one synthesized beam ensuring no overlap between adjacent beams. Each beam corresponds to ∼ 0.4 to 2 kpc at the distance of the galaxies (see Table 2). The galaxy NGC 5055, where the arm and interarm are not clearly distinguishable, the mean field in arm was only about 5 percent stronger than that in the interarm. For the other galaxies the mean magnetic field in the arms are stronger by 10-15 percent (see Table 4). Overall, the mean magnetic field strength in the arm is higher than that in the interarm by 12 ± 3 percent, however, in certain regions it could be higher by up to 40 percent. We note that, when magnetic fields are higher in the arms, the limited telescope resolution suppresses the observed field strength in the arms and increases that in the interarm regions. The same effect is caused by larger CRe diffusion length at λ 90 cm, so that more radio emission is observed in the interarm regions. Thus the differences seen in the magnetic field strength between arm and interarm regions are lower limits. In Figure 4, we study the variation of the relative magnetic field strength B eq ( r, d ) /B eq ( r ) , where, r is the galactocentric dis- 2.0 NGC 5236 Length (kpc) 0 2 4 6 8 10 12 14 0 5 10 15 20 0 5 10 15 NGC 5055 Length (kpc) NGC 6946 Length (kpc) tance and d is the linear distance measured from the center along the corresponding arm or interarm. In the figure, the black circles and the gray squares represent arms and interarms respectively. After correcting for the radial variation, the magnetic field do not change significantly along arm or interarm. The mean value of B eq ( r, d ) /B eq ( r ) in the arm is found to be 1 . 03 ± 0 . 03 and 0 . 97 ± 0 . 02 in the interarms for all the galaxies combined.", "pages": [ 6, 7 ] }, { "title": "3.3 Radial scale lengths", "content": "It is believed that CRe originates from supernova explosions of OB stars found in HII complexes, i.e., regions of star formation. These CRe then propagate away to larger distances in galaxies giving rise to larger radial distribution of synchrotron emission than that of CRe sources and gas (see Tabatabaei et al. 2007; Beck 2007). The total intensity radio maps at λ 90 cm appears to be significantly smoother than that at λ 20 cm (see Basu et al. 2012a). The former mainly originates from older ( ∼ 10 8 yr) population of CRe that diffuses farther away from their formation sites than that at λ 20 cm. We estimate the exponential scale length of nonther- mission at λ 20 cm ( I nt , 20cm ) and λ 90 cm ( I nt , 90cm ), total equipartition magnetic fields and surface mass density of total gas ( Σ gas ) to explore the effect of diffusion of CRe. Σ gas is computed from atomic and molecular hydrogen surface mass density (see Appendix A). The scale lengths ( l 0 ) were obtained by fitting a function f ( r ) = f 0 exp( -r/l 0 ) to the radial profiles shown in left panel of Fig. 6 leaving aside the central bulge. For the ringed galaxy NGC 4736, the scale lengths were computed leaving aside the ring. The scale lengths obtained are summarized in Table 3. The scale length of the nonthermal emission ( l nt ) at λ 90 cm was found to be higher than that at λ 20 cm. This is caused due to higher diffusion scale lengths of low energy ( ∼ 1 . 5 GeV) CRe at λ 90 cm as compared to higher energy ( ∼ 3 GeV) CRe at λ 20 cm (Basu et al. 2012b) in a typical galactic magnetic field of ∼ 10 µ G. In the simple case of energy dependent diffusion of CRe, the diffusion length ( l diff ) after time τ is given by, l diff ∼ ( D τ ) 0 . 5 . Here, D is the diffusion coefficient assumed to be constant, and is ∼ 10 28 cm 2 s -1 . We assume the diffusion time to be same as the synchrotron cooling timescales ( t syn ) given by, t syn = 8 . 35 × 10 9 ( E CRe / GeV) -1 ( B eq /µG ) -2 yr , where, E CRe is the energy of the electrons. The expected diffusion length for the galax- Arm Interarm Interarm Arm Interarm ) r ( q e B / ) l , r ( q e B 1.5 1.0 0.5 0.0 ) r ( q e B / ) l , r ( q e B ) r ( q e B / ) l , r ( q e B 1.4 1.2 1.0 0.8 0.6 1.4 1.2 1.0 0.8 0.6 Arm ies at λ 90 cm and λ 20 cm are ∼ 1 . 4 kpc and ∼ 1 kpc respectively, i.e, l diff , 90cm /l diff , 20cm = ( E CRe , 90cm /E CRe , 20cm ) -0 . 5 ∼ 1 . 4 . CRe can also propagate by the streaming instability at the velocity of Alfv 'e n wave in the ionized galactic medium and the propagation distance is given by, l A = v A t syn . Here, v A is the Alfv 'e n velocity assumed to be ∼ 50 km s -1 . In this scenario, the propagation distance at λ 90 cm and λ 20 cm are ∼ 1 kpc and ∼ 2 kpc respectively, i.e., l A , 90cm /l A , 20cm = ( E CRe , 90cm /E CRe , 20cm ) -1 ∼ 2 . From our data, the ratio of scale length of nonthermal emission at λ 90 cm and λ 20 cm, i.e, l nt , 90cm /l nt , 20cm are 1 . 77 ± 0 . 38 , 1 . 37 ± 0 . 29 , 1 . 92 ± 0 . 17 , 1 . 64 ± 0 . 25 and 1 . 31 ± 0 . 15 for NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946 respectively. For 3 of the galaxies the increase in the estimated l nt between λ 90 and λ 20 cm is larger than that expected from simple diffusion estimates and is consistent with streaming with Alfv 'e nic velocity. The scale length of nonthermal emission at λ 20 cm for NGC 6946 is similar to what was found by Walsh et al. (2002) and Beck (2007). The nonthermal scale length is related to scale length of magnetic field ( l B ) as l B = l nt (3 + α nt ) under the assumption of equipartition of energy between magnetic field and cosmic ray particles. For the galaxy NGC 5236, l B is found to be comparatively smaller than other galaxies and is only ∼ 3 times than that of l nt at λ 20 cm and ∼ 1.9 times at λ 90 cm. This is perhaps the effect of magnetic field strength being overestimated towards the inner parts of the galaxy, where α nt /lessorequalslant 0 . 5 (see Section 2.1). Thus the estimated l B for NGC 5236 is lower than the actual value. The scale length of the magnetic field for NGC 6946 is found to be slightly higher than what was estimated by Beck (2007). This is likely to be caused due to their assumption of a constant α nt throughout the galaxy and use of nonthermal emission at λ 20 cm which has a smaller l nt as compared to our λ 90 cm maps. The scale length of total gas surface density ( l gas ) is found to be smaller than that of the nonthermal emission. However, l gas is close to l nt at λ 20 cm and much smaller than l nt at λ 90 cm, suggesting the λ 20 cmnonthermal emission is a better tracer of star forming activity than at λ 90 cm, wherein the later mostly traces the older population of CRe which are well mixed.", "pages": [ 7, 8 ] }, { "title": "4.1.1 Slope of the radio-FIR correlation", "content": "Simulations of MHD turbulence in the ISM revealed, under conditions of equipartition, the magnetic field ( B ) and the gas density ( ρ gas ) are coupled as B ∝ ρ κ gas , where κ ∼ 0 . 4 -0 . 6 (see e.g., Fiedler & Mouschovias 1993; Groves et al. 2003). The slope of the well known radio-far infrared (FIR) correlation was used to determine κ for four of the galaxies at scales of ∼ 1 kpc (Basu et al. 2012b) using synchrotron emission at λ 90 cm and λ 20 cm and FIR emission at λ 70 µ m. The estimated value of κ was found to be 0 . 51 ± 0 . 12 , indicating energy 'equipartition' among magnetic field and kinetic energy of gas due to turbulent motions. However, in this method equipartition between magnetic field and cosmic ray particles is assumed a-priori. The validity of this assumption can be checked from the dispersion seen in the radio-FIR correlation and our estimated values of the magnetic fields.", "pages": [ 8 ] }, { "title": "4.1.2 Dispersion of the radio-FIR correlation", "content": "Dispersion of quantity ' q ' defined as log 10 ( I FIR /I nt ,ν ) is widely used as a measure of the tightness of the radio-FIR correlation. Where, I FIR is the FIR flux density and I nt ,ν is the nonthermal radio flux density at frequency, ν . The far infrared flux density ( I FIR ) can be written as, I FIR ∝ n UV Q ( λ, a ) B ( T dust ) , where n UV is the number density of the UV photons responsible for dust heating, Q ( λ, a ) is a wavelength (here λ = 70 µ m) dependent absorption coefficient of dust grains of radius a (Draine & Lee 1984; Alton et al. 2004). B ( T dust ) is the Planck function for dust emitting at temperature T dust . The flux density at radio frequency ν , can be written as I nt ,ν ∝ n CRe ,ν B 1+ α nt , where, n CRe ,ν is the number density of cosmic ray electrons (CRe) emitting at a frequency ν and B is the actual magnetic field. From the above, the ratio of FIR and radio flux density is, I FIR /I nt ,ν ∝ ( n UV /n CRe ,ν )(1 /B 1+ α nt ) , assuming same dust properties throughout the galaxy and T dust is seen to remain constant throughout the galaxy (see e.g. Tabatabaei et al. 2007; Basu et al. 2012a). Hummel (1986) showed that the cumulative frequency distribution of I FIR /I nt ,ν and that of B 1+ α nt eq follows each other indicating energy 'equipartition' between magnetic field and cosmic ray particles to hold good and thus B eq is close to ☛ Arms ✎ Interarms ✒ 10 10 10 10 10 10 10 10 10 10 ) 3 10 10 m c g 10 r e ( 10 ty si n 10 e d 10 y g 10 r e n 10 e 10 10 10 10 10 10 10 10 10 10 -10 -11 -12 -13 -14 -15 -16 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -10 -11 -12 -13 -14 -15 -16 -10 -11 -12 -13 -14 -15 -16 -10 -11 -12 -13 -14 -15 -16 NGC 5055 0 2 4 6 8 10 12 0 1 2 3 4 5 6 0 2 4 6 8 10 NGC 4736 NGC 6946 Distance (kpc) B . In our case, for each of the galaxy, we determine the quantities I 70 µ m /I nt , 20cm and B 1+ α nt eq within a region of ∼ 1 kpc and normalized them with their respective median values. In Fig. 5 we plot these median normalized cumulative distribution function of I 70 µ m /I nt , 20cm (shown in gray) and B 1+ α nt eq (shown in black) for all the galaxies together. The left and right panels show the distribution in the arm and interarm regions respectively. In the arm regions, α nt do not change significantly, we assumed a constant value of 0 . 8 . However, α nt varies significantly in the interarm regions, and we have used the observed values of α nt for each region from Basu et al. (2012a). At λ 20 cm, the dispersion in the quantity I 70 µ m /I nt , 20cm is similar to the dispersion in B 1+ α nt eq for both arms and interarm regions determined at spatial scales of ∼ 1 kpc. Thus at scales of ∼ 1 kpc, the variations in I 70 µ m /I nt , 20cm , i.e., dispersion seen in the quantity 'q' is caused due to variations in the magnetic field, where the magnetic field is represented by B eq . Thus, B eq (or a constant multiple of it) is a good representative of the actual magnetic field, B . However, at λ 90 cm, the dispersion in I 70 µ m /I nt , 90cm is ∼ 20 percent higher than that of B 1+ α nt eq for the interarm regions. At λ 90 cm the low energy ( ∼ 1 . 5 GeV) CRe propagate to farther distances from the arms into the interarms, which has the effect of increasing the dispersion. We note that equipartition assumption is valid only at scales larger than the diffusion length, which is better fulfilled at λ 20 cm than at λ 90 cm.", "pages": [ 8, 9 ] }, { "title": "4.2 Energy density in magnetic field and gas", "content": "Magnetic energy is expected to be in equipartition with ISM turbulent energy (Crutcher 1999; Cho & Vishniac 2000; Groves et al. 2003). In Section 3, we found that the magnetic field falls off as a function of galactocentric distance and had a larger scale length than that of the gas surface density. Here, we compare the magnetic field energy density ( U mag = B 2 eq / 8 π ) with that of the ISM energy density from kinetic energy of gas due to turbulent motions ( U turb , gas ), thermal energy density of warm ionized gas ( U ion ) and total neutral (atomic; U th , HI + molecular; U th , H 2 ) gas at spatial scales of 0.4 - 0.9 kpc, except for NGC 5236 for which the spatial resolution is ∼ 1 . 2 kpc (see Appendix A for details). U turb , gas is estimated from the surface mass density maps of atomic and molecular hydrogen, using U turb , gas = 1 . 36( U turb , HI + U turb , H 2 ) . The factor 1.36 is to account for the presence of Helium and U turb , HI , H 2 = (1 / 2)(Σ HI , H 2 /h HI , H 2 ) v 2 turb , where, Σ HI , H 2 are the surface mass density of atomic (HI) and molec- NGC 5236 NGC 4736 Umag Uturb,gas Uion Uth,H2 Uth,HI ular (H 2 ) gas. v turb is the velocity of the turbulent gas, assumed to be ∼ 9 km s -1 for HI and ∼ 6 km s -1 for H 2 (van der Kruit & Shostak 1982; Combes & Becquaert 1997; Sellwood & Balbus 1999; Kasparova & Zasov 2008) and h HI , H 2 are the line of sight depth of atomic and molecular gas assumed to be ∼ 400 pc and ∼ 300 pc respectively. The surface mass densities were calculated using moment-0 maps of CO and HI line emission (see Appendix A for details). The thermal energy densities of warm ionized gas and neutral gas were computed using, U th = 3 2 〈 n 〉 kT . Here, 〈 n 〉 is the mean number density, k is the Boltzmann constant and T is the temperature. For the warm ionized gas, the mean number density of thermal electrons 〈 n e 〉 was calculated from the emission measure ( EM ) maps, such that, 〈 n e 〉 ≈ [ EM f d /h ion ] 1 / 2 . EM was determined from dereddened H α maps using Equation 9 in Valls-Gabaud (1998) (see Basu et al. 2012a, for details). We assumed a constant filling factor ( f d ) of ∼ 5 percent and scale height ( h ion ) of the ionized medium as 1 kpc (Wang et al. 1997; Hoopes et al. 1999). The temperature, T e was assumed to be 10 4 K. To estimate the energy density of molecular ( U th , H 2 ) and atomic ( U th , HI ) gas, the number densities were determined from the corresponding surface mass density maps assuming a constant scale height as discussed above. We assumed a constant temperature of ∼ 50 K for molecular gas and ∼ 100 K for atomic gas. In Figure 6 (left panel) we study the variation of energy density of magnetic field ( U mag ; shown in black diamonds), kinetic energy of total gas ( U turb , gas ; shown in blue triangles) and thermal energy density of ionized gas ( U ion ; shown in red triangles), molecular gas ( U th , H 2 ; shown in gray circles) and atomic gas ( U th , HI ; shown in gray squares) as a function of galactocentric distance for the galaxies NGC 4736, NGC 5055, NGC 5236 and NGC 6946. The energy density of the warm ionized gas is found to be about two orders of magnitude lower than that of magnetic field and kinetic energy of turbulent gas. For neutral gas, the thermal energies are about 3 orders of magnitude less. The energy density of magnetic field, turbulent energy of total gas, thermal energy of warm ionized gas and neutral gas matches well with Beck (2007) for the galaxy NGC 6946. However, due to simplistic assumption for α nt ∼ 1 . 0 throughout the galaxy, the total magnetic field may be underestimated in Beck (2007), specially in the inner regions where α nt is smaller than 0.7. In Figure 6 (right panel) we study the variation of the ratio of magnetic field energy density and total ISM gas energy density ( U gas ) estimated at spatial scales of ∼ 0.4-1.2 kpc depending on the distance of the galaxies. U gas was computed as, U gas = U turb , gas + U ion +1 . 36( U th , HI +2 U th , H2 ) . The black and gray symbols denote arms and interarms respectively. The blue squares show the radially averaged value of the ratio determined within annulus of one beam width. Pixels more than 4 σ in gas density were considered for this analysis. The ratio of total ISM gas energy density and magnetic field are found to be almost constant throughout the galaxies except for a systematic offset from unity due to our assumed constant values of scale heights and velocity dispersion. After dividing by the mean values of the ratio for each of the galaxies, the ratio has a dispersion less than ∼ 30% for arms and interarms of the galaxies, indicating energy/pressure balance between magnetic field and gas. For the galaxy NGC 6946, we observe that the ratio of the interarm regions to be higher than that in the arms by a factor 1 . 47 ± 0 . 11 , i.e, magnetic field energy dominates over the turbulent gas energy. Overall, this difference is not significant for other galaxies, but for an adja- t arm-interarm, the ratio in the interarm could be higher by more than 50% than the arm. From our data we found that as we move from the center to the disk to the edge of the galaxies, magnetic field energy systematically dominates over the total ISM energy density. However, this trend disappears once we consider only the high signal-to-noise regions ( > 4 σ ) as is shown by the blue points in Fig. 6 (right panel). Magnetic fields in galaxies are amplified by two major processes, firstly the small scale fields are amplified by field line stretching and twisting due to turbulent motions of gas (small-scale dynamo) and secondly, large-scale amplification due to large-scale dynamo action. In the previous case, magnetic field and gas are closely coupled such that the magnetic field is amplified by transfer of kinetic energy from gas. The fields can be amplified up to equipartition levels (Groves et al. 2003; Cho & Vishniac 2000). This would result in small scale turbulent fields with low degree of polarization in regions of high gas density, i.e, the spiral arms. This is indeed observed in many of the galaxies. In NGC 6946 only 1-5% of the emission is polarized in the inner spiral arms (Beck 2007). In M51, turbulent magnetic field dominates in arms (Fletcher et al. 2011). In NGC 5236, low degree of uniform magnetic field is seen in the spiral arms in inner parts of the galaxy (Neininger et al. 1993). In NGC 4736, comparatively lower degree of polarization is seen in the star forming ring than in the outer parts (Chy ˙ z y & Buta 2008). Our results show that in the gaseous arms, the magnetic field energy density and the energy density in gas to be similar and do not vary by more than 30 percent throughout the galaxies. This indicates that in the arms the magnetic field is perhaps amplified by field line stretching due to turbulent gas motions driven by star formation. In the interarms and towards the outer parts, many galaxies show higher degree of ordered magnetic field caused by large scale dynamo action (see e.g. Beck et al. 1996; Beck 2007; Chy ˙ z y & Buta 2008, and references therein). For NGC 6946, where ordered fields are observed in the interarm regions, it is thought that finite time dynamo relaxation causes a phase shift between magnetic and gas/star forming spiral arms, such that magnetic arms lags (Chamandy et al. 2013). Magnetic field energy dominating over the turbulent gas energy thus helps maintaining this field orderness in interarm regions and outer parts of the galaxies. For this galaxy, Walsh et al. (2002) found the regular magnetic field (using polarized emission at λ 6 cm) to trace regions of low star formation efficiency and coincides with the regions where the ratio was found to be significantly higher, suggesting insufficient energy in turbulent gas to amplify the turbulent magnetic field.", "pages": [ 9, 10 ] }, { "title": "5 SUMMARY", "content": "We have measured total magnetic fields in five nearby normal galaxies, NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946, assuming equipartition of energy between cosmic ray particles and magnetic fields. In this study, magnetic fields were probed at sub-kpc scales except for NGC 1097, for that it was 2.8 kpc.", "pages": [ 10 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank the referee Rainer Beck for important comments which considerably improved the presentation of the paper. We thank Wilfred Walsh for providing us the CO J=3 → 2 moment-2 map for NGC 6946. We would like to thank Adam Leroy for providing us the FITS files of the moment-0 CO J=2 → 1 maps. We thank Dipanjan Mitra for useful discussions and Visweshwar Ram Marthi for going through the manuscript. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work is based (in part) on observations made with the Spitzer Space Telescope , which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "Walter, F., Brinks, E., de Blok, W. J. G., et al., 2008, AJ, 136, 2563 Wang, J., Timothy, M. H., Lehnert, M. D., 1997, ApJ, 491, 114 Young, J., et al., 1989, ApJS, 70, 699 Zweibel, E. G., Heiles, C., 1997, Nature, 385, 131", "pages": [ 12 ] }, { "title": "APPENDIX A: ATOMIC AND MOLECULAR SURFACE GAS DENSITY", "content": "Four of the galaxies (NGC 4736, NGC 5055, NGC 5236 and NGC 6946) studied in this work were observed as a part of THINGS (Walter et al. 2008) to trace HI. We used natural weighted moment0 HI maps to calculate the surface density of atomic gas using the equation, Here, i is the inclination angle ( 0 · for face-on) of the galaxy and I HI is the line integrated intensity. The above equation includes a factor 1.36 to account for the presence of Helium. CO is commonly used as a tracer for molecular gas. We used CO J=2 → 1 moment-0 maps from the HERACLES (Leroy et al. 2009) for three of the galaxies, namely NGC 4736, NGC 5055 and NGC 6946. These maps has an angular resolution of 13.4 arcsec, better than the resolution of the nonthermal maps. Assuming a constant CO-to-H 2 conversion factor, X CO = 2 × 10 20 cm -2 (K km s -1 ) -1 and a line ratio of 0.8 for converting CO J=2 → 1 flux to CO J=1 → 0 flux, the molecular gas surface density was calculated using, Here, I CO J=2 → 1 is the line integrated intensity. Note that, for NGC 6946, the cos i factor was missing in Beck (2007) by mistake (Rainer Beck, private communication). However, this would not change their conclusions significantly. For the galaxy NGC 5236 we used CO J=1 → 0 moment-0 map from the NRAO 12-m telescope (Crosthwaite et al. 2002) to calculate the molecular gas density. This map has an angular resolution of 55 arcsec. The line integrated flux density ( S CO J=1 → 0 ) was converted into molecular gas mass ( M H 2 ) using the formula from Young et al. (1989). Here, D is the distance to the galaxy. The mass was then converted to surface density by diving by the linear area for each pixel.", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "Unveiling the nature of cosmic dark matter (DM) is an urgent issue in cosmology. Here we make use of a strategy based on the search for the imprints left on the cosmic microwave background temperature and polarization spectra by the energy deposition due to annihilations of the most promising DM candidate, a stable weakly interacting massive particle (WIMP) of mass m χ = 1 -20 GeV. A major improvement with respect to previous similar studies is a detailed treatment of the annihilation cascade and its energy deposition in the cosmic gas. This is vital as this quantity is degenerate with the annihilation cross-section 〈 σv 〉 . The strongest constraints are obtained from Monte Carlo Markov chain analysis of the combined WMAP7 and SPT data sets up to /lscript max = 3100. If annihilation occurs via the e + -e -channel, a light WIMP can be excluded at the 2 σ confidence level as a viable DM candidate in the above mass range. However, if annihilation occurs via µ + -µ -or τ + -τ -channel instead we find that WIMPs with m χ > 5 GeV might represent a viable cosmological DM candidate. We compare the results obtained in this work with those obtained adopting an analytical simplified model for the energy deposition process widely used in the literature, and we found that realistic energy deposition descriptions can influence the resulting constraints up to 60%. Key words: dark matter", "pages": [ 1 ] }, { "title": "C. Evoli 1 /star , S. Pandolfi 2 and A. Ferrara 3", "content": "16 October 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "According to the widely accepted Λ Cold Dark Matter (ΛCDM) cosmology, the Universe is mostly made of dark components, i.e. dark energy (75% of the mass-energy budget) and dark matter (DM; 20%); these components largely dominate over baryons (Komatsu et al. 2011). The situation is then rather unsatisfactory as the nature of the dark components is far from being established and it stands as one of the most crucial issues in cosmology. The most promising DM interpretation is in terms of a thermal relic density of stable weakly interacting massive particles (WIMPs). An appealing feature of such a scenario is that the annihilation cross-sections predicted by the electroweak scale automatically provide the right DM density after freeze-out (Bertone & Silk 2010). This argument applies equally well to particles with 1 -20 GeV masses as to those with masses more traditionally associated with supersymmetric neutralinos ( m χ ∼ 40 -1000 GeV). In the recent years, pieces of evidence have been accumulating in favour of DM in the form of ∼ 10 GeV WIMPs. In fact, a relatively light DM particle with an annihilation cross-section consistent with that predicted for a simple thermal relic ( 〈 σv 〉 T ∼ 10 -26 cm 3 s -1 ) and a distribution in the halo of the Milky Way consistent with that predicted from simulations could accommodate the indirect detection of gamma-rays from the Galactic Centre, the synchrotron emission from the Milky Way radio filaments and the diffuse synchrotron emission from the inner galaxy (the so-called 'WMAP Haze' 1 (Finkbeiner 2004; Hooper, Finkbeiner & Dobler 2007; Hooper & Linden 2011a,b; Dobler et al. 2010). At the same time it would be compatible with claims of low-energy signals from DM direct detection exper- iments as DAMA/LIBRA, CoGeNT, and CRESST-II. In particular, the striking detection of annual modulation observed by DAMA/LIBRA (now supported by CoGeNT) appears inconsistent with all known standard backgrounds. Note, however, that (a) other experiments, such as CDMS and XENON100, have not confirmed the result of the direct detections, and (b) indirect detection features might have alternative astrophysical explanations (Bernabei et al. 2008; Biermann et al. 2010; Akerib et al. 2010; CDMS II Collaboration et al. 2010; Bernabei et al. 2010; Crocker & Aharonian 2011; Aalseth et al. 2011a,b; Aprile et al. 2011; XENON100 Collaboration et al. 2012; Guo & Mathews 2012). A phenomenological model of light DM particle able to accommodate the collection of indirect and direct observations should require that DM annihilates primarily into leptons with a cross-section close to 〈 σv 〉 T . Moreover, approximately 20% of annihilations must also proceed to hadronic final states in order to yield a spin-independent, elastic scattering cross-section ( ≈ 10 -41 cm 2 ) with nucleons compatible with the direct detection (see Hooper 2012, for a detailed review). The light DM hypothesis implies a larger cosmic number density of such particles ( n DM ∝ Ω DM h 2 /m DM ); in addition, the annihilation rate ( ∝ n 2 DM ∝ (1 + z ) 6 ) increases dramatically at early cosmic times. These two facts imply that the annihilation energy deposition might profoundly affect the thermal and ionization history of the intergalactic medium (IGM) 2 prior to reionization. In turn, this modified evolution with respect to the standard recombination scenario can in principle leave detectable signatures in the cosmic microwave background (CMB) anisotropy power spectrum 3 . Determining the amplitude of this effect is the chief goal of the present study. The effects of the DM annihilation around the redshift of the last scattering surface (LSS) have been discussed in Padmanabhan & Finkbeiner (2005) and are only briefly summarized here. The extra free-electrons resulting from the DM energy cascade scatter CMB photons, thus thickening the LSS and in principle shifting the position of the peaks in the temperature-temperature (TT) power spectrum. In practice, reasonable electron density excesses yield corrections to the positions of the peaks that can be safely ignored here. More importantly, oscillations on scales smaller than the LSS width are damped in the TT and EE spectra in a manner inversely proportional to their wavelength. Such DM annihilation effects on the TT spectrum are degenerate with variations of the slope ( n s ) and amplitude ( A s ) of the primordial power spectrum, and, to a lesser extent, with the baryon (Ω b h 2 ) and DM (Ω DM h 2 ) density parameter. Polarization spectra are generated via Thomson scattering of the local quadrupole in the temperature distribution. As the broadening of the LSS increases the intensity of the quadrupole moment, the EE spectrum is enhanced on large scales. Furthermore, it can be shown (i.e. Padmanabhan & Finkbeiner 2005) that the quadrupole is dominated by the free-streaming from the dipole perturbation that is π/ 2 out of phase of the monopole. A thicker LSS boosts the fractional contribution from the monopole, thus slightly shifting the peaks of the EE and TE spectra. A key aspect of these calculations is that only a fraction of the released energy is finally deposited into the IGM in the form of heating and H/He ionizations. However, earlier studies (Padmanabhan & Finkbeiner 2005; Mapelli, Ferrara & Pierpaoli 2006; Galli et al. 2009) have used a simplified description of such processes, based on the hypothesis that a redshift-independent fraction of the DM rest-mass energy is absorbed by the IGM. More recently, Slatyer, Padmanabhan & Finkbeiner (2009); Galli et al. (2011); Hutsi et al. (2011) have reassessed the energy deposition problem including various energy-loss mechanisms in a more realistic way. This approach, based on semi-analytical solutions lacks an implementation of low-energy atomic processes that determine the actual absorption channel (e.g. heating, ionization, excitations) and because of this they have to rely on the results of Chen & Kamionkowski (2004). To fill this gap here we build upon our previous work (Vald'es, Evoli & Ferrara 2010) in which we developed the Monte Carlo Energy Deposition Analysis ( MEDEA ) code which includes bremsstrahlung and inverse Compton processes, along with H/He collisional ionizations and excitations, and electron-electron collisions. MEDEA enables us to compute the energy partition into heating, excitations and ionizations as a function of the primary initial energy, the gas ionization fraction and the redshift. MEDEA has been recently improved (Evoli et al. 2012) to include the energy cascade from particles generated by primary leptons/photons using the most up-todate cross-sections and extending the validity of the model to unprecedented high ( ∼ TeV) energies (see Shull 1979; Shull & van Steenberg 1985; Furlanetto & Stoever 2010). In addition, arbitrary initial energy distribution of electrons, positrons and photons can be assigned. These improvements make MEDEA suitable for studying the IGM energy deposition for some of the most popular DM candidates (Evoli et al. 2012). With this greatly improved physical description we aim at computing the signatures left in the CMB spectrum by annihilating light DM.", "pages": [ 1, 2 ] }, { "title": "2 METHOD", "content": "In this Section we compute the energy input of DM annihilations in the IGM. This approach is similar in spirit to a number of recent works (Padmanabhan & Finkbeiner 2005; Galli et al. 2009; Hutsi, Hektor & Raidal 2009; Slatyer, Padmanabhan & Finkbeiner 2009; Galli et al. 2011; Hutsi et al. 2011; Natarajan 2012); however, we improve upon them by a more accurate description of the energy deposition channels.", "pages": [ 2 ] }, { "title": "2.1 Modified ionization history", "content": "For the reasons given in the Introduction, we concentrate on light DM candidates that annihilate mainly in leptonic channels. In Fig. 1 we show the annihilation spectra of a 10 GeV DM particle for the different annihilation channels, computed using the public code DarkSUSY . The muonic and tauonic channels produce a leptonic pair whose prompt annihilation gives rise to an energy spectrum of primary electrons or positrons with kinetic energy from 10 GeV down to few tens of MeV; annihilation in the electron channel produces an electron/positron pair in which both the two primary leptons have a kinetic energy which is the mass of the annihilation particle. The total energy density input from DM annihilations is: (1) where ρ c = 3 H 2 0 / 8 πG is the critical density of the universe today, Ω DM is the DM density contribution to the critical density, m DM is the mass of the DM particle and 〈 σv 〉 is the thermally averaged product of the cross-section and relative velocity of the annihilating DM particles. Moreover we have defined B ≡ 〈 σv 〉 / 3 × 10 -26 cm 3 s -1 . Note that equation 1 is valid only for DM Majorana particles. In the light of the earlier works of Cirelli, Iocco & Panci (2009) and Hutsi et al. (2011) we neglect the role of structure formation in the calculation of the energy deposition. In fact, haloes with density higher than the background could in principle boost the average annihilation rate; however, their formation starts at a relative low redshift ( z < ∼ 100) when the ionization rate due to DM annihilation is already negligible. By introducing the mean number density of hydrogen nuclei n H ≈ 1 . 9 × 10 -7 (1 + z ) 3 cm -3 and the parameter equation 1 becomes: It has been pointed out that 〈 σv 〉 could be somewhat boosted by the Sommerfeld effect (e.g. Galli et al. 2009; Slatyer, Padmanabhan & Finkbeiner 2009). Although it is easy to implement this process in this scheme we have not considered it here as it depends strongly on the DM model chosen (van den Aarssen, Bringmann & Goedecke 2012). Moreover, over the parameter space considered by most studies, this effect can also be approximated as a constant boost to the annihilation rate over the redshift range of interest and then applied to our final results. To derive the DM-modified cosmic ionization/thermal history, we need to include the above heating (and corresponding ionization) rate into the relevant detailed balance equations. To this aim, we have modified the publicly available code 4 RECFAST (Seager, Sasselov & Scott 1999), part of the CAMB (Lewis, Challinor & Lasenby 2000) code, by adding the following terms: where f ion , H , f ion , He ( f h ) are energy deposition fractions into H or He ionizations (heating) including those induced by Ly α photons on atoms in the excited states. A key point to take from equations 4 -6 is that energy deposition fractions are fully degenerate with the parameter we aim to constrain, i.e. 〈 σv 〉 . To partly alleviate this difficulty, a possible strategy, first proposed by Ripamonti, Mapelli & Ferrara (2007), is to determine the lowest possible bound by assuming f h = f ion = 1. More often, constraints have been derived by using the Chen & Kamionkowski (2004) prescription for f i . Based on the results of Shull & van Steenberg (1985), these authors pointed out that when the gas is mostly neutral, energy is evenly distributed among ionizations, excitation and heating; for a fully ionized medium, almost all of the energy goes instead into gas heating. A linear interpolation is used for intermediate ionization values: This approximation is too crude to be used for highprecision predictions as clearly shown by the comparison with fully fledged Monte Carlo simulations (Vald'es et al. 2007; Furlanetto & Stoever 2010). Moreover, for primary energies > ∼ 1 MeV, inverse Compton energy losses on the CMB 4 Recently other similar codes (e.g. CosmoREC, HyREC) have improved the precision of the results implementing a more detailed description of the atomic structure (Hutsi et al. 2011; Giesen et al. 2012). However, given the current precision of CMB experiments, these corrections do not affect our conclusions. become important and introduce a significant redshift dependence of the fractions. These processes have been carefully modelled in Evoli et al. (2012) and here we use their results for f i . Note that the latter assume that photon energy deposition occurs locally, which is not true in general (see Slatyer, Padmanabhan & Finkbeiner 2009). In Appendix A, we show that this approximation is very accurate in the energy range of interest here. The energy depositions calculated with MEDEA are shown in Fig. 2 for different annihilation channels of a 10 GeV DM particle mass. Such curves show a dependence on the annihilation channel since different initial spectral distributions involve different energy loss mechanisms. For computational speed-up purposes, we have derived handy fitting formulae, given in Appendix B, to the MEDEA numerical results.", "pages": [ 2, 3, 4 ] }, { "title": "2.2 MCMC analysis", "content": "To obtain a constraint on the annihilation cross-section of light DM candidate, we have performed a Monte Carlo Markov chain (MCMC) analysis using the publicly available CosmoMC package (Lewis & Bridle 2002). We consider here a flat ΛCDM model with the canonical six parameters plus an additional seventh one, 〈 σv 〉 . Therefore, the theoretical model we adopt is described by the following set of parameters: where ω b ≡ Ω b h 2 and ω DM ≡ Ω DM h 2 are the baryons and CDM density parameters, θ s is the ratio between the sound horizon and the angular diameter distance at decoupling, τ is the optical depth, n s is the scalar spectral index and A s is the amplitude of the primordial spectrum. The flat priors assumed for these parameters are shown in Tab. 1. Our basic data set is the 7-yr WMAP temperature and polarization data (Komatsu et al. 2011; Larson et al. 2011). We consider purely adiabatic initial conditions and we impose spatial flatness. We also fixed the primordial fractional abundance of helium to the standard observed nominal value of Y He = 0 . 24. We refer to this basic data set as ' WMAP7 '. For each case we run five chains; convergence diagnostic tests are performed using the Gelman and Rubin 'variance of chain mean/mean of chain variances' R-1 statistics. We consider the chains to be converged only if R -1 < 0 . 03. The 68 and 95 per cent confidence level (c.l.) one- and twodimensional constraints are obtained after marginalization over the remaining 'nuisance' parameters. We have tested that varying H 0 instead of θ s , as suggested in Galli et al. (2009), our results are found to be affected by less than 5 per cent. In addition to the WMAP7 data set we also consider the case ' CMB ALL+SPT '. In this larger data set we include, in addition to the WMAP data, the CMB temperature and polarization data from QUaD (Brown et al. 2009), and the recent SPT (Keisler et al. 2011) data. The inclusion of the QUaD experiment (a) enlarges the multipole range considered for the temperature, allowing us to probe the smallscale region 500 /lessorequalslant /lscript /lessorequalslant 2500, and (b) adds information on the E - and B -mode polarization. Moreover, the SPT experiment pushes the dynamic range of CMB observations to larger multipoles with the respect of WMAP7, measuring with a better accuracy the damping tail of the CMB angular power spectrum. We consider data up to /lscript = 3100. For the SPT experiment, it is necessary to account for foreground contributions by adding three extra parameters representing the amplitude of the SZ, A SZ , clustering, A C , and shot-noise, A P , signal from point sources. We used for each foreground component the proper template provided by Keisler et al. (2011). When deriving our constraints we marginalize over these three nuisance parameters. To compute the likelihood of the data we have properly modified the CosmoMC package in order to make use of the routine supplied by the WMAP team for the WMAP7 data set, publicly available at the LAMBDA website 5 , and of the likelihood code provided by the SPT team (Keisler et al. 2011) for the SPT experiment. As we already discussed in the introduction, the inclusion of small-scale CMB measurements can greatly help in breaking the degeneracy with the other cosmological parameters, and in particular with n s , thus improving the constraints on the DM sector parameters. Moreover, the addition of the SPT data to the WMAP data improves the constraints on the ratio of the sound horizon to the angular diameter distance parameter θ s by nearly a factor of 2 (Keisler et al. 2011), thus narrowing the allowed range of the other parameters. We adopt the standard parametrization for the reionization, considered as an instantaneous process occurring at some redshift z r , with z r < 32. Such a choice leads to a one-to-one relation between z r and the adopted e.s. optical depth τ . As a caveat, we note that Pandolfi et al. (2011) showed that a more realistic reionization modelling might affect the cosmological parameters that are more degenerate with the DM annihilation cross-section, thus introducing an additional source of uncertainty(see however also Moradinezhad Dizgah, Gnedin & Kinney 2012).", "pages": [ 4, 5 ] }, { "title": "3 RESULTS", "content": "DM-modified ionization and thermal histories for a 10 GeV WIMP annihilating into muons on top of a ΛCDM model are shown in Fig. 3; the corresponding TT (EE) spectrum is shown in Fig. 4 (Fig. 5). Qualitatively similar conclusions can be drawn for the other channels. The energy released in the form of electrons and positrons from the annihilation of DM particles delays and quenches the recombination processes, thus resulting in a freeze-out relic electron fraction a factor of a few larger, depending on the value of 〈 σv 〉 . For the same reason, the temperature drop with time is less pro- d. As a consequence of the higher ionization rate, the CMB normalization value is smaller. We follow the procedure described in Section 2.2 to get constraints on the cosmological parameters in equation (9) and we compare them with those obtained by the WMAP team from their 7-yr data. We present in Fig. 6 the 2 σ c.l. constraints on the DM annihilation cross-section 〈 σv 〉 as a function of the DM mass. Differently to (e.g. Galli et al. 2009) our results cannot be given as a single number due to the mass dependence of the energy deposition fractions (see Section 2.1). A detailed comparison with their results will be given in Section 3.1. The main conclusion is that only DM candidates lighter than /lessorequalslant 10 GeV annihilating via the e + -e -channel can be excluded as a dominant component of the DM energy density. The constraints are stronger, as expected, if we include in the present analysis the recent SPT data set with /lscript max = 3100 6 and the polarization data. In this case the electron channel is excluded in the entire mass range (up to 20 GeV), where the other two channels can be excluded for masses < ∼ 5 GeV. We have verified that the stronger constraints come mainly from the SPT data inclusion, since the polarization data alone improve the constraints by < 3%. Currently polarization data alone are not of sufficient quality to robustly constrain DM parameters. Future experiments, specifically devoted to measure polarization at smaller scales like Planck (Tauber et al. 2010), PolarBear (Anthony 2012) and CMBPol (Zaldarriaga et al. 2008) are expected to significantly improve the situation. The CMB constraints we find are weaker than the constraints obtained by the Fermi experiment using the signal in the diffuse isotropic gamma emission from the Galaxy (Abdo et al. 2010) and from a combined analysis of the Milky Way satellites (Ackermann et al. 2011; Baushev, Federici & Pohl 2012; Cholis & Salucci 2012). Comparing the 10 GeV case of annihilation channel in muons and that in taus, the inferred maximum cross-section from Fermi falls below the thermal value. However, in their analysis the rather uncertain distribution of DM in galaxies must be specified, while the present approach is free from any such hypothesis. In Table 3 we report the 68% c.l. constraints on the cosmological parameters for the 10 GeV muon annihilation channel for the WMAP7 and CMB ALL+SPT cases, and the WMAP7 alone data set, i.e. a minimal ΛCDM model without annihilating DM ('WMAP7 (Standard)'). The onedimensional posterior probability for Ω b h 2 , Ω DM h 2 and n s for the three data set cases considered is also shown in Fig. 7. The strongest shift occurs for the baryon density Ω b h 2 which in the minimal, six-parameter, standard case is Ω b h 2 = 0 . 0226 ± 0 . 0006, whereas, after the inclusion of the annihilating DM, becomes Ω b h 2 = 0 . 0224 ± 0 . 0006 in the case of WMAP7 and Ω b h 2 = 0 . 0217 ± 0 . 0004 in the case of CMB ALL+SPT . This lower baryon density required results from the increased number of electrons produced DM annihilations; the two factors combine to give the same optical final depth needed to match the CMB data. The constraints on the DM density are only barely affected by the introduction of the DM annihilation, while instead the constraints on the scalar spectral index of primordial perturbations are shifted to higher values. Similarly to the case of Ω b h 2 , but in the opposite direction, the extra energy injected by the DM annihilation leads to a damping of the tail of CMB power spectrum, so that n s has to be increased in order to compensate for this effect and still provide a good fit to the data. Note that in the case of WMAP7 , the introduction of DM annihilation makes the Harrison-Zel'dovich value for the scalar spectral index n s = 1 compatible with the data within two standard deviations, while instead when also the SPT data set is added the scale invariant power spectrum is again ruled out by the data.", "pages": [ 5, 6 ] }, { "title": "3.1 Simplified energy deposition model", "content": "As we have stressed already, using a correct description of the energy deposition fractions is crucial to derive reliable DM constraints. Here we intend to quantify this statement by comparing our results with the constraints obtained using an approximated energy deposition model. This is summarized by the following expressions: where x ≡ x H + f He x He is a convenient variable to be used in RECFAST and (a similar expression is valid for the He) where C H and C He are the Peebles factors as given in Wong, Moss & Scott (2008). As in Galli et al. (2011) we have multiplied these formulae for the f abs ( z ) given by Slatyer, Padmanabhan & Finkbeiner (2009) for the DM annihilation in electrons or muons at 1 GeV. In Fig. 2, we show the corresponding energy depositions as a function of redshift for the muon channel and we compare with what is obtained from the Monte Carlo simulations. It is evident that this simplified approach over predicts the energy deposition for almost the entire redshift range. We have verified that using the analytic expression in eq.s 10-12, the derived constraints at 1 GeV are found to be coincident with the results reported in table II by Galli et al. (2011) either for the muon or the electron channel. In Fig. 8, we show the relative differences between our results and the results obtained adopting the simplified model. We compare the case in which only WMAP7 data are used. In the range m DM = 1 -20 GeV, the differences can be quoted between 10 and 30 per cent for the electron channel, and between 20 and 60 per cent for the muon channel. The constraints we get always tend to be weaker than those given by Galli et al. (2011): the difference originates from the inclusion of the low-energy processes inducing a net energy-loss (i.e. energy not going into heating, ionization or excitation). As explained in the introduction, decreasing the energy deposition fractions makes the constraints weaker.", "pages": [ 6 ] }, { "title": "4 SUMMARY AND DISCUSSION", "content": "We have investigated the imprints left on the CMB temperature and polarization spectra by the energy deposition due to annihilations of one of the most promising DM candidates, a stable WIMP of mass m χ = 1 -20 GeV annihilating into leptons. A major improvement with respect to previous similar studies is a detailed treatment of the annihilation", "pages": [ 6 ] }, { "title": "8 C. Evoli, S. Pandolfi and A. Ferrara", "content": "cascade and its energy deposition in the cosmic gas. This is vital as this quantity is degenerate with 〈 σv 〉 . We performed an MCMC analysis using a modified version of the CosmoMC code and CMB data from the WMAP, QUaD and SPT experiments. By further marginalizing over the cosmological parameters of the background cosmology, we obtain the constraints on the annihilation cross-section for each annihilation channel. The strongest constraints are obtained by combining all the available data sets up to /lscript max = 3100. If annihilation occurs via the e + -e -channel, a light WIMP can be excluded as a viable DM candidate in the above mass range. However, if annihilation occurs via µ + -µ -or τ + -τ -channel instead, we find that WIMPs with m χ > 5 GeV cannot be ruled out at 2 σ c.l. to provide the cosmologically required DM content. We have compared our results with the constraints obtained by assuming a simplified energy deposition model, such as the one profusely used in the recent literature, and we found that realistic energy deposition descriptions can influence the resulting constraints up to 60 per cent. How- ever, at the present stage it was not possible to disentangle the effects of the on-the-spot approximation used in the current analysis from the effects of adopting more realistic low-energy deposition fractions and we postpone to a forthcoming analysis a more detailed comparison between the two approaches. We expect that a better understanding of the energy deposition by DM annihilation will be relevant in particular with the upcoming Planck 7 data, with their better sensitivity, which allow a better constraining of this additional source of ionization.", "pages": [ 8 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank C. L. Bianco for his precious help. We also thank Silvia Galli, Fabio Iocco, Luca Maccione, Pasquale Dario Serpico and Tracy Slatyer for useful comments and discussions. CE and SP acknowledge a Visiting Grant from SNS where part of this work has been carried out. CE acknowledges support from the Helmholtz Alliance for Astroparticle Physics funded by the Initiative and Networking Fund of the Helmholtz Association. The Dark Cosmology Centre is funded by the Danish National Research Foundation.", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "Aalseth C. E. et al., 2011a, Physical Review Letters, 106, 131301 -, 2011b, Physical Review Letters, 107, 141301 Abdo A. A. et al., 2010, JCAP, 4, 14 Ackermann M. et al., 2011, Physical Review Letters, 107, 241302 Akerib D. S. et al., 2010, Phys. Rev. D, 82, 122004 Anthony A. E., 2012, in American Astronomical Society Meeting Abstracts, Vol. 219, American Astronomical Society Meeting Abstracts 219, p. 212.04 Aprile E. et al., 2011, Physical Review Letters, 107, 131302 Baushev A. N., Federici S., Pohl M., 2012, Phys. Rev. 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A. et al., 2010, A&A, 520, A1 Vald'es M., Evoli C., Ferrara A., 2010, MNRAS, 404, 1569 Vald'es M., Ferrara A., Mapelli M., Ripamonti E., 2007, MNRAS, 377, 245 van den Aarssen L. G., Bringmann T., Goedecke Y. C., 2012, Phys. Rev. D, 85, 123512 Wong W. Y., Moss A., Scott D., 2008, MNRAS, 386, 1023 XENON100 Collaboration et al., 2012, arXiv:1207.5988 Zaldarriaga M. et al., 2008, arXiv:0811.3918", "pages": [ 8, 9 ] }, { "title": "APPENDIX A: LOCAL DEPOSITION", "content": "We have assumed in this paper that photon energy deposition occurs locally, which is not true in general (see Slatyer, Padmanabhan & Finkbeiner 2009). In the following, we show that this approximation is accurate in the energy range of O (1) GeV electrons. CMB photons gain energy as they are inverse Compton scattered by energetic leptons. At each scattering event, a CMB photon with mean energy E γ, CMB will be upscattered to an energy equal to: where γ is the Lorentz factor for the lepton. At epochs in which energy deposition is important ( z /lessorequalslant 1000) such upscattered photons are subsequently mainly downgraded by Compton scattering with thermal electrons (Chen & Kamionkowski 2004; Slatyer, Padmanabhan & Finkbeiner 2009). To estimate the efficiency of this mechanism, we compare in Fig. A1 the Compton cooling time, t -1 cool = ( dlnE/dt ) = cσ T n e (1+ z ) 3 /epsilon1g ( /epsilon1 ), of a photon upscattered by a 1 GeV electron (see equation A1) with the Hubble time, t H = H -1 ( z ), where /epsilon1 ≡ E γ /m e c 2 and g ( /epsilon1 ) is the classical Klein-Nishina cross-section. It is evident that the local deposition assumption, requiring t H /greatermuch t cool , can be safely applied in the redshift range of interest here ( z ∼ 600 as confirmed by the principal component analysis performed by Finkbeiner et al. (2012)). Note however that ICS produces a broad spectrum of photons, and photons produced at lower energies with respect to the peak energy of equation A1 cool more slowly than their higher energy counterparts. This can in part explain the differences between the results we obtain with the two deposition models described in Section 3.1.", "pages": [ 9, 10 ] }, { "title": "APPENDIX B: FITTING FORMULAE", "content": "Below are the fitting formulae to the numerically derived energy depositions of electrons and positrons in the various channels. where: The values of the parameters are given in Tab. B1. Moreover, we provide an updated version with respect to Evoli et al. (2012) for the energy deposition fractions that can be used for energies below the IC threshold: E th = ((1+ z ) / 21) -1 / 2 MeV: where the values of the parameters are given in Tab. B2.", "pages": [ 10 ] } ]

[ { "title": "ABSTRACT", "content": "Recent observations in galaxies and clusters indicate dark matter density profiles exhibit core-like structures which contradict to the numerical simulation results of collisionless cold dark matter. The idea of self-interacting cold dark matter (SICDM) has been invoked to solve the discrepancies between the observations and numerical simulations. In this article, I derive some important scaling relations in galaxies and clusters by using the long-range SICDM model. These scaling relations give good agreements with the empirical fittings from observational data in galaxies and clusters if the dark matter particles are only slightly self-interacting. Also, there may exist a universal critical optical depth τ c that characterizes the core-like structures. These results generally support the idea of SICDM to tackle the long-lasting dark matter problem. Key words: Dark matter, galaxies, clusters", "pages": [ 1 ] }, { "title": "M. H. Chan /star", "content": "Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong Shatin, New Territories, Hong Kong, China Accepted XXXX. Received XXXX", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The nature of dark matter remains a fundamental problem in astrophysics and cosmology. The rotation curves of galaxies and the mass profile probed by the hot gas in clusters indicate the existence of dark matter. It is commonly believed that dark matter is collisionless and becomes nonrelativistic after decoupling. Therefore, they are regarded as cold dark matter (CDM). The CDM model can provide excellent fits on large scale structure observations such as Ly α spectrum (Croft et al. 1999; Spergel and Steinhardt 2000), 2dF Galaxy Redshift Survey (Peacock et al. 2001) and Cosmic Microwave Background (Spergel et al. 2007). However, on the cluster and galactic scales, the CDM model shows discrepancies from observations. N-body simulations based on the CDM theory predict that the density profile of the collisionless dark matter halo should be singular at the center ( ρ ∼ r α ). Navarro et al. (1997) first obtained α = -1 (the NFW profile). Later, different values of α ranging from -0 . 75 to -1 . 5 were obtained (Moore et al. 1999; Klypin et al. 2001; Taylor and Navarro 2001; Col'ın et al. 2004; Diemand et al. 2005). Recently, high resolution numerical simulation indicates α = -0 . 8 for r ≈ 120 pc and α = -1 . 4 for r ≈ 2 kpc (Stadel et al. 2009). Nevertheless, observations show us core-like structures instead of singular density profile in many clusters and galaxies. For example, H α observations indicate cores", "pages": [ 1 ] }, { "title": "c © XXXX RAS", "content": "present in over a hundred of disk galaxies and dark matter dominated galaxies (Salucci 2001; Borriello and Salucci 2001). Later, de Blok et al. (2003) get a mildly cuspy slope α = -0 . 2 ± 0 . 2 based on modelling the presence of realistic observational effects. In cluster scale, observational data from gravitational lensing also show that cores exist in some clusters (Tyson et al. 1998; Newman et al. 2011). In particular, Sand et al. (2008) get α = -0 . 45 ± 0 . 2 by the combination of gravitational lensing and dynamical data of clusters MS2137-23 and Abell 383. Clearly, observations do not support the numerical small-scale predictions by the CDM model. This discrepancy is known as the core-cusp problem (de Blok 2010). In addition, computer simulations predict that there should exist thousands of small dark halos or dwarf galaxies in the Local Group if the dark matter particles are collisionless (Cho 2012). However, observations of the Local Group only reveal less than one hundred galaxies (Spergel and Steinhardt 2000). Such discrepancy is known as the missing satellites problem (Cho 2012). Many theories have been invoked to solve the corecusp problem and the missing satellites problem. One of the most spectacular idea is that the dark matter is not cold. The existence of keV sterile neutrinos, as a candidate of warm dark matter (WDM), has been proposed to solve the discrepancies (Xue and Wu 2001). However, recent observations tend to reject the keV sterile neutrinos to be the major component of dark matter since the observational bound of sterile neutrino mass in Lyman-alpha forest contradicts to that in x-ray background (Abazajian and Koushiappas 2006; Viel et al. 2006; Seljak et al. 2006). Also, the WDM model alone cannot get a good agreement on the large scale power spectrum (Spergel and Steinhardt 2000; Boyarsky et al. 2009). The WDM model is likely to be ruled out in standard cosmology. Therefore, the success of the CDM model on large scales suggests that a modification of the dark matter properties may be the only approach to solve the discrepancies (Spergel and Steinhardt 2000). Spergel and Steinhardt (2000) proposed that the conflict of observations and simulations can be reconciled if the CDM particles are selfinteracting. Later, Burkert (2000) performed the numerical simulation of the self-interacting cold dark matter (SICDM) and showed that core-like structures can be produced. On the other hand, the analysis of the metallicity distributions of globular clusters indicates that the existence of the SICDM is able to solve the missing satellites problem (Cˆot'e et al. 2002). The earliest estimated range of the crosssection per unit mass of the self-interacting dark matter particle is σ/m = (0 . 45 -450) cm 2 g -1 (Spergel and Steinhardt 2000). This ratio has been estimated several times by some model dependent observations of clusters and galaxies and numerical simulations. For example, Randall et al. (2008) and Bradaˇc et al. (2008) obtained σ/m < 0 . 7 cm 2 g -1 and σ/m < 4 cm 2 g -1 respectively by using the observational data from the clusters 1E 0657-56 and MACS J0025.41222. On the galactic scales, Ahn and Shapiro (2005) and Koda and Shapiro (2011) show that σ/m ∼ 100 cm 2 g -1 can explain the core-like structures. However, gravitational lensing and X-ray data indicate that the cores of clusters are dense and ellipsoidal where SICDM model predicts that to be shallow and spherical (Loeb and Weiner 2011). Therefore, the dark matter crosssection may either be smaller than expected or depend on velocity. Nevertheless, Peter et al. (2012) show that the discrepancies can still be solved even if the cross-section is velocity-independent. The latest numerical simulations with SICDM indicate that the cross-section per unit mass should be σ/m ∼ 0 . 01 -0 . 1 cm 2 g -1 in order to produce the reported core sizes and central densities of galaxies and clusters (Buckley and Fox 2010; Rocha et al. 2012; Peter et al. 2012; Zavala et al. 2013). In this article, I will show in another way that the slightly long-range interaction of dark matter can naturally generate some model independent scaling relations in galaxies and clusters which agree with the observations. Lastly, I will comment on this small interaction of dark matter.", "pages": [ 1, 2 ] }, { "title": "2 OPTICAL DEPTH OF THE DARK MATTER PARTICLES", "content": "In SICDM model, the size of a core in a structure depends on the self-interacting rate of the dark matter particles. This rate is closely related to a physical quantity 'optical depth of the dark matter particles' τ . The optical depth for dark matter is defined as τ ≡ nσd , where d is the distance travelled by a dark matter particle and n is the mean number density of the dark matter particles. Therefore, the optical depth within the core radius r c is given by τ = nσr c . The dark matter particles can be considered as collisonless if τ ≈ 0. Spergel and Steinhardt (2000) propose that τ ≈ 1 within the core, which corresponds to the 'photosphere' of the dark matter. However, this optical depth is too large to match the observational data. Here, we assume that the size of the core is characterized by a critical optical depth τ c such that nσr c = τ c , where 0 /lessorequalslant τ c /lessorequalslant 1. Since the core mass is given by M c = 4 πmnr 3 c / 3, we have The above equation indicates a rough scaling relation M c ∝ r 2 c if τ c is a constant. This relation is generally consistent with the recent result in galaxies obtained by Gentile et al. (2009): M c = 72 +42 -27 πr 2 c M /circledot pc -2 .", "pages": [ 2 ] }, { "title": "3.1 Baryonic Tully-Fisher relation", "content": "The orbital speed in a galaxy is given by where M and R are the total enclosed mass and radius of luminous matter respectively. From Eq. (2), the observed flat rotation curves in most galaxies give M/R ≈ M c /r c . By combining Eqs. (1) and (2), we get The density profile of the SICDM can be approximately given by the Burkert profile (Burkert 1995; Rocha et al. 2012): where ρ 0 is the central density of dark matter. Therefore, the integrated mass profile is given by where f ( r ) = ln[( r 2 + r 2 c ) /r 2 c ] + 2ln[( r + r c ) /r c ] -2 tan -1 ( r/r c ). The size of luminous matter R can be regarded as the radius r max where the rotation curve peaks in the simulations, ie. R ≈ r max . Since the numerical simulations indicate that r max ≈ 3 r c (Rocha et al. 2012), by Eq. (5), the integrated total mass to core mass ratio is about M/M c ≈ 5. Assume that the ratio of total baryonic mass to total mass is nearly a constant for all galaxies ( M b /M ≈ Ω b / Ω m ≈ 0 . 17, where Ω b and Ω m are the cosmological density parameters of baryonic matter and total matter respectively), the total baryonic mass of a galaxy is If τ c and σ/m are constant for all galaxies, we have M b ∝ V 4 . This scaling relation is indeed the baryonic Tully-Fisher relation (Tully and Fisher 1977; McGaugh 2005, 2012). Latest observations indicate M b = (47 M /circledot km -4 s -4 ) V 4 (McGaugh 2012). If σ/m = 0 . 1 cm 2 g -1 , we get τ c = 0 . 005. In fact, Mo and Mao (2000) have already shown that the TullyFisher relation can be obtained by assuming a particular form of cored density profile. Here, I use another independent and simpler way to show that the Tully-Fisher relation is consistent with the SICDM scenario. Furthermore, from Eq. (1), we have ρ 0 r c = τ c ( σ/m ) -1 , which would be a constant if τ and σ/m are constants. Surprisingly, recent analysis indicates that ρ 0 r c = 141 +82 -52 M /circledot pc -2 , which is a constant for a large sample of dwarf and late-type galaxies (Gentile et al. 2009). If τ = 0 . 005 and σ/m = 0 . 1 cm 2 g -1 , we get ρ 0 r c ≈ 240 M /circledot pc -2 , which is generally closed to the empirical fits from observations.", "pages": [ 2, 3 ] }, { "title": "3.2 Size-Temperature relation in clusters", "content": "Reiprich and Bohringer (2001) studied more than 100 clusters' hot gas profiles and probed the total mass of each cluster. The mass profile of a cluster can be approximately given by (Reiprich and Bohringer 2001) where β is the parameter ranging from 0 . 4 -1 . 1 in the King's β -model (King 1972), T is the hot gas temperature and m g is the mean mass of a hot gas particle. Here, we have used the fact that the hot gas profiles are nearly isothermal in most clusters (Reiprich and Bohringer 2001). From Eq. (7), the central density of the dark matter is given by Since the central density of hot gas is just 10 -26 g cm -3 (Mohr et al. 1999), which is much less than the total central density 10 -23 g cm -3 , the effect of the baryons at the centre is ignored. By combining Eqs. (1) and (8), we get In fact, r c represents the core sizes of both total matter (dominated by dark matter) and baryonic matter (Reiprich and Bohringer 2001). Therefore, the size of the hot gas in cluster can be characterized by r c . If τ c and σ/m are constant for all clusters, we have a scaling relation r c ∝ T , which agrees with the empirical fits R ' = 0 . 5( T/ 6 keV) 1 . 02 Mpc from observational data of some nearby clusters (Mohr et al. 2000; Sanders 2007), where R ' is the isophotal size of a cluster. For a 10 15 M /circledot cluster, r c ≈ 300 kpc (Rocha et al. 2012), which is ≈ 0 . 7 R ' . If σ/m = 0 . 1 cm 2 g -1 , by using Eq. (9) and the mean β for all clusters, we have τ c = 0 . 006.", "pages": [ 3 ] }, { "title": "3.3 Mass-Temperature relation in clusters", "content": "Besides the Size-Temperature relation, we can also obtain a scaling relation of the total cluster mass and hot gas temperature. At large radii, the hot gas in clusters may not be isothermal. The total cluster mass will be closed to the Burkert mass profile in Eq. (5): where we have assumed that R 200 ≈ 15 r c and R 200 is the radius when the mean total mass density equals to 200 times cosmological critical density. By putting Eqs. (8) and (9) into the above equation and assuming M b /M ≈ Ω b / Ω m , we have Since the hot gas mass dominates the baryonic mass in most clusters, the total hot gas mass M g in a cluster is closed to the total baryonic mass M b . This scaling relation M g ≈ M b ∝ T 2 , again, agrees with the empirical fitting from clusters M g / 10 14 M /circledot = 0 . 017( T/ 1 keV) 2 (Mohr et al. 1999; Sanders 2007). By putting all the known numerical values and σ/m = 0 . 1 cm 2 g -1 into Eq. (11), we get τ c = 0 . 002.", "pages": [ 3 ] }, { "title": "4 DISCUSSION", "content": "In this article, I show that the long-range interaction of CDM can naturally obtain some important scaling relations, including the baryonic Tully-Fisher relation for galaxies ( M b ∝ V 4 ), the Size-Temperature relation ( r c ∝ T ) and Mass-Temperature relation ( M g ∝ T 2 ) in clusters. These scaling relations get remarkably good agreements with the empirical fits from observations. If the cross-section per unit mass is ≈ 0 . 1 cm 2 g -1 , the characteristic critical optical depth τ c ≈ 0 . 002 -0 . 006, which is the nearly the same values in different scaling relations for galaxies and clusters. Moreover, we can get ρ 0 r c ≈ 240 M /circledot pc -2 , which is a constant for all galaxies. This result is generally consistent with the recent analysis from the observations of dwarf and late-type galaxies (Gentile et al. 2009). It means only a slight dark matter interaction is enough for producing core-like structures. Therefore, when the central density is high enough such that τ = τ c , a core would be produced. It may explain why some clusters do not exhibit core-like structures as their central densities are too low such that τ /lessorequalslant τ c within a resolvable radius. In the past decade, it is believed that the dark matter cross-section is velocity-dependent (Koda and Shapiro 2011; Loeb and Weiner 2011). Nevertheless, recent results in simulations show that it is possible to have a velocityindependent cross-section σ/m ≈ 0 . 1 cm 2 g -1 (Rocha et al. 2012; Peter et al. 2012). In this model, the scaling relations derived may support this idea and enable us to measure this small cross-section by using observational data. Although only a small window of constant cross-section is remained (Zavala et al. 2013), our results provide more evidences to support the SICDM scenario, which can successfully address the dark matter problem, core-cusp problem and the missing satellite problem. To conclude, the derived scaling relations by using the SICDM scenario can get good agreements with observations. It generally supports the idea of the SICDM and the velocity-independent dark matter cross-section. More simulations and observations will be needed to confirm the existence of the universal critical optical depth τ c , which characterizes the core-like structures in galaxies and clusters.", "pages": [ 3 ] } ]

[ { "title": "ABSTRACT", "content": "This paper explores a mathematical technique for deriving dynamical invariants (i.e. constants of motion) in time-dependent gravitational potentials. The method relies on the construction of a canonical transformation that removes the explicit time-dependence from the Hamiltonian of the system. By referring the phase-space locations of particles to a coordinate frame in which the potential remains 'static' the dynamical effects introduced by the time evolution vanish. It follows that dynamical invariants correspond to the integrals of motion for the static potential expressed in the transformed coordinates. The main difficulty of the method reduces to solving the differential equations that define the canonical transformation, which are typically coupled with the equations of motion. We discuss a few examples where both sets of equations can be exactly de-coupled, and cases that require approximations. The construction of dynamical invariants has far-reaching applications. These quantities allow us, for example, to describe the evolution of (statistical) microcanonical ensembles in time-dependent gravitational potentials without relying on ergodicity or probability assumptions. As an illustration, we follow the evolution of dynamical fossils in galaxies that build up mass hierarchically. It is shown that the growth of the host potential tends to efface tidal substructures in the integral-of-motion space through an orbital diffusion process. The inexorable cycle of deposition, and progressive dissolution, of tidal clumps naturally leads to the formation of a 'smooth' stellar halo. Key words: galaxies: haloes - Galaxy: evolution - Galaxy: formation - Galaxy: kinematics and dynamics", "pages": [ 1 ] }, { "title": "Dynamical invariants and diffusion of merger substructures in time-dependent gravitational potentials", "content": "Jorge Peñarrubia 1 , 2 glyph[star] 1 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 2 Ramón y Cajal Fellow, Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía, 18008, Granada, Spain 6 September 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The description of dynamical systems out of equilibrium remains an outstanding problem in Physics and Astronomy. Hamilton was among the first to attack it via the construction of 'perturbed' Hamiltonians for systems that are close to an equilibrium state where ( q , p ) are the coordinates of a particle in configuration space, t is the time and glyph[epsilon1] glyph[lessmuch] 1. In perturbative methods solutions to the equations of motion are calculated iteratively from low to high order. Unfortunately, the trajectories of particles through phase space can be rarely expressed analytically, limiting the applicability of this method to the simplest of cases (e.g. Binney & Tremaine 2008). An improved perturbation theory is obtained by expressing the Hamiltonian in terms of action-angle variables. This technique is particularly attractive for Hamiltonians that are completely separable. In these systems the actions ( J ) associated with the timeindependent term ( H 0) are conserved along the path of a particle glyph[star] motion and define the surface of a torus in phase-space, while angle variables vary linearly with time and provide the coordinates of a particle on the torus. Yet, this approach has its own drawbacks. For example, the analytical expression of action variables is only possible for a few cases of astronomical interest, namely the Keplerian, harmonic and isochronic potentials (e.g. Binney 2010, 2012a,b). Furthermore, in time-varying potentials actions do not remain constant but oscillate with an amplitude | ∆ J / J 0 | ∝ glyph[epsilon1] T 0, where glyph[epsilon1] ≡ ˙ Φ / Φ 0 is the time-derivative of the gravitational potential and T 0 is the period associated with the motion on the torus. When averaged over several orbital periods, actions are conserved at order O ( glyph[epsilon1] T 0) 2 . Thus, identifying glyph[epsilon1] with the growth rate of the potential in Equation (1) shows that perturbation theories can only be applied to systems that evolve in an adiabatic regime. Perturbative methods are deterministic , i.e. they rest upon solutions to the dynamical equations of motion. A fundamentally different description of many-body systems approaching dynamical equilibrium is provided by the construction of statistical ensembles, in which trajectories of particles are replaced by a probability distribution of finding sets of particles in a given phase-space volume. Although mechanical statistics successfully describe the gross evolution of systems evolving under a rapidly-varying grav- itational field, such as those driving 'violent relaxation' processes (Lynden-Bell 1967; Tremaine et al. 1986; Ponzten & Governato 2013), it was early realized that deterministic and statistical descriptions of systems subject to long-range forces do not lead to the same physical behaviour (e.g. Hertel & Thirring 1971; see Padamanabhan 1990 for a review). Indeed, gravitationally bound objects have negative specific heat (Antonov 1961; Lynden-Bell & Lynden-Bell 1977; Padmanabhan 1989; see Lynden-Bell 1999 for a review), a quantity that must be positive definite in ensembles where the energy of individual particles is allowed to fluctuate probabilistically about a time-average value, as in the canonical and grand canonical distributions. As a consequence gravitating systems must be described by microcanonical ensembles, in which the energy of individual particles is kept fixed. In order to derive the microcanonical distribution it is commonly assumed that all individual microstates on a given energy surface in phase-space are equally probable (the so-called ergodic hypothesis). This assumption guarantees that the time-averaged properties of microcanonical ensembles can be directly derived from a phase-space average over all possible microstates. However, the equivalence between time average and average over ensembles only arises when the system can visit all the possible microstates, many times, during a long period of time. In the case of ensembles out of dynamical equilibrium, which contain transient microstates by definition, the ergodic assumption may lead to a biased description of the system. This paper explores a mathematical tool for deriving the dynamical evolution of microcanonical ensembles in time-dependent gravitational potentials which does not rely on ergodicity or probability assumptions. Instead, a canonical transformation is constructed (§2) that removes the explicit time-dependence from the Hamiltonian of the system. By referring the phase-space locations of the particle ensemble to a coordinate frame in which the potential remains 'static', the dynamical effects introduced by the time evolution disappear. It follows that dynamical invariants (i.e. quantities that are conserved along the phase-space path of a particle) can be straightforwardly constructed by expressing the integrals of motion for the static potential in the transformed coordinates. The main difficulty of this technique reduces to solving the differential equations that define the canonical transformation, which are typically coupled with those that define the trajectory of particles through phase space. Section 2.3 discusses a few examples of astronomical interest where both sets of equations can be de-coupled. The construction of invariants allows us to describe the macroscopical (statistical) properties of large ensembles of gravitating particles through a simple time averaging of microscopic (deterministic) equations. As an illustration, we study the thermodynamics of cold tidal substructures orbiting in a time-dependent potential in Section 3. In particular, we follow the evolution of entropy, temperature and specific heat, and compare the results against those derived from mechanical statistics. In Sections 3.1 and 3.2 we use dynamical invariants to describe the evolution of dynamical fossils in galaxies that build up mass hierarchically. This is a timely issue given that Gaia (Perryman et al. 2001) is expected to uncover a large number of accreted substructures in the integral-of-motion space (Helmi & de Zeeuw 2000; Brown et al. 2005; Gómez et al. 2010; Sharma et al. 2011; Mateu et al. 2011; although see Valluri et al. 2012). Given that integrals are not conserved quantities in hierarchical models of galaxy formation, we put special emphasis on understanding the diffusion of tidal substructures in the integral-of-motion space. Our analytical results are illustrated by means of restricted N -body models in Section 3.3. Section 3.5 discusses the detectability of tidal substructures. The conclusions are laid out in Section 4.", "pages": [ 1, 2 ] }, { "title": "2 DYNAMICAL INVARIANTS", "content": "The method for constructing dynamical invariants proposed below is conceptually simple. The goal is to find a coordinate transformation in which the explicit time-dependence of the potential vanishes, so that the integrals of motion in the transformed coordinates become the desired dynamical invariants. For simplicity, our calculations are derived in the mean-field limit, thus ignoring the granularity of N -body systems.", "pages": [ 2 ] }, { "title": "2.1 Newtonian formulation", "content": "Considering a universe in which Newton's constant G decreases, Lynden-Bell (1982) found a coordinate transformation that recovers standard equations of motion (see §2.3.2). This Section generalizes Lynden-Bell's arguments for any system with a timedependent gravitational potential. Let us first write the equations of motion of a particle subject to a time-varying force F ( r , t ) as The simplest distance transformation that one can introduce is the following so that the left-hand term in Equation (2) becomes Now we define a new time coordinate so that Equation (4) becomes For f = R -2 the velocity term vanishes, and the equation of motion becomes Weare still free to choose the time-dependent scaling function R ( t ). One would like to use this freedom to identify Equation (7) with the equations of motion in a time-independent potential, i.e. From Equations (7) and (8) the scaling factor must be a solution of the following differential equation Clearly, if F is a conservative force ( ∇× F = 0) then F ' is also conservative. Therefore, it is possible to define a time-independent scalar potential Φ ' = -∫ F ' d r ' , so that in the transformed coordinates the energy ( I ) becomes an exact dynamical invariant (i.e. a constant of motion) where R ( t ) is a solution of (9) and Φ ( r , t ) = -R ∫ F ( R r ' , t )d r ' . Note also that the angular momentum L remains invariant under the transformation d r ' / d τ = R ˙ r -˙ R r , i.e. In general it is straightforward to show that all classical integrals of the equations of motion of a Newtonian potential Φ ' ( r ' ) reappear as the constants of motion of Φ ( r , t ). This result also applies to integrals derived numerically, although obtaining these quantities is usually difficult (e.g. Bienaymé & Trevon 2013 and references therein).", "pages": [ 2, 3 ] }, { "title": "2.2 Hamiltonian formulation", "content": "Statistical mechanics provides a powerful tool in order to understand the physical behaviour of gravitating systems composed of many particles. For such analysis it is useful to generalize the results obtained in §2.1 using the Hamiltonian formalism. The Hamiltonian of a system with ν -degrees of freedom can be written as where ( q 1 , ..., q ν ; p 1 , ..., p ν ) are the coordinates of a particle in configuration space. The equations of motion are The fact that the dynamical invariants derived in §2.1 correspond to an energy in a new coordinate system suggests that there must exist a canonical transformation ( qi , pi ) → ( q ' i , p ' i ) that removes the explicit time dependence from the Hamiltonian. To find such a transformation we first consider an intermediate Hamiltonian ˆ H and a time-dependent generating function Q , so that ˆ H ( q ' , p ' , t ) = H ( q ' , p ' , t ) + ∂ Q ( q ' , p ' , t ) / ∂ t (see also Lewis & Leach 1982; Struckmeier & Riedel 2001). Following Equation (10), the goal is to find a generating function that yields the transformations p ' i = Rpi -˙ Rqi , and qi = Rq ' i . It is straightforward to show that the function is the desired one given that Before we calculate the new Hamiltonian ˆ H = H + ∂ Q / ∂ t recall that our goal is to find a dynamical invariant that is conserved along the phase-space path of a particle motion , i.e. the subset of the 6 N dimensional phase space on which the equations of motion (13a) and (13b) are fulfilled. This means that along the phasespace path all terms in Equation (14) that depend on the particle trajectory are functions of t only, so that the canonical transformation yields the following Hamiltonian where ˆ Φ ( q ' , t ) ≡ RR 3 ∑ i (1 / 2) q ' 2 i + R 2 Φ ( R q ' , t ). The time still appears explicitly in Equation (16) through R ( t ) and ˆ Φ ( q ' , t ). This dependence can be eliminated in two steps. First, through a re-scaling of the time coordinate. Choosing f = R -2 in Equation (5) yields the equations of motion d q ' i / d τ = R 2 ∂ ˆ H / ∂ p ' i = p ' i and d p ' i / d τ = -R 2 ∂ ˆ H / ∂ q ' i . Hence, the Hamiltonian H ' = R 2 ˆ H = ∑ i 1 2 p ' 2 i + ˆ Φ ( q ' , t ) has a time-dependence through ˆ Φ only. Next, as in §2.1, the freedom in the time-dependent scaling function R ( t ) can be used to remove the explicit time-dependence from the potential ˆ Φ so that Note that Equation (9) is recovered by differentiating with respect to ∂ / ∂ q ' i on both sides of Equation (17), which demonstrates that the Newtonian and Hamiltonian formalisms provide equivalent invariants. For example, writing Equation (16) as shows that the Hamiltonian H ' ( q ' , p ' ) contains no explicit time dependence and is therefore a dynamical invariant. To generalize the Newtonian invariant given by Equation (10) one can simply express the Hamiltonian (18) in terms of the original canonical coordinates through the transformation (15a) and (15b), i.e. where R ( t ) is a solution of (17), and [ q ( t ) , p ( t )] obey the equations (13a) and (13b). For potentials which do not explicitly depend on time (also known as autonomous systems ) R = 1 is a solution of Equation (17), and the total energy becomes an integral of motion.", "pages": [ 3 ] }, { "title": "2.3 Examples", "content": "The construction of analytical invariants is only possible in systems where the differential equations that define the coordinate transformation can be de-coupled from the equations of motion. In practice this condition is met when the phase-space coordinates do not appear explicitly in Equation (9) or (17). Sections 2.3.1 and 2.3.2 discuss a few cases where exact dynamical invariants can be found using the transformation given by Equations (3) and (15), while Section 2.3.3 shows that approximate invariants can be analytically calculated for slowly-varying powerlaw forces. For more complex systems the construction of analytic invariants will generally require coordinate transformations tailored to the specific scale and/or symmetry of the gravitational field.", "pages": [ 3 ] }, { "title": "2.3.1 Time-dependent harmonic oscillator", "content": "The case of an harmonic potential which varies with time has a broad range of applications in Quantum Mechanics (see Kaushal 1998 for a review). In Astronomy this potential arises in systems with an homogeneous mass distribution, like in the cores of elliptical galaxies (e.g. Binney & Tremaine 2008). For systems with spherical symmetry 1 the force in Equa-", "pages": [ 3 ] }, { "title": "4 Jorge Peñarrubia", "content": "tion (2) can be written as F = -ω 2 ( t ) r . Equations (8) and (9) become where ω 0 is a constant. The pair of Equations (20a) and (20b) corresponds to an Emarkov (1880) system, wherein the information about the time-evolution of the potential is carried by the auxiliary Equation (20b) (see Ray & Reid 1982). From Equation (8) the energy can be written as IHO = 1 / 2(d r ' / d τ ) 2 + 1 / 2 ω 2 0 r ' 2 . After substituting Equation (20b) into (10) the energy integral of a time-dependent harmonic oscillator reduces to where R ( t ) is a solution of Equation (20b).", "pages": [ 4 ] }, { "title": "2.3.2 Dirac's Cosmology", "content": "On Dirac's large-number hypothesis (Dirac 1938) 2 where t is the time since the big bang, mp and me are the proton and electron masses, and e is the electron charge. In a Universe where the properties of elementary particles remain constant G = G 0 / ( H 0 t ), where G 0 and H 0 are constants. The force term in Equation (2) can be simply written as F ( r , t ) = F ( r )( H 0 t ) -1 . Lynden-Bell (1982) showed that the choice R = H 0 t in Equation (7) naturally leads to Equation (8). After carrying the inverse transformation of coordinates the energy written as is an exact invariant. Here Φ is the Newtonian potential in the Dirac's cosmology, i.e. Φ ( r , t ) = -∫ d r F ( r , t ) with G = G 0 / ( H 0 t ).", "pages": [ 4 ] }, { "title": "2.3.3 Time-dependent power-law forces", "content": "The examples outlined above correspond to time-evolving forces with known constants of motion. Unfortunately, rare are the cases where the trajectory of the particle does not appear explicitly in R ( t ) (see Lewis & Leach 1982 and Feix et al. 1987 for further examples related to the harmonic oscillator). Fortunately, it is relatively simple to construct approximate invariants that de-couple from the equations of motion for systems that orbit in a slowly-varying gravitational potential. Although the resulting invariants are not exact constants of motion, we shall see below that their evolution is remarkably slow even when forces change on relatively short time scales. As in §2.1 and §2.2, the energy integral in the new coordinates turns out to be an invariant of the system. Let us consider a time-dependent power-law force Here the radius expressed in Cartesian coordinates is r 2 = x 2 / a 2 + y 2 / b 2 + z 2 / c 2 , with a , b and c being constant dimension-less quantities. Equations (7) and (8) lead to the auxiliary equation where µ 0 is a constant. For forces that change slowly, that is glyph[epsilon1] ' ≡ glyph[epsilon1] T 0 ≡ ( ˙ µ / µ 0) T 0 glyph[lessmuch] 1, where T 0 is the radial period of the orbit at t = t 0, it is straightforward to find approximate solutions for Equation (25) using a perturbative approach. At first order the function glyph[negationslash] is a solution of Equation (25) for r ' = 0. By substituting Equation (26) into Equation (10) the energy in the new coordinate system becomes where Φ 1 can be written as glyph[negationslash] Before we study perturbations at a higher order it is worth examining the solution implied by Equation (26) in more detail. Let us consider the fully analytical case of a time-dependent spherical harmonic oscillator ( n = 1) with frequency ω 2 ( t ) = ω 2 0 (1 + glyph[epsilon1] ' t / T 0), with glyph[epsilon1] ' = 0 . 01 and T 0 = 2 π / κ 0 = π / ω 0 = π , where κ 0 = 2 ω 0 is the radial frequency of an oscillator. Fig. 1 shows the variation of energy (black solid line) of an orbit with initial energy E 0 = 0 . 58 and angular momentum L 0 = 0 . 4, which correspond to orbital apo and pericentres r apo = 1 and r peri = 0 . 4, respectively. In an adiabatic regime the orbital energy of power-law potentials varies as (e.g. Pontzen & Governato 2012) which for n = 1 and µ / µ 0 = (1 + glyph[epsilon1] ' t / T 0) yields E ad / E 0 = (1 + glyph[epsilon1] ' t / T 0) 1 / 2 ≈ 1 + 0 . 01 t / (2 T 0) (magenta dotted line). Note that the time-average of the adiabatic energy is 〈 E ad 〉 ≈ 〈 E 〉 , showing that the adiabatic approximation is indeed accurate. However, deviations from the adiabatic approximation are clearly visible when the particle is not located at either orbital peri- or apocentre. This behaviour can be easily understood using our dynamical invariants. Through Equation (10) one can express the energy as From Equations (26) and (29) the energy evolution calculated at first order is where the invariant is set to the initial value of the energy, I = E 0, and µ = ω 2 . Thus, deviations from the adiabatic approximation in Fig. 1 oscillate in phase with the radial motion of the particle and are proportional to ˙ ω / ω . Similarly, Fig. 1 shows that the radial action Jr (dashed lines) oscillates with an amplitude ∼ O ( glyph[epsilon1] ' ). The harmonic potential allows for an analytical expression of the radial action, which can be written as (e.g. Goodman & Binney 1984) where L is the angular momentum of the particle. Inserting Equation (31) in (32) yields The first term Jr , 0 = ( E 0 -ω 0 L ) / (2 ω 0) is a constant, while the righthand term oscillates in phase with the radial motion of the orbit. Notice, however, that the term accompanying ( r · ˙ r ) vanishes when averaged over a full orbital revolution. Therefore, the timeaverage 〈 Jr 〉 = 1 / t ∫ t 0 d t ' Jr ( t ' ) is conserved at order O ( glyph[epsilon1] ' 2 ) for t glyph[greatermuch] T 0 = 2 π / ω 0. In contrast the evolution of the approximate invariants In is at order O ( glyph[epsilon1] ' 2 ) along the phase-space path of the particle motion . This is a remarkable result given that R 1 is a solution of Equation (25) at order O ( glyph[epsilon1] ' ). To understand the higher-order behaviour of R ( t ) let us construct a function R ≈ R 1 + δ R 2, where R 1 is a solution of Equation (26) and δ glyph[lessmuch] 1, and R 2 a residual function. Inserting R in Equation (25), isolating the terms proportional to δ , and neglecting those at higher order yields the second-order differential equation Note that for n = 1 this is the equation of a harmonic oscillator with frequency 2 √ µ 0, which also corresponds exactly to the radial frequency of the time-independent potential associated with Equation (24), i.e. κ = 2 ω 0 = 2 √ µ 0. Indeed, a similar result is obtained for the Keplerian case ( n = -2) if we approach r ' ≈ a , where a is the semi-major axis of the orbit. In this case we find that R 2 follows a cycle with a frequency √ µ 0 / a 3 , or a period T = 2 π √ a 3 / µ 0, which corresponds to the radial period of a Keplerian orbit. In general, Equation (34) shows that R 2 oscillates approximately in phase with the radial motion of the particle about the centre of the power-law force field. Hence, if the time dependence of the force evolves slowly, the averaged contribution of the terms accompanying R in Equation (10) can be safely neglected. As a result we find that in slowly-varying potentials ( glyph[epsilon1] ' glyph[lessorsimilar] 1) the approximate invariants given by Equations (27) and (28) are accurate at order O ( glyph[epsilon1] ' 2 ), even if the scale factor R 1 is a solution of Equation (25) at order O ( glyph[epsilon1] ' ). This result is illustrated in Fig. 2, where the mean variation of | ∆ I / I 0 | is plotted against the rate of potential change 2 / (3 + n ) glyph[epsilon1] ' . The factor 2 / (3 + n ) ensures that the fractional change of energy is approximately the same for all the orbits considered here. Note that on average I varies by a small amount, | ∆ I / I 0 | glyph[lessorsimilar] 0 . 1, even in cases where the adiabatic approximation does not hold and the potential evolves on a time-scale comparable to the radial period of the orbit, i.e. glyph[epsilon1] T 0 ∼ 1.", "pages": [ 4, 5 ] }, { "title": "3 AN APPLICATION: ACCRETED SUBSTRUCTURES IN TIME-DEPENDENT GALACTIC POTENTIALS", "content": "Hierarchical theories of structure formation propose that galaxies form through the accretion of smaller, gravitationally-bound bodies (White & Rees 1978). A natural prediction from this scenario is the presence of dynamical fossils in the present-day configuration space of the Milky Way. The integral-of-motion space may offer the best chances to uncover the hierarchical build-up of our Galaxy. Here accreted stars are expected to distribute in tight clumps rather than homogeneously, reflecting the fact that they were originally bound to lowmass systems which did not form in situ. However, this approach has a strong limitation: as the Galaxy grows hierarchically, so does its overall gravitational potential. Under such circumstances none of the integrals of motion is conserved. Hence, a natural question arises as to how tidal substructures evolve in the integral-of-motion space under a time-dependent potential. Below we attempt to tackle this issue using the dynamical invariants constructed in the previous Sections.", "pages": [ 6 ] }, { "title": "3.1 Orbital diffusion", "content": "Before we attempt to answer this question in detail it is worthwhile to study simple models that share the essential features of dynamical fossils in a time-dependent potential. For example, let us consider two particles moving in a spherical potential Φ ( r , t ) = µ ( t ) h ( r ) which varies slowly with time. These particles do not interact gravitationally, so their motion is entirely governed by Φ . We construct our experiment so that at time t = t 0 both particles move on the same orbit but are located at different radii. Hence ∆ E ( t 0) = E 1( t 0) -E 2( t 0) = 0, that is v 2 1 / 2 + µ ( t 0) h ( r 1) = v 2 2 / 2 + µ ( t 0) h ( r 2). We now integrate their orbits forward in time until both particles exchange their radial location. At that particular time, say t = t 1, their energies are E 1( t 1) ≈ v 2 2 / 2 + µ ( t 1) h ( r 2) and E 2( t 1) ≈ v 2 1 / 2 + µ ( t 1) h ( r 1), where it is assumed that the potential varies so slowly as to leave the orbital velocity at both radii unchanged. It is straightforward to show that at t = t 1 the energies of the two particles differ by the amount ∆ E ≈ [ µ ( t 1) -µ ( t 0)][ h ( r 2) -h ( r 1)]. This simple model illustrates two interesting features of tidal substructures evolving in time-dependent potentials that will become more obvious below. First, notice that ∆ E depends on the relative position of the particles. For tidal debris composed of many particles distributed over a large phase-space volume the growth of the host potential must necessarily lead to a progressive diffusion of orbital energies 3 . Over time this process may smooth out (and perhaps even efface) pre-existing clumps in the integral-of-motion space. Also, the fact that the length of tidal tails oscillates between peri- and apocentre, namely stretching over a large range of galactocentric distances as the progenitor moves close to orbital perincentre and piling-up at orbital apocentre (see Dehnen et al. 2004 for a beautiful illustration of this cycle), suggests that the diffusion process, rather than being gradual, may follow a cyclic evolution in phase with the radial motion of the progenitor system 4 . Sections 3.3 and 3.4 show that dynamical invariants provide a useful tool to understand the intricacies of this complex process.", "pages": [ 6 ] }, { "title": "3.2 Entropy evolution", "content": "The simplistic model of §3.1 assumes that all particles moving on a tidal stream follow the same orbit. In reality, tidal streams have an involved orbital structure (e.g. Küpper et al. 2010; Eyre & Binney 2011) whose complexity increases in proportion to the mass of the progenitor system (Peñarrubia et al. 2006; Choi et al. 2007). Let us define π ( E , r , ˙ r , t ) as the probability to find a star with an energy E at a phase-space location ( r , ˙ r ) at the time t . The probability function is normalized so that ∫ ∫ ∫ π ( E , r , ˙ r , t )d E d 3 r d 3 v = 1. To measure the orbital scatter introduced by the time dependence of the host potential it is useful to define the entropy associated with the energy distribution as Stars that are strongly clumped in the integral-of-motion space will have low values of H E , whereas the entropy of stars moving on loosely correlated orbits will be comparable to that of the smooth stellar background. The time-evolution of the entropy can be calculated through the energy invariants constructed in §2.1 and §2.2. Using Equation (30) and expanding the energy distribution at order O ( glyph[epsilon1] ' ) the probability distribution becomes where the prime denotes derivative with respect to energy, i.e. π ' = d π / d E , evaluated at E = R -2 I . Neglecting the terms O ( glyph[epsilon1] ' 2 ) and after some algebra the entropy defined in Equation (35) becomes where H I is the entropy associated with the invariant energy distribution and has, therefore, a constant value. The right-hand term of Equation (37), H osc, depends on the distribution of stars in phase space and has in general no analytical expression. However, its time evolution is well defined. Let us illustrate the evolution of H osc by considering stream particles with an energy distribution that is initially separable in space, an assumption which in less inaccurate for tidal debris from low-mass progenitors evolving in static potentials (see Peñarrubia et al. 2012). Under this approximation all particles on a given energy surface I have equal probability to move with a radial velocity ˙ r · e r , where e r is the radial unit vector. It is then straightforward to show that the integral ∫ π ' (1 + ln π )d I = 0, and thus H osc = 0. However, as stream particles spread out on their orbital paths a correlation between the radial velocity and the relative location in phase-space will arise as a result of the orbital diffusion process discussed in Section 3.1. During the early phases of the stream evolution most particles move on orbits that remain close to that of the progenitor. Thus, at early times the term H osc fluctuates in phase with the radial motion of the progenitor system. If the host potential evolves adiabatically the orbital periods of the stellar orbits are much shorter than the time-scale of the potential evolution, i.e. T 0 glyph[lessmuch] ( ˙ Φ / Φ ) -1 = glyph[epsilon1] -1 . Under the assumption that ˙ R / R remains approximately constant during a full orbital revolution, the oscillations of H osc are cyclic and the time-averaged evolution of the entropy becomes where t glyph[greatermuch] T 0. As tidal streams progressively fill the phase-space volume available to their orbits the number of particles moving toward apocentre ( r · ˙ r > 0) tends to approach that moving away from it ( r · ˙ r < 0). Hence, the probability of finding a particle in the energy interval ( E , E + d E ) at a given time t becomes independent of the sign of ( r · ˙ r ). By symmetry the right-hand term of Equation (37) tends to lim ∑ ( r · ˙ r ) → 0 H osc = 0, and through comparison with Equation (38) we find that the entropy of the microcanonical ensemble tends toward its time-averaged value as substructures mix in phase space. Henceforth we shall consider that tidal debris have reached a state of dynamical equilibrium if H E -〈H E 〉 = 0. Note that in this limit the entropy evolves as lim t →∞ ∆ H E = 〈 ∆ H E 〉 ≈ -2ln R , independently of the initial particle distribution in the integral of motion space. Perhaps the most remarkable result inferred from Equation (38) is the increasing entropy of substructures orbiting in a growing potential ( R < 1). In cosmologically-motivated galaxy models, which have a triaxial shape and only admit one integral of motion (the orbital energy), this result suggests that the dynamical signatures of accretion may be erased as a result of the hierarchical growth of the host galaxy. The inescapable consequence of the dynamical deposition, and progressive dissolution, of tidal clumps in the integral-of-motion space is the formation of a smooth stellar halo. A completely smooth galaxy, however, will never emerge from this process because the same mechanism that removes tidal substructures, i.e. the potential growth through the merger of smaller bodies, is also responsible for the formation of new ones. In galaxies with shrinking potentials ( R > 1) the evolution of entropy follows the reverse trend. Here tidal clumps tend to become more prominent with time. Examples of systems which have a decreasing potential may be found, for example, in globular clusters or satellite galaxies losing mass to tides 5 .", "pages": [ 6, 8 ] }, { "title": "3.3 N -body models in a logarithmic potential", "content": "It is worth illustrating the above results by running restricted N -body models of unbound substructures evolving in a timedependent logarithmic potential. Let us construct an idealized tidal stream model composed of N glyph[star] particles which do not interact gravitationally among themselves and follow a Gaussian energy distribution, i.e. where Es ( t 0) = v 2 s / 2 + Φ ( r s , t 0) is the mean orbital energy and σ s ( t 0) the energy dispersion at t = t 0. Following Küpper et al. (2012) all particles are placed initially at orbital apocentre with a common velocity vector, v s = ( vr , vt ), where the radial component is vr = ˆ r · v s = 0, and the tangential component is vt = √ 2[ Es -Φ ( r s , t 0)]. Note that the eccentricity of the orbits is set by our choice of vt . With this set-up it is straightforward to generate a sample of apocentres (e.g. using a rejection method) so that the initial energy distribution follows Equation (39). The host galaxy is modelled as an isothermal sphere whose potential evolves linearly with time, i.e. Hence, setting t 0 = 0, this potential varies from µ (0) = 1 -glyph[epsilon1] f , to µ ( t f ) = 1 within an integration time t f = 10 T 0, where T 0 is the radial period of an orbit with energy Es (0) and angular momentum Ls = rvt = vt . Note that glyph[epsilon1] f can be either positive or negative. The energy invariant can now be easily calculated by choosing n = -1 in Equation (27) and (28), and computing the scale factor from Equations (26) and (40) as Fig. 3 shows snap-shots of the time-evolution of a tidal stream with an energy dispersion σ s (0) = 0 . 05 orbiting on an eccentric orbit ( vt = 0 . 24 vc [0], where vc [0] = √ µ [0] is the circular velocity of the host at t = 0) in a potential that varies at a rate glyph[epsilon1] f = + 0 . 3 (two uppermost panels) and glyph[epsilon1] f = -0 . 3 (two lower-most panels). The projection of the particles onto the orbital plane shows that particles progressively spread out on the orbital path of the 'progenitor' system, which follows an orbit with energy Es and angular momentum Ls , leading to the formation of tail-like structures. Because the dynamical time scales as t dyn ∝ µ -1 / 2 the formation of tails accelerates in potentials that grow with time, and slows down in potentials that shrink with time. The second row of panels show the evolution of the energy distribution of a tidal substructure orbiting in a growing potential. These models illustrate the 'dissolution' of tidal substructures through the orbital diffusion process outlined in §3.1. The bottom row shows that in shrinking potentials the diffusion process appears to reverse , i.e. the energy distribution of tidal debris tends to become narrower with time. The evolution of the entropy associated to the above models is shown in Fig. 4 (blue dotted-dashed lines). As expected from Equation (38), the entropy of tidal debris oscillates about a timeaverage value 〈 ∆ H E 〉 = -2ln R , where R is given by Equation (41) (magenta dotted lines). The amplitude of the oscillations decreases as the stream particles approach dynamical equilibrium. However, the damping process is considerably slower for substructures that are dynamically 'cold' ( σ s = 0 . 01, green long-dashed lines), or if the tidal stream particles orbit in a decreasing potential ( glyph[epsilon1] f < 0). This suggests that not all substructures may reach dynamical equilibrium within a Hubble time. It is also worth noting that the entropy associated to the energy invariants (red dashed line) remains remarkably constant throughout the evolution of these models. The accuracy of our energy invariant, I , can be estimated directly from Fig. 2. Measuring the fractional variation of the logarithmic potential as glyph[epsilon1] = glyph[epsilon1] f / t f = 0 . 03 T -1 0 , we find | ∆ I / I 0 | glyph[lessorsimilar] 10 -4 and | ∆ H I | glyph[lessorsimilar] 10 -3 .", "pages": [ 8 ] }, { "title": "3.4 Thermodynamics of tidal substructures", "content": "Statistical mechanics provide an alternative physical description of the macroscopic properties of gravitating systems in dynamical equilibrium. In classical thermodynamics the probability of finding a particle in the energy interval ( E , E + d E ) at a given time t can be calculated as π th( E , t ) = g ( E , t ) f ( E , t ), where g ( E , t ) is the volume of phase space of the constant energy surface E = H ; and f ( E , t ) is the distribution function. For simplicity let us again adopt the dispersion of an isothermal sphere as σ 2 = 〈 v 2 〉 / 3, Equation (43) becomes µ = 3 T = (3 m / KB ) σ 2 . The phase-space distribution corresponding to the potential (40) is (Binney & Tremaine 2008) Thus, from Equations (42) and (44) the probability π th( E , t ) can be written as with B chosen so that the normalization of the probability function is ∫ π th( E , t )d E = 1. Let us now compare the entropy derived using the standard methods of equilibrium statistical mechanics and that resulting from the construction of dynamical invariants. Substituting Equation (45) into (35) and changing the integration variable to I = ( µ 0 / µ ) E , so that π th( E )d E = π th( I )( µ 0 / µ )d I , we find that the entropy associated to π th evolves as It is straightforward to show that the time-averaged entropy of tidal streams corresponds to the thermodynamical entropy of the host. For logarithmic potentials, n = -1, Equation (26) becomes R = ( µ / µ 0) -1 / 2 . Comparison of Equations (46) and (38) shows that ∆ H E , th = 〈 ∆ H E 〉 . Hence, both descriptions of entropy become identical in the limit of dynamical equilibrium, i.e. when the number of particles on an energy surface E = H moving outwards is equal to that moving inwards, lim ∑ ( r · ˙ r ) → 0 H E = H E , th. Fig. 4 can now be re-interpreted in terms of thermodynamical temperatures. Comparison of Equation (38) and (43) shows that the temperature of a logarithmic potential evolves as where brackets denote average over time. In a hierarchical galaxy formation framework this implies that galaxies heat up as they build up mass through the accretion of smaller bodies. In contrast, the entropy, and thus the temperature, of tidally-stripped objects drops progressively as they lose mass to tides 6 . Thermodynamically these systems behave as if they were in contact with hot and cold thermal baths, respectively 7 . Although the thermal bath analogy is helpful, it fails to provide a correct description of the dynamical evolution of tidal substructures that have not yet spread out on their orbital paths and are, therefore, out of dynamical equilibrium. Indeed, Fig. 4 shows that, far from changing monotonically as one would expect for systems in contact with a thermal bath, the temperature of tidal streams fluctuates about that of the host galaxy. In terms of statistical mechanics this implies that the amount of energy required to change the temperature (i.e. the heat capacity) varies along the orbital path. potential (40), which corresponds to a self-gravitating isothermal sphere in dynamical equilibrium. In this potential both functions g ( E , t ) and f ( E , t ) have analytical expressions. The density of states is where rm ( E , t ) denotes the radius at which E = Φ at time t (e.g. Binney & Tremaine 2008); and A = 8 / 9 π 2 √ 6 π . Using the classical definition of thermal entropy, S = ln g ( E , t ), the temperature of the sphere can be calculated as If the system is composed of particles with mass m with mean kinetic energy 〈 1 / 2 mv 2 〉 , the temperature is typically measured as 3 / 2 KB T = 〈 1 / 2 mv 2 〉 , where KB is Boltzmann's constant (e.g. Feynman 1963). Therefore, by defining the (one-dimensional) velocity This odd property can be easily understood through the energy invariants. Defining the stream energy as Es = ∫ ∫ ∫ d E d 3 r d 3 v π ( E , r , ˙ r , t ) E and using Equation (30) it is straightforward to show that at the early stages of the stream evolution, i.e. when the phase-space distribution of stream particles remain close to the phase-space location of the progenitor system, this quantity becomes where the scale factor R has been expressed in terms of the thermodynamical definition of temperature given by Equation (43), i.e. R = ( µ / µ 0) -1 / 2 = ( T / T 0) -1 / 2 ; and Is is the invariant energy of the stream. From Equation (48) the first-order variation of the heat capacity is For tidal substructures that are energetically bound ( Is < 0) the heat capacity has a negative sign. Throughout the orbit of tidal substructures the heat capacity oscillates about a constant value 〈 C 〉 = Is / T 0. In growing potentials ( ˙ T / T 0 > 0) the quantity C -〈 C 〉 is negative toward pericentre and positive toward apocentre. In shrinking potentials ( ˙ T / T 0 < 0) the cycle reverses. Note also that the right-hand term of Equation (49) is proportional to ˙ T / T 2 . Therefore, the fluctuations in temperature of tidal streams are bound to damp out as the temperature of the host rises. In contrast, if the temperature of the host drops the right-hand term of Equation (49) grows with time and the convergence toward dynamical equilibrium cannot be guaranteed (see Fig. 4). The reason why the thermodynamical definition of entropy does not reproduce this peculiar behaviour can be traced back to the definition of density states itself. Equation (42) presumes that the particles of an ensemble distribute throughout volumes of constant energy surfaces E = H . Maximization of entropy S is thus equivalent to the maximization of the phase volume available to those particles. However, Fig. 3 shows that tidal substructures violate this assumption. Indeed, the phase-space volume filled by tidally-stripped particles fluctuates with time and, in general, it takes several orbital revolutions until the particles spread out on the available phasespace volume of the orbit. Not surprisingly we find that the temperature of tidal streams approaches the thermodynamical value in the limit of dynamical equilibrium, that is Note, however, that for low-mass progenitors this limit is reached on time scales longer than the age of the Universe!", "pages": [ 8, 9, 10 ] }, { "title": "3.5 Smooth vs. clumpy stellar halo", "content": "The previous Sections have laid out the evolution of dynamical fossils in a time-dependent potential. The fate of stellar substructures that form through the merger of small bodies is to be effaced by the hierarchical growth of the host potential. Hence, in a hierarchical galaxy formation framework the same dynamical mechanism that leads to the proliferation of tidal substructures, i.e. the accretion of gravitationally-bound systems, is also responsible for their progressive removal. The 'smooth' galactic component arises as an inescapable by-product of this cycle. Given that all substructures are on average equally affected by orbital diffusion (see §3.2), whether or not dynamical fossils can be detected in the present-day configuration space will mainly depend on three (typically unknown) factors, namely, the time-dependence of the Milky Way potential, the 'age' of tidal substructures (defined as the look-back time since these stars were tidally stripped from the progenitor system), and the initial distribution of tidal debris in the integral-of-motion space. A quantitative description of the evolution of tidal substructures appears, therefore, an impossibly difficult task. However, it is feasible to construct simple toy models that share the essential features of these systems and hence offer useful insight into the problem at hand. Let us begin by adopting a cosmologically-motivated mass growth for our host galaxy. In numerical (collisionless) simulations of structure formation the average mass evolution of galactic haloes follows a relatively simple function, where zc = c 0 / 4 . 1 -1 is the formation redshift and c 0 is the virial concentration at z = 0 (Wechsler et al. 2002). Adopting a fiducial Milky Way mass of µ 0 = 10 12 M glyph[circledot] and using the mass-concentration relationship observed in cosmological simulations (Macciò et al. 2007) yields zc glyph[similarequal] 1 . 44. Hence, in the concordance cosmology it takes 7.1 Gyr ( z = 0 . 85) for this model to double its mass. How is this mass distributed throughout the Galaxy? According to galaxy formation models the shape of the Galaxy is expected to vary with radius. At large radii the potential is typically dominated by a triaxial dark matter halo. In the inner-most regions the assembly of the baryonic components renders a close-to-axisymmetric potential shape (Kazanztidis et al. 2010). The relative orientation between the principal axes of the halo and the spin vector of the Milky Way is still poorly understood. Although it is generally assumed that discs are aligned with one of the principal axes, it is also possible to find tilted configurations that are dynamically stable (e.g. Binney 1978; Velázquez & White 1999; Dubinski & Chakrabarty 2009). To complicate this picture further, recent numerical simulations suggest that the disc-halo orientation may change repeatedly throughout the formation of spiral galaxies (e.g. Debattista et al. 2013). Therefore, in cosmologically-motivated potentials none of the components of the angular momentum may be conserved. Here we shall bypass these theoretical uncertainties by considering a set of spherical power-law models that covers the range of potentials of astrophysical interest, i.e. with force-indices between n = -2 (point-mass) and n = 1 (homogeneous density distribution), and adopting a logarithmic potential ( n = -1) as our fiducial model for the Milky Way. This assumption allows us to concentrate on the energy evolution of tidal substructures in a growing potential, without worrying about the possible existence of other integrals of motion 8 . In these potentials it can be easily shown through Equations (26), (50), and (51) that the average temperature of tidal substructures evolves as where T 0 is measured at the redshift when the particles become tidally unbound from the progenitor system. The detection of tidal clumps in the integral-of-motion space is limited to substructures that are currently much colder than the smooth Milky Way background, i.e. T glyph[lessmuch] T h = ( m / KB ) σ 2 h ; where σ h = 220 / √ 2kms -1 is the fiducial velocity dispersion of our Galaxy model at z = 0. In a growing potential this condition puts a strong constrain on the maximum 'age' of the substructures. Given that dynamical fossils have an energy dispersion that correlates with the dynamical mass of the progenitor system (e.g. Peñarrubia et al. 2006), it is useful to express temperatures in terms of mean kinetic energies. Through Equation (52) the condition of detectability then becomes where σ s is the velocity dispersion of the progenitor system at z = z age. Hence, lacking additional information (e.g. metal abundances, see Sheffield et al. 2012), the remnants of systems accreted at z age ∼ z smooth would be hardly distinguishable from the smooth stellar halo of the Milky Way. Fig. 5 shows the value of z smooth as a function of the velocity dispersion of the progenitor system. Focusing on the logarithmic potential (dashed-dotted line), which has a flat velocity curve and provides the closest representation of the Galaxy, shows that tidal debris of massive satellite galaxies such as LMC and SMC ( σ s glyph[greaterorsimilar] 80kms -1 ) rapidly dissolves in the stellar halo of the host. The detection of tidal debris associated to the tidal disruption of LMC- ype galaxies is thus limited to the most recent ( z age glyph[lessorsimilar] 0 . 6) merger events. In contrast, Fig. 5 suggests that it may be possible to identify a large number of tidal clumps associated to the accretion of dwarf spheroidal galaxies ( σ s glyph[lessorsimilar] 12kms -1 ; see Walker et al. 2009). Tidal debris from low-mass globular clusters ( σ s glyph[lessorsimilar] 2kms -1 ) provide a clear-cut target for the search of substructures in the integral-ofmotion space 9 . What is the impact of the Milky Way formation on the stellar halo? Dissipational processes lead to a steepening of the potential in the central regions of the Galaxy, wherein one would expect to find the largest concentration of tidal substructures. According to Fig. 5 the formation of the Galaxy will tend to accelerate the diffusion of tidal substructures that originated from early accretion events. On the other hand, the presence of a disc also enhances mass loss of satellite galaxies and stellar clusters (D'Onghia et al. 2010; Peñarrubia et al. 2010; Zolotov et al. 2012). Thus, the formation of the Milky Way favours the growth of both the 'clumpy' and 'smooth' stellar halo components.", "pages": [ 10, 11 ] }, { "title": "4 SUMMARY AND CONCLUSIONS", "content": "This work introduces a general technique for constructing dynamical invariants (a.k.a. constants of motion) in time-dependent gravitational potentials. The method rests upon the derivation of a system of coordinates in which the explicit time-dependence is removed from the Hamiltonian. After carrying out the inverse transformation the integrals of motion admitted by the gravitational potential become dynamical invariants in the original coordinates. By construction, dynamical invariants are conserved quantities along the phase-space path of a particle motion. In practical terms this means that the differential equations that define the coordinate transformation and those that determine the motion of particles through phase-space are coupled. However, in a few exceptional cases both sets of equations can be de-coupled, thus allowing the derivation of exact invariants. This is the case, for example, of the harmonic potential (see Feix et al. 1987) as well as Dirac's cosmology, where Newton's constant G varies as the reciprocal of the time (Lynden-Bell 1982). In a regime where the mean field varies slowly it is possible to derive approximate invariants for power-law forces, F ( ξ , t ) = -µ ( t ) ξ n , where glyph[epsilon1] ≡ ˙ µ / µ 0 glyph[lessorsimilar] T -1 0 , and T 0 is the radial period of an orbit. Numerical tests show that these quantities are conserved at order | ∆ I / I 0 | glyph[lessorsimilar] 0 . 1( glyph[epsilon1] T 0) 2 for the range of powerlaw forces of astronomical interest ( -2 glyph[lessorequalslant] n glyph[lessorequalslant] 1). This technique offers advantages over standard perturbation methods. For example, while actions are only conserved in systems that evolve adiabatically, dynamical invariants stay constant independently of the time scale for change in the potential. Except for a few rare cases that admit exact invariants (see above), the construction of analytical invariants is only possible for scale-free potentials that vary slowly. However, it is worth noting that approximate invariants remain accurate even outside the adiabatic regime ( glyph[epsilon1] T 0 glyph[lessorsimilar] 1), as shown in Fig. 2. In general, for scaled potentials the transformation R ( t ) is coupled to the trajectory in phase-space of individual particles and needs to be computed numerically. The derivation of dynamical invariants yields tight constraints on the dynamical evolution of collisionless systems. For example, invariants can be used to describe the evolution of the microcanonical distribution of gravitating systems without relying on ergodicity or probability assumptions. As an illustration, we consider the case of tidal streams orbiting in a logarithmic potential whose circular velocity can either grow or drop linearly with time. Restricted N -body simulations show that tidal tails exhibit fluctuations in entropy, temperature and specific heat that damp out as these systems approach dynamical equilibrium. This behaviour can not be described by the canonical distribution, which evolves toward a suitable equilibrium configuration through maximization of entropy (e.g. Penrose 1979). For gravitating systems this is equivalent to maximizing the phase-space volume available to the particle ensemble (Padmanabhan 1990). However, substructures that have not mixed in phase space violate this condition, as the distribution function oscillates in phase with the radial motion of their orbits. These systems also violate the ergodic hypothesis, which assumes that the time-averaged properties of microcanonical ensembles can be derived from a phase-space average over all possible microstates. In contrast, dynamical invariants allow us to describe the statistical properties of tidal tails through a simple time averaging of deterministic equations. We show that the equivalence between the micro and macrocanonical descriptions only emerges as tidal tails progressively fill the phase-space volume available to their orbits and a state of dynamical equilibrium is reached. Merger substructures tend to diffuse in the integral-of-motion space throughout the growth of the host potential. In galaxies that build up mass hierarchically, a smooth stellar halo emerges as the inescapable by-product of the deposition and progressive dissolution of dynamical fossils. Given the stochasticity of merger trees, substructures in the stellar halo are expected to cover a continuous spectrum of temperatures. Attempts to quantify the amount of substructure in the stellar halo of our Galaxy (e.g. Bell et al. 2008; Starkenburg et a. 2009; Schlaufman et al. 2010; Xue et al. 2011) are biased toward the coldest and youngest substructures and must therefore be taken as lower limits. For example, using cosmologically-motivated models we estimate that the detection of tidal debris associated to massive satellites (i.e. LMC-type galaxies) is limited to the most recent events, z age glyph[lessorsimilar] 0 . 6, in gross agreement with the results derived from N -body models of structure formation (e.g. Font et al. 2008; Johnston et al. 2008). This suggests that the majority of substructures identifiable as dynamical fossils in the present-day configuration space likely originate from the tidal stripping of low-mass objects, such as dwarf spheroidals and stellar clusters. A noteworthy remark refers to the active role that baryons may play in the formation of stellar haloes. Dissipational processes in the host galaxy accelerate both the disruption rate of gravitationally-bound objects and the 'dissolution' of tidal substructures through a steepening of the central potential. Further applications of dynamical invariants to gravitating systems approaching an equilibrium state will be explored in separate contributions.", "pages": [ 11, 12 ] }, { "title": "5 ACKNOWLEDGEMENTS", "content": "This work has greatly benefited from the comments and suggestions of Douglas Heggie, John Peacock, Andrew Pontzen and Matt Walker. The generous input of James Binney regarding the analysis of actions is appreciated. Also, a word of thanks to the anonymous referee for his/her very useful comments.", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "We develop a novel abundance matching method to construct a mock catalog of luminous red galaxies (LRGs) in the Sloan Digital Sky Survey (SDSS), using catalogs of halos and subhalos in N -body simulations for a Λ -dominated, cold dark matter model. Motivated by observations suggesting that LRGs are passively-evolving, massive early-type galaxies with a typical age > ∼ 5 Gyr, we assume that simulated halos at z = 2 ( z 2 -halo) are progenitors for LRG-host subhalos observed today, and we label the most tightly bound particles in each progenitor z 2 -halo as LRG 'stars'. We then identify the subhalos containing these stars to z = 0 . 3 (SDSS redshift) in descending order of the masses of z 2 -halos until the comoving number density of the matched subhalos becomes comparable to the measured number density of SDSS LRGs, ¯ n LRG = 10 -4 h 3 Mpc -3 . Once the above prescription is determined, our only free parameter is the number density of halos identified at z = 2 and this parameter is fixed to match the observed number density at z = 0 . 3 . By tracing subsequent merging and assembly histories of each progenitor z 2 -halo, we can directly compute, from the mock catalog, the distributions of central and satellite LRGs and their internal motions in each host halo at z = 0 . 3 . While the SDSS LRGs are galaxies selected by the magnitude and color cuts from the SDSS images and are not necessarily a stellar-mass-selected sample, our mock catalog reproduces a host of SDSS measurements: the halo occupation distribution for central and satellite LRGs, the projected auto-correlation function of LRGs, the cross-correlation of LRGs with shapes of background galaxies (LRG-galaxy weak lensing), and the nonlinear redshift-space distortion effect, the Finger-of-God effect, in the angle-averaged redshift-space power spectrum. The mock catalog generated based on our method can be used for removing or calibrating systematic errors in the cosmological interpretation of LRG clustering measurements as well as for understanding the nature of LRGs such as their formation and assembly histories. Key words: cosmology: theory - galaxy clustering - galaxy formation - cosmology: largescale structure of the Universe", "pages": [ 1 ] }, { "title": "Understanding the nature of luminous red galaxies (LRGs): Connecting LRGs to central and satellite subhalos", "content": "1 , 2 /star 3 4 4 , 5 Shogo Masaki † , Chiaki Hikage , Masahiro Takada , David N. Spergel , and Naoshi Sugiyama 1 , 3 , 4 18 October 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Galaxy redshift surveys are one of the primary tools for studying the large-scale structure in the Universe (Davis & Huchra 1982; de Lapparent et al. 1986; Kirshner et al. 1987; York et al. 2000; Peacock et al. 2001). Over the coming decade, astronomers will have even larger surveys including BOSS 1 (Dawson et al. 2013), WiggleZ 2 (Blake et al. 2011), VIPERS 3 , FMOS 4 , HETDEX 5 , BigBOSS 6 (Schlegel et al. 2009), Subaru Prime Focus Spectrograph (PFS 7 ; Ellis et al. 2012), Euclid 8 , and WFIRST 9 . The upcoming generation of galaxy redshift surveys is aimed at understanding cosmic acceleration as well as measuring the composition of the Universe via measurements of both the geometry and the dynamics of structure formation (Wang et al. 1999; Eisenstein et al. 1999; Tegmark et al. 2004; Cole et al. 2005). On large scales, galaxies trace the underlying distribution of dark matter, and their clustering correlation is a standard tool to extract cosmological information from the measurement. Because of their relatively high spatial densities and their intrinsic bright luminosities, luminous red galaxies (LRGs) are one of the most useful tracers (Eisenstein et al. 2001; Wake et al. 2006). Measurements of the clustering properties of LRGs have been used to measure the baryon acoustic oscillation (BAO) scale (Eisenstein et al. 2005; Percival et al. 2007; Anderson et al. 2012) as well as to constrain cosmological parameters (Tegmark et al. 2004; Cole et al. 2005; Reid et al. 2010; Saito et al. 2011). Our lack of a detailed understanding of the relationship between galaxies and their host halos complicates the analysis of large-scale clustering data. The halo occupation distribution (HOD) approach or the halo model approach has provided a useful, albeit empirical, approach to relating galaxies to dark matter (see e.g., Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001, for the pioneer works). In these approaches, the distribution of halos is first modeled for a given cosmological model, e.g. by using N -body simulations, and then galaxies of interest are populated into dark matter halos. The previous works have shown that, by adjusting the model parameters, the HOD based model well reproduces the auto-correlation functions of LRGs measured from the Sloan Digital Sky Survey 10 (SDSS) (Zehavi et al. 2005; Zheng et al. 2007; Wake et al. 2008; Reid & Spergel 2009; White et al. 2011). It has been shown that LRGs reside in massive halos with a typical mass of a few times 10 13 h -1 M /circledot . However, the HOD method employs several simplified assumptions. For instance, the distribution of galaxies is assumed to follow that of dark matter in their host halo and the model assumes a simple functional form for the HOD. An alternative approach is the so-called abundance matching method. The abundance matching method directly connects target galaxies to simulated subhalos assuming a tight and physicallymotivated relation between their properties, e.g., galaxy luminosity and subhalo circular velocity, without employing any free fitting parameters (e.g., Kravtsov et al. 2004; Conroy et al. 2006; Trujillo-Gomez et al. 2011; Reddick et al. 2012; Masaki et al. 2013; Nuza et al. 2012). However, it is not still clear whether the method can simultaneously reproduce different clustering measurements such as the auto-correlation function and the galaxy-galaxy weak lensing (Neistein & Khochfar 2012). Most of the previous studies use only the auto-correlation function to test their abundance matching model. In this paper, we develop an alternative approach to the abundance matching method for constructing a mock catalog of LRGs. Motivated by observations suggesting that LRGs are passive, massive early-type galaxies, which are believed to have formed at z > 1 (Masjedi et al. 2008; Carson & Nichol 2010; Tojeiro et al. 2012), we assume that the progenitor halos for LRG-host subhalos are formed at z = 2 . We identify massive halos at this redshift, de- fine the innermost particles of each progenitor halo as hypothetical 'LRG-star' particles, follow the star particles to lower redshifts, and then identify subhalos at z = 0 . 3 containing these star particles. We adjust the number of halos identified as LRG progenitors at z = 2 to match the observed number density of the SDSS LRGs, ¯ n LRG /similarequal 10 -4 h 3 Mpc -3 (also see Conroy et al. 2008; Seo et al. 2008, for a similar-idea based approach when connecting galaxies to halos). With this method, we can directly trace, from the simulations, how each progenitor halo at z = 2 experiences merger(s), is destroyed or survives at lower redshift as well as which progenitor halos become central or satellite subhalos (galaxies) in each host halo at z = 0 . 3 . Thus, our method allows us to include assembly/merging histories of the LRG-progenitor halos. Our method is solely based on a mass-selected sample of progenitor halos at z = 2 and does not have any free fitting parameter because the mass threshold is fixed by matching to the number density of SDSS LRGs. We compare statistical quantities computed from our mock catalog with the SDSS measurements: the HOD, the projected auto-correlation function of LRGs, the LRG-galaxy weak lensing and the redshift-space power spectrum of LRGs. Even though our method is rather simple, we show that our mock catalog remarkably well reproduces the different measurements simultaneously. The structure of this paper is as follows. In Section 2, we describe our method to generate a mock catalog of LRGs by using N -body simulations for a Λ -dominated cold dark matter ( Λ CDM) model as well as the catalogs of halos and subhalos at z = 2 and z = 0 . 3 . In Section 3, we show the model predictions on the relation between LRGs and dark matter, and compare with the SDSS measurements. Section 4 is devoted to discussion and conclusion.", "pages": [ 1, 2 ] }, { "title": "2.1 Cosmological N -body simulations", "content": "Throughout this paper we use two realizations of cosmological N -body simulations generated using the publicly-available Gadget2 code (Springel et al. 2001b; Springel 2005). For each run, we employ a flat Λ CDM cosmology with Ω m = 0 . 272 , Ω b = 0 . 0441 , Ω Λ = 0 . 728 , H 0 = 100 h = 70 . 2 km s -1 Mpc -1 , σ 8 = 0 . 807 and n s = 0 . 961 using the same parameters and notation as in the the WMAP 7-yr analysis (Komatsu et al. 2011). Our simulation of larger-size box, which we hereafter call 'L1000', employs 1024 3 dark matter particles in a box of 1 h -1 Gpc on a side. The L1000 simulation allows for a higher statistical precision in measuring the correlation functions from the mock catalog. To test the effect of numerical resolution on our results, we also use a higher resolution simulation that employs 1024 3 particles in a box of 300 h -1 Mpc . We call the smaller-box simulation 'L300'. The mass resolution for the simulations (mass of an N -body particle) is 7 × 10 10 h -1 M /circledot or 1 . 9 × 10 9 h -1 M /circledot for L1000 or L300, respectively. The initial conditions for both the simulation runs are generated using the second-order Lagrangian perturbation theory (Crocce et al. 2006; Nishimichi et al. 2009) and an initial matter power spectrum at z = 65 , computed from the CAMB code (Lewis et al. 2000). We set the gravitational softening parameter to be 30 and 8 h -1 kpc for the L1000 and L300 runs, respectively. We use the friends-of-friends (FoF) group finder (e.g., Davis et al. 1985) with a linking length of 0 . 2 in units of the mean interparticle spacing to create a catalog of halos from the simulation output and use the SubFind algorithm (Springel et al. 2001a) to identify subhalos within each halo. In this paper, we use halos and subhalos that contain more than 20 particles. Each particle in a halo region is assigned to either a smooth component of the parent halo or to a satellite subhalo, where the smooth component contains the majority of N -body particles in the halo region. Hereafter we call the smooth component a central subhalo and call the subhalo(s) satellite subhalo(s). For each subhalo, we estimate its mass by counting the bounded particles, which we call the subhalo mass ( M sub ). We store the position and velocity data of particles in halos and subhalos at different redshifts. To estimate the virial mass ( M vir ) for each parent halo, we apply the spherical overdensity method to the FoF halo, where the spherical boundary region is determined by the interior virial overdensity, ∆ vir , relative to the mean mass density (Bryan & Norman 1998). The overdensity ∆ vir /similarequal 268 at z = 0 . 3 for the assumed cosmological model. The virial radius is estimated from the estimated mass as R vir = (3 M vir / 4 π ¯ ρ m 0 ∆ vir ) 1 / 3 , where ¯ ρ m 0 is the comoving matter density.", "pages": [ 2, 3 ] }, { "title": "2.2 Mock catalog of LRGs: connecting halos at z = 2 to central and satellite subhalos at z = 0 . 3", "content": "LRGs are very useful tracers of large-scale structure as they can reach a higher redshift, thereby enabling to cover a larger volume with the spectroscopic survey (Eisenstein et al. 2001, 2005). LRGs are passively-evolving, early-type massive galaxies, and their typical ages are estimated as ∼ 5 Gyrs (Kauffmann 1996; Wake et al. 2006; Masjedi et al. 2008; Carson & Nichol 2010). This implies that LRGs, at least a majority of the stellar components, were formed at z > ∼ 1 (Masjedi et al. 2008). Motivated by this fact, we here propose a simplest abundance-matching method for connecting LRGs to dark matter distribution in large-scale structure as follows. Our method rests on an assumption that progenitor halos for LRG-host subhalos today are formed at z = 2 , which is closer to the peak redshift of cosmic star formation rate (Hopkins & Beacom 2006). Our choice of z = 2 is just the first attempt, and a formation redshift can be further explored so as to have a better agreement with the SDSS measurements (see Section 4 and Appendix A for a further discussion). (1) We select halos from the simulation output at z = 2 as candidates of the progenitor halos (hereafter sometimes called z 2 -halo). In doing this, we select the z 2 -halos in descending order of their masses (from more massive to less massive) until the comoving number density becomes close to that of SDSS LRGs at z = 0 . 3 , which we set to ¯ n LRG = 10 -4 h 3 Mpc -3 . More precisely, we need to identify more halos having the number density of /similarequal 1 . 3 × 10 -4 h 3 Mpc -3 at least, because about 30% of z 2 -halos, preferentially in massive halo regions at z = 0 . 3 , experience mergers from z = 2 to z = 0 . 3 for the assumed Λ CDM model (see below for details). (2) We trace the 30% innermost particles of each z 2 -halo particles to lower-redshift simulation outputs until z = 0 . 3 , where the innermost particles are considered as 'LRG star' particles and defined by particles within a spherical boundary around the mass peak of each z 2 -halo (see Figure 1). (3) We perform a matching of the star particles of each z 2 -halo to central and satellite subhalos at z = 0 . 3 (hereafter z 0 . 3 -subhalo). If more than 50% of the star particles are contained in a z 0 . 3 -subhalo, we define the subhalo as a subhalo hosting LRG at z = 0 . 3 . (4) We repeat this procedure in descending order of masses of z 2 -halos until the comoving number density of the matched z 0 . 3 -subhalos (LRG-host subhalos) is closest to the target value, ¯ n LRG = 10 -4 h 3 Mpc -3 . The minimum mass of LRG-progenitor halos at z = 2 is about 6 × 10 12 h -1 M /circledot for the L1000 run (which contains about 90 N -body particles for the). However, we need to a priori determine some model parameters before implementing to the simulation halo/subhalo catalogs: the formation redshift of LRG-progenitor halos, z form = 2 , and the fractions '30%' or '50%' for the star particles or the matching particles, respectively. Rather than exploring different combinations of the model parameters to have a better fit to the SDSS measurements, we will below study the ability of our mock catalog to predict various statistical quantities of LRGs, by employing our fiducial choices of the parameters ( z form = 2 , f star = 0 . 3 and f match = 0 . 5 ). In Appendix A, we study how variations of the model parameters change the mock catalog. The brief summary of the results is the change of each parameter affects only the smallscale clustering signals, which are sensitive to the fraction of satellite LRGs, and does not largely change the clustering signals at large scales in the two-halo regime. In our method, central and satellite subhalos are populated with LRG galaxies under a single criterion: if a subhalo at z = 0 . 3 is a descendant of the z 2 -halo, the subhalo is included in the matched sample. On the other hand, the standard abundance matching method often uses different mass proxies of central and satellite subhalos when matching subhalos to the target galaxies (in the order of their stellar masses or luminosities). For instance, the mass of a central subhalo is assigned by a maximum circular velocity of the bounded N -body particles, computed from the output at the target redshift ( z = 0 . 3 in the LRG case), while the mass of a satellite subhalo is assigned by the maximum circular velocity from the simulation output at the 'accretion' epoch before the subhalo started to accrete onto the main host halo (Conroy et al. 2006), which allows one to estimate the mass of each satellite subhalo before being affected by the tidal stripping during penetrating the main halo. Thus the standard abundance matching method is computationally more expensive in a sense that it requires many simulation outputs at different redshifts in order to trace the accretion/assembly history of each subhalo. To be fair, we below compare our method with the standard abundance matching method for some statistical quantities of LRGs. Some of the LRG-host halos at z = 0 . 3 , especially massive halos, contain multiple LRG-subhalos in our mock catalog (see the example in the lower panel of Figure 1). We often call such systems 'multiple-LRG systems' in the following discussion (also see Reid & Spergel 2009; Hikage et al. 2012a). We refer to the LRGhost halos, which host only one LRG inside, as 'single-LRG systems'. The average halo masses for the single- and multiple-LRG systems are found from the L1000 run to be ¯ M vir = 4 . 8 × 10 13 and 1 . 5 × 10 14 h -1 M /circledot , respectively. The fraction of the multiple-LRG systems among all the LRG-host halos is about 8% in the L1000 run. Because we assumed that most stars of each LRG are formed until z = 2 and the total stellar mass scales with the mass of z 2 -halo, we define the brightest LRG (BLRG) in each multiple-LRG system by the LRG-subhalo that corresponds to the most massive z 2 -halo among all the progenitor z 2 -halos in the system, while we call the smallest z 2 -halo the faintest LRG (FLRG). Note that we also refer to an LRG in a single-LRG system as BLRG. A BLRG in a single-LRG system is not necessarily a central galaxy in the parent halo at z = 0 . 3 (in other words, the central subhalo does not correspond to any LRG-progenitor halo at z = 2 ). Similarly, a central LRG in a multiple-LRG system is not necessarily a BLRG, i.e. the most massive z 2 -halo, although the central subhalo is the most massive subhalo in the parent halo by definition. Table 1 summarizes properties of the LRG-host halos com- puted from the L1000 and L300 mock catalogs. To estimate statistical uncertainties of each quantity, we divided the L1000 catalog into 27 sub-volumes (the side length of each sub-volume is 333 h -1 Mpc ) and computed the mean and rms of the quantity 11 . Hence the error quoted for each entry of the L1000 run corresponds to the sample variance scatter for a volume of [333 h -1 Mpc] 3 . The L1000 result with the error bar can be compared with the L300 result, because of the similar volumes of the sub-divided L1000 catalog and L300 run ( 333 3 and 300 3 [ h -1 Mpc] 3 , respectively). The L1000 and L300 results agree with each other to within 2 σ for the quantities except for the fraction of satellite LRGs for all the LRG-host halos. The disagreement for the satellite LRG fraction is probably due to the numerical resolution, because the L1000 simulation may miss some less-massive LRG progenitor-halos at z = 2 , which are identified in the L300 run, due to lack of the numerical resolution and such small z = 2 -halos preferentially become satellite LRGs at z = 0 . 3 (also see below and Appendix A). In Table 1, we also compare the mock results with the measurement results from the SDSS DR7 LRG catalog in Hikage et al. (2012a). The SDSS results were derived using the different clustering measurements, the LRG-galaxy lensing, the LRG redshiftspace power spectrum, and the LRG-photometric galaxy crosscorrelation to constrain the properties of LRG-host halos. To be conservative, we here quote the measurement result that has largest uncertainties among the three measurements. The table shows that the mock catalog fairly well reproduces the SDSS results within the error bars. Although one may notice sizable disagreement for the single-LRG systems, especially for the fraction of halos hosting satellite LRGs ( f sat -LRG ) or the fraction of central BLRGs ( q BLRG cen ), the SDSS measurements are not yet reliable for the single-LRG systems, as reflected by the large error bars and stressed in Hikage et al. (2012a). Hence, this requires a further careful study. Figure 1 shows snapshots of the N -body particle distribution in the L1000 run outputs at different redshifts, for the regions where multiple- or single-LRG systems are formed at z = 0 . 3 . The figure illustrates how each LRG-progenitor halo is defined at z = 2 , how the innermost particles are assigned as 'star' particles, and how the star particles are traced to lower redshifts and how LRGprogenitor halos merge with each other and become to reside in central and satellite subhalos at the final redshift z = 0 . 3 . Our method allows us to directly include the merging and assembly histories of LRG-progenitor halos. Although the number density of LRG-host subhalos is set to the density of LRGs as we described above, the figure shows that more LRG-progenitor halos or subhalos survive at higher redshift than at z = 0 . 3 . Hence our method has a capability to study what kinds of halos or subhalos at higher redshift are progenitors for the SDSS LRGs (see Section 4 for a further discussion). Figure 2 shows how each LRG-progenitor halo at z = 2 loses or gains its mass due to mass accretion, merger and/or tidal stripping when it becomes an LRG-host subhalo at z = 0 . 3 , computed using the catalogs of halos and subhalos in the z = 2 and z = 0 . 3 outputs of L1000 run. Note that the halo mass shown in the x -axis, M FoF ( z = 2) , is the FoF mass, the sum of FoF particles in each halo region at z = 2 . First, the figure shows that we need to select the LRG-progenitor halos at z = 2 down to a mass scale of about 6 × 10 12 h -1 M /circledot . Some subhalos for satellite LRGs lose their masses due to tidal stripping as implied in Figure 1, while subhalos for central LRGs gain their masses due to mass accretion and/or merger. Comparing the left and right panels manifests that multiple-LRG systems tend to reside in more massive LRGprogenitor halos at z = 2 and become more massive LRG-host halos at z = 0 . 3 , and that the mass difference between subhalos for central and satellite LRGs is larger in multiple-LRG systems, implying a larger difference between their luminosities (see Hikage et al. 2012a, for a similar discussion). Thus our method is primarily based on the masses of LRGprogenitor halos at z = 2 (see Figure 2) and the connection with central and satellite subhalos in the parent halos at z = 0 . 3 . On the other hand, LRGs in the SDSS catalog are selected based on the magnitude and color cuts from the SDSS imaging data (primarily gri ), and are not necessarily a stellar-mass-selected sample, although their stellar masses are believed to have a tight relation with the host halo masses. Nevertheless, we will show below that the mock catalog perhaps surprisingly well reproduces the different SDSS measurements. Since LRGs in our mock catalog reside in relatively massive halos at z = 2 , with masses M FoF > ∼ 6 × 10 12 h -1 M /circledot (Figure 2), as well as in massive parent halos at z = 0 . 3 , our method does not necessarily require a high-resolution simulation. A simulation with 1024 3 particles and 1 h -1 Gpc size on a side seems sufficient, which allows for a relatively fast computation of the N -body simulation as well as an accurate estimation of statistical quantities of LRGs. This is not the case if one wants to work on the abundance matching method for less massive galaxies or more general types of galaxies (e.g., Trujillo-Gomez et al. 2011; Reddick et al. 2012; Masaki et al. 2013).", "pages": [ 3, 4, 5 ] }, { "title": "3.1 Halo occupation distribution and properties of satellite LRGs", "content": "First, we study the halo occupation distribution (HOD) for LRGs in Figure 3, where the HOD gives the average number of LRGs that the halos at z = 0 . 3 host as a function of host-halo mass. Here we consider the HODs for central and satellite LRGs which reside in central and satellite subhalos in the LRG-host halos, respectively. Again we should emphasize that our method does not assume any functional forms for the HODs, unlike done in the standard HOD method, and rather allows us to directly compute the HODs from the mock catalog. Even if LRG-progenitor halos are selected from halos at z = 2 by a sharp mass threshold, our mock catalog naturally predicts that the central HOD has a smoother shape around a minimum halo mass, as a result of their merging and assembly histories from z = 2 to z = 0 . 3 . To be more precise, the central HOD is smaller than unity ( 〈 N cen 〉 < 1 ) for host halos with M vir < ∼ 10 14 h -1 M /circledot , meaning that only some fraction of the halos host a central LRG. On the other hand, most of massive halos host at least one LRG and can host multiple LRGs inside. Conversely, the fraction of massive halos, which do not host any LRG, is 1.3% for halos with masses M vir /greaterorequalslant 1 × 10 14 h -1 M /circledot , while all halos with M vir /greaterorequalslant 2 × 10 14 h -1 M /circledot have at least one LRG inside. To test validity of our mock catalog, we compare the HODs with the SDSS measurement in Reid & Spergel (2009, hereafter RS09), where the HOD was constrained by using the Countsin-Cylinders (CiC) method for identifying multiple LRG systems from the SDSS DR7 LRG catalog with the aid of halo catalogs in N -body simulations. Although RS09 employed the slightly different cosmological model and redshift ( z = 0 . 2 ) from ours ( z = 0 . 3 ), we employed the same best-fit parameters in RS09 to compute the LRG HOD for this figure. To be more precise, due to limited constraints from the SDSS LRG catalog, especially for low-mass host-halos, RS09 assumed the fixed form for the central HOD: with M min = 5 . 7 × 10 13 h -1 M /circledot and σ log M = 0 . 7 , in order to obtain meaningful constraints on the satellite HOD. The central HODfor low-mass host-halos is difficult to constrain, because lowmass host-halos of LRGs are observationally difficult to identify. Therefore, we do not think that the difference for the central HODs is significant, and needs to be further carefully studied. On the other hand, the satellite HOD in RS09 is almost perfectly recovered by our mock catalog, where RS09 assumed the functional form for the satellite HOD to be given by 〈 N sat ( M ) 〉 = 〈 N cen ( M ) 〉 [( M -M cut ) /M 1 ] α and then constrained the parameters ( M cut , M 1 , α ) from the SDSS LRG catalog. The hatched region is the range at each host-halo mass bin that is allowed by varying the model parameters within the 1 σ confidence regions. Our results confirm that parent halos of ∼ 10 15 h -1 M /circledot have up to several LRGs inside, as first pointed out in RS09. The L300 result, the simulation result with higher spatial resolution, gives similar results to the L1000 results, showing that the numerical resolution is not an issue in studying the satellite HOD. Even though SDSS LRGs are selected by the magnitude and color cut, not by their masses, our method seems to capture the origin of SDSS LRGs; mass-selected halos at z ∼ 2 are main progenitors of LRGs, and their subsequent assembly and merging histories determine where LRGs are distributed within the host halos at lower redshift. Furthermore, to be comprehensive, we also compare our method with the standard abundance matching method in Conroy et al. (2006). In this method, the mass proxy of each subhalo is assigned by the maximum circular velocity V cir computed from the member N -body particles. More specifically, the central subhalo mass is assigned by V cir at the LRG redshift z = 0 . 3 , while the satellite subhalo mass is estimated by V cir from the simulation output at its 'accretion' epoch when the subhalo started to accrete onto the parent halo at z = 0 . 3 (more exactly, the circular velocity is estimated from the last output when the 'subhalo' was identified as an 'isolated' halo before the accretion) (see also Masaki et al. 2013). This prescription for satellite subhalos allows for a better assignment of the subhalo mass so that it avoids the effect by tidal stripping during accreting onto the parent halo. We use the L300 run outputs at 44 different redshifts from z = 10 to trace the merging and assembly history of each subhalo till z = 0 . 3 . Then, assuming that the stellar masses of LRGs trace the subhalo masses, we match the z = 0 . 3 subhalos to LRGs in descending order of the mass proxies ( V cir ) until the number density is closest to the target value, ¯ n LRG = 10 -4 ( h Mpc -1 ) 3 . The curves labeled as 'Standard ( V acc )' show the central and satellite HODs measured from the mock catalog of the V cir -based abundance matching method. The satellite HOD is in a nice agreement with our method, while the central HOD from the abundance matching method displays a sharper cut-off than in our method. Again we do not yet know the genuine cut-off feature of the central HOD due to lack of the measurement constraints. We will below further compare our method with the abundance matching method for other statistical quantities of LRGs. One motivation of this paper is to understand the physics of the nonlinear redshift-space distortion, i.e. the Finger-of-God (FoG) effect, in the redshift-space power spectrum of LRGs. The FoG effect is caused by internal motion of satellite LRG(s) in LRG-host halos (Hikage et al. 2012b,a). In the following, we study several quantities relevant for the FoG effect; the fraction of satellite LRGs, the radial profile of satellite LRGs inside the parent halos and the internal velocities of satellite LRGs (see Hikage et al. 2012b, for details of the theoretical modeling). Figure 4 shows how much fraction of LRG-host halos at z = 0 . 3 host satellite LRG(s) inside, as a function of the halo mass. Note that we excluded halos that do not host any LRG in this statistics, but included the single-LRG systems hosting one LRG as a satellite galaxy when computing the numerator of the fraction (in this case, the central subhalo of the parent halo does not correspond to any LRG-progenitor halo at z = 2 ). The error bars around the solid curve are Poisson errors, estimated using the number of halos in each mass bin. The figure shows that more massive halos have a higher probability to host satellite LRG(s). About 20% of parent halos with M vir /similarequal 10 14 h -1 M /circledot host satellite LRG(s). We naively expect that BLRG, the most massive LRGprogenitor halo at z = 2 among LRG-progenitor halo(s) accreting onto the same parent halo at z = 0 . 3 , becomes a central galaxy. The solid curves in Figure 5 show the fraction of BLRGs to be a satellite galaxy in LRG-host halos at z = 0 . 3 as a function of the halo mass, computed using all the LRG-host halos. For halos with M vir > ∼ 10 14 h -1 M /circledot , there is up to 10% probability for its BLRG to be a satellite galaxy. The dashed curves are the similar fraction, but computed us- ing only the multiple LRG systems. This sample is intended to compare with the recent result in Hikage et al. (2012a) (also see Table 1). In this case, the fraction of satellite BLRGs is higher for host halos with smaller masses, with larger error bars. This can be explained as follows. Most of low-mass host-halos with masses < ∼ 10 14 h -1 M /circledot are single-LRG systems as can be found from Figure 3, and only a small number of such halos are multiple-LRG systems, causing larger Poisson error bars at each mass bin. We have found from the simulation outputs that such low-mass halos of multiple LRG systems (mostly the systems with 2 LRGs) tend to display a bimodal mass distribution due to ongoing or past major merger, where the BLRG and other (mostly central) LRG tend to have the small mass difference. As a result, such low-mass multiple-LRG systems have a higher chance to host the BLRG as a satellite LRG. On the other hand, the fraction of halos with satellite BLRG converge to the solid curve with increasing the host-halo mass, because most of such massive halos are multiple-LRG systems. For multiple LRG systems with mass of M vir /similarequal 10 14 h -1 M /circledot , about 30% of BLRGs are satellite galaxies. Recently, Hikage et al. (2012a) studied the multiple-LRG systems defined from the SDSS DR7 catalog by applying the CiC technique as well as the FoF group finder method to the distribution of LRGs in redshift space. Then they used the different correlation measurements, the redshift-space power spectrum, the LRG-galaxy lensing and the cross-correlation of LRGs with photometric galaxies, to study properties of satellite LRGs. From the lensing analysis, they found that the multiple-LRG systems has a typical halo mass of M vir /similarequal 1 . 5 × 10 14 h -1 M /circledot (with a roughly 10% statistical error), and that 37 ± 21 % of BLRGs in the multiple-LRG systems appear to be satellite galaxies 12 . Our mock catalog shows a fairly good agreement with the SDSS results, for the average halo mass and the fraction of satellite BLRGs (also see Table 1). In Figure 6, we study the average radial profile of satellite LRGs. In this calculation, we employ only the host halos containing satellite LRG(s), and estimate the radial profile by stacking the radial distribution of satellite LRG(s) in units of the radius relative to the virial radius of each halo. We use the mass peak of the smooth component as the halo center. The average profile p off is normalized as where r off is the distance from the density maximum of the smooth component. The average mass of the host halos is M vir /similarequal 1 . 31 × 10 14 or 1 . 24 × 10 14 h -1 M /circledot for the L1000 or L300 run, respectively, while the average virial radius R vir /similarequal 1 . 07 or 1 . 06 h -1 Mpc in the comoving unit. Compared to the dark matter profile, the radial profile of satellite LRGs clearly displays a flattened profile. The typical off-center radius, where the profile starts to be flattened, is found to be about 400 h -1 kpc because R vir /similarequal 1 h -1 Mpc , which is in a good agreement with the result for the multiple systems in Hikage et al. (2012a). The radial profile also shows a decline at the smaller radii. Thus our result is not consistent with the assumption often used in a standard HOD method that the radial profile of member galaxies follows the dark matter profile (see Berlind & Weinberg 2002, for the improved HOD method including a possible variation in the radial profile of member galax- s). However, the L300 run shows no satellite LRG at small radii r off /R vir /lessorequalslant 0 . 1 , except for the innermost bin. Thus the satellite LRGs at the small radii are mainly from most massive hosthalos, which do not exist in the smaller box simulation, L300. Although the mock catalogs show a sharp rise at the innermost bin r off /R vir /similarequal 0 . 06 ( r off /similarequal 60 h -1 kpc) , which may indicate merging LRGs to the central subhalo in the less massive halos, the scatters are large even for the L300 run, so the result is not significant. Nevertheless, it is worth mentioning that the satellite LRG distribution in our mock catalog seems to show a similar profile to the profile of most massive subhalos in cluster-scale halos in Gao et al. (2012) (see Figures 15 and 16 for the profile). These features in the radial profile of massive subhalos may be as a result of dynamical friction, tidal stripping and merger to the central subhalo. However, the L300 and L1000 results show some difference at the small scales, so a further careful study is needed to derive a more robust conclusion, by using high-resolution simulations as well as a larger number of the realizations. Figure 7 shows the average radial profile of internal motions of satellite LRGs in the parent halos, where the bulk motion of each parent halo (the average velocity of N -body particles belonging to the smooth component of the halo) is subtracted from the velocity of each LRG-host subhalo. We considered only the host halos with satellite LRG(s) as in Figure 6. The curves, labelled as 〈 v off ,r 〉 , are the average radial velocities for all the satellite LRGs with respect to the halo center. The average velocity is negative, reflecting the coherent infall motion towards the halo center, and the infall velocity is larger with increasing radius up to the virial radius. The average radial velocity becomes zero on average at the halo center. These support that the LRG-host subhalo approaches to the halo center due to dynamical friction. On the other hand, the curves, labelled as σ off ,r , are the average velocity dispersions of satellite LRGs. The velocity dispersion has greater amplitudes with decreasing the radius, reaching to σ off ,r /similarequal 500 km s -1 . For comparison, the horizontal dotted line shows the average virial velocity dispersion, σ vir ≡ √ GM vir / 2 R vir = 521 km s -1 , among the satellite LRG-host halos in the L1000 run. The combination of 〈 v off ,r 〉 and σ off ,r implies that satellite LRGs gradually approach to the halo center due to dynamical friction and have an oscillating motion around the halo center. Again the amplitude of the velocity dispersion, σ off ,r /similarequal 500 km s -1 , is in nice agreement with the recent measurement in Hikage et al. (2012a), where they found the velocity dispersion of about 500 km s -1 for satellite LRGs in the multiple-LRG systems by combining the different correlation measurements from the SDSS DR7 LRGs. In Section 3.4 we will further discuss how satellite LRG-subhalos affect the redshift-space power spectrum due to the FoG effect.", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "3.2 Projected correlation function", "content": "Next we study the projected auto-correlation function of LRGs, w p ( R ) , defined as where R is the projected separation between two LRGs in the pairs used for the correlation measurement in units of the comoving scale, π is the separation parallel to the line-of-sight and ξ gg ( r ) is the three-dimensional correlation function. Following Zehavi et al. (2005), π max is set to be 80 h -1 Mpc . The projected correlation function is not affected by the redshift-space distortion effect due to peculiar velocities of LRGs. In Figure 8, we compare the projected correlation function measured from our LRG mock catalog with the SDSS measurements (Zehavi et al. 2005; Masjedi et al. 2006). In the SDSS measurements, Zehavi et al. (2005) used an LRG sample in the magnitude range of -23 . 2 < M g < -21 . 2 and with the mean redshift 〈 z 〉 /similarequal 0 . 3 . Masjedi et al. (2006) used the same sample to extend the measurement down to very small scale, below R = 500 h -1 kpc , by taking into account various observational effects such as the fiber collision. Note that the cosmological model employed in the measurement is slightly different from the model we assumed for our simulations. The figure shows that our mock catalog remarkably well reproduces the projected correlation function of LRGs, to within 30% accuracy in the amplitude, over a wide range of separation radii, which arise from correlations between LRGs within the same host halo and in different host halos, the so-called oneand two-halo regimes, respectively 13 . Comparing the results for the L1000 and L300 runs reveals that the correlation function for L1000 has a smaller amplitude at R < 0 . 7 h -1 Mpc than that for L300. Thus the L1000 run implies a systematic error due to the lack of numerical resolution at the small scales. The L300 result shows a better agreement with the SDSS measurement in Masjedi et al. (2006). The small-scale clustering arises mostly from correlation between LRGs in the same multiple-LRG system, so that numerical resolution seems important to resolve these small subhalos (also see below for a further discussion). As in Figure 3, the dotted curve gives the result from the standard abundance matching method, which shows almost similarlevel agreement with the SDSS measurements to our method. Thus, since the abundance matching method rests on the higherresolution L300 outputs at 44 different redshifts (in our case), our method can provide a much computationally-cheap, alternative approach to making a mock catalog of LRGs. Furthermore, for comparison, the dot-dashed curve shows the correlation function, if the abundance matching is done by using the maximum circular velocity at the LRG redshift ( z = 0 . 3 ) for each satellite subhalo as its mass proxy, instead of the velocity at the accretion epoch. The result shows a significant discrepancy with the SDSS measurements or our method and the standard abundance matching method, especially at small radii. The disagreement means that the circular velocity at z = 0 . 3 is not a good mass proxy for satellite subhalos when matching the subhalos to LRGs, because it misses satellite subhalos in the multiple-LRG systems. To be more precise, mass (circular velocity) of each satellite subhalo tends to be underestimated due to the tidal stripping, then tends to be not selected by the abundance matching, and instead other isolated, less-massive halos tend to be selected. This reduces the clustering signals at small scale due to less contribution from satellite subhalos and also reduces the clustering signal at large scales due to a smaller bias for such low-mass halos. Thus detailed features of the correlation functions at different scales are sensitive to the contribution of satellite LRGs as well as the low-mass threshold of central HOD in Figure 3 (also see Appendix A). Note that an explicit implementation of the abundance matching method to LRGs is the first time, and the result in Figure 8 highlights the importance of proper assignment of subhalo masses in the abundance matching method.", "pages": [ 9, 10 ] }, { "title": "3.3 LRG-galaxy weak lensing", "content": "Correlating the positions of LRGs with shapes of background galaxies, the so-called LRG-galaxy weak lensing, is a powerful means of probing the average dark matter distribution around the LRGs (Mandelbaum et al. 2006, 2012). The LRG-galaxy lensing measures the radial profile of differential surface mass density defined as The profile Σ( R ) is the average surface mass density around the LRGs defined as where ¯ ρ m 0 is the mean background mass density today, and ξ gm ( r ) is the three-dimensional cross-correlation between LRGs and the surrounding matter. In Eq. (4), ¯ Σ( < R ) is the surface mass density averaged within a circular aperture of a radius R . Our use of the mean mass density today ( ¯ ρ m 0 ) is due to our use of the comoving units. Figure 9 shows that the average mass profile measured for all LRGs in the mock catalog is in good agreement with the SDSS measurement in Mandelbaum et al. (2012), to within 30% level in the amplitude. Note that, to obtain the average mass profile from our mock catalog, we stacked all N -body particles around all the LRG-host subhalo in the simulation, including the particles outside dark matter halos. The lensing signal at the radii smaller than about 1 h -1 Mpc arises from the mass distribution within the same halo, while the signal at the larger scale arises from the mass distribution surrounding the host halos - the oneand two-halo terms, respectively (e.g. see Oguri & Takada 2011). The mock catalog well reproduces both the signals of different scales. The average halo mass inferred from the SDSS measurement is ¯ M vir /similarequal 4 . 1 × 10 13 h -1 M /circledot (Hikage et al. 2012a) (see also Table 1). Furthermore, the standard abundance matching method shows a similar-level agreement with the SDSS measurement, similarly to Figure 8. Hikage et al. (2012a) also used the SDSS LRG catalog to study the weak lensing for the multiple-LRG systems. When making the lensing measurements, they used three different proxies for the halo center of each multiple-LRG system, the BLRG, FLRG and the arithmetic mean position of member LRGs (hereafter 'Mean'). By comparing the lensing signals for the different centers, they constrained the average radial profiles of satellite BLRGs and FLRGs, finding about 400 h -1 kpc for a typical offset radius from the true center. Figure 10 shows that the mock catalog predictions are in remarkably good agreement with the SDSS measurements for the different centers. Since these lensing signals are from the exactly same catalog of the multiple-LRG systems, the differences between the different measurements should be due to the off-centering effects of the chosen centers. As nicely shown in Hikage et al. (2012a), the lensing signals for the BLRG and FLRG centers can be well explained by a mixture of the central and satellite BLRGs or FLRGs in the sample. The lensing signals for the FLRG center have smaller amplitudes due to the larger dilution effect because of a larger fraction of satellite (off-centered) FLRGs than in the BLRG centers. On the other hand, the Mean center does not have any galaxy (subhalo) at its position, and therefore the Mean center always has an off-centering effect from the true center in each LRG system. This causes decreasing powers of the lensing signal at the smaller radii than the typical off-center radius. The lensing signals at some radii for the FLRG and Mean centers show some discrepancy from the mock catalog, but we do not think that the disagreement is significant. The average masses inferred from the SDSS measurement and the mock catalog for the multiple-LRG halos agree within about 30%; ¯ M vir /similarequal 1 . 46 or 1 . 52 × 10 14 h -1 M /circledot , respectively (also see Table 1). As can be shown in Figures 8, 9 and 10, our mock catalog of LRGs well reproduces both the SDSS measurements for the auto-correlation function of LRGs and the LRG-galaxy weak lensing simultaneously. As recently discussed in Neistein & Khochfar (2012) (also see Neistein et al. 2011), the abundance matching method has a difficulty to reproduce these measurements with the same model, although they considered the spectroscopic sample of SDSS galaxies, rather than focused on LRGs. Thus the agreements of our mock catalog show a capability of our method to predict different statistical quantities of LRGs by self-consistently modeling, rather than assuming, the fractions of satellite LRGs among different halos and the radial distribution of satellite LRGs in the parent halos (also see Masaki et al. 2013, for a recent development on the extended abundance matching method based on the similar motivation).", "pages": [ 10, 11 ] }, { "title": "3.4 Redshift-space power spectrum of LRGs", "content": "Another observable we consider is the redshift-space power spectrum of LRGs. The FoG effect due to internal motion of galaxies is one of systematic errors to complicate the cosmological interpretation of the measured power spectrum. The FoG effect involves complicated physics inherent in the evolution and assembly processes of galaxies, so is very difficult to accurately model from the first principles. One way to reduce the FoG contamination is to remove satellite galaxies from the region of each multiple-LRG system, and to keep only one galaxy (LRG in our case), ideally the central galaxy, because the central galaxy is supposed to be at rest with respect to the parent halo center and does not suffer from the FoG effect. For example, Reid et al. (2010) developed a useful method for this purpose; first, reconstruct the distribution of halos from the measured distribution of LRGs by identifying multipleLRG systems based on the CiC and FoF group finder method, and then keep only one LRG for each multiple-LRG system. However, the chosen LRG is not necessarily the central galaxy (more exactly, they used, as the halo center proxy, the arithmetic mean of member LRGs or the center-of-mass of different CiC groups without any luminosity or mass weighting), so there may generally remain a residual FoG contamination in the measured LRG power spectrum as pointed out in Hikage et al. (2012a). In the left panel of Figure 11, we study the FoG effect on the redshift-space power spectrum, caused by the off-centering effect of LRGs in our mock catalog. Following the method in Reid et al. (2010) and Hikage et al. (2012a), we study the redshiftspace power spectrum for LRG-host halos, instead of the power spectrum for LRGs. To compute the power spectrum of halos, we need to specify the halo center in each LRG-host halo. For singleLRG systems, we use the LRG position as the halo center proxy. For multiple-LRG systems, we employ different proxies of halo center for each system as done in Figure 10 for the LRG-galaxy lensing; BLRG, FLRG or the arithmetic mean (Mean), where the Mean center is computed in redshift space taking into account redshift space distortion due to peculiar velocities of LRG-subhalos. The figure shows the angle-averaged redshift-space power spectra for the different centers, relative to the power spectrum for the mass center of each LRG-host halo (the mass center of N -body particles of the host halo). Note that, for the power spectrum measurement, we used the exactly same catalog of LRG-host halos, and the different power spectra differ in the chosen halo center of each multiple-LRG system. Hence, the difference between the different spectra should be from the off-centering effects of the chosen centers in the multiple-LRG systems. Interestingly, the spectra for BLRG, FLRG and Mean centers all show smaller amplitudes with increasing wavenumber, as expected in the FoG effect. To be more precise, the power spectrum of FLRG center shows the strongest FoG effect, because a larger fraction of FLRGs are satellite galaxies than BLRGs (see Table 1). These results can be compared with Figure 2 in Hikage et al. (2012a). It can be found that the mock catalog qualitatively reproduces the SDSS measurements: the spectra of BLRG and Mean centers are similar, and the spectrum for FLRG shows the stronger FoG suppression. subhalo position, in each LRG-host halo. In this part of the analysis, the power spectrum is measured by using the position of a central galaxy in each host halo. Again note that BLRG is not necessarily a central subhalo (galaxy) as shown in Figure 5. The power spectrum for the potential minimum has a smaller amplitude than that of the mass center of host halo, implying that the potential minimum is moving around the mass center in each halo. Comparing the spectra for the potential minimum and the BLRG center shows that the BLRG spectrum has a smaller amplitude than the spectra for the potential minimum or the mass center by a few % in the fractional amplitude up to k /similarequal 0 . 3 h Mpc -1 . The few %-level FoG contamination would be okay for a current accuracy of the power spectrum measurement, but will need to be carefully taken into account for a higher-precision measurement of upcoming redshift surveys. For comparison, the three dots-dashed curve shows the effect on the real-space matter power spectrum caused by massive neutrinos, where we assumed m ν, tot /similarequal 0 . 1 eV for the total mass of neutrinos, close to the lower limit on the neutrino mass for the inverted mass hierarchy. For the normal mass hierarchy, the lower limit on the total mass is about 0.05 eV, and the amount of the suppression is about half of the result of 0 . 1 eV in Figure 11. The lowest curve in the figure shows the difference of the real-space matter power spectrum when taking account of massive neutrinos relative to the spectrum for the mass-less neutrino cosmology. In the right panel of Figure 11, we also show the redshiftspace power spectrum derived by using all the LRGs in the catalog. Note that we properly subtracted the shot noise contamination from the measured power spectra by using the number densities of LRGs or LRG-host halos. In this case, the power spectrum ratio shows greater amplitudes with increasing wavenumber rather than the FoG suppression. That is, the LRG power spectrum shows a greater clustering power or greater bias than in the LRGhost halo spectrum. The scales shown here, the scales greater than a few tens Mpc, are much larger than a virial radius of most massive host-halos and the 1-halo term arising from clustering between two LRGs in the same host-halo should not be significant at these scales. Hence, the greater amplitudes in the LRG power spectrum would be due to a more weight on more massive halos, because satellite LRGs preferentially reside in more massive halos that have larger biases. Since the effect of different linear bias should cause only a scale-independent change in the power spectrum ratio, the change in the LRG power spectrum should be from a stronger nonlinear bias of such massive halos, even though the FoG suppression should be more significant for such halos. In fact, a combination of the perturbation theory of structure formation and halo bias model seems to reproduce such a non-trivial behavior in the power spectrum amplitudes (Nishizawa et al. 2012). The results in the figure imply that including satellite LRGs in the power spectrum analysis complicates the interpretation of the measured power spectrum, thereby causing a bias in cosmological parameters. These subtle effects need to be well understood if we are going to use power spectrum measurements to place unbiased constraints on cosmological parameters such as the neutrino mass. In Figure 12, we study how the residual FoG effect varies with masses of LRG-host halos. To study this, we divide the LRG halos into two sub-samples by masses of the LRG-halos smaller and larger than the median, and measured the fractional power spectra for each sub-sample relative to the halo sample. As expected, the FoG effect is larger for the sub-sample containing more massive halos, because of the higher fraction of satellite BLRGs as well as the larger velocity dispersion (larger halo mass). The right panel shows the similar results, but obtained only by using the single-LRG halos. First of all, the single LRG systems have a smaller FoG effect, because of the smaller fraction of satellite BLRGs (Figure 5 and Table 1) as well as the smaller velocity dispersions for the lower-mass host-halos. Among the single-LRG halos, more massive halos have relatively a larger FoG contamination, but only by a few percent at k < ∼ 0 . 35 h Mpc -1 in the amplitude. Thus, the use of single-LRG systems may allow a cleaner interpretation of the measured power spectrum, yielding a more robust, unbiased constraint on cosmological parameters.", "pages": [ 11, 12, 13 ] }, { "title": "4 DISCUSSION AND CONCLUSION", "content": "In this paper, we have developed a new abundance-matching based method to generate a mock catalog of the SDSS LRGs, using catalogs of halos and subhalos in N -body simulations. A brief summary of our method is as follows: (1) identify LRG-progenitor halos at z = 2 down to a certain mass threshold until the comoving number density of the halos become similar to that of the SDSS LRGsat z = 0 . 3 (2) trace the merging and assembly histories of the LRG 'star particles', the 30% innermost particles of each z = 2 -LRG-progenitor halo that are gravitationally, tightly-bounded particles, and (3) at z = 0 . 3 , identify the subhalos and halos hosting the LRG 'star' particles. If a subhalo at z = 0 . 3 contains more than 50% of the star particles of any progenitor halo, we assign the subhalo at z = 0 . 3 as an LRG-host subhalo. We should emphasize that our method does not employ any free fitting parameter to adjust in order for the model to match the measurements, once the mass threshold of the LRG-progenitor halos is determined to match the number density of SDSS LRGs. Thus, by assuming that a majority of stellar components of LRG is formed at z = 2 , we can trace the assembly and merging histories of LRGs over a range of redshift, z = [0 . 3 , 2] ; for example, we can directly trace which LRGs become central or satellite galaxies in the LRG-host halos at z = 0 . 3 . The novel aspect of our approach is that the abundance matching of halos to a particular type of galaxies (LRGs in this paper) is done by connecting the halos and subhalos at different redshifts ( z = 2 and z = 0 . 3 in our case), while the standard method is done for the same or similar redshift to the redshift of target galaxies. In addition, central and satellite subhalos are populated with galaxies under a single criterion: if a subhalo at z = 0 . 3 is a descendant of the z 2 -halos, the subhalo is included. The standard abundance matching uses different quantities for central and satellite subhalos, e.g., the circular velocities at the galaxy redshift and at the accretion epoch, respectively. Using the mock catalog, we have computed various statistical quantities: the halo occupation distribution, the projected correlation function of LRGs, the mean surface mass density profile around LRGs (which is an observable of the LRG-galaxy weak lensing), and the redshift-space power spectrum of LRGs. We showed that the mock catalog predictions are in a good agreement with the measurements from the SDSS LRG catalog (Figures 3, 8, 9, 10 and 11). Thus our method seems to capture an essential feature of LRG formation in terms of a hierarchical structure formation scenario of Λ CDMmodel. In the SDSS sample, about 5% of the halos contain multiple LRGs. In our simulation, 8% of the halos contain multiple LRGs. This modest deviation may be due to our simplified assumptions. First, we assumed that LRG progenitors are formed at a single epoch, z = 2 . This is too simplified assumption as LRG formation almost certainly took place over a range of redshifts. Including a broader period of formation of LRG-progenitor halos may improve the model prediction. Second, although LRGs are observationally selected by magnitude and color cuts, our definition of the LRG-progenitor halos at z = 2 is solely based on their masses. The agreements between our mock catalog and the SDSS measurements support that the matching based on the LRG-progenitor halo masses seems fairly reasonable to mimic a population of LRGs. However, the model may be further refined by combining masses of the progenitor halos with other indicators when matching to LRGs. For instance, using the maximum circular velocity of each halo instead of its mass may improve the model accuracy. Another potential improvement would be to introduce some stochasticity into the relationship between halo mass and inclusion in the LRG sample. Variation in star formation histories should introduce scatter into the halo mass/galaxy luminosity relation. We have done a preliminary study where we introduce some scatter and find that this lowers the characteristic halo mass, which leads to smaller bias parameters, and obviates the disagreement between theory and observation for the projected correlation function or the lensing mass profile in Figures 8 and 9. Another simplifying assumption was our neglect of satellite subhalos in the parent halo at z = 2 in the abundance matching procedure. We naively expect that subhalos at z = 2 merge into central subhalos from z = 2 to z = 0 . 3 due to dynamical friction, so we used the simplest method as the first attempt. However, including the subhalos at z = 2 for the abundance matching may improve an accuracy of the mock catalog. We plan to explore these improvements in a future work. Our mock catalog or more generally our abundance matching method offers several applications to measurements. First, Masjedi et al. (2008) showed that, by using the small-scale clustering signal and the pair counting statistics, LRGs are growing by about 1.7% per Gyr, on average from merger activity from z = 1 to z ∼ 0 . 3 . Our method directly traces how each LRG-progenitor halo acquires the mass from other LRG-progenitor halos by major or minor mergers from z = 2 to z = 0 . 3 . Hence, we can compare the prediction of our mock catalog with the measurement for the mass growth rate of LRGs. By using the constraint, we may be able to further improve the mock catalog. Second, our method can predict how the distribution of LRGprogenitor evolves in relative to dark matter distribution as a function of redshift. Thus, our mock catalog can be used to predict various cross-correlations of LRG positions with other tracers of largescale structure. As one such example, in this paper, we have studied the LRG-galaxy weak lensing measured via cross-correlation of LRGs with shapes of background galaxies, and have shown a remarkably good agreement between our model and the SDSS measurements. Another cross-correlation that has been studied in the literature is a cross-correlation of LRGs with a map of cosmic microwave background (CMB) anisotropies, which probes the stacked Sunyaev-Zel'dovich (SZ) effect (Hand et al. 2011; Sehgal et al. 2013) or the lensing effect on the CMB. Since every massive halos always host at least one LRG (100% of halos with M vir /greaterorequalslant 2 × 10 14 h -1 M /circledot in our mock catalog), the cross-correlation is a powerful cross-check of the SZ signals. Our mock catalog can predict how the stacked SZ signals change for different catalogs of LRGs such as an inclusion of satellite LRGs and multiple-LRG systems, which may be able to resolve some tension between the observed LRG-CMB cross-correlation signal and the theoretical expectation (Sehgal et al. 2013). Third is an application of our method to LRGs or massive red galaxies at higher redshift than z = 0 . 3 . The SDSS-III BOSS survey is now carrying out an even more massive redshift survey of SDSS imaging galaxies. The magnitude and color cuts used for the BOSS survey are designed to efficiently select galaxies at 0 . 4 < z < 0 . 7 , and are different from the SDSS-I/II LRG selection. The BOSS galaxies are called the 'constant mass' (CMASS) galaxies. The majority of CMASS galaxies are early-type galaxies, but are not exactly the same population as LRGs. In addition, the comoving number density of CMASS galaxies is higher than that of LRGs by a factor of 3. Hence, it would be interesting to apply the method developed in this paper to the CMASS galaxies. Figure 1 shows an interesting indication of our mock catalog: more LRGprogenitor halos survive in the z = 0 . 5 output than at z = 0 . 3 , because the halos do not have enough time to experience merging due to the shorter time duration from z = 2 to z = 0 . 5 than to z = 0 . 3 . Hence, our mock catalog naturally predicts a higher number density of LRG-progenitor halos at higher redshift than at z = 0 . 3 , and may be able to match some of the BOSS galaxies without any fine tuning. Since the BOSS survey will provide us with a higher-precision clustering measurement and therefore has the potential to achieve tighter cosmological constraints, it is critically important to use an accurate mock catalog of the CMASS galaxies in order to remove or calibrate various systematic errors inherent in an unknown relation between the CMASS galaxies and dark matter. We hope that our method is useful for this purpose and can be used to attain the full potential of the BOSS survey or more generally upcoming redshift surveys for precision cosmology. This is our future study and will be presented elsewhere.", "pages": [ 13, 14 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank Kevin Bundy, Surhud More and David Wake for useful discussion and valuable comments. We also thank Rachel Mandelbaum, Zheng Zheng and Idit Zehavi for kindly giving us their SDSS measurement results. We appreciate Takahiro Nishimichi for kindly providing us with the second-order Lagrangian perturbation theory code. We also thank Naoki Yoshida for valuable advices for numerical techniques. We are grateful to the anonymous referee for helpful comments. SM acknowledges support from a Japan Society for Promotion of Science (JSPS) fellowship (No. 23-6573). This work was supported by JSPS Grant-in-Aid for Scientific Research Numbers 22340056 (NS), 23340061 (MT), 24740160 (CH) and NASA grant ATP11-0034 and ATP11-0090 (DS). MT greatly thanks Department of Astrophysical Sciences, Princeton University and members there for its warm hospitality during his visit, where this work was done. The authors acknowledge KobayashiMaskawa Institute for the Origin of Particles and the Universe, Nagoya University for providing computing resources useful in conducting the research reported in this paper. This research was in part supported by the Grant-in-Aid for Nagoya University Global COE Program, 'Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos', from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, by JSPS Core-to-Core Program 'International Research Network for Dark Energy', by Grant-in-Aid for Scientific Research from the JSPS Promotion of Science, by Grant-inAid for Scientific Research on Priority Areas No. 467 'Probing the Dark Energy through an Extremely Wide & Deep Survey with Subaru Telescope', by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, by the FIRST program 'Subaru Measurements of Images and Redshifts (SuMIRe)', CSTP, Japan, and by the exchange program between JSPS and DFG.", "pages": [ 14 ] }, { "title": "REFERENCES", "content": "Anderson L. et al., 2012, MNRAS, 427, 3435 Berlind A. A., Weinberg D. H., 2002, ApJ, 575, 587 Blake C., et al., 2011, MNRAS, 415, 2876 Springel V., Yoshida N., White S. D. M., 2001b, NewA, 6, 79 Tegmark M., et al., 2004, ApJ, 606, 702 Tojeiro R. et al., 2012, MNRAS, 424, 136 Trujillo-Gomez S., Klypin A., Primack J., Romanowsky A. J., 2011, ApJ, 742, 16 Wake D. A. et al., 2006, MNRAS, 372, 537 Wake D. A. et al., 2008, MNRAS, 387, 1045 Wang Y., Spergel D. N., Strauss M. A., 1999, ApJ, 510, 20 White M. et al., 2011, ApJ, 728, 126 York D. G. et al., 2000, AJ, 120, 1579 Zehavi I. et al., 2005, ApJ, 621, 22 Zheng Z., Coil A. L., Zehavi I., 2007, ApJ, 667, 760", "pages": [ 14, 15 ] }, { "title": "APPENDIX A: VARIANTS OF OUR ABUNDANCE MATCHING METHOD", "content": "Although we have developed the simplest abundance matching method by connecting halos at z = 2 to subhalos at z = 0 . 3 , the method rests on some simplified settings or assumptions that are not obvious: (1) the formation redshift of LRG-progenitor halos is set to a single epoch of z = 2 , (2) we defined the 'LRGstar-particles' by the innermost 30% particles of each z = 2 -progenitor halo, and (3) the 'matching' fraction of the star particles to each subhalo at z = 0 . 3 is set to more than 50% (if a subhalo at z = 0 . 3 contains more than 50% of the star particles of any z = 2 -progenitor halo, the subhalo is called as the LRG-host subhalo). In summary, for our fiducial method, we have used as described in detail in Section 2. In this Appendix, we study how the results are changed if varying each of these parameters to other values. In doing this, we use the L300 run because it has a higher resolution than L1000 run and allows us to better resolve less massive halos or sub-halos from the z = 2 or z = 0 . 3 simulation outputs.", "pages": [ 15 ] }, { "title": "A1 LRG-progenitor halo formation redshift: z form", "content": "Throughout this paper, we employed z form = 2 for the formation redshift of LRG-progenitor halos, as the first attempt, motivated by the fact that LRGs typically have old ages > ∼ 5 Gyr . Figure A1 shows how the mock catalog of LRGs is changed if varying the formation redshift to z form = 1 or 3 from our fiducial choice z form = 2 . Here, to assess the difference, we show the projected correlation functions obtained from the mock catalogs. The figure shows that the formation redshift of z form = 1 or 3 gives lower or higher amplitudes at small separations < ∼ 1 h -1 Mpc , which is in the one-halo term regime, than our fiducial case of z form = 2 . Nevertheless, the encouraging result is that the large-separation correlation function in the two-halo regime is not largely changed. These changes can be understood as follows. For the case of z form = 1 , the LRG-progenitor halos have a shorter duration since the formation and each halo has a smaller chance to experience subsequent mergers than in the case of z form = 2 , the progenitor in turn has a smaller chance to be included in multiple-LRG systems being a satellite subhalo at z = 0 . 3 , which decreases clustering power in the one-halo regime. Thus Figure A1 implies that a choice of z form > ∼ 2 is reasonable. The upper panel of Figure A2 shows how the change of z form alters the HOD as in Figures 3. As we discussed above, the change to z form = 3 from z form = 2 leads to a smoother HOD shape around the cutoff halo mass for the central LRG HOD extending down to less massive halos as well as to an increase of satellite LRGs. The choice of z form = 1 leads to opposite effects. On the other hand, the lower panel shows that the projected mass profile of LRG-host halos is almost unchanged by the change of z form . Thus an accuracy of the mock catalog can be improved, especially for predicting the small-scale clustering, by further introducing a variation of the formation epochs as additional parameter.", "pages": [ 15, 16 ] }, { "title": "A2 The fraction of LRG-star particles in z = 2 -LRG-progenitor halo: f star", "content": "To trace the LRG-progenitor halos from z = 2 to z = 0 . 3 , we defined the LRG-star particles in each z = 2 LRG-progenitor halo by the 30% innermost particles of the FoF member particles, f star = 0 . 3 , assuming that the stars are formed at the central region of each halo. However, the fraction 30% is an arbitrary choice. Figure A3 shows how the projected correlation and the HOD are changed by varying to f star = 0 . 2 or 0 . 4 from the fiducial choice of f star = 0 . 3 . Note that, when matching each z = 2 -progenitor halo to a subhalo at z = 0 . 3 , we imposed the condition that a subhalo at z = 0 . 3 should have more than 50% of the star particles in each of the mock catalogs. Figure A3 shows that the change of f star alters the correlation function at the small separation, in the one-halo regime. This is also found from the lower panel, which shows that the change alters the satellite HOD. If we use f star = 0 . 4 from 0 . 3 , it tends to include, in the star particles, more loosely-bounded member particles in each z = 2 -progenitor halo, the star particles tend to be stripped by tidal interaction or merger with other halos, and then it is in turn difficult to satisfy the 50% matching condition to z = 0 . 3 subhalo. Thus this results in a smaller number of the survived satellite LRG-halos at z = 0 . 3 , which causes to reduce the correlation amplitude at the small scales. The opposite is true for the case of f star = 0 . 2 , because it leads to a larger number of the survived satellite LRG-halos at z = 0 . 3 than in the case of f star = 0 . 3 . Nevertheless, the encouraging result is the correlation function at large scales in the two-halo regime is not sensitive to the variation of f star . We have found that the projected mass profile is not changed as in the lower panel of Figure A2.", "pages": [ 16 ] }, { "title": "A3 The matching fraction between z = 2 -LRG-progenitor halos and z = 0 . 3 -subhalos: f match", "content": "Finally, we study how the mock catalog is changed by varying the matching fraction to f match = 0 . 3 or 0 . 7 from our fiducial choice f match = 0 . 5 . Figure A4 shows that the change of f match alters the projected correlation function at small scales in the one-halo regime and the satellite HOD, similarly to Figures A1 and A3. Again, if increasing the matching fraction to f match = 0 . 7 from 0 . 5 , it leads to a less number of the matched satellite subhalos at z = 0 . 3 and in turn leads to a decreased power in the correlation function. On the other hand, the large-scale correlation is more robust to the change of f match similarly to the previous figures. We have again found that the projected mass profile is not changed as in the lower panel of Figure A2.", "pages": [ 16, 17 ] } ]

[ { "title": "ABSTRACT", "content": "The next generation of galaxy surveys will observe millions of galaxies over large volumes of the universe. These surveys are expensive both in time and cost, raising questions regarding the optimal investment of this time and money. In this work we investigate criteria for selecting amongst observing strategies for constraining the galaxy power spectrum and a set of cosmological parameters. Depending on the parameters of interest, it may be more efficient to observe a larger, but sparsely sampled, area of sky instead of a smaller contiguous area. In this work, by making use of the principles of Bayesian Experimental Design, we will investigate the advantages and disadvantages of the sparse sampling of the sky and discuss the circumstances in which a sparse survey is indeed the most efficient strategy. For the Dark Energy Survey (DES), we find that by sparsely observing the same area in a smaller amount of time, we only increase the errors on the parameters by a maximum of 0.45%. Conversely, investing the same amount of time as the original DES to observe a sparser but larger area of sky we can in fact constrain the parameters with errors reduced by 28%. Key words: cosmology", "pages": [ 1 ] }, { "title": "P. Paykari 1 /star and A. H. Jaffe 2", "content": "1 Laboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, SAp/SEDI, Service d'Astrophysique, CEA Saclay, F-91191 GIF- SUR-YVETTE CEDEX, France. 2 Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The measurements of the cosmological parameters heavily rely on accurate measurements of power spectra. Power spectra describe the spatial distribution of an isotropic random field, defined as the Fourier transform of the spatial correlation function. The perturbations in the universe can be described statistically using the correlation function ξ ( r ) between two points, which depends only on their separation r (when isotropy is assumed) 1 ; where δ ( x ) = ( ρ ( x ) -¯ ρ ) / ¯ ρ measures the continuous overdensity, where ρ ( x ) is the density at position x and ¯ ρ is the average density. The power spectrum P ( k ) , which is the Fourier transform of the correlation function, is enough to define the perturbations completely when the perturbations are assumed uncorrelated Gaussian random fields in the Fourier space. Power spectra (or correlation functions) are what the surveys actually measure, from which cosmological parameters are inferred. These spectra are normally a convolution of the primordial power spectrum (which measures the statistical distribution of perturbations in the early universe) and a transfer function which depends on the cosmological parameters. Hence accurate measurements of the power spectra from surveys are very important for accurate measurements of the cosmological parameters. The most important observed spatial power spectrum for cosmology is the galaxy power spectrum; the Fourier transform of the galaxy correlation function, which was first formulated by Peebles (1973). A galaxy survey lists the measured positions of the observed galaxies. As proposed by Peebles, these positions are modelled as a random Poissonian point source, where the galaxy density is modulated by the fluctuations in the underlying matter distribution and the selection effects. The selection function of the survey is described by ¯ n ( x ) , which is the expected galaxy density at position x in the absence of clustering. The fluctuations in the underlying matter density are given by δ ( x ) , as described previously. The the galaxy number over-density n ( x ) , which is the observed quantity, is related to the matter over-density via the bias b (Kaiser 1984) - galaxies trace dark matter up to this b factor. We define the galaxy power spectrum P g ( k ) as where P p ( k ) is the primordial power spectrum P p ( k ) = A s k n s -1 . The transfer function T ( k ) further depends upon the cosmological parameters (e.g., the matter density Ω m , the scalar spectral index, n s , etc.) responsible for the evo- tion of the universe. The bias b relates the galaxy power spectrum to the matter power spectrum, as explained above. This power spectrum is very rich in terms of constraining a large range of cosmological parameters. On large scales this spectrum probes structure which is less affected by clustering and evolution. Hence these scales are still in the linear regime and have a 'memory' of the initial state. The information from these regimes are, therefore, the cleanest since the Big Bang and any knowledge on these large scales would shed light on the physics of early universe and hence the primordial power spectrum. On intermediate scales the spectrum provides us with information about the evolution of the universe since the Big Bang; for example the matterradiation equality which is responsible for the peak of the galaxy spectrum. The matter-radiation equality is a unique point in the history of the evolution, giving information about the amount of matter and radiation in the universe. On relatively small scales there is a great deal of information about galaxy clustering via the Baryonic Acoustic Oscillations (BAO) which encode a characteristic scale; the sound horizon at the time of recombination. Therefore, measuring the galaxy power spectrum on a large range of scales can help us constrain the cosmological parameters responsible for the evolution of the universe as well as the ones of its initial state. Accurate measurements of the galaxy power spectrum depend on two main factors; the Poisson noise and the cosmic variance. To overcome the Poisson noise, surveys aim to maximise the number of galaxies observed. The impressive constraints on cosmological parameters from previous and current surveys, such as the 2dF (Croom et al. 2004) and SDSS (Adelman-McCarthy et al. 2008), has motivated even more ambitious future surveys such as DES (The Dark Energy Survey Collaboration 2005) and Euclid (Laureijs 2009), aiming to observe millions of galaxies over large volumes of the universe. Considering the large investments in time and money for these surveys, one wants to ask what is really the optimal survey strategy! In this work we want to investigate this exact questions and find the optimal strategy for galaxy surveys such as DES and Euclid. In this era of cosmology where the statistical errors have reduced greatly and are now comparable with systematics, observing, for example, a greater number of galaxies may not necessarily improve our results. We need to devise more strategic ways to make our observations and take control of our systematics. For example, to investigate larger scales, it may be more efficient to observe a larger, but sparsely sampled, area of sky instead of a smaller contiguous area. In this case we would gather a larger density of states in Fourier space, but at the expense of an increased correlation between different scales - aliasing. This would smooth out features on these scales and decrease its significance if any observed. Here, by making use of Bayesian Experimental Design we will investigate the advantages and disadvantages of the sparse sampling and verify if a complete contiguous survey is indeed the most efficient way of observing the sky for our purposes. The parameter of interest here is the galaxy power spectrum itself and a set of cosmological parameters that depend on this spectrum. Some previous work on sparse sampling includes Kaiser (1986) and Blake et al. (2006); Kaiser (1986) shows that measuring the large scale correlation function from a com- plete magnitude-limited redshift survey is actually not the most efficient approach. Instead, sampling a fraction of galaxies randomly, but to a fainter magnitude limit, will improve the constraints of the correlation function measurements significantly, for the same amount of observing time. Blake et al. (2006) have shown that a sparse-sampling (achieved by a non-contiguous telescope pointings or, for a wide-field multi-object spectrograph, by having the fibres distributed randomly across the field-of-view) is preferred when the angular size of the sparse observed patches is much smaller than angular scale of the features in the power spectrum (the acoustic features).", "pages": [ 1, 2 ] }, { "title": "2 BAYESIAN EXPERIMENTAL DESIGN AND FIGURE-OF-MERIT", "content": "Bayesian methods have recently been used in cosmology for model comparison and for deriving posterior probability distributions for parameters of different models. However, Bayesian statistics can do even more by handling questions about the performance of future experiments, based on our current knowledge (Liddle et al. 2006; Trotta 2007a,b). For example, Parkinson et al. (2007) use a Bayesian approach to constrain the dark energy parameters by optimising the Baryon Acoustic Oscillations (BAO) surveys. By searching through a survey parameter space (which includes parameters such as redshift range, number of redshift bins, survey area, observing time, etc.) they find the optimal survey with respect to the dark energy equation-of-state parameters. Here we will use this strength of Bayesian statistics for optimising the strategy to observe the sky for galaxy surveys. There are three requirements for such an optimisation; 1. specify the parameters that define the experiment which need to be optimised for an optimal survey; 2. specify the parameters to constrain, with respect to which the survey is optimised; 3. specify a quantity of interest, generally called the figure of merit (FoM), associated with the proposed experiment. The choice of the FoM depends on the questions being asked, as will be explained later in the text. We then want to extrimise the FoM subject to constraints imposed by the experiment or by our knowledge about the nature of the universe. Below, we will explain the procedure. Assume e denotes the different experimental designs that we can implement and M i are the different models under consideration with their parameters θ i . Assume that experiment o has been performed, so that this experiment's posterior P ( θ | o ) forms our prior probability function for the new experiment. The FoM will depend on the set of parameters under investigation, the performed experiment (data) and the characteristics of the future experiment; U ( θ, e, o ) . From the utility we can build the expected utility E [ U ] as where ˆ θ i represent the fiducial parameters for model M i . This says: If a set of fiducial parameters, ˆ θ , correctly describe the universe and we perform an experiment e , then we can compute the utility function for that experiment, U ( ˆ θ, e, o ) . However, our knowledge of the universe is described by the current posterior distribution P ( ˆ θ | o ) . Averaging the utility over the posterior accounts for the present uncertainty in the parameters and summing over all the available models would account for the uncertainty in the underlying true model. The aim is to select an experiment that extremises the utility function (or its expectation). The utility function takes into account the current models and the uncertainties in their parameters and, therefore, extremising it takes into account the lack of knowledge of the true model of the universe. One of the common choices for the FoM is some form of function of the Fisher matrix, which is the expectation of the inverse covariance of the parameters in the Gaussian limits (We will explain in the next section how a Fisher matrix is obtained in more detail.). One can refer to the Dark Energy Task Force (DETF) FoM, that use Fisher-matrix techniques to investigate how well each model experiment would be able to restrict the dark energy parameters w 0 , w a , Ω DE for their purposes. Three common FoMs, which we will be using as well, are trace of the Fisher matrix (or its log ) and is proportional to sum of the variances. This prefers a spherical error region, but may not necessarily select the smallest volume. determinant of the Fisher matrix (or its log ), which measures the inverse of the square of the parameter volume enclosed by the posterior. This is a good indicator of the overall size of the error over all parameter space, but is not sensitive to any degeneracies amongst the parameters. where P ( θ | ˆ θ, e, o ) is the posterior distribution with Fisher matrix F and P ( θ | o ) is the prior distribution with Fisher matrix Π . The entropy forms a nice compromise between the A-optimality and D-optimality. Note that these are the utility functions, not the 'expected' utility functions. In our current models of the universe, we do not expect a significant difference between the parameters of the same model. However, this will be investigated in a future work, where we will explicitly use expected utility functions. In the next section we will explain how a Fisher matrix is formulated.", "pages": [ 2, 3 ] }, { "title": "3 FISHER MATRIX ANALYSIS", "content": "The Fisher matrix is generally used to determine the sensitivity of a particular survey to a set of parameters and has been largely used for optimisation (and forecasting). Consider the likelihood function for a future experiment with experimental parameters e , L ( θ | e ) ≡ P ( D ˆ θ | θ, e ) , where D ˆ θ are simulated data from the future experiment assuming that ˆ θ are the true parameters in the given model. We Taylor expand the log-likelihood around its maximum value: where the first term is a constant and only affects the height of the function, the second term describes how fast the likeli- hood function falls around the maximum. The Fisher matrix is defined as the ensemble average of the curvature of the likelihood function L (i.e., it is the average of the curvature over many realisations of signal and noise); where the second line is appropriate for a Gaussian distribution with correlation matrix C determined by the parameters θ i , and L is the likelihood function. The inverse of the Fisher matrix is an approximation of the covariance matrix of the parameters, by analogy with a Gaussian distribution in the θ i , for which this would be exact. The Cramer-Rao inequality 2 states that the smallest frequentist error measured, for θ i , by any unbiased estimator (such as the maximum likelihood) is 1 / √ F ii and √ ( F -1 ) ii , for non-marginalised and marginalised 3 one-sigma errors respectively. The derivatives in Equation 6 generally depend on where in the parameter space they are calculated and hence it is clear that the Fisher matrix is function of the fiducial parameters. The Fisher matrix allows us to estimate the errors on parameters without having to cover the whole parameter space (but of course will only be appropriate so long as the derivatives are roughly constant throughout the space). So, a Fisher matrix analysis is equivalent to the assumption of a Gaussian distribution about the peak of the likelihood (e.g. Bond et al. 1998). It also makes the calculations easier. For example, if we are only interested in a subset of parameters, then marginalising over unwanted parameters is just the same as inverting the Fisher matrix, taking only the rows and columns of the wanted parameters and inverting the smaller matrix back. It is also very straightforward to combine constraints from different independent parameters: we just sum over the Fisher matrices of the experiments (remember Fisher matrix is the log of the likelihood function). We further note, as in all uses of the Fisher matrix, that any results thus obtained must be taken with the caveat that these relations only map onto realistic error bars in the case of a Gaussian distribution, usually most appropriate in the limit of high signal-to-noise ratio and/or relatively small scales, so that the conditions of the central limit theorem obtain. As long as we do not find extremely degenerate parameter directions, we expect that our results will certainly be indicative of a full analysis, using simulations and techniques such as Bayesian Experimental Design (Trotta 2007c).", "pages": [ 3 ] }, { "title": "3.1 Fisher Matrix for Galaxy Surveys", "content": "We follow the approach of Tegmark (1997) to define the pixelisation for galaxy surveys. First we define the data in pixel i as 2 It should be noted that the Cramer-Rao inequality is a statement about the so-called 'Frequentist' confidence intervals and is not strictly applicable to 'Bayesian' errors.", "pages": [ 3 ] }, { "title": "4 P. Paykari and A. H. Jaffe", "content": "where n ( x ) is the galaxy density at position x and ¯ n is the expected number of galaxies at that position. The weighting function, ψ i ( x ) , which determines the pixelisation (and is sensitive to the shape of the survey as you will see later), is defined as a set of Fourier pixels where V is the volume of the survey. Here we have divided the volume into sub-volumes, each being much smaller than the total volume of the survey, but being large enough to contain many galaxies. This means ∆ i is the fractional overdensity in pixel i . Using this pixelisation we can define a covariance matrix as where C S and C N are the signal and noise covariance matrices respectively and are assumed independent of each other. The signal covariance matrix can be defined as By equating the number over-density ( n ( x ) -¯ n ) / ¯ n to the continuous over-density δ ( x ) = ( ρ ( x ) -¯ ρ ) / ¯ ρ we obtain where ˜ ψ i ( k ) is the Fourier transform of ψ i ( x ) and the window function W ij ( k ) is defined as the angular average of the square of the Fourier transform of the weighting function. With the same approach, the noise covariance matrix -which is due to Poisson shot noise - is given by The design of the survey will shape the form of the weighting function in Equation 9, which will be discussed in the next section. This prescription gives us a data covariance matrix for a galaxy survey. What we actually need is a Fisher matrix for the parameters we are interested in. For this we will use Equation 6 above, which defines the Fisher matrix of parameters in terms of the inverse of the data covariance matrix and its differentiation with respect to the parameters of interest. We are interested in the galaxy power spectrum and hence the differentiation of the covariance matrix in Equation 6 is taken with respect to the bins of this power spectrum. As the noise covariance matrix does not depend on the power spectrum, we only need to differentiate the signal covariance matrix in Equation 12. Taking the galaxy power spectrum as a series of top-hat bins where P B is the power in each bin, the differentiation takes the form We insert this and the inverse of the data covariance matrix into Equation 6 to get a Fisher matrix for the galaxy power spectrum bins. To get a Fisher matrix for the cosmological parameters one can use the parameters Jacobian where F ab is the galaxy spectrum Fisher matrix and F αβ is the Fisher matrix for the cosmological parameters λ α and λ β .", "pages": [ 4 ] }, { "title": "4 SURVEY DESIGN", "content": "We will investigate the FoM of a sparse design to that of a contiguous survey, which we have chosen to be similar to that of the Dark Energy Survey (DES).", "pages": [ 4 ] }, { "title": "4.1 Dark Energy Survey (DES)", "content": "The Dark Energy Survey (DES) 4 (The Dark Energy Survey Collaboration 2005) is designed to probe the origin of the accelerating universe and help uncover the nature of dark energy. Its digital camera, DECam, is mounted on the Blanco 4-meter telescope at Cerro Tololo Inter-American Observatory in the Chilean Andes. Starting in December 2012 and continuing for five years, DES will catalogue 300 million galaxies in the southern sky over an area of 5000 square degrees and a redshift range of 0 . 2 < z < 1 . 3 . In the next section we will explain how we 'sparsify' the DES survey for our purposes. Here, we use a flat-sky approximation. Euclid, with a survey area of 20 , 000 square degrees should be treated on the full sky and is not investigated here. Nonetheless we expect qualitatively similar results to DES.", "pages": [ 4 ] }, { "title": "4.2 Sparse Design", "content": "For simplicity, we will design the sparsely sampled area of the sky as a regular grid of n p × n p square patches of size M × M - Figure 1. We therefore define the structure on the sky as a top-hat in both x and y directions where x i and y j mark the centres of the patches in our coordinate system. In the z direction we use the step function, which is defined as: With this design the weight function in equation 9 takes the form: where q = k i -k , q x = q sin θ cos φ , q y = q sin θ sin φ , q z = q cos φ and dµ = d cos θ . The volume V is the total sparsely sampled volume, M is the size of the observed patch on the surface of the sky and L is the observed depth. The last equality in the above equation uses the Dirichlet Kernel which can be used due to the symmetry of the design. The window function, defined in Equation 12, now takes the form Note that there are two scales that control the behaviour of the window function; one is the size of the patches, M , and the other is their distance from one another, x i . We will investigate the influence of both of these scales on the FoM by trying two different configurations, discussed in the next section. In case of the contiguous sampling of the sky where we are observing through a contiguous square, the window function takes the form of one single big patch, as shown below which is a square cylinder.", "pages": [ 4, 5, 6 ] }, { "title": "4.3 Sparsifying DES", "content": "We divide the total area of DES into small square patches, as explained in the design of the mask previously. There are two ways to sparsify this area; In this setting the size of the patches are kept fixed at 60 Mpc, and the area is sparsified by placing the patches further and further from one another. Here the total observed area is constant, while the total sampled area increases as the patches are put further and further. This configuration is shown in Figure 3. Now, the length of time for the survey remains the same, but is spread out over a larger area of sky. Note that the areas we consider here are small enough that the flat sky approximation is valid. Also note that in all the above setting we keep the number of bins of the galaxy power spectrum constant at n bin = 60 . In reality we should let the total volume of the survey choose the binning of the power spectrum via k min = (2 π/V ) 1 / 3 = dk , and hence the number of the bins n bin . However, if n bin changes from case to case it will be unfair to compare D-optimality and Entropy as they will have different units as n bin changes. To have a fair comparison between the cases we keep n bin constant.", "pages": [ 6 ] }, { "title": "5 RESULTS", "content": "We have chosen a geometrically flat Λ CDM model with adiabatic perturbations. We have a five-parameter model with the following values for the parameters: Ω m = 0 . 214 , Ω b = 0 . 044 , Ω Λ = 0 . 742 , τ = 0 . 087 and h = 0 . 719 , where H 0 = 100 h km -1 Mpc -1 . The FoM used are where Π is the prior Fisher matrix, which we have chosen to be that for a SDSS-LRG-like survey. The posterior Fisher matrix is F = L + Π , where L is the likelihood Fisher matrix, which is the current sparse survey we have designed. The utility functions above are defined so that they need to be maximised for an optimal design.", "pages": [ 6 ] }, { "title": "5.1 Constant Total Area", "content": "Figures 4 shows the FoM for both the galaxy power spectrum bins on the left and the cosmological parameters on the right. In both cases, the Entropy, A-optimality and Doptimality all increase with f . This is as expected as a contiguous sampling of the sky captures all the information and should be the best to constrain cosmology. The top panels in the Figure show A-optimality for the bins on the left and the cosmological parameters on the right. In both cases, A increases with f and reaches its maximum at f = 1 for DES. Note that A-optimality is a measure of the errors of the parameters only - it is a measure of the trace of the Fisher matrix. Therefore, it is does not account for the correlations between parameters. Although A increases with f for both the bins and the parameters, note that this increase is very small. To see the amount of change in each of the elements of the power spectrum Fisher matrix as f increases, look at the top panel of Figure 5. This shows the diagonal elements of the Fisher matrix F for galaxy power spectrum bins for the different f . The elements are all on top of each other and indeed the gain obtained by increasing f is very small. The middle panels of Figure 4 show D-optimality, which again increases with f for both the bins and the parameters. Note that, D-optimality is a measure of the determinant of the Fisher matrix and therefore takes the correlation between the parameters into account. The correlation between the parameters is indeed very important; one disadvantage of the sparse sampling is the correlation it induces between the parameters due to aliasing. To see this effect, look at the bottom panel of Figure 5, where the row of the Fisher matrix that corresponds to the middle bin of the power spectrum is shown. Going away from the peak in both direction, the elements show the correlation between the different bins and the middle one. As f decreases and we get more and more sparse, the power in the off-diagonal elements of the Fisher matrix increases, meaning there is more aliasing. The DES survey, as a full contiguous survey, has the least aliasing, while the sparsest survey has the most. The rise towards the small k (large scales) is due to sample variance. Looking at the correlations and the errors in the Fisher matrix of the spectrum one notes that the decrease in D- optimality for sparser surveys is mostly due to the increased correlation between the bins rather than the the increased errors; as we saw in the top panel of this Figure the decrease in the errors are negligible. In general we conclude that total aliasing induced by sparsity is small and the loss in the constraining power of the survey due to this aliasing is negligible. Hence, overall, little is gained by observing the sky more contiguously. The bottom panels in Figure 4 show the Entropy for the bins and the parameters. Again, E increases with f and reaches its maximum for DES. The Entropy measures the total size of the errors of the parameters in the Fisher matrix as well as their correlation. Hence it is a good compromise of A- and D-optimality. It measures the total information gain of the survey relative to a prior survey. Having an SDSS-likesurvey as our prior, and taking into account both the errors and the correlation between the parameters, the contiguous DES survey has the largest gain compared to the sparse surveys. However, note that this gain is again very small. Figure 6 shows the relative loss in the marginalised errors of each of the cosmological parameters with respect to DES. The largest loss for a sparse observation of the sky is on the spectral index with δ Ω Λ / Ω Λ ∼ 0 . 45% and the smallest is for Ω c with a loss of δ Ω c / Ω c ∼ 0 . 15% . The non-marginalised errors show a qualitatively different behaviour, where n s has the largest and Ω Λ has the smallest loss.", "pages": [ 6, 7 ] }, { "title": "5.2 Constant Observed Area", "content": "Figure 7 shows the FoM for the power spectrum bins and the cosmological parameters. In this case the Entropy, Aoptimality and D-optimality all decrease with f . And the overall changes in all the FoM are much larger than the ones seen in the previous scenario for both the bins and the parameters. The top panel of Figure 8 shows the diagonal elements of the Fisher matrix of the bins. As we sparsify the survey these elements increase, and hence better constrain the spectrum. The bottom panel in the Figure shows the row of the Fisher matrix that corresponds to the middle bin of the spectrum. Going away from the peak, the elements show the correlation between the different bins and the middle one. For DES the middle bin has a correlation with the close neighbouring bins. However, the correlation decreases as we go away from the peak. Towards small k (large scales) it starts to increase again due to sample variance. As f decreases and we get more and more sparse, the middle bin has a sharper drop (due to the larger total size of the sur- vey) i.e., less correlation with neighbouring bins. However, there is more aliasing between distant bins. Also, there are peaks (i.e., larger correlations) at certain scales which are related to the distances between the patches, which changes case by case. The DES survey, as a full contiguous survey, has indeed the least aliasing, while the sparsest survey has the most. Note that in this case the sparsity is obtained by placing the observed patches further and further away from each other. As the sparsity increases as the patches are placed further, the total size of the survey is greatly increased, which seems to make up for the aliasing that the sparse design has induced. Overall we gain a great deal by spending the same amount of time on larger but sparsely sampled area. Figure 9 shows the relative gain in the marginalised errors of each of the cosmological parameters with respect to DES. The largest gain for a sparse observation of the sky is on Ω Λ with δ Ω Λ / Ω Λ ∼ 27% and the smallest is for Ω c with a gain of δ Ω c / Ω c ∼ 7% . Again, a qualitatively different scenario is seen for the non-marginalised errors; Ω b has the largest gain due to sparsity, and h has the smallest.", "pages": [ 7, 8, 9 ] }, { "title": "6 CONCLUSION", "content": "In this work we have investigated the advantages and disadvantages of sparsely sampling the sky as opposed to a contiguous observation. By making use of Bayesian Experimental Design, we have defined our Figure of Merit as different functions of the Fisher matrix. These FoM capture different aspects of the parameters of interest such as their overall variance, the correlation between them or a measure of both as in Entropy. By optimising these functions we investigate an optimal survey design for estimating the galaxy power spectrum and a set of cosmological parameters. We have compared a series of sparse designs to a contiguous design of DES. We split the area of the DES survey into small square patches and sparsify the survey in two ways: (i) by shrinking the size of the patches while they are kept at a constant position. In this case the total sampled area of the survey is constant while the observed area (and the survey observing time) shrinks. This means the total information gained from the survey reduces in each case. In this scenario all the three FoM (A-optimality, D-optimality and Entropy) increase with f , both for the power spectrum bins and the cosmological parameters. This is expected as a contiguous sampling should capture all the information and constrain cosmology the best. However, we note that this increase with decreasing sparsity is very small for both the bins and the cosmological parameters. Looking at the variance and the covariance of the parameters, we note that the slight degrading of the surveys due to sparsity is mostly because of the increased correlation between the bins - aliasing - rather than the the increased errors. In general we conclude that total aliasing induced by sparsity is small and the loss in the constraining power of the survey because of it is negligible. Hence, overall, little is gained by observing the sky more contiguously. Indeed the largest loss in terms of the errors of the cosmological parameters is of the order of ∼ 4 . 5% in the sparsest case. (ii) by keeping the size of the patches constant, but placing them further and further from one another. In this scenario the observed area (and observing time) is kept constant, while sparsifying means larger and larger total sampled area. This means the total information gained from the survey in each case is the same. Therefore, there are the two competing factors; one is the increase in the total sampled area as the survey is sparsified and the other is aliasing induced due to the larger and larger sparse mask on the sky. In this case all FoM decrease with f , and the change in the FoM is much larger than the ones seen in the previous scenario. As we sparsify the survey the decrease in errors makes up for the increased aliasing induced and hence cause a general improvement in constraining power of the survey. Overall we gain a great deal by spending the same amount of time on larger but sparsely sampled area. Indeed we gain as much as ∼ 27% on the sparsest survey, which is a significant improvement. Weconclude that sparse sampling could be a good substitute for the contiguous observations and indeed the way forward for future surveys. At least for small areas of the sky, such as that of DES, sparse sampling of the sky can have less cost and less observing time, while obtaining the same amount of constraints on the cosmological parameters. On the other hand we can spend the same amount of time but sparsely observe a larger area of the sky. This greatly improves the constraining power of the survey. In this work we have chosen square observation patches, which may be the worst shape in terms of the correlation they induce. Yet another constraint in this design is the fixed and determined positions of the patches which cause a loss of information at certain scales. The advantage of this approach has been its analytical formalism, which has made it possible to understand the important factors in the sparse sampling. For future work we will investigate an optimal shape foe the patches and have a numerical approach where these patches are randomly distributed on the sky. This causes an even loss of information on all scales and is expected to improve results greatly.", "pages": [ 9, 10, 11, 12 ] }, { "title": "REFERENCES", "content": "Adelman-McCarthy J. K., et al., 2008, Astrophys. J. Suppl., 175, 297 Tegmark M., 1997, Physical Review Letters, 79, 3806 The Dark Energy Survey Collaboration 2005, ArXiv Astrophysics e-prints Trotta R., 2007a, MNRAS, 378, 72 Trotta R., 2007b, MNRAS, 378, 819 Trotta R., 2007c, MNRAS, 378, 819", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "It has been established that self-gravitating disc-satellite interaction can lead to the formation of a gravitationally unstable gap. Such an instability may significantly affect the orbital migration of gap-opening perturbers in self-gravitating discs. In this paper, we extend the two-dimensional hydrodynamic simulations of Lin & Papaloizou to investigate the role of the perturber or planet mass on the gravitational stability of gaps and its impact on orbital migration. We consider giant planets with planetto-star mass ratio q ≡ M p /M ∗ ∈ [0 . 3 , 3 . 0] × 10 -3 (so that q = 10 -3 corresponds to a Jupiter mass planet if M ∗ = M /circledot ), in a self-gravitating disc with disc-to-star mass ratio M d /M ∗ = 0 . 08, aspect ratio h = 0 . 05, and Keplerian Toomre parameter Q k 0 = 1 . 5 at 2.5 times the planet's initial orbital radius. These planet masses correspond to ˜ q ∈ [0 . 9 , 1 . 7], where ˜ q is the ratio of the planet Hill radius to the local disc scale-height. Fixed-orbit simulations show that all planet masses we consider open gravitationally unstable gaps, but the instability is stronger and develops sooner with increasing planet mass. The disc-on-planet torques typically become more positive with increasing planet mass. In freely-migrating simulations, we observe faster outward migration with increasing planet mass, but only for planet masses capable of opening unstable gaps early on. For q = 0 . 0003 (˜ q = 0 . 9), the planet undergoes rapid inward type III migration before it can open a gap. For q = 0 . 0013 (˜ q = 1 . 5) we find it is possible to balance the tendency for inward migration by the positive torques due to an unstable gap, but only for a few 10's of orbital periods. We find the unstable outer gap edge can trigger outward type III migration, sending the planet to twice it's initial orbital radius on dynamical timescales. We briefly discuss the importance of our results in the context of giant planet formation on wide orbits through disc fragmentation. Key words: planetary systems: formation, planetary migration, protoplanetary discs", "pages": [ 1 ] }, { "title": "Ryan Cloutier /star and Min-Kai Lin †", "content": "Canadian Institute for Theoretical Astrophysics, 60 St. George Street, Toronto, ON, M5S 3H8, Canada 8 October 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "From the discovery of 'hot Jupiters' (e.g. 51 Peg b, Mayor & Queloz 1995) to long-period giant planets (e.g. HR 8799b, Marois et al. 2008), the wide range of observed exoplanet orbital radii suggest that orbital migration due to gaseous disc-satellite interaction may play an important role in planet formation theory. Since its initial development (Goldreich & Tremaine 1979, 1980), disc-satellite interaction has been studied with the inclusion of increasingly complex physics. For a review of the theory and recent advancements, see Kley & Nelson (2012) and Baruteau & Masset (2013). A less well-explored area is the interaction between a planet and large-scale instabilities in the disc. For example, recent works have shown that disc gaps induced by a planet can be dynamically unstable under appropriate conditions (Koller et al. 2003; Li et al. 2005; Lin & Papaloizou 2010, 2011b). Gap-opening requires a sufficiently massive planet (Lin & Papaloizou 1986) and/or low viscosity disc (Rafikov 2002; Dong et al. 2011; Duffell & MacFadyen 2013). In a self-gravitating disc, planet gaps may become gravitationally unstable (Meschiari & Laughlin 2008; Lin 2012b) even if the initial disc is Toomre stable (Toomre 1964). The result of this instability is the development of large-scale spiral arms associated with the gap edge, which exert significant torques on the planet (Lin & Papaloizou 2011a). This gravitational edge instability, and its impact on orbital migration, has been less appreciated. However, it may be rele- vant to planet formation theories requiring a self-gravitating disc (e.g., Boss 1997; Nayakshin 2010, 2013). We remark that in the 'tidal downsizing' theory, Nayakshin discusses gap-opening in massive discs, a situation that we consider in this work. In a previous study, Lin & Papaloizou (2012, hereafter LP12) simulated the orbital migration of a gap-opening giant planet in self-gravitating discs which became gravitationally unstable only in the presence of a planet gap. They found a gravitationally unstable outer gap edge induced outward orbital migration. LP12 fixed the planet mass in their simulations. However, since the instability is associated with the gap, and the gap structure depends on the planet mass, we expect the gravitational stability of planet gaps to also depend on the planet mass. The present study is a natural follow-up to LP12 in which we investigate the role of planet mass. This paper is organized as follows. In the next subsection, we review the basic properties of the gravitational instability associated with planet gaps, and further explain the motivation for our study. We describe our disc-planet models and numerical methods in § 2. We present results for fixedorbit simulations in § 3 and freely-migrating simulations in § 4. We show it is possible to for edge modes to counter-act inward migration, but not indefinitely. We find the unstable gap edge easily triggers rapid outward migration. An example of this phenomenon is discussed in § 5. We summarize in § 6 with a discussion of possible applications of our results to giant planets on wide orbits, and important caveats of our models.", "pages": [ 1, 2 ] }, { "title": "1.1 Gravitational instability of planet gaps", "content": "Gaps induced by giant planets are associated with extrema in the disc potential vorticity (PV) profile η = κ 2 / 2ΩΣ, where κ is the epicycle frequency, Ω is the rotation rate and Σ is the surface density. It is well-known that the presence of PV extrema permits dynamical instability (e.g. Papaloizou & Pringle 1985; Li et al. 2000; Lin & Papaloizou 2010). In the case of a planet gap, local max( η ) and local min( η ) results from PV generation and destruction across spiral shocks induced by the planet (Koller et al. 2003; Li et al. 2005; Lin & Papaloizou 2010). The PV profile of a planet gap resembles its Toomre parameter profile Q = c s κ/πG Σ (Lin & Papaloizou 2011a), where c s is the sound-speed and G is the gravitational constant. Thus, we may also associate planetary gap instabilities with max( Q ) or min( Q ). Fig. 1 shows typical Q -profiles for gaps induced by giant planets. Only the outer disc is shown since this is where instability is most prominent in our models. The horizontal axis is plotted in units of Hill radii r h away from the planet's orbital radius (defined later). The co-orbital region for massive planets, within which fluid particles execute horseshoe turns upon encountering the planet, is approximately within 2 . 5 r h of the planet's orbital radius (Artymowicz 2004b; Paardekooper & Papaloizou 2009; Lin & Papaloizou 2010). So the local max( Q ) is located just inside the co-orbital region, while the local min( Q ) is located just outside. In a self-gravitating disc, Lin & Papaloizou (2011a) showed that there is a gravitational instability associated with the local Q -maximum at a planetary gap edge 1 . By association we mean that the co-rotation radius r c of the unstable mode coincides with or is close to that of the PV or Q maximum. The local max( Q ) is situated just inside the gap edge (Fig. 1). Thus, the instability presents nonaxisymmetric disturbances in the planet's co-orbital region. In their disc models where the instability is associated with the outer gap edge, LP12 found the passage of an associated spiral arm leads to a net positive torque applied to the planet, because the spiral arm supplies material to execute inward horseshoe turns upstream of the planet. This interaction is explicitly illustrated in Fig. 8 of LP12. Fig. 1 shows that the gap profile is a function of planet mass. It is not obvious how this affects the positive torques due to the gap gravitational instability described above. Increasing the planet mass makes the gap edges sharper, which should favour instability and increase spiral mode amplitudes, and lead to stronger torques. On the other hand, a larger planet mass opens a deeper gap with lower surface density (reflected by the larger Q -value in Fig. 1). The latter effect should decrease the magnitude of disc-on-planet torques originating from the co-orbital region. The purpose of this paper is to clarify, through hydrodynamical numerical experiments, the role of planet mass on the gravitational stability of gaps and the subsequent orbital migration due to the interaction with the instability. Lin & Papaloizou (2011a) found that Saturnian mass planets ultimately migrated inward, despite brief phases of outward migration induced by the gap edge instability. However, LP12 simulated 2-Jupiter mass planets and found sustained outward migration. It is of interest to examine the possibility suggested in LP12: zero net migration due to a balance between the outward migration induced by the instability and the tendency for inward type II migration (Lin & Papaloizou 1986).", "pages": [ 2 ] }, { "title": "2 DISC-PLANET MODEL", "content": "We consider a two-dimensional (2D) gaseous self-gravitating protoplanetary disc of mass M d orbiting a central star of mass M ∗ . Embedded within the disc is a planet of mass M p . We use ( r, φ ) plane polar co-ordinates centered on the star. The disc-planet model is the same as that in LP12, but we describe it here for ease of reference. We adopt units such that G = M ∗ = 1. The disc has radial extent r ∈ [ r i , r o ] = [1 , 25]. The strength of disc self-gravity is characterized by specifying Q k 0 ≡ Q k ( r o ), where is the Keplerian Toomre parameter for thin discs. In Eq. 1, c iso is the isothermal sound-speed defined below, Ω k ( r ) = √ GM ∗ /r 3 is the Keplerian orbital frequency. For all our simulations, the disc surface density is initialized with Σ( r ) ∝ r -3 / 2 for r /greatermuch r i , such that Q k 0 = 1 . 5. Our disc model is gravitationally stable to axisymmetric perturbations according to the Toomre criterion (Toomre 1964). It should be emphasised that the outcome of the simulations depends on the initial Q k 0 . Our choice of Q k 0 = 1 . 5 favours the instability-induced outward migration as seen in LP12 (where outward migration slows down for higher Q k 0 , and is not observed within their simulation timescale of 100 P 0 for Q k 0 = 2). The surface density normalization Q k 0 = 1 . 5 gives a disc mass of M d = 0 . 08 M ∗ . This is ∼ 8 times larger than the traditional minimum-mass Solar nebula (MMSN, Weidenschilling 1977), but comparable to the more massive MMSN constructed by Desch (2007). We adopt a locally isothermal equation of state so that the vertically integrated pressure p = c 2 s Σ. Without the planet the sound-speed c s = c iso ≡ hr Ω k , where h is the disc aspect-ratio. We fix h = 0 . 05. The sound-speed is modified close to the planet when it is introduced (see § 2.1.2). We also impose a constant kinematic viscosity ν = 10 -5 r 2 i Ω k ( r i ). This corresponds to an alpha-viscosity of order 10 -3 , which is typical for disc-planet simulations.", "pages": [ 3 ] }, { "title": "2.1 Planet configuration", "content": "The main parameter that we vary is the planet mass M p ≡ qM ∗ , where q is the planet-to-star mass ratio. We consider q ∈ [0 . 3 , 3] × 10 -3 , but will be primarily interested in cases with q /similarequal 10 -3 . If M ∗ = M /circledot then q = 0 . 001 corresponds to a Jupiter-mass planet and q = 0 . 0003 corresponds to a Saturn-mass planet. The position of the planet is denoted r p = ( r p , φ p ). The planet is introduced on a circular orbit of radius r p 0 = r p ( t = 20 P 0 ) = 10 where P 0 = 2 π/ Ω k ( r p 0 ). This corresponds to Q k ( r p 0 ) = 2 . 77, and the mass within | r -r p 0 | /lessorsimilar 2 . 5 r h is initially /similarequal 10 M p for q = 10 -3 . The planet mass is ramped up from zero to its full value over 10 P 0 . Thus the planet is fully introduced into the disc by t = 30 P 0 . Orbital migration is allowed for t > 30 P 0 , if at all. The planet's gravitational potential is softened with a softening length /epsilon1 p = 0 . 6 H . Accretion onto the planet is neglected, but since we compute the full self-gravity of the gas, material gravitationally bound to the planet effectively increases its mass.", "pages": [ 3 ] }, { "title": "2.1.1 Gap opening criteria", "content": "Crida et al. (2006) showed that gap opening by a planet depends specifically on its mass, the disc scale-height H = hr , and the viscosity of the disc. Their criterion for gap opening as a function of q , H , and ν is where r h = ( q/ 3) 1 / 3 r p is the Hill radius of the planet and Ω p ≡ Ω( r p ). This criteria is useful in determining which of our parameter survey values q , are able to induce a gap in the disc. Specifically, we can solve for the critical gap-opening mass, q c for which the left-hand side of Eq. 2 equals unity. Doing so, we find that q c = 5 × 10 -4 . We therefore expect all planet masses with q > q c to open gaps. The case with q = 3 × 10 -4 is somewhat smaller than q c , but we will find even a partial gap can be gravitationally unstable.", "pages": [ 3 ] }, { "title": "2.1.2 Equation of state (EOS)", "content": "In the presence of the planet, we adopt the following prescription for the sound-speed: where Ω 2 kp = GM p /d 3 p with d 2 p = | r -r p | 2 + /epsilon1 2 p and h p is a dimensionless parameter. Note that c s → c iso as d p →∞ . Eq. 3 is taken from Pepli'nski et al. (2008a), and is used here to increase the disc temperature near the planet relative to c iso . The magnitude of this increase is controlled by h p . This temperature increase mitigates accumulation of gas near r = r p . This would occur if we set c s = c iso (implying the disc temperature is unaffected by the planet), which may lead to spurious torques arising from gas near the planet due to the diverging potential and limited resolution. Physically, we expect gas near the planet to heat up as it falls into the planet potential. The appropriate value for h p depends on detailed thermodynamics occurring near the planet, which would depend on planet mass. However, since use of this EOS in the present study is motivated by numerical considerations, we simply choose h p to ensure c s /c iso > 1 everywhere. In practice, we choose h p = 0 . 5 for all planet masses except for q = 3 × 10 -4 , for which h p = 0 . 65 was needed.", "pages": [ 3 ] }, { "title": "2.2 Numerical simulations", "content": "We evolve the disc-planet system using the FARGO code (Masset 2000; Baruteau & Masset 2008). FARGO solves the 2D hydrodynamic equations using a finite-difference scheme similar to the ZEUS code (Stone & Norman 1992), except with a modified azimuthal transport algorithm which circumvents the time-step limitation set by the inner disc boundary. The self-gravity solver is described in Baruteau & Masset (2008). When allowed to respond to disc forces, the planet's motion is integrated with a fifth-order Runge-Kutta scheme. Indirect potentials are included to account for the non-inertial reference frame. The disc indi- rect potential is not expected to play a significant role because our discs are not very massive (cf. Adams et al. 1989; Shu et al. 1990; Kratter et al. 2010a). The disc is divided in to N r × N φ cells in radius and azimuth, respectively. The radial grid is logarithmically spaced while the azimuthal grid is uniformly spaced. We use ( N r , N φ ) = (512 , 1024) for fixed-orbit simulations ( § 3) and ( N r , N φ ) = (1024 , 2048) for simulations where the planet is allowed to migrate ( § 4). In the latter case, the resolution is increased in order to resolve regions close to the planet, where co-orbital torques arise and were found to be responsible for the outward migration seen in LP12. We then apply open boundaries in the radial direction and periodic boundary conditions in azimuth. We initialize the disc azimuthal velocity v φ from centrifugal balance with stellar gravity, self-gravity and pressure forces. The initial radial velocity is v r = 3 ν/r .", "pages": [ 3, 4 ] }, { "title": "3 FIXED ORBIT SIMULATIONS", "content": "We first examine how the gap edge spiral instability ('edge modes' hereafter) and the associated disc-planet torques depend on planet mass. To focus on these issues we neglect orbital migration and hold the planet on a fixed circular orbit throughout the simulations presented in this section. It is important to keep in mind that fixed-orbit simulations suppress disc-planet torques due to orbital migration. Such torques can be expected for giant planets in massive discs undergoing type III migration (Masset & Papaloizou 2003; Pepli'nski et al. 2008a). Nevertheless, we find these numerical experiments useful to aid the interpretation of freelymigrating cases considered later.", "pages": [ 4 ] }, { "title": "3.1 Gap evolution", "content": "Edge modes are associated with local PV maxima located just inside the gap (see Fig. 1). The presence of such instabilities will therefore be reflected in the gap properties such as gap depth. This is a useful quantity to examine because edge modes have a 'gap-filling' effect. This signifies material being brought into the co-orbital region of the planet, which can subsequently provide a torque (LP12). In our disc models the Toomre stability parameter decreases with radius, which favours instability of the outer gap edge rather than the inner gap edge. We therefore focus on the gap structure in r > r p . Following LP12, we define the radius of the outer gap edge r rout > r p such that δ Σ( r out ) = 0, where is the one-dimensional relative surface density perturbation and 〈·〉 φ denotes azimuthal average. The outer gap depth is defined to be the average value of δ Σ in r ∈ [ r p , r out ] and is denoted 〈 δ Σ 〉| r out r p . This is a negative quantity because the gap is a surface density deficit, but for convenience we will refer to the magnitude of the outer gap depth simply as the gap depth. Fig. 2 shows the evolution of the gap depth for a range of planet masses q ∈ [0 . 3 , 3 . 0] × 10 -3 . Increasing q results in deeper gaps. Local self-gravity is expected to be less important for deeper gaps because the surface density is relatively low compared to Σ( t = 0). This should discourage gravitational instabilities. However, increasing q produces steeper gap edges which are expected to favour edge modes. In fact, Fig. 2 indicates the latter effect is more important, as discussed below. An important feature in Fig. 2 is that the gap depth does not decrease monotonically. This signifies instability. In some cases a 'spike' appears in the evolution where the gap fills up temporarily, which is attributed to the protrusion of edge mode spiral arms into the gap (LP12, see also Fig. 3). Thus the rate of gap-opening can be slowed down by edge modes. This is explicitly shown in Fig. 2. As a result, the gap is not completely cleared even for q = 0 . 002 ( δ Σ ∼ -0 . 7) which would be expected if no instabilities develop. We find edge modes set in earlier with increasing planet mass. The spiral arms develop at t ∼ 50 P 0 for q = 3 × 10 -4 , which is consistent with previous work (Lin & Papaloizou 2011a). For q ∈ [0 . 6 , 2 . 0] × 10 -3 the spiral arms develop at t ∼ 40 P 0 . An example of the edge modes is shown in Fig. 3. The instability is also stronger with increasing q . This is reflected in Fig. 2 by the steep increase in 〈 δ Σ 〉| r out r p at t /similarequal 40 P 0 for 10 3 q = 0 . 8 , 1 . 0 , and 2.0 (the 'spikes'). By contrast, such gap-filling effect is only just noticeable for q = 3 × 10 -4 with a slight decrease in the rate at which the gap deepens around t ∼ 50 P 0 (the small bump), i.e. when instability sets in. We observed fragmentation in the run with q = 0 . 003. In this case the spiral wake induced by the planet undergoes fragmentation. There is no well-defined gap and and large-scale coherent edge mode spirals were not identified (cf. Fig. 3). Planet-induced fragmentation of self-gravitating discs have been previously examined by Armitage & Hansen (1999) and Lufkin et al. (2004). We do not consider this fragmenting case further for fixed-orbit or migration simulations.", "pages": [ 4, 5 ] }, { "title": "3.2 Torque measurements", "content": "In this section we compare the disc-on-planet torques Γ tot measured from the above simulations. Here, we also apply an exponential envelope to taper off torque contributions from within the planet's Hill sphere. This avoids potential numerical artifacts from this region because of the low-resolution adopted, and allows us to focus on the effect of edge mode spiral arms, which typically reside in r /greaterorsimilar r p +2 r h (LP12). The previous section showed that edge modes develop earlier and are stronger with Jovian planetary gaps than sub-Jovian gaps. Fig. 4 compares the evolution of the total torque between these two cases. Oscillatory torques signify the presence of edge modes, which occurs almost immediately for q = 10 -3 , but takes longer for q = 3 × 10 -4 . We also find the average amplitude of the torque oscillations to be larger with increasing planet mass as a consequence of stronger instability. This is despite the fact that increasing q lowers the average surface density in the gap region. For q = 3 × 10 -4 , the torque is at first negative due to the planetary wakes (or differential Lindblad torque), until the point at which edge modes develop ( t ∼ 50 P 0 ). The contrast between the torque evolution before and after the onset of instability in Fig. 4 shows that edge modes significantly modify disc-planet torques. That is, the development of edge modes may render the Lindblad torques negligible compared to those provided by the edge mode spirals. Fig. 4 also shows that the amplitude of the torque oscillations is largest when the instability first sets in (it saturates later). This suggests that in their initial stages of development, edge mode spirals can give the planet a 'kick', which can influence the nature of the subsequent orbital migration. However, instantaneous torques in the presence of edge modes may be positive or negative, so it is difficult to predict the direction of migration based solely on Fig. 4. It takes ∼ 10 P 0 longer for the q = 3 × 10 -4 gap to become unstable than the q = 10 -3 gap. Although this is a small difference, for giant planets in massive discs considered in this paper, type III migration may be applicable (Masset & Papaloizou 2003), but is suppressed in fixed-orbit simulations. The timescale for type III orbital migration is on the order a few 10's of P 0 (Pepli'nski et al. 2008b). This suggests that edge modes may be irrelevant if they do not develop early on, because type III migration could occur before edge modes develop. We confirm this for sub-Jovian planets in § 4.", "pages": [ 5 ] }, { "title": "3.2.1 Time-averaged torques", "content": "Here, we compute time-averaged torques in order to remove the oscillatory behaviour observed above and compare discplanet torques for a range of q with unstable gaps. We define the running time-averaged torque as and we set t 0 = 30 P 0 . We plot the running time-averaged torque evolution in Fig. 5 , while Table 1 compares the values of average torques at the end of the simulation. Fig. 5 shows that edge modes cause the average torque to become more positive with time. The case with q = 3 × 10 -4 has noticeably more negative torque values than cases with larger q . Increasing the planet mass, and hence the instability strength, typically results in more positive discon-planet torques (see Table 1). However, the trend is not a clean function of q . This may be due to the fact that increasing planet mass also increases the magnitude of the differential Lindblad torque, which is negative.", "pages": [ 5, 6 ] }, { "title": "3.3 Implications from fixed-orbit simulations", "content": "The above numerical experiments show that for the adopted disc parameters, gaps opened by giant planets in selfgravitating discs are unstable to edge modes even for a Saturnian-mass planet ( q = 3 × 10 -4 ), which is expected to open a partial gap at most (over sufficiently long timescales). However, edge modes develop later with decreasing q . The overall effect of increasing q is to increase the strength of the edge mode instability. This is despite the increased tidal torque lowering the gap surface density, an effect that should discourage gravitational instabilities. This effect is, however, outweighed by the increased sharpness of gap edges due to increased q , which favour edge modes. Given that the edge mode-modified torques oscillate rapidly between positive and negative values, the simulations here cannot be used to predict the direction of orbital migration in the presence of edge modes. Nevertheless, we expect inwards migration to be become less favourable with increasing q based on time-averaged torques, which display a more positive torque with increasing planet mass.", "pages": [ 6 ] }, { "title": "4 SIMULATIONS WITH A FREELY MIGRATING PLANET", "content": "In this second set of calculations, we allow the planet to respond to the disc forces after its potential is fully introduced. These simulations permit type III migration, for which the torque results from fluid crossing the planet's orbital radius and the torque magnitude is proportional to the migration rate (Masset & Papaloizou 2003). Such torques originate close to the planet. Accordingly we double the grid resolution in both directions from the previous experiments ( § 2.2). We examine cases with 10 3 q = 0 . 3 , 0 . 8 , 1 . 0 , 1 . 3 , 1 . 5 and 2.0. Our specific aim is to see whether or not it is possible to balance the tendency for inward migration with the positive torques induced by the edge modes. We find the resulting migration is broadly consistent with torque measurements made above. In particular, only planet masses that induce strong instability during gap formation experience the effect of positive torques brought upon them by edge modes. Otherwise, the planet falls in on dynamical timescales. In fact, only for the cases q /greaterorequalslant 1 . 3 × 10 -3 did we find the situation originally envisioned for our study: a planet residing in a gap with large-scale edge mode spiral arms at the edge over long ( ∼ 100 P 0 ) timescales (such as Fig. 3). We will focus on such cases later, but first briefly review the smaller planet mass runs.", "pages": [ 6 ] }, { "title": "4.1 Inward migration of sub-Jovian mass planets", "content": "We find rapid inward migration for planet masses less than about Jovian ( q /lessorsimilar 10 -3 ). Fig. 6 shows two examples with q = 3 × 10 -4 and q = 8 × 10 -4 . This case undergoes inward migration after release, and falls in within ∼ 15 P 0 . This timescale is consistent with type III migration (Masset & Papaloizou 2003; Pepli'nski et al. 2008b). The occurrence of type III migration for q = 3 × 10 -4 is further evidenced in Fig. 7 where we plot the surface density during the inward migration. The front-back density asymmetry reflects strong co-orbital negative torques originating from fluid crossing the planet's orbital radius by executing outward horseshoe turns from the inner disc ( r < r p ) to the outer disc ( r > r p ) behind the planet ( φ < φ p ). Note the inward migration timescale for q = 3 × 10 -4 is comparable to that needed for edge modes to develop when the planet was held on fixed orbit. However, the planet migrates significantly before a sufficiently deep gap can be opened to induce instability. Therefore edge modes are irrelevant in this case.", "pages": [ 6, 7 ] }, { "title": "4.1.2 q = 8 × 10 -4", "content": "The orbital evolution for q = 8 × 10 -4 displays a more complicated behaviour. In this case we found a disturbance at the outer gap edge has already developed at planet release. It caused the planet to further interact with the outer gap edge, scattering fluid inward, resulting in rapid outward migration on a timescale of ∼ 10 P 0 . Fig. 8 shows snapshots during the initial outward migration. The increase in r p during this phase is ∼ 6 Hill radii (measured at the initial orbital radius of r = 10). So the planet is 'kicked out' of its original co-orbital region. However, we find the planet eventually undergoes inward type III migration after reaching a maximum orbital radius of r /similarequal 1 . 4 r p 0 . This may be because the edge dis- turbance responsible for the initial kick becomes ineffective as the planet has migrated out of its gap. Negative Lindblad torques then initiated inward type III migration. The outward-inward migration seen here is similar to that observed in Lin & Papaloizou (2011a) where a planet scatters off an edge mode spiral arm.", "pages": [ 7 ] }, { "title": "4.2 Outward migration of Jovian planets", "content": "We find planet masses with q /greaterorsimilar 10 -3 induced sufficiently strong edge instabilities early on to counter-act the tendency for inward migration (Specifically to that of inward type III migration as observed for q = 3 × 10 -4 .). Several examples of such cases are shown in Fig. 9. The case with q = 2 × 10 -3 was considered in LP12 and is reproduced here for reference (with a longer simulation time). Simulations with 10 3 q = 1 . 3 , 1 . 5 were performed to explore the possibility of a torque balance to achieve zero net migration. We see from the figure that this is not possible, and we discuss this issue in more detail later. Fig. 9 shows that, for q /greaterorequalslant 0 . 0013, orbital migration proceeds, on average, outward more rapidly with increasing planetary mass. The run with q = 10 -3 does not fit this trend, however. We examine these cases separately below.", "pages": [ 7 ] }, { "title": "4.2.1 q = 10 -3", "content": "In this case the planet experiences an initial kick similar to q = 0 . 0008 in the previous section. However, with q = 0 . 001 the initial outward migration in t = [30 , 35] P 0 corresponds to about 1.4 Hill radii (at r = 10), implying the planet remains in its gap afterwards. This is shown in the left panel of Fig. 10. The smaller kick may be due to the increased planet inertia: the fluid mass within the planet's Hill sphere M h plus M p is 0 . 0017 M ∗ for q = 0 . 001 and 0 . 0012 M ∗ for q = 0 . 0008 (at planet release). The planet, having remained inside the gap, does not experience the rapid inward type III migration observed previously. However, we find the outer disc ( r > r p ) became highly unstable and dynamic (including transient clumps), as shown in the right panel of Fig. 10. We suspect this is attributable to the passage of the spiral arm upstream of the planet (at r = r p +2 . 5 r h and φ -φ p = 0 . 5 π ) across the planet-induced outer wake. We did not identify large-scale edge mode spiral arms to persist after the initial outward kick, because it led to a large increase in the effective planet mass ( M h ∼ 0 . 0037 M ∗ by t = 40 P 0 ), which strongly perturbs the outer disc directly through the planet-induced wake (becoming unstable itself). The planet nevertheless migrates outwards secularly because the inner gap edge is on average closer to the planet than the outer gap edge.", "pages": [ 8 ] }, { "title": "4.2.2 q /greaterorequalslant 1 . 3 × 10 -3", "content": "For these cases, Fig. 9 displays outward migration on two distinct timescales: a relatively slow phase over several 10's of P 0 followed by an outward kick towards the end of the simulation. The mechanism responsible for the first phase was described in LP12. In summary, the passage of outer edge mode spirals by the planet supplies material to execute inward horseshoe turns upstream of the planet. This applies a positive co-orbital torque, and overcomes the negative differential Lindblad torque on average. We find the net outward migration during this phase is roughly linear in time. As Fig. 9 shows, decreasing q and hence instability strength, the migration rate is reduced during this phase. This is de- spite the expectation that lowering q should permit higher surface densities within the gap. The second phase of outward migration is very fast and almost monotonic. For q = 0 . 002, r p increases by about 30% in t ∈ [110 , 120] P 0 . For q = 0 . 0013 and 0 . 0015, r p increases by more than 60% over a similar timescale. We will examine this phase in more detail in the next section. The extent of outward migration during the second phase is much larger than the initial kicks observed in the previous section with smaller q . We caution that the simulation should not be trusted close to and after max( r p ) has been attained (i.e. after t ∼ 120 P 0 for q = 0 . 0013), because the planet's coorbital region approaches the outer disc boundary. It is interesting to note that, had we not simulated the case with q = 0 . 0013 beyond t /similarequal 100 P 0 , the first phase would suggest little net migration. A snapshot is shown in Fig. 11. However, it still enters the second phase due to interaction with the gap edge as discussed next. Thus, it appears unlikely that a state may be reached in which positive torques due to edge modes balance against the negative differential Lindblad torques, keeping the planet at a fixed orbital radius (which was hypothesized in LP12).", "pages": [ 8 ] }, { "title": "5 RAPID OUTWARD TYPE III MIGRATION INDUCED BY EDGE MODES", "content": "A common feature in simulations with a freely migrating planet with mass q /greaterorequalslant 0 . 0013 is the second phase of rapid outward migration. For reference, we show the orbital evolution for q = 0 . 0013 in Fig. 12. The semi-major axis a and eccentricity e were calculated assuming a Keplerian orbit about the central star. The orbit remains fairly circular ( e /lessorsimilar 0 . 1) and maintains a /similarequal 10 for t /lessorsimilar 80 P 0 , but experiences a strong outward kick at t ∼ 100 P 0 . This kick appears almost spontaneously. To understand its origin, we examine the fiducial case q = 0 . 0013 in more detail in the proceeding section. This simulation was also performed at a lower resolution ( N r × N φ = 512 × 1024), which exhibited the same qualitative behaviour as the high resolution ( N r × N φ = 1024 × 2048) results discussed here. We remark that the characteristic kick may also occur in less massive discs. Specifically, we performed low resolution simulations of a q = 0 . 0013 planet migrating in discs with Q k 0 = 1 . 7 , 2 . 0. In a disc with Q k 0 = 1 . 7 the planet gets kicked outwards at t ∼ 110 P 0 similar to the fiducial case ( Q k 0 = 1 . 5 at low resolution) but at a slightly later stage in the planet's evolution. In a disc with Q k 0 = 2 . 0 the planet remains at r p ∼ 10 for the length of the simulation ( t ∈ [30 , 150] P 0 ). This is similar to the behaviour of our fiducial case prior to the characteristic kick, which suggests the q = 0 . 0013 planet in the Q k 0 = 2 . 0 disc may eventually experience the same kick evident in Fig. 12.", "pages": [ 9 ] }, { "title": "5.1 Interaction with the outer gap edge", "content": "We first show that the kick is associated with the planet migrating into the outer gap edge. We define the dimensionless outer gap width w , out as where we recall r out is the radius of the outer gap edge ( § 3.1). This definition is only valid when the planet resides in the gap ( w out > 0). The outer gap width leading up to the kick is plotted in Fig. 13, along with r p ( t ). For t /lessorsimilar 110 P 0 , w out oscillates on orbital time-scales with an average value of 4. The oscil- lations are due to the periodic passage of non-axisymmetric over-densities associated with the outer gap edge mode by the planet. There is a tendency for inward migration due to negative Lindblad torques except when an edge mode spiral is approaching the planet from upstream, by supplying some material to execute inward horseshoe turns. It is evident from the outward migration shown in Fig. 13 that the positive torques supplied by the edge mode spiral overcome negative torques on average. The planet migrates outward with respect to the outer gap edge, so w out decreases. Notice w out stops oscillating after t /similarequal 115 P 0 , and the kick commences. At this point w out /similarequal 2. That is, the kick occurs when the outer gap edge enters the co-orbital region of the planet. (Recall the co-orbital region of a giant planet is | r -r p | /lessorsimilar 2 . 5 r h .) Afterwards r p rapidly increases and w out → 0, implying the planet has exited the gap. Fig. 14 shows the interaction between the planet and the outer gap edge during this kick. In this case, rather than passing by the planet, the bulk of the edge mode overdensity executes an inward horseshoe turn upstream of the planet, i.e. the planet scatters fluid comprising the outer gap edge inwards. The resulting azimuthal surface density asymmetry about the planet is that which characterizes outward type III orbital migration (Pepli'nski et al. 2008b). An under-dense(over-dense) region behind(ahead of) the planet implies a strong positive co-orbital torque. This configuration persists until the planet migrates close to the outer disc boundary. We conclude that the outer edge mode spiral arms act as a natural 'trigger' for outward type III migration.", "pages": [ 9 ] }, { "title": "5.2 Growth of effective planet mass", "content": "We also observe a significant increase in the effective planet mass after the planet enters outward type III migration. We define M /epsilon1 as the fluid mass contained within a radius /epsilon1r h of the planet. For /epsilon1 = 1, M /epsilon1 equals M h defined previously. Note that M /epsilon1 includes fluid gravitationally bound to the planet and orbit-crossing fluid, the latter being responsible for type III migration. In Fig. 15 we plot M /epsilon1 /M p for /epsilon1 = 0 . 3 , 0 . 5 , and 1 . 0. For t /lessorsimilar 110 P 0 there is negligible mass contained within the planet's Hill radius. This mass rapidly increases as the planet interacts with the edge mode spiral arm ( t /similarequal 115 P 0 ). Notice M /epsilon1 /greaterorequalslant 0 . 5 increases more rapidly than M /epsilon1 =0 . 3 , suggesting a significant flux of orbit-crossing fluid. The planet migrates to a maximum orbital radius max( r p ) /similarequal 18 at t /similarequal 120 P 0 . At this point M /epsilon1 =0 . 3 /similarequal 0 . 6 M p and the fluid within the Hill radius exceeds the planet mass, M h /similarequal 1 . 3 M p . The rapid increase in planet inertia may attribute to stopping type III migration (Pepli'nski et al. 2008c). However, since the planet's co-orbital region approaches the disc boundary ( r p +2 . 5 r h /greaterorsimilar 0 . 85 r out ), boundary conditions may come into effect (e.g., the lack of disc mass exterior to the planet to sustain type III migration). Nevertheless, we expect the effective planet mass to generally increase if it undergoes outward type III migration because the Hill radius r h scales with orbital radius. In this sense, edge modes can indirectly increase the effective planet mass.", "pages": [ 9, 10 ] }, { "title": "6 SUMMARY AND DISCUSSION", "content": "We have performed hydrodynamic simulations of gapopening satellites (planets) in self-gravitating discs. Our aim was to examine the role of planet mass on the development of gravitational instability associated with the gap, and its subsequent effect on orbital migration. We first considered simulations where the planet was held on a fixed orbit. Outer gap edge modes developed for planet-to-star mass ratios q ∈ [0 . 3 , 2 . 0] × 10 -3 . Despite being a gravitational instability, we find edge modes developed earlier and are stronger with increasing q , which corresponds to deeper gaps with lower surface density. This is because edge modes are fundamentally associated with potential vorticity maxima resulting from planet-induced spiral shocks, which become sharper with increasing planet mass (Lin & Papaloizou 2010). We also find disc-on-planet torques typically become more positive with increasing q . This is consistent with positive torques being supplied by the outer gap edge mode spirals (LP12). However, in simulations where orbital migration is allowed, we found edge modes were only relevant for planet masses that opened an unstable gap early on. This is because the condition required for edge modes to develop - a massive disc - also favours type III migration, which operates on dynamical timescales. We found for q = 0 . 0003, the planet immediately underwent inward type III migration, having no time to open a gap. This initial inward type III migration could be avoided due to initial conditions (Artymowicz 2004a). Pepli'nski et al. (2008c) demonstrated outward type III migration of giant planets when they are initially placed at the edge of an inner cavity. Thus, an appropriate choice of initial surface density profile may prevent a Saturnian mass planet ( q = 0 . 0003) to undergo immediate rapid inward migration as seen in our simulations, and allow it to remain at approximately the same orbital radius for at least ∼ 20 orbits. Then edge modes will become relevant, because our fixedorbit simulations indicate that even the partial gap opened by q = 0 . 0003 is unstable. For q /greaterorequalslant 0 . 0013 we find net outward migration with an increasing rate with planet mass and therefore gap instability. Although we found it was possible, by decreasing q , to achieve almost zero net migration, this phase only lasted a few 10's of orbital periods. The planet eventually interacted with an outer gap edge mode spiral and underwent outward type III migration. We conclude that the scenario hypothesized by LP12 - inwards type II migration begin balanced by the positive torques due to outer gap edge mode spirals -is a configuration that is unlikely to persist beyond a few tens of orbital periods.", "pages": [ 10, 11 ] }, { "title": "6.1 Relation to previous studies", "content": "Recent hydrodynamic simulations which focus on the orbital migration of giant planets in massive discs have been motivated by the possibility of planet formation through gravitational instability of an initially unstructured disc (Boss 2005, 2013; Baruteau et al. 2011; Michael et al. 2011). Such disc models are already gravitationally unstable without a planet. These studies employed models in which the planet does not significantly perturb the gravito-turbulent disc, and no gap is formed. By contrast, the gravitational instability in our disc model is caused by the planet indirectly through gap formation. Detailed disc fragmentation simulations have shown that, while most clumps formed through gravitational instability are lost from the system (e.g. by rapid inward migration), in some cases it was possible to form a gap-opening clump (Vorobyov & Basu 2010; Vorobyov 2013; Zhu et al. 2012). These authors did not specifically examine gap stability, but we note several interesting results that may hint the presence of edge modes. In Vorobyov & Basu (2010), a clump is seen to migrate outward. Just prior to this outward migration, their Fig. 1 shows a spiral arm upstream to the planet, and is situated at a radial boundary between low/high disc surface density (but a gap is not yet well-defined). This spiral arm may have contributed to a positive torque on the clump. Their Fig. 1 also suggest an eccentric gap. This is probably due to clump-disc interaction (Papaloizou et al. 2001; Kley & Dirksen 2006; Dunhill et al. 2013), but eccentric gaps have also been observed in disc-planet simulations where edge modes develop and saturate (Lin & Papaloizou 2011a, Fig. 12). Vorobyov (2013) improved upon Vorobyov & Basu (2010) by including detailed thermodynamics. No edge mode spirals are visible from their plots. However, they found that in all cases where a gap-opening clump is formed, the clump migrates outward. The origin of the required positive torque was not identified. It is conceivable that destruction of the outer gap edge, perhaps due to the edge spiral instability, could result in the clump being on average closer to the inner gap edge than the outer gap edge. This will lead to outwards type II migration. Zhu et al. (2012) also simulated the fragmentation of massive discs with realistic thermodynamics. In one case where a gap-opening clump formed, further clump formation was observed at the gap edge. The gap-opening clump first migrated outward, but ultimately falls in. This outward migration may be due to the interaction with the unstable gap edge, similar to our simulations.", "pages": [ 11 ] }, { "title": "6.2 Giant planets on wide orbits", "content": "Our results have important implications for some models seeking to explain the observation of giant planets on wide orbits. Such examples include the 4 giant planets orbiting HR 8799 between 15-68AU (Marois et al. 2008, 2010), Fomalhaut b at 115AU from its star (Kalas et al. 2008; Currie et al. 2012), and AB Pic b at 260AU(Chauvin et al. 2005). Disc fragmentation has been proposed as an in situ mechanism to form long-period giant planets (Dodson-Robinson et al. 2009; Boss 2011; Vorobyov 2013). Vorobyov showed that this was possible if the clump opened a gap to avoid rapid inward migration (Baruteau et al. 2011). Then gravitational gap stability becomes an issue that should be addressed. Our results indicate an unstable gap can help to prevent inward migration, but there is the danger that it may scatter the planet if edge modes persist. The gravitational edge instability is therefore a potential threat to clump survival; in addition to other known difficulties with the disc fragmentation model (see Kratter et al. 2010b). On the other hand, our simulations reveal a way for a single giant planet to migrate outward, by opening a gravitationally unstable gap and letting it trigger rapid outward type III migration. Our simulations indicate that such outward migration will increase the planet's effective mass, which may contribute to a circumplanetary disc. Indeed, circumplanetary discs associated with planets on wide orbits have been observed (Bowler et al. 2011). In our models it is not clear whether the rapid increase in effective planet mass discussed in § 5.2 could slow down or halt the triggered rapid outward type III migration. We cannot address this possibility because of the finite disc domain in our models. Conversely, type III migration is self-sustaining (Masset & Papaloizou 2003), so giant planets that underwent type III migration, initiated by this 'trigger' mechanism, could be found at large orbital radii.", "pages": [ 11, 12 ] }, { "title": "6.3 Caveats and future work", "content": "Our study is subject to several caveats that should be clarified in future work: Initial conditions. Our simulations with q = 0 . 0008 and q = 0 . 001 displayed very different results for orbital migration, despite having similar planet mass. We identified the planet to interact with a disturbance at the outer gap edge at planet release. This initial kick provided a strong co-orbital torque. It can be interpreted as a brief phase of type III migration, which is known to depend on initial conditions. A more extensive exploration of numerical parameter space is needed to assess the importance of this initial kick. Specifically the resolution of simulations involving these particular planet masses may be important for whether or not the initial interaction between the planet and a disturbance at the outer gap edge occurs. Thermodynamics. The locally isothermal equation of state implicitly assumes efficient cooling. This favours gap formation and gravitational instabilities. We expect the gap to become gravitationally more stable if the disc is allowed to heat up. Numerical simulations including an energy equation, which adds another parameter to the problem - the cooling time - will be presented in our follow-up paper on the gravitational stability of planet gaps in non-isothermal discs. Disc geometry. The thin-disc approximation is expected to be valid for gap-opening perturbers, since their Hill radius exceeds the disc thickness by definition. Indeed, threedimensional (3D) simulations carried out by Lin (2012a) also revealed outward migration of a giant planet due to a gravitationally unstable gap. For partially gap-opening planets (e.g. the q = 0 . 0003 case in our models), 2D works less well, but our simulations indicate they nevertheless open unstable gaps. Whether or not this remains valid in 3D, needs to be addressed.", "pages": [ 12 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This project was initiated at the CITA 2012 summer student programme. RC would like to thank CITA for providing funding throughout the project and the use of the Sunnyvale computing cluster. Computations were also performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. The authors thank C. Matzner, K. Kratter and the anonymous referee for comments and suggestions.", "pages": [ 12 ] }, { "title": "REFERENCES", "content": "Adams F. C., Ruden S. P., Shu F. H., 1989, ApJ, 347, 959 Armitage P. J., Hansen B. M. S., 1999, Nature, , 402, 633 Artymowicz P., 2004a, in Astronomical Society of the Pacific Conference Series, Vol. 324, Debris Disks and the For- mation of Planets, L. Caroff, L. J. Moon, D. Backman, & E. 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[ { "title": "ABSTRACT", "content": "In this paper we present a new method to extract cosmological parameters using the radial scale of the Baryon Acoustic Oscillations as a standard ruler in deep galaxy surveys. The method consists in an empirical parametrization of the radial 2-point correlation function, which provides a robust and precise extraction of the sound horizon scale at the baryon drag epoch. Moreover, it uses data from galaxy surveys in a manner that is fully cosmology independent and therefore, unbiased. A study of the main systematic errors and the validation of the method in cosmological simulations are also presented, showing that the measurement is limited only by cosmic variance. We then study the full information contained in the Baryon Acoustic Oscillations, obtaining that the combination of the radial and angular determinations of this scale is a very sensitive probe of cosmological parameters, able to set strong constraints on the dark energy properties, even without combining it with any other probe. We compare the results obtained using this method with those from more traditional approaches, showing that the sensitivity to the cosmological parameters is of the same order, while the measurements use only observable quantities and are fully cosmology independent. Key words: data analysis - cosmological parameters - dark energy - large-scale structure of the universe", "pages": [ 1 ] }, { "title": "Precise Measurement of the Radial Baryon Acoustic Oscillation Scales in Galaxy Redshift Surveys", "content": "E. S'anchez 1 glyph[star] , D. Alonso 2 , F. J. S'anchez 1 , J. Garc'ıa-Bellido 2 , I. Sevilla 1 1 Centro de Investigaciones Energ'eticas, Medioambientales y Tecnol'ogicas (CIEMAT), Madrid, Spain 2 Instituto de F'ısica Te'orica (UAM-CSIC), Madrid, Spain 12 June 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Finding the physical origin of the accelerated expansion of the Universe is one of the most important scientific problems of our time, and is driving important advances in XXIst century cosmology. Several observational probes to study the nature of the mysterious dark energy, which powers that expansion, have been proposed. Among them, the measurement of the scale of the Baryon Acoustic Oscillations (BAO) in the galaxy power spectrum as a function of redshift is one of the most robust, since it is insensitive to systematic uncertainties related to the astrophysical properties of the galaxies. Moreover, it provides information about dark energy from two different sources: the angular diameter distance, through the measurement of the BAO scale in the angular distribution of galaxies, and the Hubble parameter, through the measurement of the BAO scale in the radial distribution of galaxies. There are some measurements of the BAO scale in the purely radial direction (Gazta˜naga et al. 2009; Kazin et al. 2010; Xu et al. 2013), but most of them use mainly the monopole of the 3-D correlation function (Eisenstein et al. 2005; Hutsi 2006a,b; Percival et al. 2007; Padmanabhan et al. 2007; Okumura et al. 2008; S'anchez et al. 2009; Anderson et al. 2012). This has been the traditional way of determining the BAO scale, trying to optimize the sensitivity when the number of galaxies in the survey is not very high, paying the price of introducing model dependence in the measurement through the use of a fiducial model. However, the new galaxy surveys which are already taking data or those proposed for the future do not suffer from this problem and new and more robust methods can be used. In this paper we propose a new method to extract the evolution of the radial BAO scale with the redshift, and explain how to use it as a standard ruler to determine cosmological parameters. We use data from galaxy surveys in a manner that is fully cosmology independent, since only observable quantities are used in the analysis and therefore, results are unbiased. A method based on the same idea for the measurement of the angular BAO scale was described in S'anchez et al. (2011) and provided the measurement of the angular BAO scale presented in Carnero et al. (2012). Here we present how to extract the radial BAO scale using a similar approach. The method is designed to be used as a strict standard ruler, and provides the radial BAO scale as a function of the redshift, but we do not intend to give a full description of the radial correlation function. This approach is more robust against systematic effects, and in fact we demonstrate that the measurement is only limited by cosmic variance, since the associated systematic errors are much smaller than the purely statistical errors.", "pages": [ 1, 2 ] }, { "title": "2 GALAXY CLUSTERING AND OBSERVABLES", "content": "One of the main statistical probes of the properties of the matter distribution in the Universe is the 2-point correlation function, ξ ( r ), which is defined as the excess joint probability that two point sources ( e.g. galaxies) are found in two volume elements dV 1 dV 2 separated by a distance r compared to a homogeneous Poisson distribution (Peebles 1980). If the fluctuations on the matter density field are Gaussian, this function contains all the information about the large scale structure of the Universe. However, what is observationally accesible is the distribution of galaxies in angle-redshift space, not directly the matter distribution in real space. For each galaxy we determine its angular position in the sky and its redshift. To obtain ξ ( r ) we need to convert the measured redshift to a comoving distance, for which a cosmological model is needed. Therefore, the 3-D correlation function is not observable in a cosmology independent way for a given galaxy survey. Moreover, the observational techniques to obtain the angular position in the sky and the redshift are completely different and independent. Consequently, if we want to keep the measurement completely free of any theoretical interpretation, we should measure, on the one hand, the angular correlation function as a function of the angular separation of galaxy pairs, and on the other hand, the radial correlation function as a function of the redshift separation of galaxy pairs, and then extract the BAO scale from each function. Both of them are defined in complete analogy with the 3D function, as the excess of probability with respect to a homogeneous Poisson distribution, and both are observable in a cosmology independent way, providing, together, information about the full 3-D large scale structure that can be described in any cosmological model. These aspects have been recently pointed out in Bonvin & Durrer (2011), and here we present a practical implementation of a cosmology independent measurement, in this case for the radial BAO scale. Several effects must be taken into account when the large scale structure of the Universe is studied from galaxy surveys, like bias, redshift space distortions and non-linear corrections. In this paper we take all of them into account, both in the theoretical calculations and in the observational analysis. For this particular measurement, the influence of the non-linear effects is relatively small, since we are analyzing large scales, around the BAO scale. Also the influence of bias is small, because the analysis method is designed to minimize its impact on the final result. However, redshift space distortions are very important, since we are measuring along the line of sight.", "pages": [ 2 ] }, { "title": "3 THE SOUND HORIZON SCALE", "content": "The BAO are a consequence of the competition between gravitational attraction and gas pressure in the primordial plasma, which produced sound waves. Differences in density created by these sound waves leave a relic signal in the statistical distribution of matter in the Universe, defining a preferred scale, the sound horizon scale at the baryon drag epoch, which is a robust standard ruler from which the expansion history of the Universe can be inferred. The BAO scale is very large, ∼ 100 Mpc/h, which poses an important challenge for observations, since surveys must cover large volumes to map such a distance. However, the very large scale of BAO also has certain advantages, because structure formation at these scales is rather well understood, and details about the description of galaxy formation and astrophysics do not compromise the accurate measurements of the standard ruler. In practice, the strength of the BAO standard ruler method relies on the potential to relate the position of the acoustic peak in the correlation function of galaxies to the sound horizon scale at decoupling. The current measurements of the BAO scale typically use the spherically symmetric monopole contribution of the 3-D correlation function. This is a mixture of the angular and radial scales, and therefore, does not contain all the information that can be extracted from the data samples. On top of that, all analyses have been done using what is usually called a 'fiducial model' in order to determine distances. Rather than full tests of the cosmology, they should be understood as precision measurements within the context of the standard ΛCDM cosmological model. Considering this way of analyzing data, what the current results have achieved is a strong confirmation of the consistency of the current data with the standard ΛCDM model, but the analysis strategy is very limited if one wants to test other non-FLRW cosmological models against data, or at least, the whole analysis must be repeated from the very begining for every cosmological model one tries to test. On the other hand, if both the angular and the radial BAO scales are determined, which in principle is already possible with current data sets, the cosmological constraints that can be set using only the BAO scale are much stronger than the current values, since the radial and angular scales are sensitive to cosmological parameters in quite different ways, breaking degeneracies and allowing a very precise determination when combined.", "pages": [ 2 ] }, { "title": "4 METHOD TO MEASURE THE RADIAL BAO SCALE", "content": "The method we propose relies on a parametrization of the radial correlation function which allows extracting the radial BAO scale with precision. It is based on a similar approach previously proposed for the angular correlation function (S'anchez et al. 2011). The suggested parametrization was inspired by the application of the Kaiser effect to the shape we used for the angular correlation function. It is important to remark that the calculations we perform to check the goodness of the parametrization do include redshift space distortions using the Kaiser description (Kaiser 1987) and also non-linearities using the RPT approach (Crocce & Scoccimarro 2006), and we expect the result to be correct for large scales, where the neglected effects (fingers of god, mode-mode coupling) are very small. We have used a bias parameter b = 1 throughout the calculation. The possible influence of bias is studied in the systematic errors section. We use the linear power spectrum from CAMB (Lewis et al. 2000), on top of which non-linearities and redshift space distortions are then included. The full recipe to obtain the radial BAO scale as a function of redshift for a galaxy survey is as follows: and perform a fit to ξ ‖ (∆ z ) with free parameters A , B , C , D , E , F , ∆ z BAO and σ . (v) The radial BAO scale is given by the parameter ∆ z BAO . The BAO scale as a function of the redshift is the only parameter needed to apply the standard ruler method. The cosmological interpretation of the other parameters is limited, since this is an empirical description, valid only in the neighborhood of the BAO peak. (vi) Fit cosmological parameters to the evolution of ∆ z BAO with z . In order to test the goodness of this parametrization, we have computed the radial correlation function for the 14 cosmological models described in Table 1 in a redshift range from 0.2 to 1.5, always including redshift space distortions and non-linearities as explained before. Then, we have applied the method to each model and each redshift. The parametrization describes the theory very accurately, since the values of the χ 2 /ndof are close to 1, and the probabilities of the fit lie between 0.98 and 1.00 when the error in each point of the correlation function is arbitrarily fixed to 2% (or ∼ 10 -5 ) for all bin widths and cosmological models. This error corresponds to a precision much better than the cosmic variance for the full sky, for all models in the full redshift range, but the parametrization is able to recover the correct radial BAO scale for the 14 cosmologies. We have used this level of precision because the systematic errors coming from theoretical effects (non-linearities, bias, fingers of god) only affect this method if they induce a disagreement between the models and the parametrization, giving then a wrong measurement of the BAO scale. If the description is good to this level of precision we guarantee a small contribution from these systematic effects, as will be shown in the systematic errors section below. The calculation of the realistic errors is described in the next section. Some examples of these descriptions can be seen in Figure 1. Once we have verified the parametrization on the theoretical predictions, we are ready to apply it in more realistic environments. For this purpose, in the next section we apply this algorithm to a galaxy catalogue obtained from a N-body cosmological simulation. We will also compute the main systematic errors associated to this method.", "pages": [ 2, 3 ] }, { "title": "5 APPLICATION TO A SIMULATED GALAXY SURVEY", "content": "We have tested the method to recover the radial BAO scale using a large N-body simulation capable of reproducing the geometry (e.g. area, density and depth) and general features of a large galaxy survey. The simulated data were kindly provided by the MICE project team, and consisted of a distribution of dark matter particles (galaxies, from now on) with the cosmological parameters fixed to the fiducial model of Table 1. The redshift distribution of the galaxies is shown in Figure 2. The simulation covers 1/8 of the full sky (around 5000 square degrees) in the redshift range 0.1 < z < 1.5, and contains 55 million galaxies in the lightcone. This data was obtained from one of the largest N-body simulations completed to date 1 , with comoving size L box = 3072 h -1 Mpc and more than 8 × 10 9 particles ( m p = 2 . 3 × 10 11 h -1 M glyph[circledot] ). More details about this simulation can be found in Fosalba et al. (2008) and Crocce et al. (2010). The simulated catalogue contains the effect of the redshift space distortions, fundamental for the study of the radial BAO scale. Data with similar characteristics will be obtained in future large spectroscopic surveys, such as DESpec (Abdalla et al. 2012), BigBOSS (Schlegel et al. 2011) or EUCLID (Laureijs et al. 2011). It is important to note that the radial BAO determination needs a very large survey volume. We tried to extract the BAO peak from catalogs with smaller areas (200, 500 and 1000 sq-deg), finding a very small significance (or no detection at all) in most cases. This is due to the fact that the statistical error related to the cosmic variance is specially large for the radial correlation function, and therefore it can only be reduced by increasing the volume explored. Thus, we have divided the simulation into 4 redshift bins. We apply the method described in the previous section for each bin and obtain the correlation functions using the Landy-Szalay estimator (Landy & Szalay 1993). Fits are shown in Figure 3. We have used an angular pixel with of 02.5 sq-deg size, in order to retain enough number of galaxy pairs in the colliner direction. We generate the random catalogues taking into account the N ( z ) distribution in order to obtain the correct determination of the radial correlation function. This is a very important effect, since we are measuring in the radial direction and the distributions on the redshift coordinate have a large influence in measurements. The statistical significance of the BAO observation in the first bin is very low ( ∼ 1 . 4 σ ) and it is consequently not considered in the cosmology analysis. Following the same approach of S'anchez et al. (2011), we have computed the statistical significance of the detection by measuring how different from zero the F parameter of the fit is Eq. (1), using its statistical error. We have computed a theoretical estimation of the covariance matrix for the 3-D correlation function (see the Appendix A), which is then applied to the estimation of the covariance for the radial correlation function Eq. (A6), taking the corresponding value for the line of sight direction. This estimate relies on the model used for the simulation, but we expect a small variation of the error with the cosmological model. In any case, to keep the method fully model independent we have validated the calculation of the covariance matrix obtaining it with an alternative method directly from the simulation, which can also be applied to any real catalogue. We have used many realizations, dividing the total area in patches of different sizes and computing the corresponding dispersions and covariance matrices. We have then obtained the error for the total area, scaling this estimates to the full area of the measurement. Both determinations agree in the region of interest for the BAO scale measurement, as is shown in Figure 4. There is a disagreement for small scales, which is coming from the incomplete description of the non-linearities in the theoretical calculation, where the mode-mode coupling effects are neglected, and from boundary effects in the realizations. It is important to note that the radial BAO determination needs a very large survey volume. We are not able to obtain a significant observation for the first redshift bin and the contribution to the total error of the cosmic variance and the shot noise is comparable for all the other bins, which is shown in Figure 4. As already pointed out and verified by Crocce et al. (2011), in order to obtain a correct description z ∆ z ∆ D of the error we need to sum linearly the cosmic variance and the shot noise contribution in harmonic space. This is not surprising, since galaxy surveys are designed to have a determined and fixed galaxy density, and this fact correlates both contributions to the error. The measured values of the BAO scale as a function of the redshift can be seen in Figure 5 as dots. The results for 3 alternative bins shifted with respect to the nominal ones are also shown as triangles. These have not been used to obtain cosmology results, since they are fully correlated with the red ones and are only shown for illustration and verification of the correct behaviour of the analysis method. We have used in this analysis the center of the redshift bin to obtain the prediction of the model, although what is really observed is the average within the bin. We have verified that they are very close if the N ( z ) distribution is smooth, as it is in this case.", "pages": [ 3, 4, 5, 6 ] }, { "title": "5.1 Systematic Errors", "content": "We have studied the main systematic errors that affect the determination of the radial BAO scale using this method. We have found a specific systematic effect associated to the algorithm, which is coming from the size of the angular pixel. Other systematic errors are generic and will be present in any determination of the BAO scale, namely, the influence of the non-linearities, the starting and end point of the fit to the correlation function and the possible influence of the galaxy bias in the measurement.", "pages": [ 6 ] }, { "title": "5.1.1 Size of the Angular Pixel", "content": "One specific systematic error associated with this method is the possible influence that the size of the angular pixel has on the determination of the radial BAO scale, since it determines which galaxies are considered collinear in the analysis. The effect of different pixel sizes on the radial correlation function can be seen in Figure 6. The radial correlation function clearly changes on small scales (Figure 6, top), but this effect does not change the position of the BAO peak (Figure 6, bottom). The small scale effect comes from the smooth- ing of the correlation function due to the inclusion in the calculation of galaxy pairs which are not exactly collinear. For a larger angular pixel, the effect is larger. However, the scale where this effect acts is fixed by the angular pixel size, which is very far from the BAO scale, that remains, therefore, unaffected, since the parametrization is able to absorb the change in the slope of the function. In order to quantify this influence on the determination of the BAO scale as a systematic error, we have repeated the full analysis for different pixel sizes. The obtained results are shown in Figure 7. The radial BAO scale is recovered with high precision for any angular pixel size, even for sizes as large as 1 degree, which corresponds to a range of two orders of magnitude. The associated systematic error can be estimated to be δ (∆ z BAO ) = 0 . 20%, much smaller than the statistical error for the nominal pixel size of 0.25 sq-deg, which is shown as error bars.", "pages": [ 6, 7 ] }, { "title": "5.1.2 Non-linearities", "content": "Also the error due to the uncertainty in the goodness of the description provided by the parametrization for different theoretical effects (non-linearities at the scale of the BAO peak) has been computed obtaining a global error of 0.10%. This was estimated in a conservative way as the difference between the ∆ z BAO measured using linear and non-linear ξ ‖ (∆ z ), for the same redshift bins of the analysis. Nonlinearities are computed using the RPT formalism (Crocce & Scoccimarro 2006), excluding mode-mode coupling, since it only affects small scales, far enough from the BAO scale. The contribution of these uncertainties to the systematic error can be estimated as δ (∆ z BAO ) = 0 . 10%.", "pages": [ 7 ] }, { "title": "5.1.3 Galaxy Bias", "content": "At the BAO scales it is a good approximation to consider that the bias is scale independent (Cresswell & Percival 2009; Crocce et al. 2011). It affects the radial correlation function not only as an overall normalization, but also through its effect in the redshift space distortions. Bias can influence the determination of the BAO scale only through the changes in the goodness of the parametrization of the correlation function for different biases. In order to estimate the contribution of the galaxy bias to the measurement of the radial BAO scale, we have repeated the analysis with different values of the bias, to obtain the propagation of the uncertainty in the galaxy bias for the selected galaxy population to the measured value of the radial BAO scale. The influence on the peak position is small, and we can estimate the associated systematic error as δ (∆ z BAO ) = 0 . 15%. Moreover, we have tested the effect of a scale dependent bias, introducing artificially the effect in the correlation functions, using an approximate Q-model with the determination of parameters from Cresswell & Percival (2009). The bias variation with ∆ z in the fitted region of ξ ‖ (∆ z ) ranges from 1% to 6%, but the measurement of the BAO scale is insensitive to these changes. We estimate the systematic error in the presence of a scale dependent bias as δ (∆ z BAO ) = 0 . 20%.", "pages": [ 7 ] }, { "title": "5.1.4 Starting and End Point of the Fit", "content": "To compute the systematic error associated to the parametrization method, we have done some further analysis on theoretical radial correlation functions with the same bin widths and central redshifts as those used in the analysis of the MICE simulation. The error associated to the method comes from the possible influence in the obtained ∆ z BAO of the range of ∆ z used to perform the fit. To evaluate the error, we have varied this range for the 3 redshift bins where we have a significant detection of the BAO scale, and performed the fit for each range. In the decision of the range to be fitted, we have to choose a starting point at angles smaller than the BAO peak, where the physics is determined by non-linearities, and an end point after the peak, beyond which the effects of cosmic variance may be relevant. By varying these two points we can study how much the result varies with this decision. Results can be seen in Fig. 8, where the obtained ∆ z BAO is shown for different starting points and end points of the fit, for the 3 redshift bins. In all cases, the uncertainty is of the order of 0.1%, which we assign as the associated systematic error. This uncertainty is much smaller than the statistical error, including Poisson shot noise and cosmic variance, which is depicted as error bars.", "pages": [ 8 ] }, { "title": "5.1.5 Total Systematic Error", "content": "The different sources of the systematic errors are completely independent, and therefore, we can compute the total systematic error by summing quadratically these contributions, resulting on a value of δ SY S (∆ z BAO ) = 0 . 33%. There are some other potential systematic errors, the gravitational lensing magnification, which introduces a small correlation between redshift bins, or those mainly associated to the instrumental effects which could affect the used galaxy sample. However, these effects are expected to be very small and we have neglected them in this analysis.", "pages": [ 8 ] }, { "title": "5.2 Cosmological Constraints", "content": "The evolution of the measured radial BAO scale, including the systematic errors, with redshift is shown in Figure 5. The cosmological model of the simulation is the solid line. The recovered BAO scale is perfectly compatible with the true model, demonstrating that the method works. When these measurements are translated into constraints on the cosmological parameters, we obtain the results depicted in Figure 9, where the contours for 1 σ , 2 σ and 3 σ C. L. in the ( w 0 , w a ) plane are shown at the top panel and the bottom panel shows the same contours in the (Ω M , w 0 ) plane. To obtain the constraints on the cosmological parameters, we have performed a χ 2 fit to the evolution of the measured radial BAO scale with the redshift to the model, where where r S is the sound horizon scale at the baryon drag epoch and H ( z, Ω M , w 0 , w a ... ) is the Hubble parameter. We leave free those cosmological parameters which are shown in the figures, while all other parameters have been kept fixed to 1 m their values for the simulation. The cosmology of the simulation is recovered, and the plot shows the sensitivity of the radial BAO scale alone, since no other cosmological probe is included in these constraints.", "pages": [ 8 ] }, { "title": "5.3 Combination of Radial and Angular BAO Scales", "content": "We have applied the method of S'anchez et al. (2011) to determine the angular BAO scale for the same simulation. The results of this analysis are presented in the Appendix B. We have combined the results of this analysis with the radial BAO scale, using the same approach of the previous section. For the angular analysis, the BAO scale is described as where d A ( z, Ω M , w 0 , w a ... ) is the angular diameter distance. The constraints on the ( w 0 , w a ) plane coming from this determination of the angular BAO scale can be seen in Fig- ure 10 (top) as thin dashed lines, and correspond to the 1 σ , 2 σ and 3 σ C. L. contours. The sensitivity of the angular BAO scale is complementary to that of the radial BAO, shown as the thin solid lines. When combined, the contours represented by the thick solid lines are found. The same constraints for the (Ω M , w 0 ) plane are shown in the bottom panel of Figure 10. These constraints are as precise as what is usually quoted for the BAO standard ruler, which is based on the use of the monopole of the 3-D correlation function. It is important to remark that these constraints are obtained ONLY with the BAO standard ruler, independently of any other cosmological probe, which shows the real power of the standard ruler method when the full information is used. We provide also the combined result of BAO, both radial and angular, with the distance measurements from CMB using the WMAP7 covariance matrix (Komatsu et al. 2011) and assuming the measurements correspond to the MICE cosmology. The combination has been performed following the procedure as detailed in Komatsu et al. (2009). m The corresponding contours at 1 σ , 2 σ and 3 σ C. L. are presented in Figure 11 both in the plane ( w 0 , w a ) (top) and in the plane (Ω M , w 0 ) (bottom) as thick solid lines, and compared with the result using only BAO (both radial and angular), which is presented as the thin lines. As before, the other parameters have been kept fixed. There is an important improvement in the precision of the determination of the corresponding parameters in both cases, larger in the (Ω M , w 0 ) plane, showing that a precise measurement can be achieved when all the information provided by the BAO scale is included in the fit.", "pages": [ 8, 9 ] }, { "title": "5.4 Comparison with Other Methods", "content": "In order to compare our results, obtained combining radial and angular BAO scale determinations, with the standard approach of measuring the position of the sound horizon scale in the monopole of the two-point correlation func- tion, we performed on our mock catalogue the same analysis that was carried out to obtain the latest BAO detection by the BOSS collaboration (Anderson et al. 2012). For this we calculated the three-dimensional correlation function ξ ( r ) using the Landy and Szalay estimator in the three wide redshift bins used for the analysis of the radial BAO (0 . 45 < z < 0 . 75, 0 . 75 < z < 1 . 10 and 1 . 10 < z ). We chose these wide bins in order to maximize the number of pairs that contribute to the measurement of the monopole, since this is one of the key advantages of the standard method. In order to calculate ξ ( r ), redshifts must be translated into distances. We have used the true cosmology of the MICE simulation to ensure that our results will not be biased by this choice. The covariance matrix for the monopole was calculated using the Gaussian approach described by Xu et al. (2013). This calculation was, again, validated by comparing it with the errors computed from subsamples of the total catalogue, and both estimations were found to be compatible within the range of scales needed for the analysis. As is done in (Anderson et al. 2012), we fit the model to the estimated correlation monopoles. Here ξ th ( r ) is a template theoretical correlation function corresponding to the fiducial cosmological model used to translate redshifts into distances in the survey (in our case the MICE cosmology). This template was calculated from the CAMB linear power spectrum for the MICE cosmology, and corrected for nonlinearities via the RPT damping factor. We are mainly interested in the fitting parameter α , which relates real and fiducial scales: where d V ( z ) ≡ ((1 + z ) 2 d 2 A ( z ) z/H ( z )) 1 / 3 is the volumeaveraged distance defined by Eisenstein et al. (2005). Since the true cosmology was used to translate redshifts into distances, the value of α must be compatible with 1 (within errors). The statistical uncertainty in α was calculated following the same method used in Anderson et al. (2012). We have not studied the different sources of systematic errors for this measurement, and no systematic contribution has been added to the errors. On the one hand this provides a more conservative comparison with our approach, since the results quoted in section 5.3 do contain systematics. On the other hand, there exist several potential systematics that are specific for the standard method, such as the effect of the fiducial cosmology used to obtain the three-dimensional positions of the galaxies, or the choice of template used to perform the fit. Studying this effect would be extremely interesting, but we have postponed this analysis for a future work. As we have seen before, the systematic errors that are common to both approaches (bias, RSDs, non-linearities, fitting limits) are clearly subdominant compared to the statistical uncertainties. The cosmological constraints drawn in the (Ω M , w 0 ) and ( w 0 , w a ) planes from the values of α measured from the correlation functions are shown in Figure 13. The figure also shows the contours corresponding to the combination of radial and angular information, described in the previous section, for comparison. Plots show that the constraining power of both methods is very similar. There is a degenerate direction in the (Ω M , w 0 ) plane for the standard method, which coincides with the orientation of the contours for the angular BAO shown in Figure 10 (bottom panel). This is a reasonable result: most of the information in the angle-averaged BAO signature comes from the angular part, since there are two transverse dimensions and only one longitudinal. On the other hand, our combined approach seems to be able to obtain better constraints on the evolution of the dark energy equation of state. This could be due to the fact that the radial BAO enables us to measure the evolution of the expansion rate alone, which is a local quantity, unlike the angular diameter distance, which is an integrated one depending on the expansion history. Although the degenerate directions are very similar for both methods, they are not exactly the same. The reason is the different treatment of the redshift space distortions, which are ignored, to first approximation, in the standard method when the angular average is performed. However, they are fully taken into account in our proposal, where the angular and radial correlation functions have very different shapes, mainly because of the redshift space distortions.", "pages": [ 9, 10, 11 ] }, { "title": "6 CONCLUSIONS", "content": "We have developed a new method to measure the BAO scale in the radial two-point correlation function. This method is adapted to the observational characteristics of galaxy surveys, where only the angular position on the sky and the redshift are measured for each galaxy. The sound horizon scale can be recovered from the non-linear radial correlation functions to a very high precision, only limited by the volume of the considered survey, since the systematic uncertainties associated to the determination of the BAO scale are very small, around 0.3%. On the other hand, the method is fully cosmology independent, since it relies only on observable quantities and, consequently, its results can be analyzed in any cosmological model. The method has been tested with a mock catalogue built upon a large N-body simulation provided by the MICE collaboration, in the light cone and including redshift space distortions. The true cosmology is recovered within 1σ . An evaluation of the main systematic errors has been included in this study, and we find that the method is very promising and very robust against systematic uncertainties. Note that this analysis over the MICE simulations is done on dark matter particles, instead of galaxies. We believe this simplification is not an essential limitation to the method presented here, as we have shown that both the modeling and the error analysis are quite generic. We have compared the cosmological constraints obtained by combining radial and angular BAO information with those obtained by performing the standard analysis of the angle-averaged BAO signature on the same dataset and with the same fitting technique. Both methods seem to yield comparable constraints, with the advantage that our method is entirely based on purely observable quantities (redshifts and angles) and is therefore completely model-independent.", "pages": [ 11 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We acknowledge useful comments from Enrique Gazta˜naga and Pablo Fosalba that helped to improve this work. We acknowledge the use of data from the MICE simulations, kindly provided by the MICE collaboration. We also thank the anonymous referee for the comments and suggestions. Funding support for this work was provided by the Spanish Ministry of Science and Innovation (MICINN) through grants AYA2009-13936-C06-03, AYA2009-13936-C06-06 and through the Consolider Ingenio-2010 program, under project CSD2007-00060. DA acknowledges support from a JAEpredoc contract. JGB and DA acknowledge financial support from the Madrid Regional Government (CAM) under the program HEPHACOS S2009/ESP-1473-02.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "(r ,r )", "pages": [ 12 ] }, { "title": "APPENDIX A: GAUSSIAN APPROACH TO ESTIMATE THE COVARIANCE MATRIX OF THE 3-D CORRELATION FUNCTION", "content": "We have used a theoretical estimation of the covariance matrix for the 3-D correlation function, which is then applied to the estimation of the covariance for the radial correlation function. Our approach is based on assuming Gaussian statistics for the overdensity field, in analogy with the approach in Xu et al. (2013) for the monopole and with the method described in Crocce et al. (2011) for the angular correlation function. We have used the following convention for the power spectrum: where δ D and δ K are the Dirac and Kronecker deltas, V is the volume and we have taken into account the Poisson contribution to the total variance as the inverse of the number density of sources ¯ n . The anisotropic power spectrum can be estimated from a given realization of δ k by averaging over the symmetric azimuthal angle: Assuming that δ k is Gaussianly distributed we can obtain the covariance for this estimator using Wick's theorem: where, as is done in Xu et al. (2013), the effect of a nonhomogeneous number density ¯ n ( z ) has been taken into account by defining the volume-averaged variance The anisotropic power spectrum can be related to the anisotropic correlation function using the 0-order cylindrical Bessel function J 0 ( x ) through And thus the covariance matrix for ξ can be calculated as Here we have taken into account the finiteness of the bins ∆ π , ∆ σ by defining the regularized functions ¯ cos and ¯ J 0 : From this result it is straightforward to compute the covariance matrix for the radial correlation function ( σ 1 , 2 = 0) as where we have defined the projected k -space variance Here the radial coordinate π is related to the redshift separation through ∆ z ≡ π H ( z ), and the transverse width ∆ σ is related to the angular pixel resolution through ∆ θ ≡ ∆ σ/r ( z ). As can be verified on closer inspection of equation A7, the covariance grows dramatically as we decrease the pixel size. This is due to the fact that the number of galaxy pairs drops and the errors become shot-noise dominated. Throughout this work, Gaussian estimations for the covariance matrices were used. Figure A1 shows the Gaussian predictions for the reduced covariance matrix ρ ij ≡ C ij / √ C ii C jj of the radial correlation function (left panel), the angular correlation function (central panel) and the monopole (right panel). The axis ranges correspond to similar comoving scales in the three cases. Notice that, while the errors on the radial correlation function are almost diagonal, they are correlated over a large range of scales in the angular (transverse) case. For the monopole, the errors are also correlated over a large number of bins, which is a sensible result, since the monopole corresponds to an angular average over two transverse and one longitudinal directions.", "pages": [ 12 ] }, { "title": "APPENDIX B: ANGULAR BAO RESULTS FOR THE COMBINATION", "content": "The angular BAO measurements we have used for the results presented in the text have been obtained applying the method described in S'anchez et al. (2011) to the cosmological simulation used in this paper. We have used 10 redshift bins of width 0.1, starting at redshift 0.2 up to redshift 1.2. We find statistically significant results in 9 of them. Results are presented in Figure B1. These are the results we combine with the radial BAO in order to obtain the cosmological constraints presented in the text. It is important to notice that the redshifts for this galaxy sample are spectroscopic and consequently, the systematic error associated to the photometric redshift that is quoted on S'anchez et al. (2011) does not affect these measurements.", "pages": [ 13 ] } ]

[ { "title": "ABSTRACT", "content": "We study a new class of equilibrium two-parametric distribution functions of spherical stellar systems with radially anisotropic velocity distribution of stars. The models are less singular counterparts of the so called generalized polytropes, widely used in works on equilibrium and stability of gravitating systems in the past. The offered models, unlike the generalized polytropes, have finite density and potential in the center. The absence of the singularity is necessary for proper consideration of the radial orbit instability, which is the most important instability in spherical stellar systems. Comparison of the main observed parameters (potential, density, anisotropy) predicted by the present models and other popular equilibrium models is provided. Key words: Galaxy: center, galaxies: kinematics and dynamics.", "pages": [ 1 ] }, { "title": "E. V. Polyachenko, 1 /star , V. L. Polyachenko, 1 I. G. Shukhman, 2 †", "content": "1 2 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskya St., Moscow 119017, Russia Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Siberian Branch, P.O. Box 291, Irkutsk 664033, Russia Accepted Received", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Equilibrium models of spherical stellar systems are needed for observations and numerical simulations of open and globular clusters (see, e.g. Kharchenko et al. 2009, Ernst & Just 2013). On the other hand, our interest in developing a new class of radially-anisotropic models is explained by our desire to perform correct stability analysis of systems with nearly radial orbits. The gravitational potential and radial force in models of spherical stellar systems in which all stars travel on purely radial orbits are singular. This makes it impossible to apply the standard methods of the linear stability theory, and also cast doubts on some of the works on Radial Orbits Instability (ROI) (see e.g. Antonov, 1973). Even a small dispersion in the angular momentum can improve the situation, however, its presence cannot guarantee the removal of the singularity. An example is a series of models known as generalized polytropes, which remain singular despite having some dispersion (see, e.g., BisnovatyiKogan and Zel'dovich, 1969; H'enon, 1973). The potential at r ≈ 0 determines the behavior of the precession rate Ω pr at small angular momentum, which plays a significant role in the stability of the system (Polyachenko et al. 2010). For singular potentials, the precession rate is no longer proportional to the angular momentum, and very quickly (with infinite derivative) departs from zero for angular momentum near L = 0 (see, e.g., Touma and Tremaine, 1997). In this case, usual arguments concerning the mechanism of radial orbit /star E-mail: [email protected] instability which, in particular, involve the linear approximation for the precession rate (see, e.g., Palmer 1994) are not useful. Note that most works that include spectrum determination by matrix methods use models that cannot be made arbitrarily close to systems with purely radial orbits. The standard choice is Osipkov-Merritt type DFs (Osipkov, 1979; Merritt 1985). However, these DFs have restrictions on the largest possible radial anisotropy. The simplest isotropic self-gravitating polytrope F ( E ) ∝ ( -2 E ) q , where E = 1 2 ( v 2 r + v 2 ⊥ ) + Φ( r ) /lessorequalslant 0 is the energy (see, e.g., Fridman & Polyachenko 1984) can be used to construct a series of purely radial models where δ ( x ) is the Dirac delta-function, L = r v ⊥ is the absolute value of the angular momentum of a star. Generalization of (1.1) is possible by replacing the delta-functions on the distribution of the form where H ( x ) is the Heaviside step function. In the limit L T → 0 the function H ( L 2 T -L 2 ) /L 2 T becomes the delta-function, The allowed range of parameter q coincides with the range of the polytropic index in classical polytropic models: -1 /lessorequalslant q < 7 2 (see, e.g., Binney & Tremaine, 2008). Parameter L T specifies width of the phase space region over angular momentum L occupied by the model, L T /greaterorequalslant 0. If L T is less than some critical value ( L T ) iso ( q ) then radial motions dominate. ( L T ) iso ( q ) has the meaning of the maximum specific angular momentum of the particles in an isotropic self-gravitating polytrope of index q . For L T /greaterorequalslant ( L T ) iso ( q ) models no longer depend on L T and become isotropic. In contrast with the previously used models, the proposed anisotropic polytropes reach the limit of purely radial systems for a wide region of polytropic index q . Besides, relative simplicity of the models allows one to achieve good accuracy for eigenmodes and stability boundaries, which in turn can help in understanding the mechanism of ROI. In Sec. 2 we give general equations and provide profiles of the potential, density, and anisotropy for the proposed models, and for several other models commonly used for spherical systems. Sec. 3 is devoted to the study properties of the models in the limit L T = 0. Then, in Sec. 4 we explore in more details several special families of models for which the equilibrium state can be obtained analytically or stability analysis is particularly simple. Sec. 5 stresses on the orbit's precession behavior of nearly radial orbits, and on difficulties that arise in systems with purely radial orbits. In Sec. 6 we summarize the results.", "pages": [ 1, 2 ] }, { "title": "2 SOFTENED ANISOTROPIC POLYTROPES", "content": "In this paper we consider two-parametric series (parameters q and L T ) of models with DF where N = N ( q, L T ) is a constant defined by the normalization condition that the total mass of the system M = 1. For simplicity, we assume that the gravitational constant and a radius of the spherical system are equal to unity as well: G = 1, R = 1. Dependence of the DF on energy is supposed to be the same as in the classical polytropic models, The form of (2.1) suggests that an additive constant in the potential Φ 0 ( r ) is chosen in such a way that the potential is equal to zero on the sphere boundary, Φ 0 (1) = 0. Moreover, the factor ( q +1) allows to include the boundary value q = -1 in the region of available values, since lim q →-1 + F 0 ( E ) = δ ( E ) (see, e.g., Gelfand and Shilov, 1964). Then, it is convenient to define the relative potential and the relative energy of a star by Ψ( r ) = -Φ 0 ( r ) /greaterorequalslant 0, E = -E /greaterorequalslant 0, and use the DF in the form: Below, we shall refer to this models as 'softened' anisotropic polytropes or PPS polytropes. For density distribution one obtains: Here Γ( z ) denotes the Gamma function. With a newly defined function, the expression for density can be written in the compact form, where The number of solutions of equation depends on the value of L 2 T . Given L T is less than the maximum specific angular momentum in the isotropic polytrope, where L circ (0) is the specific angular momentum of the star with E = 0 in a circular orbit in the polytrope q , equation (2.5) has two solutions 0 < r 1 < r 2 < 1. A condition 2 r 2 Ψ < L 2 T is satisfied in the regions adjacent to the center, 0 /lessorequalslant r < r 1 , and to the boundary of the sphere, r 2 < r /lessorequalslant 1 (regions I and III respectively). In the region II ( r 1 < r < r 2 ), 2 r 2 Ψ > L 2 T . The dependence of ( L T ) iso ( q ) is given in Fig. 10 a in Sec. 6. For L T /lessmuch 1, the solutions r 1 , r 2 tend to 0 and 1, consequently. Since Ψ( r 1 ) /similarequal Ψ(0) + O ( r 2 1 ), r 1 /similarequal L T / √ 2Ψ(0). Similarly, Ψ( r 2 ) /similarequal -(1 -r 2 )Ψ ' (1) + O [(1 -r 2 ) 2 ], so 1 -r 2 /similarequal L 2 T / [ -2Ψ ' (1)] = L 2 T / 2. Here Ψ ' (1) = -GM/R 2 = -1 from G = R = M = 1. The Poisson equation together with boundary conditions: determine the potential Ψ and the normalization constant N = N ( q, L T ). Generally, equation (2.6) with boundary conditions (2.7) is solved numerically. However, several values of q result in analytic solutions. As is the case for isotropic polytropes, if L T > ( L T ) iso , analytic solutions exist for q = -3 2 , -1 2 , 7 2 . However, if q < -1, the distribution is unphysical (i.e. unintegrable) and the q = -3 2 case is thus unphysical. The q = 7 2 case, which corresponds to the Plummer model, on the other hand results in solutions with infinite radius, which is also must be rejected due to the boundary condition. In fact the finite radius condition restricts q < 7 2 . In the L T = 0 limit, analytical solutions are found for q = -1 2 , 1 2 , but the q = 1 2 case leads to no physically acceptable solutions with finite radius. Connecting two limits, for arbitrary L T , the equation has exact analytical solutions if q = -1 2 , for which the source term of the equation becomes linear on Ψ. In addition, if q = 1 / 2, equation (2.6) for region II, where 2Ψ r 2 > L 2 T , is analytically solvable. Hence, in the q = 1 2 case it is possible to construct approximate analytic physically acceptable solutions with finite mass and radius, if L T > 0. The solutions for L T /lessmuch 1, which ignore narrow region III (with width ∝ L 2 T ), are constructed in Appendix. and for Osipkov-Merritt (hereafter OM) models of type where A is a normalization constant and r a is the so called anisotropy radius (Osipkov, 1979; Merritt, 1985). Parameters of GP and OM models have been selected in such a way that the global anisotropy (2.12) (see below for the definition) in all the models was the same and equal to ξ /similarequal 0 . 65, that roughly corresponded to the predominance of the total radial kinetic energy of stars over the total transversal kinetic energy by factor of 3 / 2. A certain degree of freedom in choosing the parameters r a and p for the OM DFs were used to fit the potential and density profiles of the PPS polytrope. Curves of the potentials for different models almost coincide at r > r 1 ≈ 0 . 070. Difference is noticeable in the central region r < r 1 , where PPS polytropes become isotropic. However, distribution of density is significantly different: while PPS polytropes and OM models demonstrate similar behavior and finite density in the center, the generalized polytropic models show rather strong singularity, ρ ∼ r -s . The local anisotropy parameter (see, e.g., Binney & Tremain, 2008) is an important characteristics of stellar systems. Here 〈 v 2 r 〉 and 〈 v 2 ⊥ 〉 are dispersions of radial and transversal velocities respectively: where L max ( r, E ) ≡ √ 2 r 2 [Ψ( r ) -E ]. Either by direct integration of the DF or using the method of Dejonghe (1986) (see also Dejonghe & Merritt, 1992) one obtains for PPS polytropes a) r are non-monotonic. Near the boundary of sphere (region III, r 2 < r < 1) the velocity distribution again becomes isotropic, and β ( r ) decreases sharply to zero. Experiments with different values of q show that lower q give sharper changes of the anisotropy parameter at boundaries of regions I-II and II-III, although in general behavior of β ( r ) changes insignificantly. The anisotropy parameter for generalized polytropes does not depend on radius, β = 1 2 s . A system as a whole can be characterized by the parameter of global anisotropy ζ ≡ 2 T r /T ⊥ (Fridman & Polyachenko, 1984), where T r = 4 π ∫ 1 0 d r r 2 ρ ( r ) · 1 2 〈 v 2 r 〉 and T ⊥ = 4 π ∫ 1 0 d r r 2 ρ ( r ) · 1 2 〈 v 2 ⊥ 〉 are total radial and transversal kinetic energy of all stars in the system. It is convenient to redefine the anisotropy parameter as follows: Then ξ = 0 corresponds to isotropic systems (in average), while ξ = 1 implies purely radial systems. Thus, the definition of ξ is consistent with the definition of the local parameter β and we shall use it henceworth as a global characteristics for stellar models. Comparison of global anisotropy for PPS polytropes and OM model is shown in Fig. 3. A characteristic feature of OM model is that for any parameters p and r a , the value of the global anisotropy does not reach unity. In contrast, in the PPS polytropes the limit of purely radial systems exists for a wide range of parameters q : -1 /lessorequalslant q < 1 2 . This is es- and and so Profiles of local anisotropy are shown in Fig. 2. In the central region I ( r < r 1 ) PPS polytrope is isotropic ( β = 0), while beyond this radius (region II, r 1 < r ) it quickly becomes radially-anisotropic, β > 0. Note that in contrast with OM models, anisotropy profiles for PPS polytropes", "pages": [ 2, 3 ] }, { "title": "4 E. V. Polyachenko et al.", "content": "1 sential for further study of stability of systems with nearly radial orbits. Properties of models near purely radial orbits boundary L T = 0 are considered in the next section in more details. For PPS polytropes, from 〈 v 2 ⊥ 〉 +(2 q +3) 〈 v 2 r 〉 = 2Ψ (see (2.11), one also obtains or where W is the total potential energy of a self-gravitating system and Φ is the potential with the zero point given Φ( ∞ ) = 0; Ψ = Φ( R ) -Φ( r ) = -GM/R -Φ. Together with the virial theorem, 2 ( T ⊥ + T r ) + W = 0 and definition of global anisotropy parameter (2.12), this means that one can express the total kinetic and potential energy via q and ξ , Provided that ∆ ≡ 7 -2 q -6 ξ = 0, /negationslash Alternatively, at fixed q , the global anisotropy of PPS polytropes is related to the potential energy:", "pages": [ 4 ] }, { "title": "3 SOFTENED ANISOTROPIC POLYTROPES AT L T → 0", "content": "Specifics of radial and nearly radial systems is a central singularity, and therefore they require special consideration. The GP models (2.8) give purely radial orbits at s = 2. However, not every q is allowed: as it was noted by H'enon (1973) and Barnes et al. (1986), no GP exists when 2 q + 3 s /greaterorequalslant 7. Thus, GPs provide systems consisting of radial orbits only when q < 1 / 2. This also can be seen from our model equations provided that L T = 0. Substituting density into the Poisson equation and using x ≡ ln(1 /r ) as a new independent variable one obtains An asymptotic solution for x →∞ (or for r → 0) is from where we infer that such solutions are possible for q < 1 2 only. /negationslash For q = 1 2 , equation (3.1) becomes linear and has exact analytical solutions. Unfortunately, from it's two linearly independent solutions it is impossible to construct a solution which would have a finite mass and finite potential energy. However, if we admit arbitrarily small smearing, L T = 0, a solution with a finite radius is possible (see Appendix for details). The models with purely radial orbits are always singular, and the singularity is not weaker than ρ ∝ r -2 . This was first pointed out by Bouvier & Janin (1968) (see also Richstone & Tremaine, 1984). However, it is more accurate to say that the singularity may be slightly stronger or slightly weaker than r -2 : ρ ( r ) ∝ r -2 [Ψ( r )] 1 / 2+ q . Since Ψ( r ) ∝ [ ln(1 /r )] m with positive m [see (3.3)] one obtains that, for q < -1 2 the singularity is slightly weaker than r -2 . In the limit of L T → 0 + , asymptotic solution for q > 1 2 takes the form Ψ( x ) ∝ exp( x ) = 1 /r -1, i.e. models degenerate into a point (considering the adopted length unit). The normalization constant in this case tends to zero: N ( q, L T ) ∝ L 2 q -1 T at L T → 0. Global anisotropy ξ for these models is less than one, which is evident, e.g., from Fig. 3 b. It may seem that there is a contradiction: on one hand the parameter L T tends to zero, and on the other hand the parameter ξ , which characterizes the anisotropy of the system as a whole, tends to a finite limit less than one. In reality, of course, there is no contradiction. With an increase of polytropic index q the number of particles with energy E ∼ 0 decreases, and the particles with energies close to the minimum potential energy begin to dominate. For small L T , the potential well near the center is very deep, so the mass is concentrated near the center in a very small region of r /lessorsimilar O ( L 2 T ). Outside this region, the potential is actually Keplerian, Ψ( r ) = 1 /r -1. In fact, radius r = 1 is infinitely remote from the region of localization of the mass. To determine the shape of orbits trapped in this region, one should not rely only on the smallness of the angular momentum in units ( GMR ) 1 / 2 . For highly elongated orbits, the angular momentum L should be small compared to an angular momentum of a circular orbit of the same energy L circ ( E ), i.e. L/L circ ( E ) /lessmuch 1. In other words, when L T is small compared to one, orbits must not be nearly radial, and anisotropy parameter ξ is not required to be close to unity. To illustrate this we define a localization radius r LOC by the equation which is the radius where the density begins to decrease more rapidly than r -3 . The reason is that beyond this radius the gravitational force is determined primarily by the mass confined withing r LOC . From Fig. 4 it is seen that models with q = 0 . 7 (fifth curve from above given by heavy solid line) tend to it's asymptotics r LOC ∝ L 2 T already for L T ∼ 10 -5 . In fact, this behavior occurs for all values of q > 0 . 5, but in order to demonstrate this, we must consider L T orders of magnitude less than L T ∼ 10 -5 , which is difficult to implement numerically. For L T → 0 + , the global anisotropy ξ as a function of q can be obtained analytically, if q > 1 2 . The Keplerian potential of the system corresponds to a point mass, if the outer boundary R is finite. On the other hand, if the system is scaled so that the potential in the center is finite, then R → ∞ , and 'surface term' GM 2 /R in (2.16) and (2.17) becomes zero. Since all energies cannot all together vanish, it requires the determinant ∆ = 0, i.e. We see that when parameter q varies from 1 2 to 7 2 the models are transformed from a model with purely radial orbits to an isotropic one with ξ = 0. Note that for q → 7 2 , the equation (2.6) reduces to the Lane-Emden equation for any finite value L T /greatermuch δ 1 / 2 , where δ ≡ 7 2 -q /lessmuch 1. Indeed, introducing variables Ψ = ψ/δ , r = z δ , N = n 0 δ 2 L 2 T , we can express (2.6) in the form with boundary conditions: which can be replaced by homogeneous boundary conditions at the origin and at infinity. Point z 1 at which 2 ψ ( z 1 ) z 2 1 ≈ 2 z 1 = L 2 T /δ also goes to infinity, provided that L 2 T /δ /greatermuch 1. The result is the Lane-Emden equation the solution of which gives the well-known Plummer potential ψ = ( a 2 + z 2 ) -1 / 2 with a = √ 21 4 n 0 , which corresponds to the isotropic polytropic model with q = 7 2 . Our calculations give n 0 ≈ 0 . 00183, i.e, a ≈ 0 . 098.", "pages": [ 4, 5 ] }, { "title": "4 SPECIAL FAMILIES", "content": "Here we consider several special families of PPS polytropes for which the equilibrium state can be obtained analytically or stability analysis is particularly simple: q = 1 2 , q = 0, q = -1 2 , q = -1. ρ r", "pages": [ 5, 6 ] }, { "title": "4.1 Models with q = 1 2", "content": "The model with a DF is a boundary model, which in the limit L T → 0 + is turned into purely radial one, i.e. ξ ( 1 2 , 0 + ) = 1, see Fig. 3b. Designation '0 + ' emphasizes the already mentioned fact that for q = 1 2 there is no physically acceptable model with a purely radial orbits, although models with arbitrarily small but finite angular momentum dispersion are possible. Solving the Poisson equation (2.6) with density given by (2.3), it is possible to obtain potential and density profiles for different L T in the range 0 < L T < 0 . 6682 (see Fig. 5). It turns out that for small values L T it is possible even to obtain analytical expressions for the potential, density and the normalization constant N . The details of this solution are described in Appendix.", "pages": [ 6 ] }, { "title": "4.2 'Step' models, q = 0", "content": "The simplest anisotropic model allowing both energy and angular momentum to vary in finite intervals corresponds ρ to parameter q = 0: Study of the stability of such a DF is the simplest, and at the same time, the model is quite realistic. Solving the Poisson equation (2.6) it is possible to obtain profiles of the potential and density for different values of L T in the range 0 < L T < 0 . 6422 (see the Fig. 6). Dependence of global anisotropy ξ ( L T ) for this model is presented in Fig. 3. It is seen, that the limit L T → 0 exists and the global anisotropy tends to one.", "pages": [ 6 ] }, { "title": "4.3 Models with q = -1 2", "content": "This anisotropic model is of interest since it allows the exact analytical solution of the Poisson equation. For q = -1 2 , the expression for density can be simplified ρ As it was discussed above, in general, there are three regions separated by radii r 1 and r 2 ( r 1 < r 2 ). Taking into account boundary conditions (2.7), the potential can be written in the form  where k 2 = 2 N/L 2 T . To find six unknowns C 1 , C 2 , A , N , r 1 and r 2 there is a set of 6 algebraic equations: 4 conditions of continuity of the potential and it's first derivative at points r 1 and r 2 , and 2 conditions (2.5) for determining the positions of r 1 and r 2 . Table 1 gives the solutions of the model parameters for several values of L T . Corresponding potential and density profiles are shown the in Fig. 7. Note that in the purely radial model the density does not vanish on the boundary r = 1. In this case ρ = 1 / (4 πr 2 ), N = 1, i.e., radial dependence of density is the same as that of the isothermal polytropic model.", "pages": [ 6, 7 ] }, { "title": "4.4 Models with q = -1", "content": "In the limit q → -1 the PPS polytropes turn into monoenergetic models ρ with density", "pages": [ 7 ] }, { "title": "5 THE PRECESSION OF ORBITS", "content": "In this section we discuss precession of orbits and emphasize related problems that arise in systems with purely radial orbits with an example of models with q = -1 2 . Such a choice is determined by the availability of an analytical expression for the potential for the purely radial system in this family, which is Ψ = -ln r . A star azimuth gains a rotation angle ∆ ϕ during one radial period: Let α ≡ L/L circ ( E ) be a ratio of the angular momentum L to the angular momentum of a star on the circular orbit with the same energy E , L circ ( E ) = exp ( E -1 2 ). Changing the integration variable from r to x ≡ r exp( -E ), we obtain (see also Touma and Tremain, 1997): where e = exp(1). Note that in variables ( E,α ) the rotation angle is independent of energy. This is the case in all scale-free potentials such as Φ = Kr n , or Φ = K ln r . The integration in (5.1) is over all x for which the radicand is positive. An explicit expression for function g ( α ) and its asymptotic expansion for nearly radial orbits can be obtained (using the Mellin transform). After some manipulations, one finally arrives at: where 'p.v.' stands for the principal value. Its asymptotic expansion at small α is: where The precession rate Ω pr is expressed through g ( α ) using the relation: where Ω 1 , 2 ( E,L ) are radial and azimuthal frequencies For nearly radial orbits α /lessmuch 1, we have µ /lessmuch 1 and Ω 1 ( E,L ) ≈ Ω 1 ( E, 0) = √ 2 /π exp( -E ), so that the precession rate is The profiles Ω pr ( L ) for several nearly radial systems are shown in Fig. 9a. It is seen that precession rates depart quickly from zero at L = 0, and the slope is steeper for models with lower L T . Thus, the derivative /pi1 ( E ) ≡ ∂ Ω pr /∂L L =0 tends to infinity as L T → 0. [ ] This anomaly is quite typical for highly anisotropic models (including ones composed of purely radial orbits) in the class of GP, F ∝ ( -2 E ) q L -s . Since all of these models have gravitational force Ψ ' ∝ r 1 -s near the center (H'enon, 1973), it is singular for highly anysotropic DFs with s > 1. In Fig. 9b the profiles /pi1 ( E ) v.s. parameter L T for different polytropic indices q are shown.", "pages": [ 8 ] }, { "title": "6 CONCLUSION", "content": "In this paper we proposed and studied two-parameter models of anisotropic spherical stellar systems. Dependence of DFs F ( E,L ) on the energy E is adopted from the polytropic and generalized polytropic models. The dependence on the angular momentum is chosen in the form of the Heaviside function H ( L 2 T -L 2 ), that allows only stars with the angular momenta L < L T . For a given value of the polytropic index q , there is some critical value L iso of L T , above which the DFs are ergodic, and the systems are isotropic (see Fig. 10). The curve L T = L iso ( q ) determines the upper boundary for the model parameters in ( q, L T )-plane. The left and right boundaries of the permissible parameters coincide with the boundaries of the polytropic models. The left boundary is q = -1, where all stars have the same zero energy. The right boundary is a straight line q = 7 2 , where the models degenerate into the Plummer model and become isotropic for all values of L T . There is no homogeneous model (one with the density independent of radius), because the corresponding value q = -3 2 is outside the permissible interval. A natural lower boundary for possible model parameters is the horizontal axis L T = 0. However, not all of the models with L T = 0 are purely radial systems. Recall that purely radial models are models for which the global anisotropy parameter ξ = 1 (see (2.12)). Fig. 10a shows isolines ξ ( q, L T ) = const in the model's domain. The isotropic models correspond to ξ ( q, L T ) = 0. The most important feature of the proposed models is the existence of a wide region for parameter q , -1 /lessorequalslant q < 1 2 , for which the limit L T = 0 means the purely radial systems. This will enable us to use them in consistent analytic and numerical study of ROI, which cannot be performed correctly using systems with purely radial orbits only. Outside this range, q > 1 2 , L T → 0 + the potential degenerates into the Keplerian one, Ψ( r ) = 1 /r -1, and models turn into points. We show that the global anisotropy ξ ( q, 0 + ) varies linearly with polytropic index q (see (3.5)) from 1 to 0, which corresponds to transformation of models from purely radial to the isotropic ones. Comparison of the parameter domains of PPS polytropes and GP is possible, recalling the relation ξ = s/ 2 for GP. In ( q, s )-plane the boundary is a trapezoid with a vertical straight line q = -1, two horizontal straight lines s = 0 and s = 2 and a sloping side 2 q + 3 s = 7 (H'enon 1973, Barnes, etc. 1986) or 2 q +6 ξ = 7 (see Fig. 10b). The straight line s = 2 corresponds to part of the boundary q < 1 2 , L T = 0, and the sloping side corresponds to another part of x -axis: q > 1 2 , L T → 0 + . The right boundary of our model q = 7 2 corresponds to single point ( q = 7 2 , s = 0 ) in the domain for the generalized polytropes (the Plummer model). Thus, the domain boundary for PPS polytropes coincide with the domain boundary of GP. This is not surprising, since if q > 1 2 , L T = 0 + the mass of the system is localized near the center. In fact, it means that sphere radius R tends to infinity. But just the same R → ∞ occurs when reaching the boundary 2 q = 3 s = 7 in GP (see H'enon 1973). For a fixed L T , central density concentration grows with increasing of the polytropic index q . For q < -1 2 , the anisotropic models have intervals of growing density at the periphery of spheres. The Agekyan's model (1962) which is a particular case of our series at q = -1, L T = 0, also has this feature. For the model q = -1 2 , we consider the precession rates at low angular momenta for nearly radial and purely radial orbits. Features of its behavior play a significant role for stability, first of all in the emergence of ROI (Polyachenko, etc. 2011). We have shown that in the limit of the purely radial systems, the derivative of the precession rate over L at L = 0 tends to infinity. This behavior is typical for all purely radial systems. Thus, the conventional methods of stability theory cannot be applied to study ROI in models a) with purely radial orbits. Suitable systems must have DFs with at least small but finite angular momentum dispersion. Note that the generalized polytropes are also unsuitable for studying the instability by analytical methods (by solving the eigenvalue problem) because of the singular behavior of the density and the potential at s ≈ 2. In a separate work we shall present results of our study of ROI for families of models discussed above. The present work can be considered as the first step in this direction.", "pages": [ 8, 9 ] }, { "title": "ACKNOWLEDGMENTS", "content": "The authors thank the referee for providing several valuable suggestions for presentation of the material, and Dr. Jimmy Philip for editing the original version of the article that helped to improve its quality. This work was supported by Sonderforschungsbereich SFB 881 'The Milky Way System' (subproject A6) of the German Research Foundation (DFG) and by Basic Research Program OFN-17 'The active processes in galactic and extragalactic objects' of Department of Physical Sciences of RAS.", "pages": [ 10 ] }, { "title": "REFERENCES", "content": "Agekyan T. A., 1962, Vestnik Leningrad. Gos. Univ., Ser. math., mech., astr., No 1, 152 (in Russian) Antonov V. A., 1973, English translation in: de Zeeuw, T., ed. Proc. IAU Symp. 127, Structure and Dynamics of Elliptical Galaxies, Reidel, Dordrecht, p. 549 Barnes J., Goodman J., Hut P., 1986, ApJ, 300, 112 Binney J., Tremain S., 2008, Galactic Dynamics: Second Edition. Princeton University Press, Princeton, NJ, USA Bisnovatyi-Kogan G. S., Zel'dovich Ya. B., 1969, Astrofizika, 5, 425 (in Russian) Bouvier P., Janin G., 1968, Publ. Obs. Gen'eve, A74, 186 Chandrasekhar C., 1939, An introduction to the study of stellar structure. Dover publications, Inc. Dejonghe H., 1986, Phys. Rep., 133, 217 Dejonghe H., Merritt D., 1992, ApJ, 391, 531 Ernst A., Just A., 2013, MNRAS, 429, 2953 Fridman A. M., Polyachenko V. L., 1984, Physics of Gravitating Systems. Springer, New York Gelfand I. M., Shilov G. E., 1964, Generalized functions. Academic Press, Inc. H'enon M., 1973, A&A, 24, 229 Kharchenko, N. V., Berczik, P., Petrov, M. I., Piskunov, A. E., Roser, S., Schilbach, E., 2009, A&A 495, 807 Merritt D., 1985, AJ, 90, 1027 Osipkov L. P., 1979, Soviet Astron. Lett. 5, 42 Palmer P. L., 1994, Stability of collisionless stellar systems: mechanisms for the dynamical structure of galaxies. Astrophysics and Space Science Library, Kluwer, Dordrecht, Boston Polyachenko V. L., Polyachenko E. V., Shukhman I. G., 2010, Astron. Lett., 36, 175 Polyachenko E. V., Polyachenko V. L., Shukhman I. G., 2011, MNRAS, 416, 1836 Richstone D., Tremaine S., 1984, ApJ, 286, 27 Touma J., Tremaine S., 1997, MNRAS, 292, 909", "pages": [ 10 ] }, { "title": "APPENDIX. Approximate analytical solution for model q = 1 2 with almost radial orbits, L T /lessmuch 1", "content": "We saw in Sec. 3 that for q /greaterorequalslant 1 2 there are no models with purely radial orbits. Now we construct a physically appropriate solution on the boundary q = 1 2 for arbitrary small but finite L T . From (2.3) and (2.6) one obtains: In the above equation, r 1 defined by L 2 T = 2 r 2 1 Ψ( r 1 ) separates two regions, I and II. In general, there is a region III adjacent to the sphere boundary (see Sec. 2), but for small L T it can be ignored since its width is of the order of L 2 T . Equation (A1) is to be solved with boundary conditions In region II (A1) is a inhomogeneous linear equation. The solution satisfying the boundary conditions at the right boundary r = 1 is where ν = √ 3 N -1 4 is a real parameter. Taking into account that for L T /lessmuch 1 radius r 1 is also very small, r 1 /lessmuch 1, and ignoring trigonometric terms in square brackets in (A2), we obtain Since the function Ψ( r ) is positive, the condition ν Λ 1 < π must be satisfied. In the region I (A1) can be written using new independent variable x ≡ r/r 1 : where Θ( x ) ≡ Ψ I ( r 1 x ) / Ψ I (1) is a new unknown function. The boundary conditions to be satisfied are: The expression for Θ ' (1) follows from the continuity of the first derivative of the potential at r = r 1 . Equation (A6) with boundary conditions (A7) can be solved numerically using standard shooting method for ν 1 ≡ ν (Λ 1 ), where Λ 1 is considered as a control parameter. Then the relation ν = ν ( L T ) (and also 3 N ( L T ) = ν 2 ( L T ) + 1 4 ) is obtained from the equality which follows straightforwardly from (A3): The dependence of the tripled normalization constant N for small L T is shown in Fig. 11. (Recall that for q = 1 2 we have ( L T ) iso ( 1 2 ) = 0 . 6682 and 3 N ( ( L T ) iso ) = 4 . 686.) It was useful to start the shooting procedure from Λ 1 /similarequal 30 which implies very small r 1 and 3 N -1 4 /lessmuch 1. For large Λ 1 , one can find an asymptotic expansion for ν by applying perturbation theory to (A6) and using factor 1 8 in the r.h.s. as a small parameter: where κ = 68 / 39. Note that from (A8) and (A3) it follows that i.e., r 1 = O ( L 4 / 3 T ), that justifies omitting trigonometric contributions of order L 2 T in derivation of (A3). For potential in the center we have an estimate This analytical solution shows that the potential become singular with L T → 0, but is remains regular as long as L T is arbitrary small, but finite.", "pages": [ 10, 11 ] } ]

[ { "title": "ABSTRACT", "content": "We report the discovery in an 80-ks observation of spatially-extended X-ray emission around the high-redshift radio galaxy TN J1388 -1942 ( z =4.11) with the Chandra X-ray Observatory . The X-ray emission extends over a ∼ 30-kpc diameter region and although it is less extended than the GHz-radio lobes, it is roughly aligned with them. We suggest that the X-ray emission arises from Inverse Compton (IC) scattering of photons by relativistic electrons around the radio galaxy. At z =4.11 this is the highest redshift detection of IC emission around a radio galaxy. We investigate the hypothesis that in this compact source, the Cosmic Microwave Background (CMB), which is ∼ 700 × more intense than at z ∼ 0 is nonetheless not the relevant seed photon field for the bulk of the IC emission. Instead, we find a tentative correlation between the IC emission and far-infrared luminosities of compact, far-infrared luminous high-redshift radio galaxies (those with lobe lengths of < ∼ 100kpc). Based on these results we suggest that in the earliest phases of the evolution of radio-loud AGN at very high redshift, the far-infrared photons from the co-eval dusty starbursts occuring within these systems may make a significant contribution to their IC X-ray emission and so contribute to the feedback in these massive high-redshift galaxies. Key words: galaxies: evolution - galaxies: high-redshift - galaxies: individual (TNJ1338 -1942) - submillimetre", "pages": [ 1 ] }, { "title": "X-ray emission around the z =4.1 radio galaxy TNJ1338 -1942 and the potential role of far-infrared photons in AGN Feedback", "content": "Ian Smail 1 /star & Katherine M. Blundell 2 1 Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE UK 2 University of Oxford, Astrophysics, Keble Road, Oxford OX1 3RH UK 19 September 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Deep X-ray observations with the Chandra X-ray Observatory and the XMM-Newton satellite have detected spatially extended X-ray emission around more than a dozen powerful radio sources at z > ∼ 2 (Carilli et al. 2002; Scharf et al. 2003; Fabian et al. 2003, 2009; Blundell et al. 2006; Erlund et al. 2006, 2008; Johnson et al. 2007; Smail et al. 2009, 2012; Laskar et al. 2010; Blundell & Fabian 2011). This emission can extend out to > ∼ 0.1-1 Mpc and usually traces the morphology of the radio emitting lobes in these systems (as well as more crudely the morphology of the giant Ly α halos often found around these galaxies, e.g. Scharf et al. 2003). The spatial coincidence of the X-ray emission with the radio lobes in most sources, along with its typical photon index of Γ eff ∼ 2 (Smail et al. 2012, hereafter S12), means that this emission is usually interpreted as arising from Inverse Comp- scattering of CMB photons by relativistic electrons with γ ∼ 1000 in the radio lobes (e.g. Carilli et al. 2002). At high redshift, z > ∼ 1, the primary source of photons for the IC emission from radio galaxies is expected to be the Cosmic Microwave Background (CMB), due to its ubiquitous nature and its strongly increasing energy density at higher redshifts, ρ CMB ∝ (1 + z ) 4 (Jones 1965; Harris & Grindlay 1979; Nath 2010). However, at lower redshifts, z < ∼ 1, it has been shown that far-infrared (IR) photons from nuclear starbursts and/or hidden QSOs in radio galaxies can also provide an important contribution to their X-ray emission through IC scattering (Brunetti et al. 1997, 1999; Ostorero et al. 2010). Inverse Compton scattering of locally-produced far-infrared photons from dusty starbursts has also been suggested as a component of the X-ray emission from low-redshift ultraluminous infrared galaxies (Colbert et al. 1994; Moran et al. 1999). Such a far-infrareddriven component of any IC X-ray emission may also occur in the more distant sources, especially because many higher- t radio galaxies are also luminous far-infrared emitters, with characteristic luminosities which rapidly increase with redshift, L IR ∝ (1 + z ) 3 (Archibald et al. 2001). As Scharf et al. (2003) showed for 4C 41.17, a radio galaxy with L IR ∼ 10 13 L /circledot at z =3.8, such extreme far-infrared luminosities mean that the far-infrared photons from the dustobscured starburst can potentially exceed the CMB photon density on scales of < ∼ 100 kpc. Thus it is possible that the luminous IC X-ray emission around compact radio sources at very high redshifts is in fact arising from a combination of two locally-produced components: γ ∼ 100 electrons in the radio lobes and far-infrared photons ( λ ∼ 100 µ m) from the intense starburst occuring within these systems. The resulting IC X-ray emission is then sufficient to ionise the gas around these galaxies, helping to form the extended Ly α halos which are frequently observed (e.g. van Breugel et al. 1998). As noted by S12, this process would represent a highly effective feedback mechanism, with the AGN and starburst activity in these galaxies combining to boost their influence on cooling gas surrounding them, and in doing so affecting evolution of the most massive galaxies seen in the local Universe. However, thus far, much of the evidence for the role of starburst-derived far-infrared photons in the IC X-ray emission around high-redshift radio sources is circumstantial (see the discussion in S12). Hence further observations are essential to test the claimed links between far-infrared and IC emission and between IC X-ray emission and the formation of extended Ly α halos. In this letter we analyse an 80-ks archival Chandra observation of TN J1338 -1942, a powerful radio galaxy at z =4.11, which is a luminous far-infrared source and exhibits a striking extended Ly α halo. We detect faint X-ray emission over a ∼ 30-kpc region which is roughly aligned with the radio lobes of this asymmetric radio source. This is one of the highest redshift detections of X-ray emission around a radio source and the highest redshift detection for a radio galaxy (extended IC X-ray emission has been reported around radio-loud Quasars at z =4.3 and 4.7, Siemiginowska et al. 2003; Yuan et al. 2003; Cheung 2004; Cheung et al. 2012). As such this system provides a potential test of the competing sources of IC production photons: either the CMB or local far-infrared photons, and their influence on the gas surrounding the radio galaxy. For our analysis we adopt a cosmology with Ω m =0.27, Ω Λ =0.73 and H 0 =71kms -1 Mpc -1 , giving an angular scale of 7.0 kpc arcsec -1 at z =4.11, a luminosity distance of 37.7 Gpc and an age of the Universe at this epoch of 1.5 Gyrs.", "pages": [ 1, 2 ] }, { "title": "2 OBSERVATIONS AND ANALYSIS", "content": "The radio source TN J1338 -1942 (R.A.: 13 38 26.11, Dec.: -19 42 32.0, J2000, hereafter TN J1338) was reported by de Breuck et al. (1999) as having redshift z =4.11. This makes it one of the highest redshift radio galaxies known in the southern hemisphere and as a result it has been the subject of extensive multi-wavelength studies (Pentericci et al. 2000; Venemans et al. 2002; Reuland et al. 2004; de Breuck et al. 2004; Zirm et al. 2005; Intema et al. 2006; Overzier et al. 2008, 2009). These studies have demonstrated that TNJ1338 is a massive, far-infrared luminous galaxy surrounded by a Ly α halo with an extent of 150 kpc × 40 kpc and that it resides in the core of an over-dense structure of Ly α emitters and Lyman-break galaxies which have formed just ∼ 1.5 Gyrs after the Big Bang. The radio galaxy is com- with a lobe diameter of ∼ 35 kpc and a very asymmetric distribution, with the brighter northern lobe lying close to the faint core emission (Fig. 1). Using the 850µ m flux of 6.9 ± 1.1 mJy from Reuland et al. (2004), along with their 450µ m limit, we estimate a 8-1000 µ m infrared luminosity of (7.4 ± 1.1) × 10 12 L /circledot assuming a modified black body spectral energy distribution with β =1.5 and a characteristic dust temperature of T d =45K (see S12). We also determine a 408-MHz luminosity of (6.5 ± 0.4) × 10 44 erg s -1 from the observed 365MHz flux of 0.72 ± 0.14 Jy, assuming a radio spectrum with frequency, ν , which is proportional to ν α with α ∼ -1 (the average spectral index between 74 MHz and 1.4 GHz is -0 . 97 ± 0 . 06, Cohen et al. 2007; Condon et al. 1998). The field of TN J1338 was observed for 82.4 ks with Chandra during 2005 in faint mode using the ACIS-S with the target placed on the nominal aim-point on the backilluminated S3 chip. The target was observed on three occasions for 32.4 ks on 2005 August 29 (ObsID: 5735), 25.2 ks on 2005 August 31 (ObsID: 6367) and a further 24.8 ks on 2005 September 3 (ObsID: 6368). We retrieved these data from the Chandra archive and analysed them in the same manner as Scharf et al. (2003) and S12. We used version 7.6.11 of the Chandra X-ray Center pipeline software for the initial data processing. We derived light-curves for all three observing sets, having masked all bright sources in the field, and found no significant flaring events in any of the datasets. We then register each set of observations by running wavdetect to generate a list of sources and matching these to the archival 4.5µ m Spitzer Space Telescope IRAC images of the field (Fig. 1) which are aligned to FK5. In this manner we confirm that the absolute astrometry of the resulting image is ∼ 0.4 '' . The final analysis step was then to merge the observations and construct images and exposure maps using the standard ASCA grade set ( ASCA grades 0, 2, 3, 4, 6) for three bands: 0.5-8.0 keV (full-band), 0.5-2.0 keV (soft-band) and 2-8 keV (hard-band). The resulting effective on-source exposure time for the combined observations was 77.5 ks and, after masking the emission from the core of the radio source (Fig. 1) by interpolation, we measure a total background-corrected count of 13 +4 -3 cts in the 0.5-8-keV band within a 30-kpc diameter aperture (4 '' ) centered on the radio galaxy nucleus. The appropriate background was determined from randomlyplaced apertures across the field. This corresponds to a net count rate of (1.7 ± 0.5) × 10 -4 cts s -1 . Converting this count rate we determine an observed, unabsorbed 0.5-8-keV flux of (1.3 ± 0.4) × 10 -15 erg s -1 , assuming Γ eff =2 as expected for IC emission 1 and a Galactic H i column density of 7.1 × 10 20 cm -2 . The corresponding observed-frame 0.58-keV luminosity is (2.2 ± 0.7) × 10 44 erg s -1 . We estimate a 2-8 keV/0.5-2 keV flux ratio of 1.1 ± 0.3, indicating a photon index for the emission of Γ eff ∼ 1, with a 30% uncertainty. Adopting this photon index to derive the observed flux would increase the estimated flux by ∼ 60% and we discuss the influence of this on our results below. 1 The weighted mean photon index for the IC emission in a sample of 10 high-redshift radio sources derived by S12 is 1.98 ± 0.07.", "pages": [ 2, 3 ] }, { "title": "3 RESULTS", "content": "We show the smoothed full-band (0.5-8 keV) X-ray map of TNJ1338 in Fig. 1. This figure also compares the X-ray emission to the radio and optical/mid-infrared morphology of the radio galaxy, using the 1.4-GHz radio map from de Breuck et al. (2004), a 8.5-GHz radio map retrieved from the VLA archive and archival Spitzer Space Telescope IRAC mid-infrared imaging and Hubble Space Telescope ( HST ) F775W ACS optical imaging (the latter has been smoothed to match the resolution of the IRAC imaging data with which it is combined). Finally, Fig. 1 also illustrates the relationship of the extended X-ray emission to the morphology of the Ly α halo around TN J1338 (Venemans et al. 2002). As Fig. 1 shows, we detect X-ray emission coincident with the core of the radio galaxy, which is merged with the northern radio lobe in the relatively low-resolution VLA 1.4GHz map (but resolved at 8.5-GHz). In addition we see very faint extended X-ray emission around TN J1338 extending out to a radius of ∼ 20 kpc and aligned within ∼ 20 · with the axis of the radio lobes. The brightest X-ray emission lies to the south of the core and we estimate the likelihood that this represents a chance alignment of an unrelated X-ray source as 0.001, which along with the absence of a counterpart in the optical or mid-infrared (Fig. 1) suggests that this extended X-ray emission is associated with the radio source. The X-ray emission could arise from thermal bremsstrahlung emission from gas in a deep potential well (cluster) around the radio galaxy. However, a cluster with such a high X-ray luminosity, ∼ 2 × 10 44 erg s -1 (60% higher if we use a photon index of Γ eff ∼ 1) similar to Coma, would be, by a large margin, the highest redshift luminous X-ray cluster: at nearly three times the redshift of the current highest-redshift, X-ray-confirmed clusters, at z ∼ 1.5. We view this as unlikely and so discard this explanation, although it remains possible that this is thermal X-ray emission driven by the growth of the radio source into the clumpy intergalactic medium surrounding the galaxy. Although comparably extended X-ray emitting jets have been found around some high-redshift radio-loud quasars (e.g. Schwartz, et al. 2000; Siemiginowska et al. 2007), we rule out a synchrotron origin for the X-ray emission as there are no compact structures (resembling the shocks of jet knots and hotspots) in the GHz-frequency radio maps (Fig. 1). Instead we suggest that this extended X-ray emission arises from the inverse Compton scattering of lower-energy photons by relativistic electrons associated with lobes of synchrotron plasma that have (recently or previously) been fuelled by the collimated jets from the radio galaxy central engine. This explanation is the same as has been proposed for extended X-ray emission around similarly luminous radio galaxies at z ∼ 2-3.8 (see § 1), although, we note that in this example the X-ray emission is not well-aligned with the radio lobe (as in the case of e.g. 3C 294, Erlund et al. 2008) and in addition it extends to a shorter distance than the lobe (as in the case of e.g. 6C 0905, Blundell et al. 2006). However, this may simply reflect the challenge of detecting faint emission from relic-synchrotron plasma around this very distant source and we return to this point in the discussion. One particularly noteworthy feature of TN J1338 is the small size of the radio source and its asymmetric morphology (best seen in the higher-resolution 8.5-GHz map - Fig 1, see also de Breuck et al. 1999). TN J1338 is more compact than any of the other high-redshift radio galaxies which exhibit IC X-ray emission in S12, likely reflecting the youth of the radio source (Blundell & Rawlings 1999). High asymmetries are seen more frequently in compact, young sources (e.g. Saikia et al. 2002) and this asymmetry may have an environmental origin (McCarthy et al. 1991), although orientation and beaming effects are also possible.", "pages": [ 4 ] }, { "title": "4 DISCUSSION", "content": "We propose that the extended X-ray emission around the z = 4 . 11 powerful radio galaxy TN J1338 arises from IC emission and we now discuss the insights this observation can provide into the photon and electron populations on which the IC process depends.", "pages": [ 4 ] }, { "title": "4.1 Which seed photons are up-scattered by the relativistic electrons?", "content": "In principle, there are two candidate seed photon fields that could be playing a role in producing IC scattered X-ray emission: CMB photons and far-infrared photons from the starbursts within the host galaxy. The former would require relativistic electrons with Lorentz factors of γ ∼ 1000 while the latter would require electrons with Lorentz factors of γ ∼ 100 to produce KeV X-ray emission. To assess the prevalence of these two contributions, we first follow S12 and consider the variation in L X / L 408 MHz with redshift, which in a naive model of a non-evolving radio galaxy should depend only on the photon density and the magnetic field in the lobes. If the X-ray emission were predominantly driven by an increasing CMB photon field, this ratio would increase with redshift (in the absence of systematic evolution in the magnetic fields or electron populations in radio galaxies with similar powers, sizes and ages). In Fig. 2 we plot L X / L 408 MHz versus redshift and include our new observations of TN J1338 along with the sample of IC-detected high-redshift radio galaxies compiled by S12. Adding TN J1338, at the highest redshift, to the S12compilation does not change their earlier conclusion: that there is no evidence of a correlation between L X / L 408 MHz and redshift (as expected from the analysis of Mocz et al. 2011). This allows us to reject a very simple model where the IC emission is driven by the CMB and all the sources have comparable powers and ages. We also see that if we separate the sources on the basis of lobe size, in the expectation that the X-ray emission in the very largest systems cannot feasibly be driven by far-infrared emission from the radio galaxy (Laskar et al. 2010), that there is still no obvious trend with redshift. We conclude that either the CMBdriven component of the IC emission is not dominant, or if it is, that any evolving contribution to the IC scattering from CMB photons is being masked by differences in the electron populations and magnetic field strengths of the lobes in different sources (e.g. Mocz et al. 2011) at the different epochs in their life cycles at which we observe them. In Fig. 2 we also plot L X / L 408MHz against L IR to discern any potential contribution from the far-infrared emission from the radio galaxy to the IC emission. In contrast to the plot versus redshift, this shows a modest correlation, which is more obvious in the smaller, higher-redshift radio galaxies. If we fit a linear relation to the more compact radio sources, those with < 100-kpc diameter lobes, we derive a gradient of 0.92 ± 0.18, consistent with unity. The median absolute dispersion around the best-fit linear trend reduces by ∼ 35 per cent when TN J1338 is included in the fit. We also use a simple Monte Carlo simulation to assess the significance of the correlation, finding a ∼ 0 . 5 per cent chance that the observed correlation of L X / L 408MHz and L IR arises by random chance. As noted earlier, our constraint on the photon index of the X-ray emission around TN J1338 is weak, if we instead adopt Γ eff ∼ 1 then the ratio of L X / L 408MHz will increase by ∼ 60%, but this has no significant effect on the correlation in Fig. 2. As suggested by S12, we therefore conclude that in the majority of more compact high-redshift radio galaxies, their luminous, dusty starbursts may be a significant source of the photons driving their IC emission.", "pages": [ 4, 5 ] }, { "title": "4.2 What is the relationship of the IC X-ray, radio and Ly α emission?", "content": "A comparison of the structures of the radio, X-ray and Ly α emission around TN J1338 shows up some striking differences, especially when compared to other previously studied z > ∼ 2 radio galaxies with IC X-ray emission (e.g. Carilli et al. 2002; Scharf et al. 2003; Blundell et al 2006; Johnson et al. 2007; Erlund et al 2006, 2008; Smail et al. 2009, 2012) and Ly α halos (Pentericci et al. 1997; Knopp & Chambers 1997; van Breugel et al. 1998). These three emission components usually have similar size-scales, but they can show morphological anti-correlations in their detailed structure. For example the small-scale IC X-ray structure around the z =3.8 radio galaxy 4C 60.07 appears to be anti-correlated with the structure of its Ly α halo (Smail et al. 2009) and a similar situation appears to hold for 4C 23.56 at z =2.48 (compare Knopp & Chambers 1997 and Blundell & Fabian 2011). We also see only the broadest correlation between the X-ray and radio structures in TN J1338 (Fig. 1): the misalignment and smaller extent of the X-ray compared to the GHz radio emission this indicates that, if the X-ray emission is IC-driven, then the current (GHz-radiating) synchrotronplasma lobes are not the source of the electrons responsible for the IC scattering. This then requires that relic synchrotron plasma, no longer radiant at GHz-frequencies, provides the relevant reservoir of relativistic electrons for the IC scattering process, such as predicted by episodic models of jet ejection in quasars and radio galaxies (Nipoti et al. 2005; Blundell & Fabian 2011). If this speculation is correct then it suggests in turn that lower-energy ( γ ∼ 100) rather than higher-energy ( γ ∼ 1000) electrons are plentiful, which is in accordance with the picture that far-infrared photons dominate the upscattering process (rather than CMB photons which must scatter from the rarer, higher-energy electrons with γ ∼ 1000). There are also mismatches between the compact scale of the radio and X-ray emission and the much more extended northern Ly α 'lobe' which spans over 15 '' (100 kpc) from the radio core (Fig. 1; Venemans et al. 2002). We have searched for extended X-ray emission associated with the Ly α lobe in the hard and soft band Chandra images and find no emission above the background in this region. Given the short cooling timescale expected for the Ly α halo (Geach et al. 2009), it appears unfeasible that it represents cooling associated with the current radio lobe or its direct IC X-ray emission, as proposed for other high-redshift radio sources (Scharf et al. 2003; S12). Very speculatively, we suggest that the Ly α halo could represent cooling associated with heating by IC scattering from an even older synchrotron plasma produced by an earlier phase of activity, whose Lorentz factors of < ∼ 100 could upscatter CMB photons into UV photons. Long-baseline ( > ∼ 1000-km) low-frequency radio observations, similar to those made possible by the most extended configurations of LOFAR, including LOFAR-UK, would be needed to search for this electron population and compare its distribution to that of the Ly α emission to test this suggestion. Until such observations are available, the lack of correspondence between the Ly α halo and the radio and Xray emission in this system remains a puzzle.", "pages": [ 5 ] }, { "title": "5 CONCLUSIONS", "content": "We analyse a deep archival Chandra X-ray observation of the powerful radio galaxy TN J1338 -1942 at z =4.11 and detect faint X-ray emission extending over a ∼ 30-kpc region around the radio core. We propose that the X-ray emission most likely arises from IC scattering, by relativistic electrons in the synchrotron plasma lobes, of sub-millimetre photons from the CMB or far-infrared photons from the dusty, starburst in this galaxy. This is one of highest redshift detections of IC X-ray emission around a radio source and the highest known for a radio galaxy. We compare the relative strength of the X-ray and radio emission in this system to other high-redshift, IC-emitting radio sources from the literature. On the basis of this and other evidence we find support for the claim that the IC emission in the majority of these systems (those with radio lobes with sizes less than ∼ 100 kpc) may be enhanced by the far-infrared photons from the dusty starbursts occuring in these active high-redshift galaxies. It has been proposed that the luminous IC X-ray emission around compact, high-redshift far-infrared-luminous radio galaxies represents a significant feedback process (Scharf et al. 2003; S12). The particle and photon emission from the lobes and starburst in these massive, radio-loud composite AGN and starbursts, would combine to produce intense X-ray emission distributed across 10-100 kpc scales, encompassing the densest regions of the gas reservoir surrounding these galaxies. The heating from these X-rays can help create an ionised halo of gas and as a result reduces the amount of material available to cool onto the galaxy and subsequently form stars. However, we stress that in TN J1338 we see little correspondence between the very extended Ly α cooling halo around the galaxy and its more compact KeV X-ray and GHz radio emission, which argues for a more complex origin for this feature.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank the anonymous referee for their constructive comments on this paper and we also thank Bret Lehmer for help and Carlos de Breuck for generously sharing his 1.4GHz VLA map. IRS acknowledges support from STFC, a Leverhulme Fellowship, the ERC Advanced Programme DustyGal and a Royal Society/Wolfson Merit Award. This work has used data from the NASA Extragalactic Database (NED), and from the NRAO VLA, HST and Spitzer archives.", "pages": [ 5, 6 ] }, { "title": "REFERENCES", "content": "Smail, I., et al., 2009, ApJ, 702, L114 Smail, I., et al., 2012, ApJ, 760, 132 Venemans, B.P., et al., 2002, ApJ, 569, L11 Yuan, W., Fabian, A. C., Celotti, A., Jonker, P. G., 2003, MN- RAS, 346, L7 Zirm, A. W., et al., 2005, ApJ, 630, 68", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Over the past 20 years reverberation mapping has proved one of the most successful techniques for studying the local ( < 1 pc) environment of super-massive black holes that drive active galactic nuclei. Key successes of reverberation mapping have been direct black hole mass estimates, the radius-luminosity relation for the H β line and the calibration of single-epoch mass estimators commonly employed up to z ∼ 7. However, observing constraints mean that few studies have been successful at z > 0 . 1, or for the more-luminous quasars that make up the majority of current spectroscopic samples, or for the rest-frame ultra-violet emission lines available in optical spectra of z > 0 . 5 objects. Previously we described a technique for stacking cross correlations to obtain reverberation mapping results at high z . Here we present the first results from a campaign designed for this purpose. We construct stacked cross-correlation functions for the C iv and Mg ii lines and find a clear peak in both. We find the peak in the Mg ii correlation is at longer lags than C iv consistent with previous results at low redshift. For the C iv sample we are able to bin by luminosity and find evidence for increasing lags for more-luminous objects. This C iv radius-luminosity relation is consistent with previous studies but with a fraction of the observational cost. Key words: quasars: general, galaxies: active, galaxies: Seyfert", "pages": [ 1 ] }, { "title": "Stacked reverberation mapping", "content": "10 April 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The inner regions of active galactic nuclei (AGN) offer a unique opportunity to study matter within a few parsecs of a super-massive black hole. Reverberation mapping is designed to study (primarily) the broad-line region (BLR) of AGN by measuring the interaction between continuum and broad-line flux variations (Blandford & McKee 1982; Peterson 1993). The physical model assumes the BLR is photoionised by a UV continuum that is emitted from a much smaller radius. Variations in the ionising continuum produce correlated variations in the broad emission-line flux after a delay that can be associated with the light travel time. Reverberation mapping of a single system requires many spectroscopic epochs of emission-line and continuum luminosity measurements. The time lag between continuum and emission-line variations is given by the peak in the cross correlation between the two light curves. To date lags have been measured for nearly 50 objects following this approach. Reverberation mapping has led to significant advances in the understanding of AGN (e.g. the radius-luminosity relation; Wandel et al. 1999; Bentz et al. 2006, stratification and kinematics of the BLR Peterson & Wandel 1999, 2000, black hole mass estimates Peterson et al. 2004 etc.). However, traditionally these campaigns have been observationally expensive as they require many observations of individual objects over a long period of time. Furthermore, these constraints have meant that even at moderate redshift ( z > 0 . 4) there are no reverberation lags measured with the exception of one tentative result at z = 2 . 2 Kaspi et al. (2007). This also means that, apart from some notable exceptions that have had multi-epoch UV spectroscopy, the vast majority of measured lags are for the H β line. In a previous paper (Fine et al. 2012) we discussed the potential gains of combining extensively-sampled time-resolved photometric surveys, specifically the PanSTARRS1 medium-deep survey (PS1 MDS), with relatively few epochs of spectroscopy on large samples of QSOs. Current multi-object spectrographs make it possible to rapidly obtain spectra of many hundreds of QSOs. While individual objects may not have a well-sampled emission-line light curves Fine et al. (2012) showed (in simulations at least) that reverberation lags could be recovered through stacking. In this paper we present the first results from a multiepoch spectroscopic survey of the PS1 MDS fields with the aim of measuring reverberation lags in stacked QSO samples. Throughout we assume H 0 , Ω M , Ω λ = 70 , 0 . 3 , 0 . 7.", "pages": [ 1, 2 ] }, { "title": "2 DATA", "content": "In Fine et al. (2012) we discussed the sources of our data and their reduction. We only repeat the main points briefly here. See Fine et al. (2012) for a more detailed description.", "pages": [ 2 ] }, { "title": "2.1 PS1 light curves", "content": "The PS1 telescope (Hodapp et al. 2004) is performing a series of photometric surveys of the northern sky. The MDS offers the best opportunity for our analysis. Images of the ten MDS fields are taken every four-five nights in the grizy -bands while not affected by the sun or moon (Kaiser et al. 2010; Tonry et al. 2012). We use psphot , part of the standard PS1 Image Processing Pipeline system (Magnier 2006), to extract point-spread-function photometry from nightly stacked images of the MDS fields. Each nightly stack is divided into ∼ 70 skycells. We calibrate each skycell separately using SDSS photometry (Fukugita et al. 1996; York et al. 2000) of moderately bright (16 < Mag. < 18 . 5) point sources ( type = 6 in the SDSS database). Each of the light curves is inspected for outliers that are removed manually, in total < 1 % of the data points were removed.", "pages": [ 2 ] }, { "title": "2.2 Hectospectra", "content": "QSOs in the MDS fields are being observed for an ongoing project to study the variability of QSOs. QSO candidates are selected for spectroscopy using photometric databases of QSOs (Richards et al. 2009; Bovy et al. 2011) to which we add point sources that correspond to X-ray sources, variability selected objects and UVX selected objects. We are surveying the MDS fields with the Hectospec instrument on the MMT. Each MDS field is tiled with seven MMT pointings. Exposures are ∼ 1.5 h in length meaning that an MDS field ( ∼ 500 QSOs) can be surveyed in ∼ 1 night of good-quality on-sky observing time. The spectra are extracted and reduced using standard Hectospec pipelines (Mink et al. 2007). They are then flux calibrated using observations of F stars in the same fields. Spectra are classified and redshifted manually using the runz code (Colless et al. 2001; Drinkwater et al. 2010). So far from 6.5 nights awarded we have observed 15 Hectospec fields (5 more than once). Further spectra were obtained from spare fiber allocations by the Pan-STARRS transient group. In total we have spectra of 2727 objects, 1228 of which are QSOs, 368 of which have > 1 spectroscopic epoch.", "pages": [ 2 ] }, { "title": "3.1 K-corrections", "content": "From the MDS we have griz -band light curves for each object in our sample. We K-correct these magnitudes using a simple model fit to the SDSS ugriz -band magnitudes. We fit a powerlaw along with a template for QSO emission lines and a Lymanα break (see e.g. Croom et al. 2009 for a similar if more detailed approach). While this model is not always a good fit to the SDSS magnitudes, it suffices for the purposes of interpolating K-corrections for this work. We K-correct all of our light curves to a single rest wavelength (1350 and 3000 ˚ A for C iv and Mg ii respectively). Note that the value of this K-correction affects any crosscovariance analysis. On the other hand cross-correlations are normalised and hence unaffected by the value of the K-correction (see section 3.3).", "pages": [ 2 ] }, { "title": "3.2 Line flux measurements", "content": "In Fig. 1 we show the redshift distribution of the 368 QSOs with > 1 spectrum. The distribution is relatively typical of optically selected QSO surveys with the majority between z ∼ 0 . 5 and 3. Over this redshift range the strongest, and most heavily studied, broad emission lines are Mg ii ( λ 2789) and C iv ( λ 1549). We will focus on these two lines throughout the rest of this work. We fit the Mg ii and C iv lines following the multipleGaussian prescription outlined in Fine et al. (2008, 2010) and each fit is manually inspected to check the reliability. Note that the standard Hectospec pipeline does not return a variance array for the spectra. However, from calculating the scatter in flat regions of the spectra it is clear that the spectral S/N is /greaterorsimilar 10 pix -0 . 5 for the majority of sources. Errors on the emission-line fluxes are therefore dominated by flux calibration errors that we find to be ∼ 5 -10 % by comparing the fiber magnitudes of calibration F stars with their SDSS photometry. Note that this will introduce correlated errors into our flux measurements since a single flux calibration is used for each Hectospec field. Manual inspection was performed to remove bad fits, sky residuals, broad and narrow-line absorption (in instances where narrow absorption lines or sky residuals did not significantly effect the emission line the effected pixels were simply masked and the line was fitted as normal) and other potential sources of contamination. The manual inspection was carried out several times with varying degrees of exclusivity. In general we found that more exclusive selection (i.e. more objects thrown out) produced he highest fidelity in our results. After this procedure we were left with 89 (Mg ii ) and 75 (C iv ) objects with a good fit to their emission lines at more than one spectroscopic epoch.", "pages": [ 2, 3 ] }, { "title": "3.3 Cross correlation analysis", "content": "The correlation coefficient r for two samples ( x , y ) is where Cov( x, y ) is the covariance between the samples and σ is the rms of the samples. Fine et al. (2012) concentrated on cross covariances rather than cross correlations primarily because the rms of the emission line fluxes is poorly constrained from just two measurements. In practice we find it considerably more favorable to use cross-correlations in our analysis. We do this since QSO-to-QSO variations require normalisation of the covariance function, and while the rms may not always be well defined it does improve the quality of the stacked results. Nevertheless, most of the results presented here for cross correlations are visible, if with decreased clarity, in the covariance functions as well. Furthermore, cross correlations are not susceptible to the step-biases we found in cross covariances in Fine et al. (2012). To calculate the stacked cross correlations we take each emission-line flux/continuum photometric observation pair for each object in our sample. We calculate the time lag τ between them and the individual i, j th term of equation 1. We bin all of the emission line-continuum data-data pairs by their lag and average over all pairs within a bin to create the stack. In practice we tried mean, variance weighted and median stacks. Each of these methods produced roughly equivalent results for the C iv stack. For Mg ii we found variance weighting gave the highest S/N in the cross correlation. Given the sparsity of solid reverberation-mapping results at z > 0 . 3, this paper's primary goal is to identify a(ny) peak in the stacked correlation functions presented here. The variance weighted mean biases the results towards the higher quality data (in practice the brighter objects) and has the effect of reducing the noise in the final stack. Because we find it gives the clearest results for Mg ii we will use the variance weighted mean to create our stacked correlation functions throughout this paper. To estimate errors on the stacks we use the field-to-field technique whereby we divided our sample into nine subsamples. We then calculated the stacked cross-correlation functions in each subsample and took the rms of the subsamples divided by √ 9 = 3 as an estimate of the error.", "pages": [ 3 ] }, { "title": "3.4 Randomised analysis", "content": "As a check on the significance of any results we obtain we also performed a randomised analysis. For this we take the spectroscopic flux measurements for each individual object and replace them with random values drawn from a Gaussian distribution with the same mean and rms as the observations (note that in most cases the mean and rms are only defined by two observations). We then perform the same stacked cross-correlation analysis with the observed photometric light curves. This process was repeated 200 times and the mean and rms of the separate realisations was recorded for comparison with our results.", "pages": [ 3 ] }, { "title": "4.1 The full C iv stack", "content": "Figure 2 shows the variance-weighted stacked crosscorrelation function for all 75 objects with more than one epoch of C iv data. The lower panel in the plot shows the number of data-data pairs that contribute to the stack in each τ bin. Since PS1 surveys the MD07 field for ∼ 6 months of each year we sample lags of ± 0 . 5 integer years poorly. Once transformed to rest-frame time delays this creates the peaks in the N pairs distribution. Each object in the C iv sample will have a different lag for its C iv line due to a variety of reasons, perhaps most importantly their different luminosities. The C iv sample spans 37 . 5 < log 10 λL λ (1350) < 39 . 5 assuming τ ∝ L 0 . 5 this gives a potential factor-of-10 range in lags within our sample. The effect of stacking objects with a broad range of lags would be to smooth our results. Despite this potential smoothing we find a significant peak in the cross-correlation function in Figure 2. The solid line and shaded area in the figure show the mean and rms from our 200 simulated cross correlations where we randomise the spectroscopic flux measurements (section 3.4). Seven of our cross correlation points lie above the rms level (and five below) indicating a significant correlation between our photometric and spectroscopic datasets. ) s y a d (", "pages": [ 3, 4 ] }, { "title": "4.2 Towards a radius-luminosity relation for C iv", "content": "To reduce the smoothing caused by the range of lags in each stack, and to aim towards a more useful result to the community (measuring the r -L relation for C iv ), we bin the sample by the i -band absolute magnitude. Figure 3 shows the stacked cross correlation for three equal size (in terms of number of objects) magnitude bins. There is some evidence for a peak in the two brighter bins, and perhaps in the faintest as well. There is also a suggestion that the peak is shifting towards longer lags in the more-luminous bins. To quantify this we fitted offset Gaussian functions to each of the cross correlations. In figure 4 we plot the centroid of the fitted Gaussian against the mean continuum luminosity at 1460 ˚ A for each bin. Errorbars in figure 4 come from the Gaussian fit for τ fit and the rms of the continuum luminosities for λL λ . The grey lines in the figure are the two fits Kaspi et al. (2007) present for their C iv r -L relation. There are competing biases that effect the location of the peak in our stacked cross correlations. For exam- ple brighter objects tend to have higher S/N photometry and spectroscopy giving them more weight in the variance weighted mean stacks. On the other hand fainter QSOs tend to be more variable potentially increasing their impact on the stacks. Although more data is clearly needed to determine its exact form, there is tantalising evidence for the existence of a C iv radius luminosity relation.", "pages": [ 4 ] }, { "title": "4.3 The Mg ii correlation", "content": "While the Mg ii sample is slightly larger than C iv we do not find this translates to a more signal in the stacked correlation function. However, in the full variance-weighted stack we do find a clear peak. The stacked correlation for the Mg ii sample is given in figure 5 along with the C iv stack for comparison. The heavy line and shaded area in the figure show the mean and rms from our 200 simulated cross correlations where we randomise the Mg ii flux measurements. Again, a significant peak is evident in the Mg ii cross correlation. The location of the Mg ii peak is at considerably longer lags than for C iv . This is consistent with results for low redshift objects where higher ionisation lines such as C iv exhibit shorter lags. We note however, that in our sample the situation is complicated by the fact that the Mg ii sample is by necessity lower redshift than the C iv , hence the objects contributing to the stack tend to be less-luminous which will bias the Mg ii lag with respect to the highz , luminous C iv sample. We have tried splitting up the Mg ii sample into magnitude bins to investigate any r -L relationship but find no convincing peaks when binned.", "pages": [ 4 ] }, { "title": "5 DISCUSSION", "content": "Reverberation mapping results are the basis for a large part of our understanding of AGN. With some notable ex- ceptions the vast majority of SMBH mass estimates for AGN come directly or indirectly from reverberation mapping. In particular single-epoch estimators (Vestergaard 2002; McLure & Jarvis 2002), that have been applied to > 100 , 000 objects have their basis in reverberation mapping, rely on the r -L relation and are calibrated against reverberation masses. As of yet there is no published r -L relation for the Mg ii line. Kaspi et al. (2007) present a relation for the C iv line based on the six objects that have had a reverberation lag measured for them. However, of these objects four have very similar luminosities and cannot be used to define the gradient of the relation. Their gradient is defined mostly by their results for S5 0836+71, the only high-redshift QSO to have a lag measured, and the dwarf Seyfert NGC 4395, an extremely under-luminous AGN (Peterson et al. 2005). In this letter we present the first results from our campaign to derive reverberation signals from stacks of objects. The technique naturally lends itself to the mapping of restframe ultra-violet lines (Mg ii and C iv ) at high redshift. We show that from the 75 objects in our C iv stack we get a clear peak in the cross correlation. When binned by luminosity there may be some evidence for a r -L relation although our data are not well constrained. However, our data are in good agreement with the C iv r -L relation from Peterson et al. (2005) and (Kaspi et al. 2007). While still at an early stage, the consistency of the Peterson results at z ∼ 0, ours at z ∼ 2, and Kaspi's at z = 2 . 2 appears to indicate little evolution in the r -L relation with redshift. We also find a peak in the stacked Mg ii sample. Comparing with the C iv results we find the Mg ii peak to be at larger τ , indicative of a stratified BLR as observed in lowredshift objects. However, our Mg ii and C iv samples are at different redshifts, with different luminosity distributions confusing a direct comparison.", "pages": [ 4, 5 ] }, { "title": "6 CONCLUSIONS", "content": "We have shown that by stacking samples of QSOs that have continuous photometric monitoring and two-or-more spectra it is possible to recover a reverberation-mapping time lag. We give stacked cross correlations for both the Mg ii and C iv lines and find a clear peak in both. The Mg ii peak is at considerably longer lags indicative of stratification of the BLR. Further more when binned by luminosity the C iv sample shows evidence for increasing lags with increasing luminosity. From these data we make an initial r -L plot for the C iv line. Although we caution that our relation is effected by significant biases we find it is consistent with previous evaluations. This paper demonstrates the potential of the stacking technique to produce reverberation-mapping results at high redshift. Potentially this technique could provide an avenue towards answering some key questions about high-redshift AGN. However, it is clear that more data are required before strong constraints can be derived from this technique.", "pages": [ 5 ] }, { "title": "7 ACKNOWLEDGMENTS", "content": "SF would like to acknowledge SKA South Africa and the NRF for their funding support. The data presented in this work came from the Pan-STARRS1 telescope and the Multiple Mirror Telescope. Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. The PanSTARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the MaxPlanck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the HarvardSmithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, and the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Bentz M. C., Peterson B. M., Pogge R. W., Vestergaard M., Onken C. A., 2006, ApJ, 644, 133 Blandford R. D., McKee C. 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[ { "title": "ABSTRACT", "content": "We investigate the condition for the formation of micron-sized grains in dense cores of molecular clouds. This is motivated by the detection of the mid-infrared emission from deep inside a number of dense cores, the so-called 'coreshine,' which is thought to come from scattering by micron ( µ m)-sized grains. Based on numerical calculations of coagulation starting from the typical grain size distribution in the diffuse interstellar medium, we obtain a conservative lower limit to the time t to form µ m-sized grains: t/t ff > 3(5 /S )( n H / 10 5 cm -3 ) -1 / 4 (where t ff is the free-fall time at hydrogen number density n H in the core, and S the enhancement factor to the grain-grain collision cross-section to account for non-compact aggregates). At the typical core density n H = 10 5 cm -3 , it takes at least a few free-fall times to form the µ m-sized grains responsible for coreshine. The implication is that those dense cores observed in coreshine are relatively long-lived entities in molecular clouds, rather than dynamically transient objects that last for one free-fall time or less. Key words: dust, extinction - infrared: ISM - ISM: clouds - ISM: evolution - turbulence", "pages": [ 1 ] }, { "title": "Hiroyuki Hirashita 1 /star and Zhi-Yun Li 2", "content": "1 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan 2 Astronomy Department, University of Virginia, Charlottesville, VA 22904, USA 2013 June 24", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Dense cores of molecular clouds are the basic units for the formation of Sun-like, low-mass stars. A fundamental question about these cores that has not been answered conclusively is: are they long-lived entities or simply transient objects that disappear in one free-fall time or less? The core lifetime is important to determine, because it affects the rate of star formation as well as the time available for chemical reactions, which in turn affect the chemical structure of not only the cores themselves but also the disks (and perhaps even objects such as comets) that form out of them (Caselli & Ceccarelli 2012). It also has implications on how the cores are formed (Ward-Thompson et al. 2007). If the cores are relatively long lived, it would favor those formation scenarios that involve persistent support against gravity from, for example, magnetic fields (Shu et al. 1987; Mouschovias & Ciolek 1999), or long mass-accumulation time (e.g. Gong & Ostriker 2011). If the core lifetime turns out comparable to the free-fall time or less, then rapid formation and collapse, through for example turbulent compression, would be preferred (Mac Low & Klessen 2004). One way to constrain the core lifetime is to compare the number of starless cores to that of young stellar objects (YSOs), whose lifetimes can be independently estimated (Ward-Thompson et al. 2007; Evans et al. 2009). Ward-Thompson et al. (2007) found that cores of 10 4 -10 5 cm -3 typically last for ∼ 2-5 free-fall times. Such estimates depend, however, on the lifetimes of YSOs, which are uncertain. Here, we explore another, completely independent, way of constraining the core lifetime, through the grain growth implied by the recently discovered phenomenon of 'coreshine.' The so-called 'coreshine' refers to the emission at midinfrared [especially the 3.6 µ m Spitzer Infrared Array Camera (IRAC) band] from deep inside dense cores of molecular clouds (Pagani et al. 2010; Steinacker et al. 2010). It is found in about half of the cores where the emission is searched for (Pagani et al. 2010). The emission is thought to come from light scattered by dust grains up to 1 µ minsize. Such grains are much larger than those in the diffuse interstellar medium (e.g. Mathis, Rumpl, & Nordsieck 1977, hereafter MRN). Since it takes time for small MRN-type grains to grow to µ m-size, the observed coreshine should provide a constraint on the core lifetime. The goal of our investigation is to quantify this constraint. Specifically, we want to answer the question: how long does it take for the grains in a dense core to grow to µ msize at a given density? Grain growth through coagulation has been studied for a long time (e.g. Chokshi, Tielens, & Hollenbach 1993; Dominik & Tielens 1997). Even before the discovery of coreshine, Ormel et al. (2009) was able to demonstrate that coagulation can in principle produce µ m-sized grains in dense cores, provided that the grains are coated by 'sticky' materials such as water ice and that the cores are relatively long-lived (see also Ormel et al. 2011). In this paper, we aim to strengthen Ormel et al. (2009)'s results by deriving a robust lower limit to the lifetimes for those cores with µ m-sized grains inferred from coreshine through a simple framework that isolates the essential physics of coagulation. We find that cores of typical density 10 5 cm -3 must last for at least a few free-fall times in order to produce µ m-sized grains. Our coagulation models are explained in Section 2 and the results are described in Section 3. We discuss the robustness and implication of the results in Section 4, and conclude in Section 5.", "pages": [ 1, 2 ] }, { "title": "2.1 Coagulation", "content": "We consider the time evolution of grain size distribution by coagulation in a dense core. We adopt the formulation used in our previous paper, Hirashita (2012) (see also Hirashita & Yan 2009), with some changes to make it suitable for our purpose. We briefly summarize the formulation here, and refer to Hirashita (2012) for further details. We assume that the grains are spherical with a constant material density ρ gr . We define the grain size distribution such that n ( a, t ) d a is the number density of grains whose radii are between a and a + d a at time t . For numerical calculation, we consider N = 128 discrete logarithmic bins for the grain radius (or mass), and solve the discretized coagulation equation. In considering the grain-grain collision rate between two grains with radii a 1 and a 2 , we estimate the relative velocity by where the grain velocity as a function of grain radius, v ( a ) , is given below in equation (3), and µ ≡ cos θ ( θ is an angle between the two grain velocities) is randomly chosen between -1 and 1 in each time-step, 1 and the cross-section by where S is the enhanced factor of cross-section, which represents the increase of cross-section by non-compact aggregates. Note that we always define the grain radius a and the grain material density ρ gr for the compact geometry, even if S > 1 , to avoid the extra uncertainty caused by the grain geometry [see also the comment in the item (iii) in Section 2.2]. We adopt the turbulence-driven grain velocity derived by Ormel et al. (2009), who assume that the driving scales of turbulence is given by the Jeans length and that the typical velocity of the largest eddies ( ∼ the Jeans length) is given by the sound speed (see also Hirashita 2012): where T gas is the gas temperature assumed to be 10 K in this paper. Thermal velocities are small enough to be neglected. The robustness of our conclusion in terms of the grain velocity is further discussed in Section 4.1. The form of equation (1) suggests that the motions of dust particles are random. This treatment is not valid in general, since 1 This treatment is different from Hirashita (2012), who represented the collisions by µ = -1 , 0, and 1. Such a discrete treatment of µ cause artificial spikes in the grain size distribution. turbulent motions are correlated. However, we do not include the full treatment of the probability distribution function of the true relative particle velocity in turbulence for the following three reasons: (i) Our simple formulation is sufficient to give a lower limit for the coagulation time-scale (Section 4.1). (ii) The probability distribution function of the true relative particle velocity in turbulence is unknown, and has only recently been investigated (Hubbard 2013; Pan & Padoan 2013). (iii) In the environments of interest to this paper, the forcing of turbulent eddies can be represented by a model where the particle motions experience 'random kicks': in this so-called 'intermediate regime' (Ormel & Cuzzi 2007), the prescription given by equation (1) is applicable. Indeed, we can confirm that the condition for the intermediate regime is satisfied as follows. The intermediate regime is defined by Re -1 / 2 < St < 1 , where Re is the Reynolds number and St is the Stokes number (Ormel & Cuzzi 2007). This condition is translated into 11( a/ 1 µ m ) 2 ( T gas / 10 K) -1 cm -3 < n H < 2 . 9 × 10 12 ( a/ 1 µ m ) 4 ( T gas / 10 K) -1 cm -3 . Since we are interested in the range of grain radius, 0 . 1 µ m /lessorsimilar a /lessorsimilar 1 µ m, the intermediate regime is applicable to the density range considered in this paper. We adopt the following coagulation threshold velocity, v ki coag , given by (Chokshi, Tielens, & Hollenbach 1993; Dominik & Tielens 1997; Yan, Lazarian & Draine 2004) where γ is the surface energy per unit area, R ki ≡ a k a i / ( a k + a i ) is the reduced radius of the grains, E /star is the the reduced elastic modulus. This coagulation threshold is valid for collision between two homogeneous spheres and would not be applicable to collisions between aggregates. At low velocities, grains stick with each other and develop a non-compact or fluffy aggregates. These aggregates stick with each other at low relative velocities, and start to deform or bounce as the relative velocities increases. Because the deformation absorbs the collision energy, the aggregates can stick with each other at a velocity larger than the above coagulation threshold. At very high velocities, cratering and catastrophic destruction will halt the growth (Paszun & Dominik 2009; Wada et al. 2011; Seizinger & Kley 2013). In this paper, we only limit the application of this threshold to compact spherical grains [i.e. cases (i) and (ii) in Section 2.2; see Ormel et al. (2009) and references therein for a detailed treatment of coagulation of aggregates.]", "pages": [ 2 ] }, { "title": "2.2 Initial condition and selection of parameters", "content": "For the initial grain size distribution, we adopt the following powerlaw distribution, which is typical in the diffuse ISM (MRN): where C is the normalizing constant, with a min = 0 . 001 µ m and a max = 0 . 25 µ m. The normalization factor C is determined according to the mass density of the grains in the ISM: where n H is the hydrogen number density, m H is the hydrogen atom mass, µ is the atomic weight per hydrogen (assumed to be 1.4) and D (0.01; Ormel et al. 2009) is the dust-to-gas mass ratio. We adopt n H = 10 5 cm -2 for the typical density of dense cores emitting coreshine (Steinacker et al. 2010), but also survey a wide range in n H . We normalize the time to the free-fall time, t ff : To isolate the key pieces of physics that determine the rate of coagulation, we examine the following three models: (iii) Maximal coagulation model: As shown by Ormel et al. (2009), the volume filling factor of the grains after coagulation is ∼ 0 . 1 because of the non-compact structure of aggregates. Thus, we adopt S = 5 [ ∼ (1 / 0 . 1) 2 / 3 ]. Like the sticky coagulation model, we do not apply the coagulation threshold. This model provides a conservative estimate for the coagulation time-scale (see Section 4.1 for discussion). Note that a and ρ gr are defined for the compact grains. In fact, the grain velocity (equation 3) also has a dependence on the volume filling factor of aggregates through a and ρ gr in such a way that the non-compact structure enhances the gas-grain coupling, leading to a lower velocity. Thus, the maximal coagulation model overestimates the grain velocity (i.e. the coagulation rate), which strengthens the case for the model being 'maximal'.", "pages": [ 2, 3 ] }, { "title": "3.1 Evolution of grain size distribution", "content": "We present the evolution of grain size distribution for n H = 10 5 and 10 7 cm -3 at t = 1 t ff , 3 t ff , and 10 t ff . The results are shown in Fig. 1 for all three models: (i) the standard silicate model, (ii) the sticky coagulation model, and (iii) the maximal coagulation model. In order to show the grain mass distribution per logarithmic radius, we show a 4 n ( a ) . In the standard silicate model shown in Fig. 1a, the grain growth stops at a ∼ 0 . 1 µ m because of the coagulation threshold: the grain velocities are too large for coagulation if a /greaterorsimilar 0 . 1 µ m. Thus, bare silicate cannot grow to µ m sizes, a result found previously by Ormel et al. (2009). We conclude that bare silicate cannot be the source of coreshine. Indeed, water ice has higher coagulation threshold, so if grains are coated by water ice, coagulation proceeds further (Ormel et al. 2009, 2011). Motivated by this, we examine the sticky coagulation model, in which there is no coagulation threshold. (The coagulation threshold of water ice is separately discussed in Section 4.1 to minimize the uncertainty in the material properties adopted.) Fig. 1b shows that grains grow beyond 0.1 µ m. For n H = 10 7 cm -3 , µ msized grains form at 10 t ff , while for the standard density n H = 10 5 cm -3 , the typical grain radius does not reach 1 µ m even at 10 t ff . In reality, aggregates are thought to form as a result of coagulation (Ossenkopf 1993). Thus, the cross-section is effectively increased compared with the spherical and compact case. Fig. 1c shows that the maximal coagulation model in which the crosssection is elevated by a factor of 5 (i.e. S = 5 ) successfully produces µ m-sized grains within 10 t ff even for n H = 10 5 cm -3 . It remains difficult, however, to produce µ m-sized grains in 3 t ff for n H = 10 5 cm -3 and in 1 t ff for n H = 10 7 cm -3 .", "pages": [ 3 ] }, { "title": "3.2 Condition for the formation of µ m-sized grains", "content": "As mentioned in Introduction, the aim of this paper is to determine the condition for the formation of µ m-sized grains thought to be responsible for the observed coreshine (Pagani et al. 2010; Steinacker et al. 2010). According to Steinacker et al. (2010), scattering dominates over absorption by an order of magnitude at λ = 3 . 6 µ m if a /greaterorsimilar 1 µ m. Since the peak of the grain size distribution in a 4 n ( a ) is well defined (see Fig. 1), we simply find the condition for the radius at the peak, a peak , reaches or exceeds 1 µ m. We also examine a more conservative criterion by using a peak = 0 . 5 µ minstead of 1 µ m, motivated in part by the fact that a ∼ 0 . 5 µ m is the grain radius at which scattering is comparable to absorption at λ = 3 . 6 µ m(Steinacker et al. 2010). We will concentrate on the maximal coagulation model with an enhancement factor for cross-section S = 5 ; the result from the sticky coagulation model with S = 1 can be obtained through a simple scaling. In Fig. 2, we show a grid of models with different core densities and times (in units of the free-fall time at the core density). The solid line marks roughly the critical time t grow at a given density n H above which µ m-sized grains are produced. It is given by where A = 5 . 5 and 3.0, respectively, if we adopt a peak = 1 µ m and 0 . 5 µ m for the criterion of micron-sized grain formation. The condition for forming µ m-sized grains is therefore t > t grow . The same condition applies to the sticky coagulation model (with S = 1 ) as well, since coagulation time is inversely proportional to the cross-section for grain-grain collision. Equation (8) can be understood in the following way. Since coagulation is a collisional process, t grow should be given by the collision time-scale, t coll = ( vSπa 2 n dust ) -1 = 4 aρ gr / (3 D µm H n H vS ) , where v and n dust are the velocity and the number density of grains, respectively (Ormel et al. 2009). The growth time-scale in terms of grain radius is t grow /similarequal 3 t coll (note that t coll is the time-scale of grain volume being doubled by coagulation). Then, t coll /t ff is evaluated by using equations (3) and (7) as t grow /t ff /similarequal 7 . 4( a/ 1 µ m ) 1 / 2 ( n H / 10 5 cm -3 ) -1 / 4 ( S/ 5) -1 ( T gas / 10 K) -1 / 4 · ( ρ gr / 3 . 3 g cm -3 ) 1 / 2 ; that is, A = 7 . 4 (5.2) for a = 1 µ m ( 0 . 5 µ m), in a fair agreement with the above numerical estimate. Thus, t grow can be understood in terms of collision time-scale, which strengthens our numerical results. Note that, to form µ m-sized grains in one free-fall time, the density n H must be of order 10 8 cm -3 or higher, even in the maximal coagulation model. In the sticky coagulation model, the required density would be higher still. Such densities are much higher than the typical core value (of order 10 5 cm -3 ). At 10 5 cm -3 , Fig. 2 and equation (8) indicate that, under reasonable conditions, it takes at least several free-fall times for the grains to grow to µ msize (see Section 4.2 for more discussion). The implication is that those dense cores detected in coreshine should be rather long-lived entities rather than transient objects that disappear in one free-fall time; the latter objects would simply not have enough time to form the µ m-sized grains responsible for coreshine.", "pages": [ 3, 4 ] }, { "title": "4.1 A lower limit to µ m-sized grain formation time", "content": "One may argue that coagulation would be faster if the grains were to collide at higher speeds than adopted in our model. However, it will be difficult for this to happen because of the existence of a coagulation threshold. As mentioned earlier, bare silicate grains al- ready acquire velocities larger than the threshold at a rather small size a ∼ 0 . 1 µ m; they do not grow beyond 0 . 1 µ m under reasonable conditions. To grow to larger sizes, the grains must be 'more sticky' than silicate, as is the case when the grains are coated with water ice (Ormel et al. 2009). For such coated grains, we can estimate the coagulation threshold for equal-sized grains from equation (4) using ρ gr = 3 . 3 g cm -3 , γ = 370 erg cm -2 , E /star = 3 . 7 × 10 10 dyn cm -2 . The result is v coag = 9 . 4 × 10 2 ( a/ 1 µ m ) -5 / 6 cm s -1 . For the micron-sized grains that we aim to form, this threshold is already smaller than the typical grain velocity v ∼ 3 . 5 × 10 3 ( a/ 1 µ m ) 1 / 2 that was used in our model. In other words, our model is already generous with the grain-grain collision speed. (Collisions at the relatively high speed that we adopted may lead to the compaction of aggregates, which should reduce the enhancement factor S for grain-grain collision cross-section and hence the rate of grain growth.). Increasing the collision speed further should not lead to faster growth to µ m-size. For this reason, we believe that the critical time t grow for the formation of µ m-sized grains estimated in equation (8) is a robust lower limit.", "pages": [ 4 ] }, { "title": "4.2 The case for long-lived dense cores", "content": "Equation (8) indicates that it takes more than ∼ 5 free-fall times to form 1 µ m-sized grains at the typical core density n H = 10 5 cm -3 if the enhancement factor for cross-section is S = 5 . If the enhancement factor is larger, the coagulation would be faster. In particular, if S = 25 , the formation of micron-sized grains may occur in a single, rather than 5, free-fall time. However, S = 25 requires the grain volume filling factor to be 25 -3 / 2 ∼ 1 per cent, which is extreme. For example, to form such a grain of a = 1 µ mwith compact spherical grains with a = 0 . 1 µ m, one need to connect 1,000 grains linearly , which is unlikely. We doubt that there is much room to increase S well beyond 5 , which corresponds aggregates of rather low volume filling factor ( ∼ 0 . 1 ) already. If the cross-section enhancement factor S is not much larger than 5, it would take several free-fall times (or more) to form µ m-sized grains at typical core densities. The long formation time would indicate that those dense cores with observed coreshine are relatively long-lived entities, rather than transient objects that form and disappear in one free- This estimate of core lifetime based on grain growth is consistent with that inferred from the number of starless cores (relative to YSOs) (Ward-Thompson et al. 2007). It is also consistent with the observational results that only a small fraction of dense cores show any detectable sign of gravitational collapse and that even those collapsing cores tend to have infall speeds less than half the sound speed (Di Francesco et al. 2007). Such slowly-evolving, relatively long-lived cores can form, for example, as a result of ambipolar diffusion in magnetically supported clouds (Shu et al. 1987; Mouschovias & Ciolek 1999), even in the presence of a strong, supersonic turbulence (Nakamura & Li 2005). They are less compatible with transient cores that are formed rapidly through fast compression by supersonic turbulence without any magnetic cushion (Mac Low & Klessen 2004), unless the core material is slowly accumulated in the post-shock region over several free-fall times (e.g. Gong & Ostriker 2011).", "pages": [ 4, 5 ] }, { "title": "4.3 Source of large grains", "content": "Large grains ( a /greaterorsimilar 0 . 1 µ m), once they are injected into the diffuse ISM, are rapidly shattered into smaller grains (Hirashita & Yan 2009; Asano et al. 2013). Thus, there should be a continuous supplying mechanism of large grains (Hirashita & Nozawa 2013). If dense molecular cores has lifetimes long enough to produce µ msized grains, they can be an important source of large grains. Including the supply of large grains from dense cores will be an interesting topic in modeling the evolution of dust in galaxies.", "pages": [ 5 ] }, { "title": "5 CONCLUSION", "content": "Motivated by recent coreshine observations, we have examined the condition for the formation of µ m-sized grains by coagulation in dense molecular cloud cores. We obtained a simple, conservative lower limit to the core lifetime t for the formation of 0.5 µ m-sized grains: t/t ff > 3(5 /S )( n H / 10 5 cm -3 ) -1 / 4 , where t ff is the freefall time at the core density n H and S the enhancement factor for grain-grain collision that accounts for aggregates. The formation time for 1 µ m-sized grains is roughly a factor of 2 longer. Since S is unlikely much larger than 5, we conclude that dense cores of typical density n H = 10 5 cm -3 must last for at least several free-fall times in order to produce the µ m-sized grains thought to be responsible for the observed coreshine. Such cores are therefore relatively longlived entities in molecular clouds, rather than dynamically transient objects.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are grateful to C. W. Ormel for comments that greatly improved the presentation of the paper. This research is supported through NSC grant 99-2112-M-001-006-MY3 and NASA grant NNX10AH30G.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Asano, R., Takeuchi, T. T., Hirashita, H., & Nozawa, T. 2013, Di Francesco, J., Evans, N. J. II, Caselli, P. et al. 2007, in Reipurth B., Jewitt D., Keil K., eds, Protostars and Planets V, University of Arizona Press, Tuscon, p. 17 Dominik, C., & Tielens, A. G. G. M. 1997, ApJ, 480, 647 Draine, B. T. 1985, in Black D. 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[ { "title": "ABSTRACT", "content": "We present a detailed analysis of spectral line profiles in Type Ia supernova (SN Ia) spectra. We focus on the feature at ∼ 3500 - 4000 ˚ A, which is commonly thought to be caused by blueshifted absorption of Ca H&K. Unlike some other spectral features in SN Ia spectra, this feature often has two overlapping (blue and red) components. It is accepted that the red component comes from photospheric calcium. However, it has been proposed that the blue component is caused by either high-velocity calcium (from either abundance or density enhancements above the photosphere of the SN) or Si II λ 3858. By looking at multiple data sets and model spectra, we conclude that the blue component of the Ca H&K feature is caused by Si II λ 3858 for most SNe Ia. The strength of the Si II λ 3858 feature varies strongly with the light-curve shape of a SN. As a result, the velocity measured from a single-Gaussian fit to the full line profile correlates with light-curve shape. The velocity of the Ca H&K component of the profile does not correlate with light-curve shape, contrary to previous claims. We detail the pitfalls of assuming that the blue component of the Ca H&K feature is caused by calcium, with implications for our understanding of SN Ia progenitors, explosions, and cosmology. Key words: line: identification - line: profile - supernovae: general - supernovae: individual: SN 2010ae - supernovae: individual: SN 2011fe", "pages": [ 1 ] }, { "title": "Ryan J. Foley 1 /star", "content": "1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Accepted . Received ; in original form", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The spectral-energy distribution (SED) of a Type Ia supernova (SN Ia) near maximum brightness is relatively similar to that of a hot star. An SN SED is predominantly a black body with line blanketing in the ultraviolet. There are also prominent spectral features associated with absorption and emission from elements primarily generated in the SN explosion. These features typically have broad P-Cygni profiles, although overlapping lines can produce larger and more complicated profiles. The exact SED of a SN Ia depends on the velocity and density structure of the SN ejecta (e.g., Branch et al. 1985). Since broad-band filters sample portions of the SED, and measurements in such filters are used to determine SN distances to ultimately measure cosmological parameters (e.g., Conley et al. 2011; Suzuki et al. 2012), understanding SN Ia spectral features is important for precise cosmological measurements. SN spectra also provide detailed information about the SN explosion, progenitor composition, circumbinary environment, and reddening law (e.g., Hoflich et al. 1998; Lentz et al. 2001; Mazzali et al. 2005a; Tanaka et al. 2008;", "pages": [ 1 ] }, { "title": "/star E-mail:[email protected]", "content": "Wang et al. 2009a; Foley et al. 2012d; Hachinger et al. 2012; Ropke et al. 2012). Furthermore, there is evidence that one can estimate the intrinsic colour of SNe Ia, and thus improve distance measurements through a better estimate of the dust reddening by measuring the ejecta velocity of SNe Ia (Foley & Kasen 2011). Ejecta velocity is measured from the blueshifted position of spectral features. For both cosmology and SN physics, it is important to have a precise understanding of SN Ia SEDs. At optical wavelengths, the two most prominent features in a maximum-light spectrum of a SN Ia are at ∼ 3750 and 6100 ˚ A, respectively. The latter is thought to be from Si II λλ 6347, 6371 ( gf -weighted rest wavelength of 6355 ˚ A), and is the hallmark spectral feature of a SN Ia. The former, at rest-frame wavelengths of ∼ 3500 - 4000 ˚ A, is generally attributed to blueshifted absorption from Ca H&K ( gf -weighted rest wavelength of 3945 ˚ A). However, the line profile of this feature is complicated, often times displaying shoulders, a flat bottom, a 'split' profile, and/or two distinct absorption components. There is broad consensus that the red component of the profile is from Ca H&K at a 'photospheric' velocity, i.e., a velocity similar to that of the ejecta at close to the τ = 2 / 3 surface, which is typically about 12000 kms -1 near maximum light. However, there is no clear consensus to the origin of the blue component. Previous studies have at- e blue absorption component to either 'highvelocity' (HV) Ca H&K absorption ( ∼ 18,000 kms -1 ; e.g., Hatano et al. 1999; Garavini et al. 2004; Branch et al. 2005, 2007; Stanishev et al. 2007; Chornock & Filippenko 2008; Tanaka et al. 2008, 2011; Parrent et al. 2012), where the absorption comes from a region at high velocity within the SN ejecta that has high-density calcium, and to Si II λλ 3854, 3856, 2863 ( gf -weighted rest wavelength of 3858 ˚ A; e.g., Kirshner et al. 1993; Hoflich 1995; Nugent et al. 1997; Lentz et al. 2000; Wang et al. 2003; Altavilla et al. 2007). Since calcium and silicon produce the strongest features in SN Ia spectra near maximum brightness, both interpretations are worth investigation. For convenience, we will generally refer to this feature as the 'Ca H&K feature.' There are several cases of clear HV material in SNe Ia. Observations showing multiple components to the Si II λ 6355 line profile (e.g., Mazzali et al. 2005b; Altavilla et al. 2007; Garavini et al. 2007; Stanishev et al. 2007; Wang et al. 2009b; Foley et al. 2012b) or strong and quickly varying HV O I λ 7774 (Altavilla et al. 2007; Nugent et al. 2011) are perhaps the cleanest way to detect HV material since there are no other strong lines just blueward of Si II λ 6355 and O I λ 7774. Other detections have been made by observing the Ca NIR triplet, often through spectropolarimetry (e.g., Hatano et al. 1999; Li et al. 2001; Kasen et al. 2003; Wang et al. 2003; Gerardy et al. 2004; Mazzali et al. 2005a), however there are several subtleties to this feature. HV features must be caused by abundance and/or density enhancements in layers of the ejecta above the SN photosphere. Two distinct 'layers' of material within a smooth density profile (i.e., an abundance enhancement) would necessarily be caused by the explosion, and observations of HV features could therefore restrict the possible explosion models. However, Mazzali et al. (2005b) suggested that abundance differences alone cannot reproduce the strength of the HV features, and therefore there must be a density enhancement. Density enhancements may be either caused by the explosion causing over-dense blobs or shells of material or by sweeping up circumbinary material (e.g., Gerardy et al. 2004; Mazzali et al. 2005a; Quimby et al. 2006). Spectropolarimetric observations have indicated that HV Ca NIR triplet features are probably caused from the explosion (e.g., Kasen et al. 2003; Wang et al. 2003; Chornock & Filippenko 2008). Because of its wavelength, it is difficult to obtain high-quality spectropolarimetric measurements of the Ca H&K feature. None the less, Wang et al. (2003) was able to make such a measurement, and the polarization spectrum suggested that the blue component of the Ca H&K feature was from Si II λ 3858 for SN 2001el. Using the large CfA sample of SN Ia spectra (Blondin et al. 2012b), Foley, Sanders, & Kirshner (2011) determined that the velocity of Si II λ 6355, v Si II , and the velocity of the red component of the Ca H&K feature, v CaH&K , at maximum light correlated with intrinsic colour, but did not correlate with light-curve shape (and thus luminosity). However, they did not find statistically significant evidence of a linear correlation between the pseudo-equivalent width of the Ca H&K feature and intrinsic colour. Using SDSS-II Supernova Survey and Supernova Legacy Survey data, Foley (2012) confirmed these trends with high-redshift SNe Ia. They also noted a slight (2.4σ significant) trend between the maximum-light v CaH&K ( v 0 CaH&K ) and host-galaxy mass. Maguire et al. (2012, hereafter M12) presented a sample of maximum-light low-redshift SN Ia spectra obtained with the Hubble Space Telescope ( HST ). After various quality cuts, the sample consisted of 16 spectra of 16 SNe Ia. These spectra covered the Ca H&K feature, but did not cover wavelengths near Si II λ 6355. Unlike Foley et al. (2011) and Foley (2012), which presumed that the red component of the Ca H&K feature was representative of photospheric calcium (and thus the wavelength of the maximum absorption of this component represented v CaH&K ), M12 fit a single Gaussian to the entire profile to measure v CaH&K . Among other claims, M12 reported a linear relationship between v CaH&K and light-curve shape (3.4σ significant). Furthermore, they claimed that after correcting for the relation between light-curve shape and v CaH&K , there is no correlation between v CaH&K and host-galaxy mass. In this paper, we examine the claims of M12 with particular scrutiny to the details of the Ca H&K profile. In Section 2, we re-examine the M12 sample. We confirm a difference in the Ca H&K line profile for SNe Ia with different light-curve shapes, but show that the difference is primarily in the blue component. We also conclude that singleGaussian fits to the Ca H&K feature give biased, unphysical velocity measurements. In Section 3, we provide simple models of calcium and silicon features in a SN Ia spectrum. Trends in the spectra indicate that the blue component of the Ca H&K feature is likely from Si II λ 3858. In Section 4, we perform further analysis with the M12 sample, finding systematic biases in single-Gaussian velocity measurements appear to be present in the M12 analysis. In Section 5, we re-examine the CfA spectral sample, providing further evidence that (1) the blue component of the Ca H&K feature is from Si II λ 3858 absorption and (2) there is no evidence for a correlation between v CaH&K and light-curve shape. Finally, we examine the spectra of the well-observed SN 2011fe and SN 2010ae, a very low-velocity SN Iax, in Section 6. Although one cannot uniquely claim that the blue component of the Ca H&K profile is from Si II λ 3858 for SN 2011fe, it must be the the case for SN 2010ae. We discuss implications of this result and conclude in Section 7.", "pages": [ 1, 2 ] }, { "title": "2 THE CA H&K LINE PROFILE", "content": "As noted above, the spectral feature at rest-frame wavelengths of ∼ 3500 - 4000 ˚ A in SN Ia spectra often has structure such as shoulders and multiple components. The main absorption is thought to be from Ca H&K (from the photosphere and possibly from a HV component) and Si II λ 3858. Wewill refer to this feature as the Ca H&K feature, although there may be additional species contributing to it. In this section, we will examine the Ca H&K feature in detail. To do this, we use the M12 spectra. After testing their claims of differences in the profile shape with light-curve shape, we examine the different results one gets depending on the method of fitting the line profile. M12 suggested that v 0 CaH&K depends on the light-cure shape (and therefore peak luminosity) of the SN. Using the WISERep database (Yaron & Gal-Yam 2012), we ob- Rest Wavelength (Å) tained most of the spectra presented by M12 1 . We exclude all SNe that M12 do not use in their final analysis, including PTF10ufj, which only has a redshift determined by SN spectral feature matching. In total, there are 14 spectra of 14 SNe Ia in the final M12 sample. In Figure 1, we present median spectra from the M12 sample. The Ca H&K feature has two clear minima (at ∼ 3720 and 3800 ˚ A, respectively) in the median spectrum. These data appear to be an excellent sample for studying the Ca H&K profile shape. We also generated median spectra for subsets of the full sample. First, we split the sample by phase. Since the velocity of SN features typically decreases monotonically with time because of the receding photosphere, one expects lower velocity features at later times. The phase-split median spectra do not appear to be significantly different from each other or the median spectrum from the full sample. This is likely the result of the M12 sample having a very narrow phase range. We also split the sample by light-curve shape. We split the sample by s = 1 . 01 to match what was done by M12. Here, we see the same result that M12 found and shows in their Figures 5 and 7. Namely, the low-stretch (corresponding to faster-declines and lower luminosity) SNe have narrower, seemingly lower velocity features than those of high-stretch SNe. Given the above difference, it is worth a detailed look at the line profiles. Despite coming from P-Cygni profiles, the line profiles appear to be similar to the sum of two Gaussians, and performing such a fit resulted in excellent matches to the profiles. The absorption component of a PCygni profile is very similar to a Gaussian, so using Gaussians to fit the absorption is a reasonable choice. In Figure 2, we display the median spectra for the full sample and the low/high-stretch subsamples. We also display the bestfitting double-Gaussian fits (after removing a linear pseudocontinuum) to each line profile. For each case, we performed a six-parameter fit, allowing the centroid, width, and height of each Gaussian to vary. The centroid of each Gaussian corresponds roughly to the characteristic velocity of that component. Similarly, the width of each feature corresponds to the velocity-width of the absorbing region for that feature. Finally, the height of each feature is roughly related to the amount of absorbing material at a given velocity. The six-parameter double-Gaussian fits to each profile are represented by the blue lines in Figure 2. We also fit the low/high-stretch subsamples with two parameters fixed and four allowed to vary. The centroid and width (the parameters related to velocity) of the redder Gaussian was fixed to match the best-fitting values for the full-sample median spectrum, and the remaining parameters (all parameters for the bluer Gaussian and the height of the redder Gaussian) were allowed to vary. These fits are represented by the red lines in Figure 2. Visually, the six-parameter fit is not a significantly better representation of the data than the four-parameter fit. The reduced χ 2 decreases by 0.10 and 0.06 when changing from the six-parameter to the four-parameter fit for the low and high-stretch subsamples, respectively. That is, the four-parameter fit has a smaller reduced χ 2 than the sixparameter fit (although only marginally smaller), and thus, the subsamples and the full sample are completely consistent with all having the same velocity for the red component. M12 argued that the difference in the red edge of the Ca H&K line profile was evidence that the subsamples have different ejecta velocities. But we have shown that simply varying the height of the redder Gaussian (and the bluer Gaussian) are sufficient to produce the red edge of the profile. That is, the apparent different in the red edge can be explained by different line strengths rather than different line velocities, and thus a difference in the red edge is not sufficient to distinguish different velocity features. We also attempted to fix the parameters of the bluer Gaussian, but that did not result in good fits. From these tests, we see that (1) the red component does not necessarily have a different centroid (and thus velocity) for the two subsamples and (2) the blue component does have a different centroid. We now turn to the difficulty of reducing these profiles to a single parameter, namely velocity. There have been two approaches to measure velocities. The first fits a single Gaussian to a line profile and ascribes the centroid of the Gaussian to the velocity of the feature. This method is used by many studies, including M12. The alternative is to measure the wavelength of maximum absorption (usually after some smoothing) to represent the velocity of the feature. This is the method described by Blondin et al. (2006) and used by Foley et al. (2011) and Foley (2012). Although there are many arguments to use either method, we will focus on the potential systematic errors of using these methods when a feature has multiple components like the Ca H&K feature. In Figure 2, we also show a single-Gaussian fit to the Ca H&K feature. Besides being a poor representation of the data, the centroid of the Gaussian is consistently intermediate to the two components. Usually, one wants to measure the photospheric velocity for a given feature. With that goal, the single Gaussian clearly fails. A single Gaussian, by measuring something intermediate to the two components, measures nothing physical. Furthermore, the centroid of the single Gaussian is significantly affected by the blue component. The single Gaussian fits for the subsamples indicate that the low-stretch SNe have significantly lower velocities than the high-stretch SNe. However, the double-Gaussian fits show that this is not the case for the photospheric component. To investigate the importance of the blue component to the measured v CaH&K from these two methods, we created artificial, but realistic, line profiles. In Figure 3, we again show the median spectrum from the M12 sample. We created a double-Gaussian line profile to mimic the profile of the median spectrum. We then varied the height of the bluer Gaussian, but left all other parameters fixed. We display several example line profiles in Figure 3. Visually, all of these line profiles appear physically possible and represented in nature. The full sample of line profiles vary from having no blue component to having a blue component that is about twice as strong as the red component. We fit single Gaussians to all artificial line profiles. We display a subset of these fits in Figure 3 (those that match the subset of profiles displayed). As expected, the stronger the blue component, the bluer the centroid of the Gaussian. In Figure 4, we show the measured v CaH&K from these Gaussian fits. Over the range we explore (from no blue component to a blue component that is twice as strong as the red component), the measured v CaH&K changes by more than 5000 kms -1 . Even when the blue component is about a third as strong as the red component, the measured v CaH&K is ∼ 1000 kms -1 different from the true v CaH&K . We also measured the wavelength of maximum absorption. This wavelength is associated with the blue component when it is stronger and quickly transition its association to the red component as the blue component becomes weaker. The measured v CaH&K for our artificial line profiles is shown in Figure 4. Although this method fails dramatically for strong blue components, the measured v CaH&K is relatively constant for line ratios less than one, with all measured velocities < 1000 kms -1 for all such cases. There is a slight bias ( ∼ 150 kms -1 ) for these measurements, some of which can be explained by increasing flux of the pseudo-continuum with wavelength. Correcting for the pseudo-continuum removes much of the bias, with the remaining bias related to the strength of the blue component. For cases where the red component is stronger than the blue component, measuring the wavelength of maximum absorption is significantly better at measuring the photospheric velocity than using a Gaussian fit to the full profile. In this regime, the wavelength of maximum absorption is only minimally affected by the strength of the blue component, while the Gaussian fit is significantly affected. In the regime of having a stronger blue component, the wavelength of maximum absorption fails. However, in this regime, the Gaussian fit also fails, producing unphysical and significantly biased results. Using the Foley et al. (2011) method of culling v CaH&K measurements that are not representative of the photospheric velocity, one should have reliable v CaH&K measurements, but will necessarily have an incomplete sample. A potential way to avoid this bias would be to perform a doubleGaussian fit. We have not investigated how this method performs with noisy data.", "pages": [ 2, 3, 4, 5 ] }, { "title": "3 SYNOW MODELS", "content": "To further understand the nature of the Ca H&K feature, we use the SN spectrum-synthesis code SYNOW (Fisher et al. 1997) to create simple SYNOW spectral models. We specifically use these models to test how temperature can affect the profile and look for trends between the Ca H&K profile shape and other spectral features. Although SYNOW has a simple, parametric approach to creating synthetic spectra, it can provide insight on basic trends in SN SEDs. To generate a synthetic spectrum, one inputs a blackbody temperature ( T BB ), a photospheric velocity ( v ph ), and for each involved ion, an optical depth at a reference line, an ex- citation temperature ( T exc ), the maximum velocity of the opacity distribution ( v max ), and a velocity scale ( v e ). This last variable assumes that the optical depth declines exponentially for velocities above v ph with an e -folding scale of v e . The strengths of the lines for each ion are determined by oscillator strengths and the approximation of a Boltzmann distribution of the lower-level populations with a temperature of T exc . We produced models consisting of only Ca II and with only Si II and Ca II to isolate their affect on the profile of the Ca H&K feature. For all models, we set T BB = 10000 K, v ph = 10000 kms -1 , v max = 80000 kms -1 , and v e = 3000 (for Si II ) and 2000 km s -1 (for Ca II ). We chose τ = 5 and 4 for Si II and Ca II , respectively. These parameters were chosen such that when T exc = 10000 K, the model Ca H&K line profile was visually similar to that of the median spectrum of the M12 sample. Keeping all other parameters fixed, we varied T exc from 5000 to 20000 K. A subset of the models spanning this range are presented in Figure 5. As seen in Figure 5, the inclusion of Si II dramatically changes the Ca H&K profile shape, making it stronger, broader, and bluer. Although the Si II λ 3858 feature may be stronger in the models than in real SN spectra, the Si II λ 6355 and the Ca H&K features appear to have reasonable strengths. There is a clear spectral progression as the temperature changes. We note that for SYNOW , T bb only changes the continuum shape of the models and does not affect the strength of features. Since SYNOW uses Ca H&K as the reference calcium line, the strength of the Ca H&K absorption by definition does not change much with T exc , and the entire calcium spectrum does not change much over the temperatures probed. Meanwhile the Si II spectrum changes significantly with varying T exc . The strength of the Ca H&K absorption within the Ca H&K feature (i.e., the strength of the red component) does change slightly with T exc because of the strength of the Si II λ 3858 emission changing the apparent Ca H&K absorption. In the red, there is the expected change in the ratio of the Si II λ 5972 and λ 6355 lines. This ratio, R (Si), is highly correlated with luminosity and light-curve shape (Nugent et al. 1995). As the Si II λ 5972 feature becomes stronger, the Si II λ 3858 feature becomes weaker. For the SYNOW models, R (Si) increases with increasing T exc , while for SNe Ia, R (Si) increases with decreasing T ; this has been previously noted (e.g., Bongard et al. 2008), and is likely the result of not simultaneously changing the opacity with T exc and/or non-local thermodynamic equilibrium effects. However, the Ca II spectrum does not change significantly with T exc and other model spectra show the same relation between Si II λ 3858 and Si II λ 5972 (e.g., Kasen & Plewa 2007; Blondin et al. 2012a). We therefore consider the qualitative changes in the spectra to be correct, although the corresponding temperatures may not be. All models show that the strength of Si II λ 3858 and Si II λ 5972 are anticorrelated; we will use this relation as the primary model prediction. We will later use R (Si) as a proxy for light-curve shape. The excitation energy for the various Si II lines also explain the correlations between the various Si II features. The Si II λ 3858, Si II λ 5972, and Si II λ 6355 features have excitation energies of 6.9, 10.0, and 8.1 eV, respectively. Because the Si II λ 3858 and Si II λ 5972 features have very different excitation energies and Si II λ 6355 has an excitation energy intermediate to the other two features, the strengths of the Si II λ 3858 and Si II λ 5972 features should change in opposite directions with changing temperature. This also explains the SYNOW results since SYNOW fixes the strength of the reference feature, Si II λ 6355. In the middle and right-hand panels of Figure 5, we show the Ca H&K feature and redder Si II complex in detail. Again, it is clear that both R (Si) and the strength of the Si II λ 3858 feature change in the way described above. A SN photosphere is, of course, more complicated than the simple SYNOW model. Specifically, as the temperature changes over the relevant range, the ionization of silicon (specifically the amount of singly and doubly ionized silicon) changes. Additionally, certain features may be saturated (and possibly for only certain temperatures). Specifically, it is thought that Si II λ 6355 is usually saturated. However, it seems unlikely that the other Si II features are saturated. Therefore, these additional complexities should not change our interpretation that the strengths of the Si II λ 3858 and Si II λ 5972 features are anti-correlated and change with temperature. We fit the Si II λ 4130, λ 5972, and λ 6355 features in each model spectrum with single Gaussians. Although the line profiles are not exactly Gaussian, the fits are reasonable approximations of the data, and the process is similar to what is done in practice. We also fit the Ca H&K feature with both a single Gaussian and a double Gaussian. We show the measured velocity in Figure 6. The measured Si II λ 4130, and λ 6355 velocities differ by at most 530 and 220 kms -1 over the entire temperature range, respectively. At the lowest T exc , the Si II λ 5972 feature is not strong enough to measure a reliable velocity, but for the other temperatures, it differs by at most 440 km s -1 . These differences are encouraging since the photospheric velocities did not change. Fitting two Gaussians to the Ca H&K feature, which should be better at recovering the true velocity (see Section 2), we see that the red component, corresponding to Ca H&K, has a measured velocity range similarly small to that of the Si II features noted above. Specifically, the maximum difference of measured Ca H&K velocities over all temperatures probed is only 290 kms -1 . However, the measured velocity for the Si II λ 3858 feature changes significantly with temperature. Over the full temperature range probed, the measured Si II λ 3858 velocity ranges from -15,550 to -19 , 380 kms -1 - a difference of 3820 km s -1 , or roughly an order of magnitude greater than that of the other features. A single-Gaussian fit performs even worse. Because of the changing Si II λ 3858 velocity and its varying strength, a single-Gaussian fit to the Ca H&K feature results in a v CaH&K range of -13,250 to -17 , 670 kms -1 over our chosen temperature range for a maximum difference of 4420 kms -1 . Since the v CaH&K measured using a single Gaussian can have dramatic differences even when there is no change in physical velocities, this is even more reason to avoid this technique. Figure 6 also shows the Si II λ 5972 to Si II λ 6355 ( R (Si)) and Si II λ 3858 to Ca H&K ratios, which we will call the Si/Ca ratio. The range for R (Si), from effectively zero (when Si II λ 5972 is difficult to discern) to ∼ 0.3, is approximately the range seen by all SNe Ia except SN 1991bg-like objects (e.g., Blondin et al. 2012b; Silverman et al. 2012). The Ca/Si ratio has a range of 1.2 to 2.3. The ratio is affected by both the strength of the Si II λ 3858 absorption and the Si II λ 3858 emission , which fills in some of the Ca H&K absorption. As noted above, the strength of the Si II λ 3858 feature can have a large affect on the Ca H&K line profile, and even dominates for many temperatures. Since Si II λ 5972 and λ 6355 do not show significant velocity differences with temperature, are relatively free of contamination from other species, and R (Si) is a good indication of light-curve shape, we can use it as a proxy for light-curve shape for our models. Figure 7 shows v CaH&K measured with a single Gaussian as a function of R (Si). The measured v CaH&K decreases in amplitude with increasing R (Si), which corresponds to decreasing stretch and luminosity. Converting stretch to R (Si), we can plot the M12 measurements on Figure 7. The M12 spectra do not cover the redder Si II features, so a direct measurement could not be made. The M12 values, which use a single Gaussian to fit the Ca H&K feature, span a similar range of v CaH&K and inferred R (Si) as the models, with the model trend going through the middle of the data values. The claimed trend of v CaH&K with light-curve shape is clear using the M12 values. However, this trend is similar to the trend generated by simply changing the temperature in the SYNOW models. We emphasize that the true v CaH&K is fixed for all models and the v CaH&K measured using a double-Gaussian fit varies only slightly over all models. The trend shown in Figure 7 is solely the result of the method for measuring the velocity. The underlying physical effect is the changing strength of Si II λ 3858 with temperature, and thus light-curve shape. There will undoubtedly be a true range of velocities in the data. We have not explored how the SYNOW models change when varying several parameters, but it is clearly possible to reproduce the M12 trend through a combination of single Gaussian fitting, an inherent velocity range, and relations between the Ca H&K profile shape with temperature and velocity.", "pages": [ 5, 6, 7 ] }, { "title": "4 THE M12 SAMPLE", "content": "With the insights of the above analysis, we re-analyse the individual M12 spectra. In Section 2, we showed that fitting the Ca H&K profile with a single Gaussian results in an imprecise, biased, and unphysical measurement of the photospheric velocity. Instead, we fit the Ca H&K profiles of the M12 spectra with two Gaussians. As suggested above, this method should provide relatively unbiased measurements of the Si II λ 3858 feature and the Ca H&K feature. We provide the best-fitting velocities for Si II λ 3858 assuming that it is 'HV' Ca H&K and Ca H&K in Table 1. We also provide the Si/Ca ratio in Table 1. Figure 8 compares the velocities measured with the double-Gaussian fit to those reported by M12. The v CaH&K is systematically lower than the M12 value for the same SNe. Similarly, the Si II λ 3858 (when treated as HV Ca H&K) is systematically higher than the M12 value for the same SNe. Performing a Bayesian Monte-Carlo linear regression on the M12-Si II and M12-Ca H&K data sets (Kelly 2007), we find that 99.8 and 86.8 per cent of the realizations have positive slopes, respectively, corresponding to 3.1σ and 1.5σ results, respectively. That is, there is significant evidence for a linear relation between the M12 v CaH&K measurements and the Si II λ 3858 velocity, but no evidence for a linear relation between the M12 v CaH&K measurements and the velocity of the red component corresponding to absorption from photospheric calcium. As expected from the results of Section 3, the M12 v CaH&K measurements are intermediate to the Si II λ 3858 and Ca H&K velocities, more closely track the Si II λ 3858 velocity than the Ca H&K velocity, and are systematically biased measurements of v CaH&K . For much of the analysis performed by M12, they corrected their measured v CaH&K to a maximum-light value, v 0 CaH&K , using a single velocity gradient for all objects derived from the v CaH&K and effective phase measurements for their sample. Using a large sample where many objects had multiple spectra obtained near maximum light, Foley et al. (2011) showed that v 0 CaH&K and the velocity gradient were highly correlated with higher-velocity SNe also having higher-velocity gradients. Taking into account this relation, they produced an equation which estimates v 0 CaH&K given v CaH&K and phase. Using the M12 velocity gradient results in differences between v CaH&K and v 0 CaH&K that can be as large as 1150 kms -1 , and the median difference is 460 km s -1 . However, if one uses the Foley et al. (2011) relation to correct the velocity measurements to have a common phase of 2.7 d (the median of the sample), then the deviation between that value and v CaH&K is at most 560 kms -1 , with a median absolute deviation of 120 km s -1 , both of which are smaller than the typical uncertainty of 670 kms -1 for the M12 v 0 CaH&K measurements. The combination of using a single velocity gradient and extrapolating to maximum light (only one spectrum in the sample has a phase before maximum brightness) introduces unnecessary additional uncertainty. Instead in all further analysis, we use the raw v CaH&K measurements, but add an additional 120 kms -1 uncertainty in quadrature to the reported uncertainty. Using our measurements of v CaH&K , we can re-examine the M12 claim that v CaH&K is correlated with light-curve shape. In Figure 9, we show the Si II λ 3858, Ca H&K, and M12 velocity measurements as a function of stretch. This figure is similar to Figure 7. Performing a Bayesian Monte-Carlo linear regression on the M12, the Si II λ 3858, and Ca H&K velocity measurements, we find that 97.7, 97.8, and 87.6 per cent of the realizations have positive slopes, respectively, corresponding to 2.3σ , 2.3σ , and 1.5σ results, respectively. We therefore find mild evidence that the M12 and Si II λ 3858 velocity measurements are linearly related to stretch. We find no statistical evidence that v CaH&K is linearly related to stretch. M12 found a 3.4σ linear relation between their v 0 CaH&K measurements and stretch. Above, our measured significance is much lower. This is partly because in the above calculation, we did not include SNe 2011by and 2011fe since we do not have the M12 spectra. Including the reported values for these SNe, there is only a minor change in the significance, changing the percentage of realizations with positive slopes to be 98.7 per cent, which is a 2.5σ result. The other difference is that above we examine v CaH&K instead of v 0 CaH&K . Performing the same analysis as M12 (using v 0 CaH&K and in- uding SNe 2011by and 2011fe), we find that 99.4 per cent of the realizations have positive slopes, which is a 2.8σ result. The difference in significance is likely in the subtleties of fitting a line. This practice is not trivial (Hogg, Bovy, & Lang 2010), but the Kelly (2007) method is generally a better choice than most options. We also performed a Kolmogorov-Smirnov (KS) test, splitting the samples by a stretch of 1.01. This is not an ideal test since there are several SNe with stretches consistent with 1.01 (and therefore could be in either group) and uncertainty in the velocity can also change the overall distribution, but it can provide an indication of a difference. The KS test resulted in p values of 0.0014, 0.0036, and 0.080 for the M12, the Si II λ 3858, and Ca H&K velocity measurements, respectively. These tests indicate that the low and high-stretch subsamples have different parent populations for both the Si II λ 3858 and M12 velocities. However, there is no statistical evidence that the low/high-stretch subsamples have different parent v CaH&K distributions. Although there is only marginal evidence that there is a linear relationship between the M12 measurements and stretch, we find a similar significance of a relationship between Si II λ 3858 velocity and stretch. Since the v CaH&K shows no evidence for a correlation with stretch and the M12 measurements are correlated with the Si II λ 3858 velocity (and at most weakly correlated with the Ca H&K velocity), the physical relationship underlying the result identified by M12 is likely the correlation between Si II λ 3858 velocity and stretch. From the SYNOW models, this relation is understood as a temperature effect (and not a real difference in photospheric velocity). Next, we examine the Si/Ca ratio. We show the Si/Ca ratio as a function of stretch for the M12 sample in Figure 10. Performing a Bayesian Monte-Carlo linear regression on the data, we find that 82.4 per cent of the realizations have positive slopes, corresponding to a 1.4σ result. Although there is no evidence for a linear relationship between stretch and the Si/Ca ratio in the M12 data, there is a slight correlation with higher stretch SNe having larger Si/Ca ratios. This is the same general trend expected from the SYNOW models, and future investigations should determine if such a trend exists. Finally, we compare the Si/Ca ratio to our measured velocities (Figure 11). There are no obvious trends (1.0 and 1.7σ ) between the Si/Ca ratio and the Si II λ 3858 or Ca H&Kvelocities. However, there is a moderate trend between the Si/Ca ratio and the M12 measurements (2.7σ ), where the M12 velocities increase with increasing Si/Ca ratio. One should expect that the single-Gaussian method (as employed by M12) should be intermediate to the Si II λ 3858 and Ca H&K velocities. The velocity should be closer to the Ca H&K velocity when the Si II λ 3858 feature is weak (small Si/Ca ratio) and closer to the Si II λ 3858 velocity when the feature is strong (large Si/Ca ratio). The behavior seen in the data is reproduced to some extent by the SYNOW models. Figure 11 also shows the SYNOW model Si/Ca ratio compared to the Si II λ 3858, Ca H&K, and single-Gaussian velocities. The Ca H&K velocity is relatively flat for the SYNOW models, while the Si II λ 3858 and single-Gaussian velocities increase with increasing Si/Ca ratio. The Ca H&K and single-Gaussian trends are similar in the data and models, but the data have higher velocities than the data (by about 700 and 2200 km s -1 , respectively). There is no obvious trend in the Si II λ 3858 data, contrary to what is seen in the models. The models are restricted to Si/Ca ratios of > 1. If we fit the data with the same restriction, the fits are more similar to the slopes seen in the models. Specifically, the trend with Ca H&K is flatter and the trend with Si II λ 3858 is stronger, although neither trend is statistically significant. Perhaps a simple linear relation is not sufficient to describe the trend between the Si/Ca ratio and Si II λ 3858 velocity.", "pages": [ 7, 8, 9, 10 ] }, { "title": "5 THE CFA SAMPLE", "content": "Although the M12 sample was used to identify some trends which were investigated above, it is limited in size. To test additional trends, we use the CfA spectral sample (Blondin et al. 2012b). Over the last two decades, the CfA SN Program has observed hundreds of SNe Ia, mostly with the FAST spectrograph (Fabricant et al. 1998) mounted on the 1.5 m telescope at the F. L. Whipple Observatory. The data have been reduced in a consistent manner (Matheson et al. 2008; Blondin et al. 2012b), producing well-calibrated spectra. These spectra often cover Ca H&K and always cover Si II λ 6355, which the M12 spectra do not cover. For SNe Ia in the sample with a measured time of maximum brightness from light curves, v Si II and v Ca H&K have been measured (Blondin et al. 2012b). Briefly, this is achieved by first generating a smoothed spectrum using an inverse-variance Gaussian filter (Blondin et al. 2006), and the wavelength of maximum absorption in the smoothed spectrum is used to determine the velocity (see Blondin et al. 2012b for details). The measurements for each spectrum have been reported by Foley et al. (2011), and measurements in all cases were obtained by Blondin et al. (2012b) The CfA sample contains 1630 v Si II and 1192 v Ca H&K measurements for 255 and 192 SNe Ia, respectively. The velocity of each absorption minimum in the Ca II H&K feature of the smoothed spectra is automatically recorded (Blondin et al. 2012b). We only examine spectra with one or two minima. If two minima are found, the higher/lower velocities are classified as 'blue'/'red.' If only one minimum is found, it is categorized as the red or lowervelocity component. Foley et al. (2011) noted that comparing all v Ca H&K measurements to their corresponding v Si II measurements, there were two distinct 'clouds' corresponding to a lower and higher velocity relative to v Si II . The higher-velocity cloud typically corresponds to the blue velocity component, although there are some red measurements in that cloud. The red measurements in the blue cloud typically have indications of a lower-velocity component, such as a red shoulder in the line profile, and it was assumed that they likely corresponded to measurements which were physically similar to the blue measurements and were simply misclassified. In Figure 12, we show the subset of CfA measurements of spectra with -1 /lessorequalslant t /lessorequalslant 4 . 5 d (chosen to match the M12 sample). This subset also shows the distinct blue/red clouds. We used the method of Williams, Bureau, & Cappellari (2010) to fit a single slope, but separate offsets to the two clouds. As a result of that fitting, there is a natural dividing line between the two clouds, and we used this line to produce cleaner subsamples. We removed every blue measurement in the red cloud, since they may be errant measurements. We also reassigned every red measurement in the blue cloud as a 'blue' measurement because of the reasons listed above. This full process is shown graphically in the three panels of Figure 12. With these clean subsets, we have a reasonable estimate of the velocities for the blue and red components of the Ca H&Kfeature. Since some SNe in the CfA sample have multiple spectra in the chosen phase range, we created samples for each velocity group where there is one measurement per SN. For each SN, we chose the measurement closest to a phase of t = 2 . 7 d, the median of the M12 sample. This resulted in samples of 66 and 67 SNe Ia with blue and red measurements (approximately one-third of the full CfA sample and about 5 times as large as the M12 sample), respectively. We present those measurements as a function of light-curve shape (specifically, ∆ m 15 ( B )) in Figure 13. There is no significant linear relation between light-curve shape and the individual velocity components. In fact, the stronger relationship of the red velocity component, which is the best representation of the Ca H&K photospheric velocity, is a 1.3 σ result in the opposite direction than the M12 relation (i.e., higher velocity for slower-declining SNe Ia). Because of this opposing trend, the CfA data are significantly inconsistent with the M12 relation. However, this result is consistent with that of Foley et al. (2011) and Foley (2012) for both Ca H&K and Si II λ 6355. In addition to the arguments detailed above, there is additional evidence in the CfA data that suggest that the blue component of the Ca H&K feature is the result of Si II λ 3858. If the blue component is predominantly from HV Ca H&K, then one would not expect a particularly high correlation between its velocity and that of Si II λ 6355. That is, the velocity of a HV calcium component could be independent of the photospheric silicon velocity. However, there is a reasonable correlation (correlation coefficient of 0.54) between the two. On the other hand, the velocities of the red and blue components are barely correlated (correlation coefficient of 0.27). Therefore, the velocity of the blue component has a larger association with the photospheric velocity of silicon than the photospheric velocity of calcium. Perhaps the best evidence that the blue component is from Si II λ 3858 absorption is presented in Figure 12. In the right panel, we plot the relation between v Si II and v CaH&K in the scenario where the v CaH&K measurement is from a misidentified Si II λ 3858 feature at the same velocity as Si II λ 6355. The line goes directly through the blue cloud, indicating that the blue component of the Ca H&K feature has a velocity consistent with that of Si II λ 6355 if it is formed by Si II λ 3858 absorption. In other words, the blue component is at the wavelength one expects by blueshifting 3858 ˚ A by v Si II . Addtionally, Blondin et al. (2012b) presented several examples of SNe where the blue component was consistent with Si II λ 3858 at the Si II λ 6355 velocity, while the red component was consistent with the velocity of the Ca NIR triplet. From the large CfA sample, we showed additional evidence that v CaH&K does not correlate with light-curve shape. The velocity of the blue component is correlated with the photospheric silicon velocity (as measured by Si II λ 6355) and relatively uncorrelated with the photospheric calcium velocity. In addition to being correlated with photospheric silicon velocity, the CfA data show that the velocity of the blue component matches the expected photospheric velocity of Si II λ 3858.", "pages": [ 10 ] }, { "title": "6 ADDITIONAL MODELING", "content": "From the above analysis of the SYNOW models, the comparison of the M12 sample to the SYNOW models, and an examination of the CfA sample, there is significant evidence that the blue component of the Ca H&K feature is predominantly from Si II λ 3858. However, additional confidence in this claim can be obtained by modeling specific SNe. In this section, we examine the two possible scenarios for the blue component of the Ca H&K feature (either HV calcium or Si II λ 3858) for two test cases: SN 2011fe and SN 2010ae.", "pages": [ 11 ] }, { "title": "6.1 SN 2011fe", "content": "SN 2011fe, which occurred in M 101 and was the brightest SN Ia in 40 years, has been incredibly well observed and extensively studied (e.g., Nugent et al. 2011; Brown et al. 2012; Chomiuk et al. 2012; Horesh et al. 2012; Margutti et al. 2012; Matheson et al. 2012; Parrent et al. 2012; Shappee et al. 2013). Here we examine a single maximum-light spectrum of SN 2011fe, obtained by HST using the STIS spectrograph (Program GO-12298; PI Ellis). The spectra were obtained on 2011 September 10 between 09:51 and 11:14 UT, corresponding to t = 0 . 0 d relative to B -band maximum brightness (M12). The observations were obtained with three different gratings, all with the 52 '' × 0 . '' 2 slit. Two exposures were obtained for each of the CCD/G230LB, CCD/G430L, and CCD/G750L setups with individual exposure times of 530, 80, and 80 s, respectively. The three setups yield a combined wavelength range of 1665 - 10,245 ˚ A. The data were reduced using the standard HST Space Telescope Science Data Analysis System (STSDAS) routines to bias subtract, flat-field, extract, wavelength-calibrate, and flux-calibrate each SN spectrum. We present the spectrum in Figure 14. We note that M12 presented a spectrum for SN 2011fe from a different phase and which only covered ∼ 2900 - 5700 ˚ A. This is the first publication of these data. This is also only the second published maximum-light SN Ia spectrum to probe below ∼ 2500 ˚ A (the first being of SN 2011iv; Foley et al. 2012c). The SN 2011fe spectrum is of extreme high quality, including in the UV. Because of its quality and wavelength coverage, we can produce a reasonable SYNOW model. We have made two attempts at fitting the SN 2011fe spectrum using SYNOW models. The first assumes that the blue component of the Ca H&K feature is from Si II λ 3858; the second assumes that it is caused by HV Ca H&K. We present our models in Figure 14 and model parameters in Table 2. When generating these models, we first attempted to fit the full spectrum with a limited number of species. These models are not optimized to fit the entire spectrum; because of potential effects other species could have on the spectral features of interest, we wanted a first-order model of the full spectrum. We then either added HV Ca II or adjusted the Si II temperature to match the blue component of the Ca H&K feature. We allow the opacity and density structures for Ca II , Si II , HV Ca II , and Na I to vary, but all other species remain the same. For the HV calcium model, we adjusted the Si II temperature to an extreme value that still fits the Si II λ 5972 feature. In this model, we do not include any Na I , and therefore Na D does not contribute at all to this feature. As a result, the Si II λ 3858 is about as weak as possible, and the HV Ca H&K is essentially as strong as possible. For the Si II λ 3858 model, we adjust the Si II temperature to an extreme value to match the blue feature in the Ca H&K feature. We then add Na I to match the strength of the feature near 5800 ˚ A. These models differ in some ways from those presented by Parrent et al. (2012) for their optical-only maximumlight SN 2011fe spectrum. Most differences are related to matching the UV region, which requires adding Co II and Cr II . Interestingly, adding these features reduces the need to include Fe II in the SYNOW model (although we cannot definitively say that it is not in the spectrum). Additionally, we are able to better model the Ca H&K feature than Parrent et al. (2012) because of the additional data blueward of the feature. Examining the SYNOW models in detail, particularly near the Ca H&K feature, the redder Si II features, and the Ca NIR triplet (see lower panels of Figure 14), we see that the models are very similar. In other words, SYNOW modeling of SN 2011fe cannot distinguish between our two scenarios; it simply has too many parameters for the data. We did not adjust the models to fit the Ca NIR triplet, with the hope that we might see signatures of HV Ca. There is a feature in the SYNOW model that is coincident with a shoulder in the SN 2011fe spectrum. However, we see a similar feature in the Si II λ 3858 model that is simply the result of a slightly different density profile for Ca II . A full spectral sequence and/or NIR spectra, which would supply additional Si II features, may provide a clear way to distinguish the models.", "pages": [ 11, 12 ] }, { "title": "6.2 SN 2010ae", "content": "With an inconclusive result from modeling SN 2011fe, we now turn to modeling SN 2010ae. SN 2010ae is a SN Iax (Foley et al. 2012a) similar to SN 2008ha (Foley et al. 2009, 2010; Valenti et al. 2009). Its spectrum is similar to that of a SN Ia, but with an extremely low ejecta velocity. This indicates that the ejecta composition, density structure, temperature, and other aspects of the explosion important for producing a particular SED are similar for SN 2010ae and SNe Ia. However, because of the low ejecta velocity, line blending is minimal. We present a near maximum-light spectrum of SN 2010ae originally presented by Foley et al. (2012a) and presumed to be obtained near maximum light in Figure 15. This spectrum only covers optical wavelengths. We dereddened the spectrum by E ( B -V ) = 0 . 6 mag to roughly match the continuum of SN 2011fe and smoothed the spectrum with a inverse-variance weighted Gaussian filter and velocity scale of 150 km s -1 . Perhaps the most important aspect of the SN 2010ae spectrum is that the Ca H&K feature is separated into two distinct features. We then attempted to produce SYNOW model spectra in a way similar to what was performed for SN 2011fe. As a starting point, we used the SN 2011fe models. We decreased v phot from 9000 kms -1 to 3000 kms -1 . We also reduced minimum and maximum velocities for each species. The details of the models are presented in Table 2. We did not change the majority of parameters for the model. As a result, the fits are not ideal. In particular, the lack of C II results in missing obvious features. Additional adjustments would certainly improve the overall fit, but this is not necessary for our purpose. However, keeping the model similar to that of a SN Ia (with mostly just adjustments to the velocity) reinforces the spectral (and compositional) similarities between SNe Iax and SNe Ia. Additionally, we changed the opacity of Si II and Ca II , and we changed the density structure of Ca II . We adjusted the opacity of Si II to roughly match the Si II λ 6355 feature. The Ca II opacity was changed to roughly match the NIR triplet. The velocity of the HV calcium and the density structure of both the HV and photospheric calcium were adjusted to match the Ca H&K feature. Ca H&K are offset by 34.8 ˚ A, which corresponds to 2640 kms -1 . The velocity difference between the two components will be present even if Ca H&K are blueshifted. For most SNe, the ejecta velocities are high enough where the two components blend together completely. But for SN 2010ae, which has an ejecta velocity similar to this separation, any Ca H&K feature will be roughly twice the width of a feature from a single line. For SN 2010ae, the blue component of the Ca H&K feature has a FWHM of 2960 kms -1 . Therefore, the Ca H&K components can barely fit within the width of the feature (with a velocity of ∼ 11200 kms -1 , about 4 times that of the photospheric velocity), but then the line can only be minimally broadened. That is unphysical, but if it were the case, then one would expect two components within the blue component, which is not seen. The only other choice is to choose a velocity which results in either Ca H or Ca K to have a minimum near 3800 ˚ A. Doing this for Ca H results in a velocity of ∼ 13000 kms -1 and a significant absorption feature at ∼ 3760 ˚ A, where no such feature exists. When assigning a velocity of ∼ 10000 kms -1 for HV calcium (such that Ca K is at ∼ 3800 ˚ A), there is no gap between the blue and red components. Neither option reproduces the observed profile for SN 2010ae. Alternatively, the Si II λ 3858 model roughly matches the spectrum of SN 2010ae. In particular, it reproduces the (now unblended) Ca H&K feature. The HV calcium model, on the other hand, does not reproduce a key aspect of the Ca H&Kfeature - its unblended nature. It is reasonably certain that Si II λ 3858 causes the absorption of the blue component of the normally blended Ca H&K feature for SN 2010ae. Furthermore, removing the HV Ca from the HV Ca model does not have two distinct features. It appears necessary to have a reasonably strong Si II λ 3858 feature to produce the emission between the two components. Since Foley et al. (2012a) showed that SNe Iax have very similar spectra to SNe Ia, except with different velocities, and since the SN 2011fe SYNOW model roughly matches the SED of SN 2010ae (with only differences in the velocity), one can extrapolate this result to SNe Ia.", "pages": [ 13, 14 ] }, { "title": "7 DISCUSSION & CONCLUSIONS", "content": "Wehave shown through a re-examination of the M12 sample, a re-examination of the CfA sample, basic SYNOW modeling, and more thorough SYNOW modeling of SNe 2010ae and 2011fe that the blue component of the Ca H&K spectral feature in near-maximum light SN Ia spectra is typically from Si II λ 3858 absorption. This was also the interpretation of Wang et al. (2003), which has spectropolarimetric observations of Ca H&K, Si II λ 6355, and the Ca NIR triplet, providing additional weight to this conclusion. Some previous claims that the component is the result of HV Ca H&K absorption may require re-examination. The Ca NIR triplet has shown HV features for some SNe, although it is also possible to reproduce some of these features with a different (but still smooth) density profile for calcium (see Section 6.1). Therefore, it is still unclear if HV calcium contributes to the Ca H&K component, how frequently it does, and if that contribution is typically blended with Si II λ 3858. The realization that the blue absorption in the Ca H&K profile is from Si II λ 3858 for most SNe Ia has far-reaching implications for our understanding of SN Ia progenitor systems and explosion models, which have interpreted the prevalence of HV calcium as an indication of specific explosion mechanisms and potentially a tracer of the environment of the progenitor system. Because the Ca H&K profile is a combination of Ca H&K and Si II λ 3858, v CaH&K should not be measured by fitting the entire Ca H&K feature with a single (Gaussian) component. Regardless of the source of the two components, we also show that if one does fit the profile with a single Gaussian component that the resulting measurements will be unphysical, inaccurate, and highly biased. However, because of the true nature of the blue component, a single Gaussian fit is particularly biased. We confirmed the M12 result that SNe in their sam- ple have different Ca H&K line profiles based on light-curve shape. However, the difference is mostly constrained to the blue component, with no evidence for a difference in velocity or width for the red component. We re-examined the claim that v 0 CaH&K is correlated with light-curve shape (M12). Using the reported M12 measurements, we do not find a statistically significant linear relation, but the KS test does indicate different parent populations for low/high-stretch subsamples. When using v CaH&K measurements from the red component of the Ca H&K profile for the M12 spectra, there is no statistically significant trend between v CaH&K and light-curve shape. An analysis of the CfA sample also showed that there is no correlation between ejecta velocity and light-curve shape, confirming the previous results of Foley et al. (2011) and Foley (2012). Instead, the underlying physical effect driving the relation between the M12 measurements and light-curve shape is likely the relation between Si II λ 3858 and temperature. This result implies that the M12 claim that v 0 CaH&K does not correlate with host-galaxy mass is not supported by data. Other claims made by M12 related to v 0 CaH&K , including correlations between v 0 CaH&K and the wavelengths or velocities of certain features, should also be re-examined. From modeling, there is some indication that the Si/Ca ratio should be a strong tracer of temperature and an indicator of light-curve shape, but this is not verified with data. There may also be a relatively low correlation between v CaH&K and the pseudo-equivalent width of the Ca H&K feature. This may be why Foley et al. (2011) did not find a relation between the pseudo-equivalent width of the Ca H&K feature and the intrinsic colour of SNe Ia. Foley et al. (2011) and Foley (2012) suggested that v 0 CaH&K could be useful for measuring the intrinsic colour of SNe Ia. However, this current analysis shows that this approach may be limited by the contamination of Si II λ 3858. At the very least, SNe with very high ejecta velocities will have a Ca H&K profile that is a blend of Si II λ 3858 and Ca H&K with no distinct components. At that point, one should be circumspect of the derived velocity. The culling technique of Foley et al. (2011) should reduce the number of spectra with velocity measurements contaminated by Si II λ 3858, but relatively low signal-to-noise ratio (S/N) spectra, galaxy contamination, and other nuisances, may reduce the viability of this option. There is a proposal to have a low-resolution ( R ≈ 75) spectrograph on WFIRST (Green et al. 2012). The main purpose of the spectrograph for SN science would be spectroscopic classification and redshift determination. Similarly, the SED Machine (Ben-Ami et al. 2012), is a proposed R ≈ 100 spectrograph to classify thousands of low-redshift SNe. Another use of these spectrographs could be to measure ejecta velocities. Assuming perfect knowledge of the SN redshift, the precision of the ejecta velocity measurement can be limited by spectroscopic resolution. To test our ability to determine ejecta velocities with different resolutions, we show artificial Ca H&K line profiles that contain two components in Figure 16. One cannot distinguish the two components of the profile at R = 50; there are ∼ 4 resolution elements in the feature, which is insufficient for a full six-parameter fit of a double-Gaussian fit. Additionally, the two components are separated by ∼ 6000 kms -1 , corresponding to R ≈ c/ 6000 km s -1 ≈ 50. At R = 75, one can start to see the effect of the two components in some spectra (i.e., flat bottoms), but the components are still not clearly separate. A resolution of 100 may be the minimal amount to clearly see the effects of multiple components. But considering additional effects such as potential [O II ] λ 3727 emission from the host galaxy contaminating the line profile, one might want a higher resolution, such as R = 200, where narrow lines should not significantly affect the overall profile shape. However, we note that Si II λ 6355 does not suffer these same problems, and R = 75 should provide accurate (and reasonably precise) measurements of the ejecta velocity. For optical spectrographs, one can easily measure v Si II to z = 0 . 3. With red-sensitive CCDs and good sky subtraction one can use optical spectrographs to measure v Si II to z ≈ 0 . 6. With NIR spectrographs, one can easily measure v Si II to z ≈ 2 (neglecting the faintness of the SNe). For the SED Machine, which aims to classify lowredshift SNe, it should also be able to measure v Si II . The proposed spectrograph on WFIRST would have a wavelength range of 0.6 - 2 µ m, which should cover v Si II to z ≈ 3, well beyond the expected redshift range of WFIRST. Rodney et al. (2012) presented an HST observer-frame NIR spectrum of a z = 1 . 55 SN Ia, SN Primo. The spectrum has a low S/N and is low-resolution ( R ≈ 130). But using the method of Blondin et al. (2006), we measure v Si II = -11200 ± 900 kms -1 at a phase of 6 ± 3 d, corresponding to v 0 Si II = -11700 ± 1000 kms -1 . This corresponds, using the Foley et al. (2011) relations, to ( B max -V max ) 0 = 0 . 00 ± 0 . 07 mag. The uncertainty in the velocity measurement is dominated by the low S/N of the spectrum, but the uncertainty in the intrinsic colour is still dominated by the uncertainty and scatter in the velocity-colour relation. None the less, SN Primo appears to be have a moderate intrinsic colour. This shows the potential of using velocity measurements for SN Ia cosmology even if the complexities of the Ca H&K profile prevents accurate measurements. The additional knowledge of the Ca H&K profile provided here is a step toward further understanding of the full SED of SNe Ia. SNe Iax, which have compositions similar to that of SNe Ia, can be exceedingly useful for determining which specific atomic transitions contribute to SN Ia spectra. Because of their low ejecta velocities, SNe Iax may provide additional insight into the specific contributions from various lines for blended SN Ia features. Similarly, additional spectropolarimetric observations of SNe Ia, and particularly those that cover both Ca H&K and the Ca NIR triplet, NIR spectra, and good spectral sequences starting at early times should produce additional insight into the formation of a SN Ia SED.", "pages": [ 14, 15 ] }, { "title": "ACKNOWLEDGMENTS", "content": "Facilities: HST(STIS) We thank D. Kasen, R. Kirshner, and J. Parrent for their comments, insights, and help. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.", "pages": [ 15, 16 ] }, { "title": "REFERENCES", "content": "Altavilla G. et al., 2007, A&A, 475, 585 Ben-Ami S., Konidaris N., Quimby R., Davis J. T., Ngeow C. 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[ { "title": "ABSTRACT", "content": "Through a large ensemble of Gaussian realisations and a suite of large-volume N -body simulations, we show that in a standard ΛCDM scenario, supervoids and superclusters in the redshift range z ∈ [0 . 4 , 0 . 7] should leave a small signature on the Integrated Sachs Wolfe (ISW) effect of the order ∼ 2 µ K. We perform aperture photometry on WMAP data, centred on such superstructures identified from SDSS LRG data, and find amplitudes at the level of 8 - 11 µ K - thus confirming the earlier work of Granett et al. (2008b). If we focus on apertures of the size ∼ 3 . 6 · , then our realisations indicate that ΛCDM is discrepant at the level of ∼ 4 σ . However, if we combine all aperture scales considered, ranging from 1 · -20 · , then the discrepancy becomes ∼ 2 σ , and it further lowers to ∼ 0 . 6 σ if only 30 superstructures are considered in the analysis (being compatible with no ISW signatures at 1 . 3 σ in this case). Full-sky ISW maps generated from our N -body simulations show that this discrepancy cannot be alleviated by appealing to Rees-Sciama (RS) mechanisms, since their impact on the scales probed by our filters is negligible. We perform a series of tests on the WMAP data for systematics. We check for foreground contaminants and show that the signal does not display the correct dependence on the aperture size expected for a residual foreground tracing the density field. The signal also proves robust against rotation tests of the CMB maps, and seems to be spatially associated to the angular positions of the supervoids and superclusters. We explore whether the signal can be explained by the presence of primordial non-Gaussianities of the local type. We show that for models with f local NL = ± 100, whilst there is a change in the pattern of temperature anisotropies, all amplitude shifts are well below < 1 µ K. If primordial non-Gaussianity were to explain the result, then f local NL would need to be several times larger than currently permitted by WMAP constraints. Key words: cosmology: observations - cosmic microwave background - large-scale structure of the Universe - galaxies: clusters: general", "pages": [ 1 ] }, { "title": "Carlos Hern'andez-Monteagudo 1 /star & Robert E. Smith 2 †", "content": "7 March 2022", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Over the last fifteen years, evidence has been mounting from various cosmological probes to support the case for an accelerating universe. It can be argued that the first compelling evidence for this arose from the study of the light curves of distant type Ia supernovae (Riess et al. 1998; Perlmutter et al. 1999). Currently, the strongest support for this picture comes from the combination of observations of the Cosmic Microwave Background radiation (hereafter CMB), and from measurements of the clustering of galaxies. From the CMB side, the Wilkinson Microwave Anisotropy Probe (hereafter, WMAP) experiment (Spergel et al. 2003, 2007; Komatsu et al. 2011) 1 provided a precise measurement of the angular size of the sound horizon at recombination, which supported the case for a spatially flat universe. On the clustering side, data from surveys like the 2-degree Field Redshift Survey (Cole et al. 2005) and the Sloan Digital Sky Survey (hereafter SDSS) (Tegmark et al. 2006) required the density in matter to be sub-critical, hence leading to the inference that ∼ 70 per cent of the current energy density of the Universe is in a form of energy that behaves like a cosmological constant, and so acts as a repulsive gravitational force. The energy density driving the accelerated expansion is unknown and so has been dubbed Dark Energy (hereafter DE). Uncovering the true physical nature of the DE is one of the main targets for many ongoing and upcoming surveys of the Universe. Standard linear cosmological theory states that if the Universe undergoes a late-time phase of accelerated expansion, then gravitational potential wells on very large scales ( > ∼ 100 h -1 Mpc) will decay. This evolution of the potential wells introduces a gravitational blueshift in the photons of the CMB that is known as the Integrated Sachs-Wolfe effect (ISW). This effect constitutes an alternative window to DE, and can be directly measured by cross-correlating CMB maps with a set of tracers for the density field, which sources the potentials (Crittenden & Turok 1996). As soon as the first data sets from WMAP were released, several works claimed detections of the ISW at various levels of significance (Scranton et al. 2003; Fosalba et al. 2003; Boughn & Crittenden 2004; Fosalba & Gazta˜naga 2004; Nolta et al. 2004; Afshordi et al. 2004; Padmanabhan et al. 2005; Cabr'e et al. 2006; Giannantonio et al. 2006; Vielva et al. 2006; McEwen et al. 2007). Subsequent analysis has led some researchers to be more cautious about interpreting these early detections (Hern'andez-Monteagudo et al. 2006; Rassat et al. 2007; Bielby et al. 2010; L'opez-Corredoira et al. 2010; Hern'andez-Monteagudo 2010; Francis & Peacock 2010). As emphasized by Hern'andez-Monteagudo (2008), the true ISW effect should only be detectable for deep galaxy surveys that cover a substantial fraction of the sky. However an erroneous interpretation of ISW cross correlation studies may be obtained from systematic errors, such as residual point source emission in CMB maps or presence of spurious galaxy auto-power on large angular scales (see also Hern'andez-Monteagudo 2010, for details on the WMAP - NVSS cross-correlation analysis.). In particular, the issue of excess power on large scales has been noted in several works (Ho et al. 2008; Hern'andez-Monteagudo 2010; Thomas et al. 2011; Giannantonio et al. 2012), but it is not yet fully accounted for. Amongst the subsequent ISW cross-correlation studies in the literature, there is the particularly puzzling work of Granett et al. (2008b, hereafter G08). Their analysis yielded one of the highest detection significances in the literature. G08 implemented the following novel approach: they produced a catalogue of superclusters and supervoids from SDSS data, and stacked WMAP-filtered data on the positions of these structures with apertures of the order ∼ 4 · . They obtained a ∼ 4 σ ISW detection. Unlike previous works on the subject, the analysis focused on a particular subset of the available large-scale structure (hereafter LSS) data. Subsequent studies have investigated the origin of the signal and assessed its compatibility with the ΛCDM scenario, (P'apai & Szapudi 2010; P'apai et al. 2011; Nadathur et al. 2012). While some works found the G08 results compatible, others found the measured amplitudes too high to be consistent (Granett et al. 2009; Nadathur et al. 2012; Flender et al. 2012; Inoue et al. 2010; Inoue 2012). Currently, the G08 results remain unexplained. In this work we shall attempt to shed new light on this problem. The paper breaks down as follows: In § 2 we give a brief theoretical overview of the ISW effect, underlining the expectation for the ISW signal from a cross-correlation anal- sis of the data used in G08. In § 3 we perform a cross-check of the G08 results. In § 4 we test whether the G08 results are consistent with expectations for the ΛCDM model. This is achieved by using a large ensemble of Gaussian Monte-Carlo realisations. We also generate full-sky nonlinear ISW maps by ray-tracing through a suite of N -body simulations. In § 5 we determine the level of significance at which the results disagree with the ΛCDM paradigm. We explore systematic errors in the foreground subtraction for WMAP. We also investigate whether the excess signal is consistent with primordial non-Gaussianities of the local type. Finally, in § 6 we summarize our findings and conclude. While this paper was in the process of submission, a parallel work on this subject from Flender et al. (2012) appeared in the internet. This work is reaching to similar conclusions to ours in some of the issues addressed in this work. Unless stated otherwise, we employ a reference cosmological model consistent with WMAP7 (Komatsu et al. 2011): the energy-density parameters for baryons, CDM and cosmological constant are Ω b = 0 . 0456, Ω cdm = 0 . 227, Ω Λ = 0 . 7274; the reduced Hubble rate is h = 0 . 704; the scalar spectral index is n S = 0 . 963; the rms of relative matter fluctuations in spheres of 8 h -1 Mpc radius is σ 8 = 0 . 809, and the optical depth to last scattering is τ = 0 . 087.", "pages": [ 1, 2 ] }, { "title": "2.1 The ISW effect", "content": "Observed CMB photons are imprinted with two sets of fluctuations: primary anisotropies, sourced by fluctuations at the last-scattering surface, and secondary anisotropies, induced as the photon propagates through the late-time clumpy Universe. The physics of the primary anisotropies is well understood (Dodelson 2003; Weinberg 2008). There are a number of physical mechanisms that give rise to the generation of secondary anisotropies (for a review see The Planck Collaboration 2006) and one of these is the redshifting of the photons as they pass through evolving gravitational potentials. The linear version of this effect is termed the Integrated Sachs-Wolfe effect (Sachs & Wolfe 1967) and its nonlinear counter-part is termed the Rees-Sciama effect (Rees & Sciama 1968). The observed temperature fluctuation induced by gravitational redshifting may be written as (Sachs & Wolfe 1967): where ˆ n is a unit direction vector on the sphere, Φ is the dimensionless metric perturbation in the Newtonian gauge, which reduces to the usual gravitational potential on small scales, the 'over dot' denotes a partial derivative with respect to the coordinate time t from the FLRW metric, χ is the comoving radial geodesic distance χ = ∫ cdt/a ( t ), and so may equivalently parameterize time. The symbols t 0 and t ls denote the time at which the photons are received and emitted, i.e. the present time and last scattering. c is the speed of light and a ( t ) is the dimensionless scale factor. On scales smaller than the horizon, relevant to our simulation boxes, the perturbed Einstein equations in Newtonian gauge lead to a perturbed Poisson equation. This enables us to relate potential and matter fluctuations (Dodelson 2003): where ¯ ρ ( t ) is the mean matter density in the Universe and the density fluctuation is defined δ ( x ; t ) ≡ [ ρ ( x , t ) -¯ ρ ( t )] / ¯ ρ ( t ). This equation may most easily be solved in Fourier space: Differentiation of the above expression gives where we used the fact that [ a 3 ( t )¯ ρ ( t )] is a timeindependent quantity in the matter-dominated epoch and Ω m ≡ Ω cdm +Ω b . In the above, we also defined H ( t ) ≡ ˙ a ( t ) /a ( t ) and Ω m ( t ) ≡ ¯ ρ ( t ) /ρ crit ( t ), with ρ crit ( t ) = 3 H 2 ( t ) / 8 πG . All quantities with a subscript 0 are to be evaluated at the present epoch. In the linear regime, density perturbations scale as δ ( k , a ) = D ( a ) δ ( k , a 0 ). Inserting this relation into Eq. (4) gives, In the matter-dominated phase, the Universe expands as in the Einstein-de Sitter case and consequently density perturbations scale as D ( a ) ∝ a . Thus, for most of the evolution of the late-time Universe the bracket, [1 -d log D/d log a ] is close to zero. In the ΛCDM model it is only at relatively late times that this term is non-zero. /negationslash Alternatively, in the nonlinear regime δ ( k, a ) = D ( a ) δ ( k, a 0 ), and this gives rise to additional sources for the heating and cooling of photons (Smith et al. 2009; Cai et al. 2009, 2010).", "pages": [ 2, 3 ] }, { "title": "2.2 Expectations for cross-correlation analysis", "content": "The expected signal for the ISW-density tracer crosscorrelation analysis has been described by several authors (Crittenden & Turok 1996; Hern'andez-Monteagudo 2008; Smith et al. 2009) and here we simply quote the main results. For a survey of galaxies (or more generically, objects that are biased density tracers) the ISW-density angular cross-power spectrum can be written: where P m ( k ) denotes the linear matter power spectrum at the present time, the super-script g refers to the objects ( galaxies in most cases, but in ours it will refer to voids or superclusters) probing the gravitational potential wells. In the above n g ( χ ) denotes the average comoving object number density; W g ( χ ) denotes the instrumental window function providing the sensitivity of the instrument to the objects at distance χ ; the j /lscript ( x ) are the usual spherical Bessel functions of order l ; and b ( χ, k ) denotes the bias of the tracer population with respect to the matter density field. This last factor may be both a function of time χ and scale k (Smith et al. 2007), but for simplicity we shall simply assume that the bias equals unity for the objects in our catalogue. In this measurement the primary CMB signal acts as noise . If we take this into account and also the variance due to the density field, then the S / N with which we expect to detect the cross-correlation for a given multipole is (e.g., Crittenden & Turok 1996): where C CMB /lscript and C g /lscript represent the auto power spectra of the temperature fluctuations and the density tracers; ¯ n g denotes the average angular object number density and hence the term 1 / ¯ n g denotes the shot-noise. In our forecast we shall neglect this term (hence the predicted significance will be slightly over optimistic). The factor f sky refers to the fraction of the sky jointly covered by both the CMB and the object surveys. The cumulative S / N for all harmonics smaller than l can then be written: where the monopole and dipole have not been included.", "pages": [ 3 ] }, { "title": "2.3 Expectations for ISW from data used in Granett et al.", "content": "We now turn to the question of what S / N should G08 have expected to find in the their data. G08 used a sample of 1.1 million Luminous Red Galaxies (hereafter LRG) selected from the SDSS data release 6 (hereafter DR6) (Adelman-McCarthy et al. 2008). This sample covered roughly 7500 square degrees and spanned a redshift range of (0 . 4 < z < 0 . 75). From this sample they identified regions as supervoids or superclusters using the algorithms ZOBOV and VOBOZ 2 , respectively, (Neyrinck 2008; Neyrinck et al. 2005). The significance of these regions was chosen to be at least at the 2 σ -level relative to a Poisson sample of points. G08 selected the largest 50 superclusters and supervoids for their analysis. Their catalogue is publically available 3 . In the left panel of Fig. 1, the dot-dashed and dotted histograms denote the redshift distributions of supervoids and superclusters, respectively. The thick solid line represents an analytic fit that we have constructed, which attempts to be a compromise between the two. Our fitting function extends to slightly lower redshifts than the G08 catalogue, and this will translate into a slightly higher ISW prediction. The middle panel of Fig. 1 presents the angular crosspower spectrum as predicted by linear theory for the void/supercluster catalogue (solid line). Recall that we are taking b = 1 for both voids and clusters. In reality the void/cluster regions will be anti-biased/biased and so the two signals will differ. However, since here we are more concerned with the S /N this simplification does not matter, especially since we are neglecting the effects of shot-noise on the cross-spectra. We also point out that our predictions for the ISW-induced cross correlation signal of superclusters and supervoids in the SDSS sample are quite similar to the predictions for the cross-correlation of AGNs ( z < 2) in the NVSS catalogue (after adopting the model of Ho et al. 2008, see the dashed line in the middle panel of Fig. 1). The right panel of Fig. 1 presents the predictions for the cumulative S / N for the SDSS supercluster and supervoid analysis. These predictions (solid black line) show that, for a ΛCDM model, one should expect no more than /similarequal 1 . 3 σ significance. In contrast, the prediction for NVSS (dashed line) is close to ∼ 5.5 (obtained after also neglecting shot noise). This is not surprising, since the G08 supervoid and supercluster catalogue is relatively shallow, spanning the redshift range (0 . 4 < z < 0 . 7), and covers only a modest fraction of the sky ( f sky /similarequal 0 . 18). The NVSS is instead significantly deeper and wider. We hence conclude that had G08 applied a standard ISW cross-correlation analysis to their data, then in the framework of the ΛCDM model, there would have been very little chance for detecting any genuine signal at high significance.", "pages": [ 3, 4 ] }, { "title": "3.1 CMB data", "content": "The WMAP experiment scanned the CMB sky from 2001 until 2010 in five different frequencies, ranging from 23 GHz up to 94 GHz. The angular resolution in each band improves with the frequency, but it remains better than one degree in all bands. The S / N is greater than one for multipoles /lscript < 919 (Jarosik et al. 2010), and in particular, on the large scales of interest for ISW studies, the galactic and extragalactic foreground residuals are below the 15 µ K level outside the masked regions (Gold et al. 2011). We concentrate our analysis on the foreground-cleaned maps corresponding to bands Q (41GHz), V (61GHz) and W (94GHz), after applying the conservative foreground mask KQ75y7, which excludes ∼ 25% of the sky. At the scales of interest, instrumental noise lies well below cosmic variance and foreground residuals, and hence will not be considered any further. The ISW is a thermal signal whose signature should not depend upon frequency and hence should remain constant in the three frequency channels. All of the WMAP data employed in this analysis were downloaded from the LAMBDA site 4 .", "pages": [ 4 ] }, { "title": "3.2 Supercluster and Supervoid data", "content": "For our tracers of the LSS, we use the same supercluster and supervoid catalogue as used by G08. As described earlier, the catalogue was constructed after applying the ZOBOV and VOBOZ algorithms to search for supervoids and superclusters in the LRG sample extracted from SDSS DR6, respectively. G08 used the 50 largest supervoids and superclusters. They claimed that this cut yielded the highest statistical significance, in that it minimized the contamination from spurious objects, whilst at the same time it provided sufficient sampling to beat down the intrinsic CMB noise.", "pages": [ 4 ] }, { "title": "3.3 Methodology", "content": "In their approach G08 have applied a top-hat compensated filter or Aperture Photometry (AP) method to the CMB map(s) positions of voids and superclusters. This filter subtracts the average temperature inside a ring from the average temperature within the circle limited by the inner radius of the ring. In order to have equal areas in both cases, the choice of the outer radius of the ring is √ 2 R , with R the inner radius of the ring. In this way, fluctuations of typical size R are enhanced against fluctuations at scales smaller or larger than such radius. Although G08 present results for apertures ranging from 3 · up to 5 · , most of the conclusions are driven from the R = 4 · choice, for which highest statistical significance is achieved: they find that AP stacks on the position of voids (superclusters) yield a decrement (increment) of ∼ -11 . 3 µ K (7 . 9 µ K) at 3.7 (2.6) σ significance level. However, it turns out that, according to G08, the typical size of clusters and voids are ∼ 0.5 · and 2 · , respectively, which seem to lie at odds with the aperture choice of 4 · . Potentials are known to extend to larger scales than densities, and it is a priori unclear which aperture radius should be used. This fact motivates a systematic study in a relatively wide range of aperture radii.", "pages": [ 4, 5 ] }, { "title": "3.4 Stacking analysis on real data", "content": "We apply the G08 method on the Q, V, and W bands of the WMAP data using the SDSS supercluster and supervoid catalogues. We have considered AP filters in 15 logarithmically spaced bins in the angular range 1 · -20 · . The filters were placed on the centers of the objects, as they are provided by the catalogue. Figure 2 displays the results for the stacked signal as a function of the AP filter aperture size in degrees. The red, green and blue symbols refer to results from the Q, V and W bands, respectively. We clearly see that there is practically no frequency dependence. The error bars are computed after repeating the analysis on 30 random sets of 50 objects placed in the un-masked region of the sky. Our findings are in good agreement with those of G08: voids and supercluster regions yield a slightly asymmetric pattern, with voids rendering amplitudes of /similarequal -11 µ K for apertures of /similarequal 3 . 6 · , and superclusters giving rise to increments of /similarequal 9 µ K at that same scale. In these two cases, the significance is about -3.3 σ and 2.3 σ , which on combination yields a combined significance ∼ 4 σ . Again, this is in good agreement with G08. Furthermore, the scale showing highest S / N is /lessorsimilar 4 · , as the significance rapidly drops for smaller and larger apertures. Intuitively, this seems to be in contradiction with the idea of ISW fluctuations being large-scale anisotropies, since in such case one would expect to attain high S / N also for moderately large ( /greaterorsimilar 5 · -10 · ) apertures.", "pages": [ 5 ] }, { "title": "4.1 Gaussian realisations", "content": "What is not clear from the analysis of the previous section, is whether the ∼ 4 σ detection from the stacked supercluster/supervoid regions is consistent with what one expects from the standard ΛCDM model (Komatsu et al. 2011). We now attempt to understand the dependence of the expected signal and its errors on the filter size. To that end, we repeat the above analysis on a set of Gaussian realisations of both the LSS distribution and the corresponding CMB temperature anisotropy distribution which would result in our ΛCDM model. The CMB maps are constructed in a two-step process: first, we generate a Gaussian map of projected density, following the angular power spectrum built upon the redshift window function W g ( r ) displayed in the left panel of Fig. 1. This density map is used for (i) constructing a supercluster and supervoid catalogue (see below), and (ii) generating an ISW component. This ISW component is correlated to the density map as predicted by Eq. 6, (see, e.g., Cabr'e et al. 2006; Hern'andez-Monteagudo 2008, for details.) Second, we generate the primary anisotropy signal at z /similarequal 1 050. This is taken to be completely uncorrelated with respect to the projected density map. This is not exactly true due to the lensing of the CMB, but it is still a very good approximation on the large angular scales of interest in this study. The simulated ISW map is then directly co-added to the primary CMB temperature map. We have checked that the cross-correlations of the simulated density and CMB maps are in direct agreement with a numerical evaluation of Eq. 6. All of the simulated sky maps were generated using the equal area pixelization strategy provided by HEALPix 5 , (G'orski et al. 2005). We take the pixel scale to be /similarequal 15 ' , corresponding to a HEALPix resolution parameter of N side = 256. For each simulated LSS map we smooth the map with a Gaussian aperture of FWHM /similarequal 2 · , and identify those peaks and troughs which exceed a given threshold ν | σ | as being associated with supervoids and superclusters, where σ in this context refers to the density field rms. The threshold ν is chosen to have an object density similar to that of superclusters and voids in the real catalogue under the SDSS footprint. Note that this Gaussian smoothing takes place only at the step of 'identifying' superstructures in the density maps. To check the dependence on the threshold choice, we bracket the preferred value of ν with two other values, one above and one below it. The final choice for the ν value set was 2.5, 2.8 and 3.2. To each simulated CMB sky, we also add a noise realization following the anisotropic noise model provided in the LAMBDA site. We then exclude pixels in accordance with the intersection of the WMAP KQ75y7 sky mask and the SDSS DR6 data footprint. Figure 3 presents the ensemble-averaged results obtained from 5000 Gaussian Monte-Carlo realisations. As expected, for the three adopted thresholds, overdensities (or positive excursions in the projected density map) yield a positive signal for the stacked aperture analysis (as displayed by the squares in the plots), whereas underdensities yield negative ones (circles in the plot). The Monte-Carlo realisations also enable us to compute the variance on the measurements. On taking the ratio of the mean signal and the rms noise, we obtain direct estimates of the S / N for each given aperture bin. The coloured solid lines in the plot present our direct measurements of the S / N . Thus we clearly see that the scatter induced by the CMB generated at z /similarequal 1050 is the dominant source of noise, keeping the S / N for each angular bin below unity. For lower thresholds there is more area covered and intuitively one would expect a higher S / N , as it seems to be the case. Our realisations also provide higher ISW amplitudes for higher thresholds, and this makes sense since deeper voids/potential wells should have a stronger impact on CMB photons. Nevertheless, in all cases typical amplitudes remain at the level of 1-2 µ K. The aperture at which the AP outputs provide the highest amplitude does not show any strong dependence upon the threshold ν , and seems to lie in a wide angle range within [3 · , 8 · ]. For apertures larger than 10 · , the S / N for the lowest threshold starts dropping slowly, and becomes half of its maximum value at an aperture of 20 · . For higher thresholds this decrease is found to be even shallower.", "pages": [ 5, 6, 7 ] }, { "title": "4.2 Generation of nonlinear ISW and density maps from N -body simulations", "content": "The previous subsection has shown that the Gaussian realizations of the ΛCDM universe are in tension with the excess signal found by G08. One weak point in the above analysis is that the density and late-time potential field are not necessarily well described by a Gaussian process, since nonlinear evolution under gravity drives the initially Gaussian distribution of density fluctuations towards one that is non-Gaussian at late times. In order to test whether nonlinear evolution could explain the excess signal seen by G08, we now turn to the challenge of constructing fully nonlinear maps of the density field and ISW effect from N -body simulations. The 8 simulations that we employ for this task are a sub-set of the zHORIZON simulations. These simulations were used in Smith et al. (2009) to calculate the expected ISW-cluster cross-power spectra. In brief, each simulation follows the gravitational evolution of N = 750 3 dark matter particles in a box of comoving size L = 1500 h -1 Mpc. The cosmological model employed was a flat ΛCDM model: Ω m 0 = 0 . 25; σ 8 = 0 . 8; n s = 1 . 0; h = 0 . 72, Ω b , 0 = 0 . 04. The transfer function for the simulations was generated using the cmbfast code (Seljak & Zaldarriaga 1996). The initial conditions were lain down at redshift z = 49 using the code 2LPT (Scoccimarro 1998; Crocce et al. 2006). Each initial condition was integrated forward using the publicly available cosmological N -body code Gadget-2 (Springel 2005). Snapshots of the phase space were captured at 11 logarithmicallyspaced intervals between a = 0 . 5 and a = 1 . 0. In order to generate full-sky nonlinear ISW maps we roughly follow the strategy described in Cai et al. (2010), but with some minor changes. Full details of how we construct our maps can be found in Appendix A. In summary, we used the density and divergence of momentum fields to solve for ˙ Φ for each snapshot. We then constructed a backward light-cone from z = 0 . 0 to z = 1 . 0 for ˙ Φ. We then pixelated the sphere using the HEALPix equal-area decomposition, taking the pixel resolution to be N side = 256, which corresponds to 786,432 pixels on the sphere. For each pixel location, we then fired a ray through the past light-cone of ˙ Φ and accumulated the line-of-sight integral given by Eq. (1). Note that we only consider the ISW signal coming from z < 1, since we do not expect a significant cross-correlation between the relatively low-redshift density slices for SDSS and the ISW from z > 1. The top panel of Fig. 4 shows one of the ISW maps that we have generated from the zHORIZON simulations. We next generated the projected density maps. These were done by first constructing the projected density map associated with each snapshot a i . The density field for a given snapshot was obtained as follows. To each snapshot a l we associate a specific comoving shell [ χ l -1 / 2 , χ l +1 / 2 ] (see Appendix A for more details). We then select all of the particles that fall into the shell for that epoch, i.e. the i th dark matter particle in the a l th snapshot, is accepted in the shell if χ l -1 / 2 < | x i -x O | /lessorequalslant χ l +1 / 2 , where x i and x O are the coordinates of the i th dark matter particle and the observer, respectively. Note that if a given value of χ is larger than L/ 2 , 3 L/ 2 , 5 L/ 2 , . . . , then we apply periodic boundary conditions to produce replications of the cube to larger distances. If the particle is accepted, then we compute the angular coordinates ( θ, φ ) for the particle, relative to the observer. Given these angular coordinates, we then find the associated HEALPix pixel and increment the counts in that pixel. The bottom panel of Fig. 4 shows the projected overdensity map for a thin redshift shell centred on z = 0 . 3. We note that it is hard, by eye, to note any apparent correspondence between the overdensity and temperature maps. At the end we have 11 density maps between z = 1 . 0 and z = 0 . 0 that form concentric shells around the observer. These shells were then co-added using the weights given by our analytic fit to the redshift distribution of superclusters and supervoids (recall the solid line in the left panel of Fig. 1). The resulting co-added all-sky density maps are then smoothed from N side = 256 to N side = 32, and the positions of the 2 n , n and n/ 2 most under- and over-dense pixels on this map are recorded. The number n corresponds to a number density of extrema that is identical to that of real voids and superclusters under the footprint of SDSS DR6. Each of those extreme pixels on the N side = 32 map is then projected back to the N side = 256 map on a subset of 64 higher-resolution pixels, out of which the position of the most under- or over-dense pixel is used as the target of the AP filter.", "pages": [ 7, 8 ] }, { "title": "4.3 Validation tests of zHORIZON derived maps", "content": "Before we apply the analysis methods of G08 to our ISW and density maps, we first test the consistency of the maps themselves. To do this, we compute the angular auto-power spectra of the ISW temperature maps for each of the 8 zHORIZON runs. The left panel of Fig. 5 presents the ensemble-averaged temperature power spectrum for the ISW effect. The measurements from the simulations are represented by the solid black points. The prediction from linear theory is given by the solid red line. For multipoles in the range 5 < /lscript < 70, the agreement between the two is excellent. At low multipoles ( /lscript < 5) the absence of power in the simulation on scales larger than L induces a low bias. Instead, on scales /lscript > 70, the non-linear evolution of potentials substantially boosts the signal relative to linear by means of the Rees-Sciama effect. We next compute the angular auto-power spectrum of the projected density contrast maps, with the projection extending from z = 0 . 1 up to z = 1 (middle panel of Fig. 5). In this case, the projected density angular power spectrum shows good agreement with the linear-theory expectations in the intermediate- and high-multipole range (5 < /lscript < 200), and some hints for power deficit on the large scales/low /lscript -s, which would probably be due to the lack of k modes beyond the box size of the simulations. Finally, the right panel of Fig. 5 compares the ISW - density cross-correlation estimated from the simulations with the linear theory. As for the other cases, there exists some power deficit at low multipoles due to finite volume effects in the simulations. At high multipoles the linear theory prediction lies above the simulations, this owes to the fact that in the deeply non-linear regime potentials do not decay, but grow with time through the Reese-Sciama mechanism, and hence this leads to a suppression of power (for further details see Smith et al. 2009). On intermediate angular scales the theoretical prediction is roughly ∼ 10 per cent higher than the simulations, although with significant scatter. This slight mismatch is likely due to the construction of the weighted projected density field, since the ISW autospectra are in excellent agreement with the simulations.", "pages": [ 8 ] }, { "title": "4.4 Aperture analysis of the zHORIZON maps", "content": "Having validated the simulated maps, we now repeat the AP analysis. Since here we have both full-sky density and temperature maps, we prefer not to apply any sky mask. Thus, these predictions will not be affected by incomplete sky coverage. We repeat the steps described in § 3 for finding the locations of the density peaks/troughs in each of the simulated smoothed maps. Then, as before, we apply the AP filters to the selected centroid positions for the 8 ISW maps from the zHORIZON simulations. Figure 6 presents the results from this analysis. The square and circular symbols denote the results for our effective supercluster and supervoid regions, respectively. The blue, green and black colours correspond to the set of extreme pixels which have half, equivalent and double the angular number density of the real supervoids and superclusters found in G08's analysis. We see that the peak of the average AP output has a temperature of the order 2 µ K, and occurs for apertures of scale ∼ 6-7 · . On comparison with the predictions from our Gaussian realisations, we find that the fully nonlinear ISW maps are in close agreement (c.f. § 4.1). The are however small differences. The peak signal is shifted to slightly larger scales for the full nonlinear case. Also, the shape of the curves obtained from the zHORIZON simulations appears smoother than in the Gaussian simulation case, which shows dips and troughs that are absent in Fig. 6. We believe that these small differences are likely a consequence of the fact that the Gaussian realisations include intrinsic CMB noise and possess a sky-mask. Actually, after applying the real sky masks to the simulated maps, we find that the peak amplitudes and the general shapes of the functions in Fig.(6) become distorted at the ∼ 10 per cent level. However, the most important point to note, is that on angular scales of the order 3-4 · , the Gaussian and fully nonlinear simulations are in close agreement: the difference induced by adopting a slightly different cosmological model should introduce changes in the ISW amplitude at the 2 per cent level, and the ISW generated beyond z = 1 seems to have little impact as well. We thus conclude this section by noting that that the excess temperature signal found by G08, and now confirmed by us in § 3, appears to be incompatible with the evolution of gravitational potentials in the standard ΛCDM model. Our results from both Gaussian realisations and ISW maps derived from the N-body sim- ions are in agreement with Granett et al. (2008a), who found no signature at the few degree scale on voids and clusters with the amplitude found on real WMAP data when producing an ISW map out of the distribution of Luminous Red Galaxies in Sloan data.", "pages": [ 8, 9 ] }, { "title": "5.1 Estimating the significance", "content": "The direct comparison of Fig. 2 and 3 reveals clear differences between the observed data and theoretical predictions. Not only is the amplitude of the maximum signal in the real data a factor of ∼ 5 times larger than the average in the Gaussian realisations, but the dependence of the signal on the filter scale shows a different shape. More quantitatively, the results from the W-band WMAP data for an aperture size of 3.6 · , are of the order ∼ 3 . 4 σ away from the supervoid simulation average, and ∼ 2 . 1 σ away from the average for the case of superclusters. In terms of probability, for an aperture scale of 3.6 · , only 5 out of the 5000 realisations possessed an ISW signal, from supervoid regions, with a temperature decrement lower than the one found in real data, and 97 of the realisations for superclusters exceeded the value obtained for the real data. Taken at face value, this analysis seems to exclude the Gaussian ΛCDM hypothesis at ∼ 4 σ significance. However, this is an a posteriori estimate, since we have neglected the fact that we also looked for a signal at other aperture scales. If, for the W-band WMAP data, we include the measurements from the 15 different angular aperture scales between 1-20 · and take into account their covariance, then the significance drops. Under the assumption of Gaussian statistics, we find that the WMAP outputs for voids produce a χ 2 voids = 23 . 5 ( n dof = 15). The corresponding figure for superclusters is χ 2 superclusters = 26 . 6 ( n dof = 15). If we treat these two constraints as being independent then their combination yields χ 2 both = 50 . 0 ( n dof = 30). In terms of probability, this means that the WMAP data have a 0.012 probability (i.e. < 2 per cent chance or ∼ 2.2 σ under Gaussian statistics) of being consistent with the evolution of gravitational potentials in the ΛCDM model. If we consider the null hypothesis of no ISW signatures expected at all (for which stacking on voids and superclusters should leave no temperature decrement/increment), then the results lie at 2 . 6 σ away from this scenario. We next study the dependence of the statistical significance on the number of substructures considered in the analysis. While the original catalogue of voids and superclusters of G08 contains 50 entries, we now repeat our tests after considering two subsamples containing only the first 30 and 40 objects. For these subsamples, the adopted values of the Gaussian threshold were ν = 2.87 and 2.97. In these cases, the pattern found for the full catalogue is reproduced: the stacked voids give a temperature decrement of /lessorsimilar -10 µ K, at ∼ 3-3 . 5 σ ; the stacked superclusters give a temperature increment of /greaterorsimilar 7 µ K, at the level of ∼ 2 . 2-2 . 4 σ . On comparison with Gaussian realisations, we find that, after considering all aperture radii, results for the first 40 superstructures are in lower tension with the outputs of Gaussian realisations (at the level of < 3 . 0 per cent or 1 . 9 σ ). This level of tension further decreases when considering only the first 30 superstructures ( ∼ 27 per cent or 0 . 6 σ ), showing that the tension of WMAP data wrt to Gaussian realisations relaxes as fewer structures are included in the analysis. This is somehow expected from the Gaussian realisations, for which the statistical significance for the ISW increases with decreasing thresholds. This is in apparent contradiction with Table 1 of G08, where it is shown that the statistical significance of their ISW measurement at a scale of 4 · decreases when increasing the number of structures from 50 (4 . 4 σ ) to 70 (2 . 8 σ ). In G08 it is argued that by considering more structures one may be diluting the signal by including unphysical structures, an extent that cannot be tested in our Gaussian maps since it is strictly associated to the algorithms identifying voids and superclusters in the galaxy catalogues. In summary, according to our Gaussian simulations, the ∼ 4 σ deviation wrt ΛCDM expectations found at ∼ 4 · aperture radius decreases to ∼ 2 . 2 σ when including different filter apertures in the range [1 · , 20 · ], and lies, in this case, 2 . 6 σ away from the null (no ISW) case. While this tension relaxes when considering fewer structures, the significance of the detected signal seems to decrease when considering more than 50 voids and superclusters (see Table 1 of G08), in an opposite trend to what is suggested by our Gaussian ISW realisations. We conclude that most of the significance of the G08 result is at odds with ISW ΛCDM predictions, both in amplitude and scale/aperture radius dependence, and that this tension considerably reduces when more aperture radii and structure sub-samples are considered in the analysis.", "pages": [ 9, 10 ] }, { "title": "5.2 Tests for systematics", "content": "Given the high level of discrepancy existing between the pattern found at 3 . 6 · aperture radius and ISW ΛCDM expectations, we next test the possibility of systematics in WMAP data giving rise to the observed signal. signal (apart from CMB) contributing to the uncertainty of the AP filter outputs and hence modifying the statistical significance found for WMAP data. We rotate in galactic longitude (in steps of 9 · ) the AP filter targets with respect to the real positions of supervoids and superclusters. In the absence of systematics, this should provide AP outputs compatible with zero. Figure 7 presents the results from this analysis. The blue circular symbols represent the supervoid regions and the red square symbols denote the superclusters. At zero rotation lag we clearly obtain a signal of higher amplitude than in any other rotation bin. We have verified that this signal does not arise as a consequence of a small subset of the supervoids/superclusters. Instead, the signal is approximately evenly distributed among all structures. From the sample of rotated bins only, the estimated significance for the 3.6 · aperture is 4.1 σ for voids, 2.7 σ for superclusters and 3.8 σ combined. This is within 1σ from the significance levels obtained with the Gaussian realisations. of Fig. 2. Actually, the profile from real data in Fig. 2 seems to be an intermediate case between the scenario depicted in Fig. 8 and the theoretical predictions of Fig. 3. However, if most of the observed amplitude at 3.6 · ( ∼ 8 -2 = 6 µ K out of the total ∼ 8 µ K observed) is to be caused by this type of contaminant, then the scale dependence of the output should accordingly be much closer to the one shown in Fig. 8, and this is not the case. Note that we have assigned ∼ 2 µ K to ISW in this estimation. If contaminants in the position of the AP targets were Poisson distributed, then the profiles obtained in Fig. 8 would approach zero faster as aperture radii increase, yet in stronger disagreement with Fig. 2. in our simulations (and this goes beyond the scope of the current work). Having said this, the stability of our results with respect to the actual density peak threshold adopted, suggests that this would not critically affect our conclusions.", "pages": [ 10, 11, 12 ] }, { "title": "5.3 Primordial non-Gaussianity", "content": "In this section we explore whether the G08 result is compatible with a non-Gaussian distribution for the primordial potential perturbations. In particular, we consider the well known local model for primordial non-Gaussianity, characterized by a quadratic correction to the Gauge invariant Bardeen's potential perturbation (see Komatsu et al. 2011, and references therein): In order to explore the observable consequences of such a modification, we have generated a set of simulated ISW maps with Gaussian initial conditions, i.e. f local NL = 0, and with non-Gaussian initial conditions f local NL = { +100 , -100 } . These maps were generated from N -body simulations seeded with Gaussian and non-Gaussian initial conditions following the methodology of AppendixA. The simulations that we employ were fully described in Desjacques et al. (2009). In brief, these were performed using Gadget-2 , and followed N = 1024 3 dark matter particles in a box of size L = 1600 h -1 Mpc. The cosmological model of the simulations was consistent with the WMAP5 data (Komatsu et al. 2009). We use a sub-set of these simulations that were used elsewhere for gravitational lensing analysis (Marian et al. 2011; Hilbert et al. 2012). The simulations were set up to have the same initial random phases for all three models, this enables us to cancel some of the cosmic variance and so permit us to better explore the model differences. where Φ G ( x ) is the Gaussian potential perturbation after matter radiation equality, scaled in terms of units of c 2 to yield a dimensionless quantity. Following standard convention, Φ NG ( x ) ≡ -Φ Newton ( x ) (i.e. the Φ in Eq. (1)). The term 〈 Φ 2 G ( x ) 〉 is subtracted to ensure that Φ NG is a mean zero field. In linear theory the typical fluctuations are of the order Φ NG ∼ 10 -5 , and so the non-Gaussian corrections are of the order ∼ 0 . 1%( f local NL / 100)(Φ G / 10 -5 ) 2 . The top panel of Fig. 9 presents the differences between the ISW temperature maps in a Universe with f local NL = +100 and f local NL = 0. The bottom panel shows the same but for the case f local NL = -100 and f local NL = 0. Note that all of the maps were smoothed with a Gaussian filter of FWHM = 1 · before being differenced. Note also that we have only included the ISW contributions between z = 0 . 0 and z = 1 . 0. Clearly, the presence of primordial non-Gaussianities can induce shifts in the temperatures of the peaks and troughs of the distribution. However, these shifts are modest for f local NL = ± 100, leading to changes that are < 1 µ K. This suggests that values of f local NL on the order of ∼ 1000, might be able to explain the AP analysis of the WMAP results. However, such large values for f local NL would be grossly inconsistent with the values of f local NL obtained from the CMB temperature bispectrum, which currently gives -10 < f local NL < 74 (95% C.L., Komatsu et al. 2011). It therefore seems unlikely that the scale-independent local model of primordial nonGaussianity is the correct explanation for the excess signal.", "pages": [ 12 ] }, { "title": "6 CONCLUSIONS", "content": "In this work we have studied the imprint of superclusters and supervoids in the temperature map of the CMB from the WMAPexperiment. Our work further explores the signature first detected in Granett et al. (2008b, G08). In § 2 we theoretically showed that if G08 had applied a standard angular cross-power spectrum analysis of the superstructures they found in the SDSS LRG data, then the expected significance for a ΛCDM model should have been < 1 . 5 σ . In § 3 we cross-checked the G08 analysis directly and found identical conclusions: on scales ∼ 3 . 6 · there was a ∼ 4 σ detection significance for excess signal associated with the supervoids and superclusters. In § 4 we performed a series of tests exploring whether these findings are consistent with the standard ΛCDM model. Gaussian Monte-Carlo realisations of the ISW effect and the LSS were unable to produce such large signals. We then investigated whether this was a consequence of our simplified Gaussian realisations. We did this by generating fully non-linear ISW maps from large volume N -body simulations. These simulated maps confirmed the findings of the simpler Gaussian realisations. In § 5 we used the Gaussian Monte-Carlo realisations to explore the significance of the deviations from the ΛCDM model found in the WMAP data. We found that for aperture photometry analysis of the maps on scales 3 . 6 · , results from WMAP data are lying about ∼ 4 σ away from ΛCDM expectations. However, on taking into account the 15 aperture scales examined, the significance of the discrepancy dropped to < 2 per cent chance (2 . 2 σ ) of the result being consistent with the ΛCDM model. In this case, results remained 2 . 6 σ away from the null (ISW-free) scenario where structures leave no signatures on the CMB at the linear level. When including fewer structures in the analysis, the tension dropped further, and results for only 30 voids and superclusters were compatible both with ΛCDM expectations (at 0 . 6 σ ) and the null (no ISW) scenario (at 1 . 3 σ ). Our simulations also suggested that the ISW significance should increase when more structures were included in the analysis, in apparent contradiction with the findings of G08. Hence, most of the detected signal appeared associated to the full set of 50 superstructures and an aperture radius of 3 . 6 · . We investigated whether the observed pattern at a radius of 3 . 6 · could be caused by a systematic error in the cleaning of foregrounds in the WMAP data. We found that if the signal were to be caused by an approximately frequencyindependent emission tracing the density field, then the resulting angular dependence would be very different to the measured shape found in the WMAP data. We next explored whether the observed signal at 3 . 6 · could be generated by primordial non-Gaussianities. We considered the local model, characterized by a quadratic correction to the primordial potential perturbations, with the coupling parameter f local NL . We found that, for f local NL positive/negative, asymmetric shifts in ISW temperature maps arise. However, for the values of f local NL = ± 100, the changes were < 1 µ K (after smoothing the maps down to degree scales). Thus values of f local NL an order of magnitude higher would be required to explain the G08 result, and they would be clearly inconsistent with current constraints on f local NL from WMAP. It is possible that the G08 result may also be explained by other more exotic scenarios, e.g., non-Gaussianity arising from the presence of a non-zero primordial equilateral or orthogonal model bispectrum (a consequence of non-standard inflationary mechanisms); alternatively it might arise as a direct consequence of modifications to Einstein's general theory of relativity, (Jain & Khoury 2010). However, more conservative scenarios involving some combination of artifacts and/or systematics cannot yet be fully discarded. In the future, we will look with interest to the results from the Planck satellite as to whether this signal represents a data artifact, or whether it constitutes a genuine challenge to the ΛCDM model and a window to new cosmological physics.", "pages": [ 12, 13 ] }, { "title": "ACKNOWLEDGMENTS", "content": "It is a pleasure to acknowledge Ra'ul Angulo, Jose Mar'ıa Diego, Marian Douspis, Benjamin Granett, S. Illic and Istv'an Szapudi for useful discussions. We also thank Laura Marian for carefully reading the manuscript. We kindly thank Vincent Desjacques for providing us with access to his non-Gaussian realisations. C.H-M. is a Ram'on y Cajal fellow of the Spanish Ministry of Economy and Competitiveness. The work of RES was supported by Advanced Grant 246797 'GALFORMOD' from the European Research Council. We acknowledge the use of the HEALPix package (G'orski et al. 2005) and the LAMBDA data base. We thank Volker Springel for making public his code Gadget-2 , and Roman Scoccimarro for making public his 2LPT code. We acknowledge the ITP, University of Zurich for providing assistance with computing resources.", "pages": [ 13 ] }, { "title": "REFERENCES", "content": "Adelman-McCarthy J. K., The SDSS Team 2008, ApJS, Hern'andez-Monteagudo C., G'enova-Santos R., AtrioBarandela F., 2006, in Mornas L., Diaz Alonso J., eds, A Century of Relativity Physics: ERE 2005 Vol. 841 of American Institute of Physics Conference Series, Is there Any Evidence for Integrated Sachs-Wolfe Signal in WMAP First Year Data?. pp 389-392 Jain B., Khoury J., 2010, Annals of Physics, 325, 1479 Jarosik N., The WMAP Team 2010, ArXiv e-prints Johnson S., Frigo M., 2008, http://www.fftw.org/ Neyrinck M. C., 2008, MNRAS, 386, 2101 Peebles P. J. E., 1980, The large-scale structure of the universe. Research supported by the National Science Foundation. Princeton, N.J., Princeton University Press, 1980. 435 p. 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B., Lahav O., 2011, Physical Review Letters, 106, 241301 Vielva P., Mart'ınez-Gonz'alez E., Tucci M., 2006, MNRAS, 365, 891 Weinberg S., 2008, Cosmology. Cosmology, by Steven Weinberg. ISBN 978-0-19-852682-7. Published by Oxford University Press, Oxford, UK, 2008.", "pages": [ 13, 14 ] }, { "title": "APPENDIX A: FULL SKY ISW MAPS FROM N -BODY SIMULATIONS", "content": "In this section we aim to construct full-sky ISW maps using a suite of N -body simulations. Our approach is similar to that described in Cai et al. (2010), but with some modifications. To be more precise, we aim to compute the line-of-sight integral Eq. (1), but taking into account the full nonlinear evolution of ˙ Φ. The steps we take to achieve this are described below.", "pages": [ 14 ] }, { "title": "A1 Determining ˙ Φ", "content": "In order to obtain ˙ Φ directly from the N -body simulations, we make use of Eq.(4), which tells us that our desired quantity can be determined from knowledge about δ ( k , a ) and ˙ δ ( k , a ). In simulations, measuring δ ( k , a ) is relatively straightforward, whereas its time derivative is more complicated. As was shown by Seljak (1996), this latter quantity may be obtained from the perturbed continuity equation (Peebles 1980): where v p ( x ; t ) is the proper peculiar velocity field. If we define the pseudo-peculiar momentum field to be, then in Fourier space we may solve the continuity equation directly to find Hence, our final expression becomes, where we defined the function Thus in order to estimate ˙ Φ, we simply require estimates of both the density field and pseudo-peculiar momentum field in Fourier space. The dark matter density field in an N -body simulation can be written as a sum of Dirac delta functions, where N is the number of particles and m l is the mass of the l th particle, and we take all particles to have equal mass. The density field averaged on a cubical lattice can then be obtained through the convolution, where x ijk labels the lattice point, W represents the dimensionless window function of the mass assignment scheme. This window function is normalized such that V W = ∫ d 3 x ' W ( x -x ' ). The filter function W that we adopt throughout is the 'cloud-in-cell' charge assignment scheme (Hockney & Eastwood 1988). Hence, our estimate for the density fluctuation is where N cell = V µ /V W is the total number of grid cells. The pseudo-momentum field may be estimated in a similar fashion. For convenience we write, where u = v p /a is the comoving peculiar velocity field. The particle momentum field is then written as This may be convolved with the mass assignment scheme to obtain the mesh averaged quantity Thus our estimate for the pseudo-momentum field is The density Fourier modes were then estimated using the publicly available FFTW routines (Johnson & Frigo 2008), and each resulting mode was corrected for the convolution with the mass-assignment window function. For the CIC algorithm this corresponds to the following operation: where and where sub-script d and g denote discrete and grid quantities, and where k Ny = πN g /L is the Nyquist frequency, and N g is the number of grid cells (Hockney & Eastwood 1988). To obtain the real space ˙ Φ( x , t ), we solved for ˙ Φ( k , t ) in Fourier space using Eq. (A4), set the unobservable k = 0 mode to zero, and inverse transformed back to real space.", "pages": [ 14, 15 ] }, { "title": "A2 Reconstructing the light-cone for ˙ Φ", "content": "We now wish to construct the past light-cone for the evolution of ˙ Φ, however we only have a finite number of snapshots of the particle phase space from which to reconstruct this. It is usually a good idea to space snapshots logarithmically in expansion factor, and for simplicity we shall now assume that to be true. The light-cone can then be constructed as follows: with ∆log a being the logarithmic spacing between two different expansion factors. The comoving distance from the observer at a 0 to expansion factor a is given by: Hence, the intervals [ χ l -1 / 2 , χ l +1 / 2 ] form a series of concentric shells centred on the observer. then ˙ Φ( x ijk , a l ) is accepted onto the new grid. Note that if a given value of χ is larger than L/ 2 , 3 L/ 2 , 5 L/ 2 , . . . , then we use the periodic boundary conditions to produce replications of the cube.", "pages": [ 15 ] }, { "title": "A3 Computing the ISW line-of-sight integral", "content": "Having constructed the backward lightcone for ˙ Φ, we may now compute the line-of-sight integral for the ISW effect through Eq. (1). In fact we use a slightly different form of this equation by transforming the integration variable from t to log a : i.e. The above expression can then be discretized to give, If we take ∆ log a to be constant, then this becomes, In evaluating the above expression, we take the number of intervals to be as large as desired, but always evaluating H ( a i ) exactly through the expression (Dodelson 2003; Weinberg 2008), which is valid in the matter dominated epoch for the ΛCDM model. Whereas, for ˙ Φ( a i ), we employ the light cone derived form from the previous subsection, but interpolate the value off the 3D mesh using a CIC like scheme. That is: where { h x , h y , h z } are the x -, y -, and z -coordinate separations of the evaluation point x and the position vector for lattice point ( i, j, k ), in units of the inter-lattice separation. In making this separation of the evolution of H ( a ) and ˙ Φ, we are effectively assuming that ˙ Φ evolves very slowly over the time interval between snapshots. This is a reasonable assumption on large scales, since, as discussed in § 2 the time derivative is close to zero for most of the evolution of the Universe and only weakly evolving away from this at later times. On smaller scales this may be a less reasonable approximation, however, we are still using the fully nonlinear gravitational potential field.", "pages": [ 15 ] } ]

[ { "title": "ABSTRACT", "content": "The application to observational data of the generalized scaling relations (gSR) presented in Ettori et al. (2012) is here discussed. We extend further the formalism of the gSR in the self-similar model for X-ray galaxy clusters, showing that for a generic relation M tot ∝ L α M β g T γ , where L , M g and T are the gas luminosity, mass and temperature, respectively, the values of the slopes lay in the plane 4 α +3 β +2 γ = 3 . Using published dataset, we show that some projections of the gSR are the most efficient relations, holding among observed physical quantities in the X-ray band, to recover the cluster gravitating mass. This conclusion is based on the evidence that they provide the lowest χ 2 , the lowest total scatter and the lowest intrinsic scatter among the studied scaling laws on both galaxy group and cluster mass scales. By the application of the gSR, the intrinsic scatter is reduced in all the cases down to a relative error on the reconstructed mass below 16 per cent. The best-fit relations are: M tot ∝ M a g T 1 . 5 -1 . 5 a , with a ≈ 0 . 4 , and M tot ∝ L a T 1 . 5 -2 a , with a ≈ 0 . 15 . As a by product of this study, we provide the estimates of the gravitating mass at ∆ = 500 for 120 objects (50 from the Mahdavi et al. 2013 sample, 16 from Maughan 2012; 31 from Pratt et al. 2009; 23 from Sun et al. 2009), 114 of which are unique entries. The typical relative error on the mass provided from the gSR only (i.e. not propagating any uncertainty associated with the observed quantities) ranges between 3-5 per cent on cluster scale and is about 10 per cent for galaxy groups. With respect to the hydrostatic values used to calibrate the gSR, the masses are recovered with deviations in the order of 10 per cent due to the different mix of relaxed/disturbed objects present in the considered samples. In the extreme case of a gSR calibrated with relaxed systems, the hydrostatic mass in disturbed objects is over-estimated by about 20 per cent. Key words: cosmology: miscellaneous - galaxies: clusters: general - X-ray: galaxies: clusters.", "pages": [ 1 ] }, { "title": "S. Ettori 1 , 2", "content": "1 2 Accepted 2013 July 22. Received 2013 July 16; in original form 2013 February 25", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "2010, Mantz et al. 2010, Reichert et al. 2011, Bohringer et al. 2012; see also a recent review in Giodini et al. 2013). The distribution of the gravitating mass in galaxy cluster is the key ingredient to use them as astrophysical laboratories and cosmological probes. In the presently favorite hierarchical scenario of cosmic structure formation, direct relations hold between observables in the electromagnetic spectrum and the depth of the cluster potential produced from a matter component expected to be dynamically cold and electromagnetically dark (see e.g. Allen, Evrard & Mantz 2011, Kravtsov & Borgani 2012). Work in recent years has focused in defining reliable X-ray proxies of the total mass in galaxy clusters. These X-ray proxies are observables which are at the same time relatively easy to measure and tightly related to total cluster mass by scaling relations having low intrinsic scatter as well as a robustly predicted slope and redshift evolution (e.g. Kravtsov et al. 2006, Maughan 2007 and 2012, Pratt et al. 2009, Stanek et al. 2010, Rozo et al. 2009 and", "pages": [ 1 ] }, { "title": "c © 0000 RAS", "content": "The X-ray properties of the intra-cluster medium (ICM) are shaped from the evidence that it emits mainly by thermal bremsstrahlung and it is hydrostatic equilibrium with the underlying gravitational potential. In this context, the self-similar scenario (e.g. Kaiser 1986, Bryan & Norman 1998) relates the integrated quantities of the bolometric luminosity, L , gas temperature, T , and gas mass, M g , to the total mass, M tot in a simple and straightforward way. By construction, the cluster mass inside a sphere with volume 4 / 3 πR 3 corresponding to a mean overdensity ∆ with respect to the critical density of the Universe at the cluster's redshift z , ρ c,z , is M tot = 4 / 3 πρ c,z ∆ R 3 ∝ E 2 z ∆ R 3 , where E z = H z /H 0 = [ Ω m (1 + z ) 3 +1 -Ω m ] 1 / 2 represents the cosmic evolution of the Hubble constant H 0 for a flat cosmology with matter density parameter Ω m . From the hydrostatic equilibrium equation (see e.g. Ettori et al. 2013), M tot is directly proportional to the TR or, using the definition above, E z M tot ∝ T 3 / 2 . The expression of the bremsstrahlung emissivity /epsilon1 ∝ Λ( T ) n 2 gas ∝ T 1 / 2 n 2 gas (the latter relation being valid for systems sufficiently hot, e.g. > 2 keV, and assuming a X-ray bolometric emission for which the cooling function Λ( T ) is ∼ T 1 / 2 ) allows us to relate the bolometric luminosity, L , and the gas temperature, T : L ≈ /epsilon1R 3 ≈ T 1 / 2 f 2 gas M 2 tot R -3 ≈ f 2 gas T 2 , where we have made use of the above relation between total mass and temperature. By combining these basic equations, we obtain that the scaling relations among the X-ray properties and the total mass are (see also Ettori et al. 2004): E z M tot ∝ T 3 / 2 ∝ E z M g ∝ ( E -1 z L ) 3 / 4 ∝ ( E z Y X ) 3 / 5 . The latter relation has been introduced from Kravtsov et al. (2006), where the quantity Y X = M g T is demonstrated to be a very robust mass proxy being directly proportional to the cluster thermal energy. Its scaling relation with M 500 is characterized by an intrinsic scatter of only 5-7 per cent at fixed Y X , regardless of the dynamical state of the cluster and with a redshift evolution very close to the prediction of self-similar model. This robustness of the M -Y X relation has been studied and confirmed in later work (see, e.g., Arnaud et al. 2007, Maughan 2007, Pratt et al. 2009 on observational data; Poole et al. 2007, Rasia et al. 2011 and Fabjan et al. 2011 on objects extracted from cosmological hydrodynamical simulations). The attempt to generalize the simple power-law scaling relations between cluster observables and total mass has become more intensive in the recent past (e.g. Stanek et al. 2010, Okabe et al. 2010, Rozo et al. 2010). In Ettori et al. (2012; hereafter E12), we have presented new generalized scaling relations with the prospective to reduce further the scatter between the observed mass proxies and the total cluster mass. Working on a set of cosmological hydrodynamical simulations, we have found a locus of minimum scatter that relates the logarithmic slopes of the two independent variables considered in that work, namely the temperature T , which traces the depth of the halo gravitational potential, and an another observable accounting for distribution of gas density which is more prone to the affects of the physical processes determining the ICM properties, like the gas mass M g or the X-ray luminosity L . In E12, we show that all the known self-similar scaling laws appear as particular realizations of generalized scaling relations. We predict also the evolution expected for the generalized scaling relations, suggesting which relations can be used to maximize the evolutionary effect, for instance to test predictions of the self-similar models itself, or, on the contrary, which relations minimize it in the case of cosmological applications. In this paper, we present and discuss the application of these X-ray generalized scaling relations on observational data to test the improvement introduced from the these relations in reconstructing the total mass in galaxy clusters. To do this, we do not define any new sample of objects but use the dataset available in the literature, analyzing them in a homogenous and reproducible way. The paper is organized as follows. In Section 2, we introduce the generalized scaling relations in the context of the self-similar model for X-ray galaxy clusters and describe how we implement the fit to the selected dataset. In Section 3, we discuss the calibration of the generalized scaling relations and present the best-fit results in terms of the values of the measured χ 2 , total and intrinsic scatter. In Section 4, we summarize our main findings.", "pages": [ 1, 2 ] }, { "title": "2 THE GENERALIZED SCALING LAWS", "content": "In E12, we have generalized the scaling relations between the total mass M tot and X-ray observables, by considering a more general proxy defined in such a way that M tot ∝ A a B b , where A is either M g or L and B = T . In doing that, we aim to minimize the scatter in the relations between total mass and observables by (i) relaxing the assumptions done in the self-similar scenario, (ii) combining information on the depth of the halo gravitational potential (through the gas temperature T ) and on the distribution of gas density (traced by M g and the X-ray luminosity) that is more affected by the physical processes determining the ICM global properties, (iii) adopting a general and flexible function with a minimal set of free parameters (3 in the general expression -the normalization and the 2 slopes- that are then reduced to 2 by linking the values of the slopes). Using a set of cosmological hydrodynamical simulations, we have found a locus of minimum scatter that relates the logarithmic slopes a and b of the two independent variables. In all cases, this locus is well represented by the lines { A = M g , B = T } ⇒ b = -3 / 2 a +3 / 2 and { A = L, B = T } ⇒ b = -2 a +3 / 2 , or, in more concise form, where d corresponds to the power to which the gas density appears in the formula of the gas mass ( d = 1 ) and luminosity ( d = 2 ). In a similar way, also the evolution with redshift of the total mass can be simply written as E z M tot ∝ E c z , with c = a and -a for A = M g and L , respectively, i.e. c = (3 -2 d ) a . The relation in equation 1 between the two logarithmic slopes allows us to reduce by one (from 3 to 2) the number of free parameters in the linear fit of the generalized scaling law between observables and total mass.", "pages": [ 2 ] }, { "title": "2.1 The generalized scaling relations in the self-similar model", "content": "The generalized scaling relations (hereafter gSR) are obtained as the extension of the self-similar model when two, or more, observables are used to recover the total gravitating mass. Indeed, the hydrostatic mass M tot is proportional to RT by definition. Using M g ∝ R 3 implies M tot ∝ M 1 / 3 g T . If we require further that the condition M tot ∝ M g (or M tot ∝ T 3 / 2 ) has to be satisfied, then the relation in equation 1 is obtained univocally. Similarly, we can infer the dependence upon the X-ray bolometric luminosity ( L ∝ T 1 / 2 M 2 g R -3 ): M tot ∝ RT ∝ L -1 / 3 M 2 / 3 g T 7 / 6 or, equivalently, ∝ L -1 M 2 g T 1 / 2 . Then, we can solve for any combination of observables to recover the relation in equation 1. It is worth noticing that these observables ( L, M g , T ) are the only ones accessible directly through the X-ray analysis: the luminosity is provided from the observed count rate once a thermal model and redshift are assumed; the gas mass is obtained as integral of the modelled (or deprojected) X-ray surface brightness; the gas temperature is constrained from the continuum of the spectral thermal model. More generally, we can write with the exponents ( α, β, γ ) satisfying, in the self-similar scenario, the equation c © 0000 RAS, MNRAS 000 , 000-000 The projections of this plane in the cartesian axes ( α, β, γ ) provide the subset of relations discussed in E12: The self-similar evolution of the equation 2 is then E z M tot ∝ ( E -1 z L ) α ( E z M g ) β T γ ∼ E /epsilon1 z , with /epsilon1 = -α + β . It is worth noticing that these gSRs reduce to the standard self-similar laws with a single observables for a proper value of the slope of equation 2 (or equation 4): one recovers M tot ∝ T 3 / 2 with ( α, β ) = (0 , 0) ; M tot ∝ M g with ( α, γ ) = (0 , 0) ; M tot ∝ Y 3 / 5 X with ( α, β ) = (0 , 3 / 5) ; M tot ∝ L 3 / 4 with ( β, γ ) = (0 , 0) ; M tot ∝ ( LT ) 1 / 2 , which is the relation corresponding to M tot ∝ Y 3 / 5 X once gas mass is replaced by luminosity, fixing ( β, γ ) = (0 , 1 / 2) . In the following analysis, we investigate particularly some projections of the gSR in equation 3, focusing our analysis on those relations that minimize the scatter in recovering the total cluster gravitating mass.", "pages": [ 2, 3 ] }, { "title": "2.2 Fitting the scaling relations", "content": "In this work, we want to compare how the assumed linear relation between logarithmic values of the observed quantities and of the gravitational mass determined through the equation of the hydrostatic equilibrium, M tot ≡ M HSE ≡ M , performs and, in particular, which is the combination of observables that minimizes the scatter in reconstructing the galaxy cluster mass. Among the relations satisfying equation 3, we focus on the most promising for our goal, M ∝ M g T and M ∝ LT , that are obtained by requiring α = 0 and β = 0 , respectively. Operationally, we adopt the following procedure. We build the variables where ' log ' indicates the base-10 logarithm, and we consider an associated error obtained through the propagation of the measured uncertainties. Then, we fit the linear function Y = n + a A + b B . The best-fit parameters are obtained by minimizing the merit function: where b is related to a through equation 1, N is the number of data points and D = N -2 are the degrees of freedom. The fit is performed using the IDL routine MPFIT (Markwardt 2008). To evaluate further the performance of the gSR with respect to the standard scaling laws, we have also estimated the total and the intrinsic scatter. Here, we define the total scatter on the logarithmic value of the mass σ M as the sum, divided by the degrees-of-freedom, of the residuals of the observed measurements with respect to the best-fit line: The two definitions of the weights w i do not change significantly the measured scatter. Hereafter, we define w i = 1 . The intrinsic scatter is a constant value σ I that is determined by adding it in quadrature to /epsilon1 i in equation 6, once the minimum χ 2 is estimated, and looking for the values that satisfy the relation where the dispersion around 1 of the reduced χ 2 , χ 2 red , is strictly valid in the limit of large D . By construction, the intrinsic scatter estimated through equation 8 translates then in a contribution (to be added in quadrature) to the relative error on the mass equals to ln(10) σ I ≈ 2 . 30 σ I .", "pages": [ 3, 4 ] }, { "title": "2.3 From the best-fit results to the total mass", "content": "From the best-fit results { n, a } obtained from the application of equations 5 and 6 to a sample where the hydrostatic masses M HSE ≡ M tot are available (see next section), it is now possible to recover an estimate of the total gravitating mass M fit where b is related to a via equation 1. /negationslash In the present analysis, the quantities A and B are estimated at R ∆ = R 500 . However, in general, they can also be observed at an arbitrary radius R 0 which is chosen, for instance, because encloses the region with the highest signal-to-noise ratio and is not expected to coincide with R ∆ . We refer to the appendix for a discussion of the case R 0 = R ∆ . The error on M fit is formally due to the sum in quadrature of the propagated uncertainty obtained from the best-fit parameters and the statistical error associated with the observed quantities. Hereafter, we only consider the former, that can be in some way considered as a systematic uncertainty related to the set of data used to calibrate the generalized scaling relations. Given the best-fit parameters { n, a } with a corresponding 2 × 2 covariance matrix Θ with elements Θ 00 = /epsilon1 2 n , Θ 11 = /epsilon1 2 a , Θ 10 = Θ 01 = cov na , the uncertainty /epsilon1 M on M fit can be written as where θ n and θ a indicate the partial derivative of M fit with respect to the best-fit parameters. We note that the third addendum in the definition of /epsilon1 M , which includes the off-diagonal element cov na , is comparable in magnitude to the other two contributions and thus cannot be neglected in the total error budget measurement.", "pages": [ 4 ] }, { "title": "3 THE CALIBRATION OF THE SCALING RELATIONS", "content": "To calibrate the (generalized) scaling laws, we decide to analyze in a homogenous and reproducible way some published dataset. We search in the literature for samples with measured set of X-ray determined total mass M HSE , temperature T and either gas mass M g or bolometric luminosity L . In the present work, we consider X-ray mass estimates obtained though the application of the equation of the hydrostatic equilibrium under the assumptions that any gas velocity is zero and that the ICM is distributed in a spherically-symmetric way into the cluster gravitational potential (see, e.g., Ettori et al. 2013). Considering that these conditions are verified more strictly in dynamically relaxed objects, we also use, when available, the information on the dynamical state of the objects, considering, for instance, if they are relaxed or with a cooling core (Cool Core -CC- objects are galaxy clusters where the X-ray core has an estimated cooling time lower than the age of the structure; in general, these systems present a Xray surface brightness map with a round shape and with no evidence of significant subclumps). For our purpose, 'CC clusters' and 'relaxed clusters' identify the same category of objects for which the hydrostatic masses are more reliable. As a result of our analysis, we discuss also any deviation in the mass reconstruction of CC/relaxed and NCC/disturbed clusters.", "pages": [ 4 ] }, { "title": "3.1 The X-ray cluster samples", "content": "We have selected samples over a wide range of masses to calibrate the gSR on group and cluster mass scales. Moreover, we have considered samples in which the extrapolation over the radial range of the observed profiles of gas density and temperature has been minimal to recover the mass at R 500 . The following samples, with the main properties listed in Table 1, are then considered: an ICM emission not spherically symmetric nor expected to be in hydrostatic equilibrium) in a temperature interval of 0.7-9 keV. All the gas density and temperature profiles of the nine clusters considered in the M12 sample extend almost to R 500 , permitting a mass estimate without any extrapolation. Arnaud et al. (2007) discuss the mass profiles in ten nearby morphologically relaxed clusters over the temperature range 2-9 keV and observed with XMM-Newton . The quoted M 500 of the seven clusters considered in the M12 sample were derived from the mass profiles measured to overdensities of about 600-700, apart from the two coolest systems (at overdensity of ∼ 1400). All the objects in the M12 sample are relaxed systems and are thus labelled as 'CC' objects (see Table B1). to at least R 1000 ≈ 0 . 7 R 500 . The adopted gas temperatures are obtained as projection of the integral of the three-dimensional profile over the radial range [0 . 15 -1] R 500 . Because this sample has been selected to have the X-ray emission centered around the central galaxy and not significantly elongated nor disturbed beyond the group core, we qualify all of them as CC/relaxed systems. All the physical quantities considered here refer to the cosmological parameters H 0 = 70 km s -1 Mpc -1 and Ω m = 1 -Ω Λ = 0 . 3 . For only one sample (S09), a conversion from an other cosmological framework has been required. In this case, we use the relations M HSE ∝ d ang and M g ∝ d 2 . 5 ang , where d ang is the angular diameter distance, to make the proper conversion. As described above, the radius of reference for the present analysis is R 500 . Note that the samples here considered use different techniques to measure it: M12 and S09 recover R 500 from the hydrostatic mass profile; M13 uses the result from the weak-lensing analysis; P09 applies the M -Y X scaling relation. Considering that we will analyze each sample independently, the use of different definitions of R 500 will permit us to test further the performance of the gSR. Note also that 6 objects (MKW4, Abell2717, Abell1991, Abell2204, Abell383, Abell2390) are in common to different samples. The quoted hydrostatic masses show differences between 0 . 2 σ and ∼ 2 σ , with the most deviant values for MKW4 and Abell383. For MKW4, the difference between the hydrostatic masses in S09 and M12 (as adopted from Vikhlinin et al. 2006), where the same Chandra dataset is used, is discussed in the Appendix of Sun et al. (2009) and is probably due to a different modelling of the gas density profile. In the case of Abell383, the difference between the values quoted in Mahdavi et al. (2013), which is based on a joint analysis of the XMM-Newton / Chandra exposures, and Vikhlinin et al. (2006), which analyze only the Chandra data, can be explained, at least partially, with both the different dataset used and the different estimate of R 500 where the total mass is evaluated. Indeed, in M13, R 500 is adopted from the result of the weak-lensing analysis and is about 7 per cent larger than in M12, implying M 500 higher by /greaterorsimilar 20 percent.", "pages": [ 5, 6, 7 ] }, { "title": "3.2 The best-fit results", "content": "To compare the performance of the gSR versus the standard relations, we focus our study on the following relations: M HSE -T , M HSE -M g , M HSE -L and the gSR M HSE -M g T , M HSE -LT . As an example, we show in Fig. 1 and 2 the best-fit lines and the distribution of the residuals for the M12 and M13 sample, respectively. The distribution of the residuals in log( M ) shows an appreciable reduction of both the median deviation and the InterQuartile-Range for the clusters in, e.g., M13. No clear improvements are noticed for M12, where the measured intrinsic scatter is already close to zero when the standard scaling laws are applied. In Fig. 3, we plot the likelihood contours obtained for a grid of values of the slopes { a, b } . These statistical constraints show the locus of the slopes preferred from the data in terms of the minimal χ 2 . This locus can be well approximated by the relation identified in the hydrodynamical simulations discussed in E12 and indicated by equation 1 (and equation 4). In the same figure, we also show the significance of the deviation from the minimum value of the χ 2 for the most interesting cases, nominally the best-fit values obtained by imposing equation 1 and the standard self-similar relations. We notice how the latter relations that make use of either the gas temperature or the gas mass only are systematically above the lowest value of χ 2 at a level of confidence > 99 per cent. Only the cases where the Y X = M g T quantity is adopted provide less significant deviations, but always in the order of 95 per cent (about 2 σ for a Gaussian distribution) or larger. The only exception is the sample S09, where the total mass can be recovered using Y X at a level of confidence of ∼ 20 per cent. However, the gSR provides always the best performance, with the significance of the deviations from the absolute minimum in the { a, b } plane ranging from only 6 per cent (S09 sample) to 99.8 per cent (M13 sample using LT ). We present the best-fit results in Tables 2, 3 and 4. For sake of completeness, we include also the case M -LM g (see equation 4), showing how this relation provides a scatter in reconstructing the total mass higher than the two other relations investigated ( M -M g T and M -LT ) and, therefore, will be not discussed further. When the standard scaling laws are used, the M13 sample shows the lowest χ 2 for the M -T relation, whereas M12 and S09 seem to prefer slightly the M -M g one. When a gSR is applied, we measure systematically a reduction of the total χ 2 , with improvements in ∆ χ 2 up to 16-325 (with 48 dof) in M13, for all the datasets analyzed here. Even in the case where a significant reduction in χ 2 is not observed (as for the M12 sample), we measure a reduction of the total scatter of /greaterorsimilar 20 per cent. The intrinsic scatter associated to the best-fit with a gSR is below 0.07 (corresponding to a relative error on the total mass lower than 16%) in all cases. In general, we measure a reduction in the χ 2 value, total and intrinsic scatter when a gSR is adopted in place of a standard selfsimilar relation (see Table 2). We use this evidence as confirmation that gSR reproduces better the distribution of the estimated hydrostatic mass used to calibrate these relations. We note also that the M13 sample presents the largest total and intrinsic scatter among the analyzed datasets. This might be related also to the use of R 500 as obtained from the weak-lensing analysis, whereas a X-ray based definition of R 500 , and therefore correlated to the quantities investigated, is adopted for the other samples. The best-fit results (Table 3) for the samples M13, M12 and S09 agree on the slope a of the M HSE -M g T relation (the errorweighted mean is a = 0 . 411 ± 0 . 050 ). The slope of the M HSE -LT relation is about 0 . 15 . On the other hand, we notice significant differences in the normalization of the M g T gSR between M12 ( n = -0 . 054 ) and M13 ( n = -0 . 095 ) that induce estimates of masses larger by about 10 per cent when the best-fit results from M12 are adopted. This can be explained by the fact that the objects in M12 are all relaxed systems, whereas the M13 sample is more heterogeneous (see also discussion in Mahdavi et al. 2013), including both relaxed and dynamically disturbed systems. The hydrostatic mass in the latter ones is indeed expected to underestimate the true mass due to an uncounted contribution from residual bulk motions of the ICM to the total energy budget (e.g. Nelson et al. 2012, Rasia et al. 2012, Suto et al. 2013). If we consider only the sub-sample of relaxed objects in M13-CC, we measure a normalization of the M HSE -M g T relation that matches (within 1 σ ) the value measured for the M12 dataset. We further confirm this evidence by quantifying it in Table 4, where we present the ratios between the estimates of the mass recovered from the best-ft gSR and the input hydrostatic masses. All the deviations are in the order of few per cent when the M HSE are recovered within the same sample and, on average, of about 10%, with a dispersion of ∼ 20%, when different sample are used. In particular, the gSR defined with M g and T and calibrated with M13 reproduces the mass estimates in M12, P09 and S09 with ( M fit -M HSE ) /M HSE = ∆ M/M HSE of -10, -4 and -7 per cent, respectively. When the gSR is calibrated with the M12 sample, the mass measurements in M13 are recovered with ∆ M/M HSE ∼ + 12 per cent and the ones in S09 and P09 with a mean ratio of + 6 per cent. Using the M13-CC sample provides similar results, with deviations in the order of + 10 per cent for the data in M13 and of few per cent the masses quoted in M12, S09 and P09. The same sub-sample induces over-estimates of the hydrostatic mass in disturbed objects (collected in the sub-sample M13-NCC) by 19 per cent on average, as produced from the M g T gSR calibrated with M12. When the LT gSR calibrated with either M13 or M13-CC is used, we measure deviations lower than 5 per cent in the reconstructed hydrostatic mass M fit of the clusters in the M13, M13-CC and, curiously, even in the M13-NCC sample. This result, which shows that the original hydrostatic masses in disturbed objects are well recovered, on average, when LT gSR is calibrated with samples containing CC systems, appears at odd with the previous evidence that M g T gSR provides values of the M fit of NCC clusters that are higher than their M HSE . To explain this, we have to consider that NCC clusters present a higher entropy level in the core with respect to the more relaxed systems (e.g. Mahdavi et al. 2013), due to the phenomena (such as merging events) that disturb their X-ray emitting plasma. As consequence of that, the global gas lu- minosity, in particular when the core is not excised as in M13, is lower, for a given mass halo, than the one measured in CC clusters used to calibrate the LT gSR. Hence, this relation will provide a M fit lower for a NCC than for a CC, almost compensating for the above-mentioned bias on the hydrostatic mass and matching the tabulated M HSE . Deviations of few per cent are also measured when the masses in P09 are reconstructed. If we consider for this sample L and T extracted over the region [0 . 15 -1] R 500 (i.e. excluding the core emission), we obtain larger deviations (in the order of -12 and -7 per cent, as mean values, using calibration provided from M13 and M13-CC, respectively), because the luminosities considered for the M13 sample are not core-excised and, therefore, are higher at a given mass. On the galaxy group scales, using the S09 sample to calibrate the M g T gSR, we measure deviations, on average, between 0 and 7 per cent for the samples M13-CC, M12 and P09, with larger values of ∼ + 15 per cent for M13. These values indicate that the gSR, although tuned to systems with mean total mass about 6-8 times lower than the ones in M12 and M13, is able to reproduce the measured M HSE in these samples, showing a bias that is due to the fact that the S09 sample is dominated by relaxed systems.", "pages": [ 7, 8, 9 ] }, { "title": "4 SUMMARYAND DISCUSSION", "content": "In this work, we have discussed the application of the generalized scaling relations presented in Ettori et al. (2012) to real data. In the context of the self-similar model for X-ray galaxy clusters, we show that a generic relation between the total mass and a set of observables like gas luminosity, mass and temperature can be written as M tot ∝ L α M β g T γ , where the values of the slopes satisfy the relation 4 α +3 β +2 γ = 3 (and M tot ≡ M HSE by the definition adopted in the present work). Some projections of this plane are particularly useful in looking for a minimum scatter between X-ray observables and hydrostatic mass: M tot ∝ A a B b , where A is either M g or L , B = T and b = 1 . 5 -(1 + 0 . 5 d ) a , with d equals to the power to which the gas density appears in the formula of the gas mass ( d = 1 ) and luminosity ( d = 2 ). We show indeed that the gSR are the most efficient relations, holding among observed physical quantities in the X-ray band, to recover the gravitating mass on both galaxy group and cluster scales, because they provide the lower values of χ 2 , total and intrinsic scatter among the studied scaling laws. The intrinsic scatter associated to the best-fit with a gSR at ∆ = 500 is below 0.07 (corresponding to a relative error on the total mass lower than 16%) in all cases. The best-fit results on the different samples considered in our analysis agree on the slope a of the M g T gSR (the error-weighted mean is a = 0 . 41 ± 0 . 05 ) and are consistent for the slope of the LT relation (the error-weighted mean is a ≈ 0 . 15 ). These values are significantly different from any adopted relations so far (e.g. M ∝ M g requires a = 1 , M ∝ T 3 / 2 needs a = 0 , M ∝ Y X is obtained for a = 0 . 6 ). This demonstrates that, still in the selfsimilar scenario, the gSR provides more flexible tool to use the X-ray observables as robust X-ray mass proxies. In particular, our best-fit results on the slope prefer a larger contribution from the gas global temperature than from the gas mass or luminosities. However, we show that the latter ones are needed to optimize the mass calibration. The combination of the constraints from the depth of the halo gravitational potential (through the gas temperature T ) and from the distribution of the gas density (traced by M g and the X-ray luminosity), that is more prone to the ongoing physical processes shaping the ICM global properties, is therefore essential to link the cluster X-ray observables to the total mass. Nonetheless, we notice a significant difference in the normalization of the M g T gSR between the fit obtained with data in Maughan (2012), that includes only relaxed systems, and that based on the Mahdavi et al. (2013) sample, that, on the contrary, is dominated (68 per cent) from disturbed objects. This difference induces estimates of masses larger by about 10 per cent when the best-fit results from Maughan (2012) are adopted and is reduced when the sub-sample of relaxed clusters from M13 is considered. Samples dominated by relaxed systems (as in M13-CC, M12, S09) provide calibrations of the M g T gSR that tend to over-estimate the hydrostatic mass in disturbed objects (M13-NCC) systematically by a mean value of 18-24 per cent. Indeed, in not-relaxed clusters, a non-thermal component is expected to contribute to the total energy budget, biasing low the estimate of the X-ray mass as traced though the hydrostatic equilibrium equation (e.g. Nelson et al. 2012, Rasia et al. 2012, Suto et al. 2013). Thus, the results quoted above seem to confirm that, in NCC systems, the total mass as estimated through the hydrostatic equation is under-estimated, on average, by 18-24 per cent. The measured bias is consistent with the results discussed in Mahdavi et al. (2013) where estimates of hydrostatic and weak lensing masses are compared. They conclude that (i) these estimates are similar in CC clusters and (ii) hydrostatic masses in NCC clusters are lower by 15-20 per cent. Using different mass proxies is definitely the most robust approach to constraint the level of mismatch on the gravitating mass between relaxed and disturbed galaxy clusters. Several observational biases can indeed play a significant role to assess the differences in mass between relaxed and disturbed objects using X-ray scaling relations only. For instance, it has been recognized that hydrostatic bias is composed from two main components, one related to the non-thermal source of extra-pressure and the other to temperature inhomogeneities in the ICM (see, e.g., discussion in Rasia et al. 2012). The acceleration of the gas becomes also a non-negligible component of the hydrostatic bias in the cluster outskirts (Suto et al. 2013, Lau et al. 2013). Moreover, during the different phases of a merger, the values of the integrated physical properties, like T and L , oscillate (see, e.g., Rowley et al. 2004, Poole et al. 2007). Only when a solid and confident knowledge is reached on the relative average variations in T , M g and L at a fixed halo mass between a CC and a NCC galaxy cluster (as classified accordingly to its observational X-ray properties), the CC-calibrated gSR can be then used to evaluate a 'correct' mass for a NCC system, where the term 'correct' indicates the value of the hydrostatic mass once a proper thermalization of the ICM occurs. On the contrary, LT gSR calibrated with M13 and M13-CC over-predicts the masses in NCC objects only by few per cent. In this case, we have to consider that the gas luminosity (as estimated over the whole cluster volume, i.e. not excluding any core emission) of NCC clusters tend to be lower than the one of relaxed objects that have been used to calibrate the gSR. This lower luminosity is the product of the higher central entropy induced from, e.g., recent mergers in disturbed, NCC systems (e.g. Rowley et al. 2004, Poole et al. 2007). For instance, by reducing the global bolometric L by a factor of 2, and considering the slope of 0.15 that appears in the gSR, a compensation of about 10 per cent is provided to the above-mentioned hydrostatic bias, permitting to recover the estimated hydrostatic mass M HSE for NCC clusters within a few per cent. Moreover, when we calibrate the gSR with galaxy groups hav- ing a mean total mass about 6-8 times lower than the most massive systems studied here, we are still able to reproduce the measured M HSE on cluster scales. A residual bias is present and due to the fact that the S09 sample used for the calibration in the present study is dominated by relaxed systems. These generalized scaling relations can be easily applied to present (e.g. XXL , Pierre et al. 2011) and future (e.g. eROSITA , Merloni et al. 2012) surveys of X-ray galaxy clusters. Either the calibrations presented here are adopted and used to infer hydrostatic masses for a sub-set of systems with measured gas temperature and gas mass or luminosity, or new calibrations are estimated as described in this work for a subsample of objects selected to be representative of the population of the observed clusters. As a by product of this study, we provide in Table B1 the estimates of the gravitating mass at ∆ = 500 for 120 objects (50 from the Mahdavi et al. 2013 sample, 16 from Maughan 2012; 31 from Pratt et al. 2009; 23 from Sun et al. 2009), 114 of which are unique entries. If we do not consider any uncertainty associated with the observed quantities, the typical relative error on the mass provided from, e.g., the M g T gSR with the considered datasets (see Table B1) ranges between 3.0 ± 1.6 per cent in M13 (with a relative uncertainty related to the residual intrinsic scatter of about 0.07 in log space, which corresponds to 16 +4 -2 per cent on the quantity /epsilon1 M /M ) and 9.6 ± 4.1 per cent (with a null intrinsic scatter) in S09. The other samples provide typical errors in of ∼ 5 per cent (M13CC: 6 ± 4 per cent and 14 +6 -4 per cent from the intrinsic scatter; M12: 5 ± 2 and almost nil contribution from the intrinsic scatter). This catalog of X-ray cluster masses can be used fruitfully, for instance, to compare results obtained with other techniques (like, e.g. lensing, galaxy velocity dispersion, caustics) or to apply statistics that want to address the presence, and the significance, of objects with extreme values in mass (e.g. Waizmann et al. 2013).", "pages": [ 9, 10 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank the anonymous referee for helpful comments that improved the presentation of the work. We acknowledge the financial contribution from contracts ASI-INAF I/023/05/0 and I/088/06/0. We thank Elena Rasia and Gianni Zamorani for useful discussions.", "pages": [ 10 ] }, { "title": "REFERENCES", "content": "Lau E.T., Nagai D., Nelson K., 2013, ApJ subm. (arXiv:1306.3993) Mahdavi A., Hoekstra H., Babul A., Bildfell C., Jeltema T., Henry J. P., 2013, ApJ, 767, 116 Mantz A. et al., 2010, MNRAS, 406, 1773 Markwardt C.B., 2008, 'Non-Linear Least Squares Fitting in IDL with MPFIT,' in proc. 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However, this is computationally expensive and can be easily avoided by modelling the radial dependence of the quantities of interest. If we consider a radial correction in the form of a power-law and using the definition of the mass associated with an overdensity ∆ within a sphere with radius R ∆ , M fit = 4 / 3 πρ c,z ∆ R 3 ∆ , we can write where ̂ ∆= 4 3 πρ c,z E z ∆ . Finally, by inverting this expression to isolate the quantity of interest R ∆ , we obtain the relation c © 0000 RAS, MNRAS 000 , 000-000 The estimated mass will be then obtained by substituting equation A3 in the definition of M fit and using equation 1. The error on M fit is formally due to the sum in quadrature of the propagated uncertainty obtained from the best-fit parameters and the statistical error associated with the observed quantities. Hereafter, we only consider the former, that can be in some way considered as a systematic uncertainty related to the set of data used to calibrate the generalized scaling relations. From the 2 × 2 covariance matrix Θ , we can write where θ n and θ a indicate the partial derivative of R ∆ with respect to the best-fit parameters. We conclude this section by quoting some simple description of the radial dependence of the observed quantities M g , L and T . By assuming that the distribution of the gas density is represented with a β -model, n gas ∝ (1 + x 2 ) -1 . 5 β , and the gas temperature profile with a functional form as in Vikhlinin et al. (2006; see also Baldi et al. 2012), and making the further assumption that R 500 is equal to 5 times the core radius r c = r/x , we measure in the range 3 /lessorequalslant x /lessorequalslant 7 the following radial behaviour", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "We study the abundances of complex carbon-bearing molecules in the oxygen-rich dustforming atmospheres of Brown Dwarfs and giant gas planets. The inner atmospheric regions that form the inner boundary for thermochemical gas-phase models are investigated. Results from D rift -phoenix atmosphere simulations, which include the feedback of phasenon-equilibrium dust cloud formation on the atmospheric structure and the gas-phase abundances, are utilised. The resulting element depletion leads to a shift in the carbon-to-oxygen ratio such that several hydrocarbon molecules and cyanopolycyanopolyynene molecules can be present. An increase in surface gravity and / or a decrease in metallicity support the increase in the partial pressures of these species. CO, CO2, CH4, and HCN contain the largest fraction of carbon. In the upper atmosphere of low-metallicity objects, more carbon is contained in C4H than in CO, and also CH3 and C2H2 play an increasingly important role as carbon-sink. We determine chemical relaxation time-scales to evaluate if hydrocarbon molecules can be a GLYPH<11> ected by transport-induced quenching. Our results suggest that a considerable amount of C2H6 and C2H2 could be expected in the upper atmospheres not only of giant gas planets, but also of Brown Dwarfs. However, the exact quenching height strongly depends on the data source used. These results will have an impact on future thermo-kinetic studies, as they change the inner boundary condition for those simulations.", "pages": [ 1 ] }, { "title": "C. Bilger, P. Rimmer, Ch. Helling", "content": "SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, Fife, United Kingdom, KY16 9SS Accepted 27 October 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Brown Dwarfs and giant gas planets have atmospheres in which carbon is usually less abundant than oxygen. This may change if the planet forms in a disk with a carbon-rich dust-gas mixture (e.g. Fortney 2012), which seems unlikely for Brown Dwarfs that form from a molecular cloud by gravitational collapse. Both kind of objects, however, have atmospheres that are so cold that clouds form from the local atmospheric gas resulting in a depletion of elements, including oxygen and perhaps carbon. The formation of mineral clouds will individually deplete the heavy and less abundant elements like O, Mg, Si, Fe, Al, Ti (Helling & Woitke 2006; Witte et al. 2009) which makes the observational determination of the element abundances di GLYPH<14> cult. The depletion of oxygen is comparably moderate, but strong enough to shift the carbon-to-oxygen ratio (hereafter C / Oratio) from an initial solar value of GLYPH<24> 0.5 to GLYPH<24> 0.7. The hypothesis of carbon-rich atmospheres in substellar objects is not new, and was inspired by observations of WASP-12b (Madhusudhan et al. 2011) and by questioning the standard planet composition (Gaidos 2000; Seager et al. 2007) earlier on. Our investigations are triggered by our finding (Sect. 4.1) that cloud formation alone causes a considerable shift in the local C / O ratio and the resulting question of how this changes the importance of carbon-bearing molecules in substellar atmospheres. Independent of the model assumptions in cloud formation is that methane (CH4) becomes more abundant than carbon monoxide (CO) at heights above roughly 1 bar in the dense atmospheres of cool Brown Dwarfs and giant gas planets (Helling et al. 2008, their Fig. 4). This is remarkable because the high binding energy of the CO-molecule causes the blocking of the carbon-chemistry in an oxygen-rich environment and vice versa. The reason is the high stability of the CO molecule due to its triple-binding between the C and the O atoms. CH4 has four single covalent C-H bindings and as a result, is much more likely to react with other gas-phase species. Hence, CH4 provides less rigid blocking of the carbon in an oxygen-rich environment. We will therefore investigate if this weakening of the carbon blocking might allow for the presence of even more complex carbon-binding molecules in the oxygen-rich atmospheres of Brown Dwarfs and giant gas planets, and this might be a GLYPH<11> ected by vertical mixing. Deviations from local chemical gas-phase equilibrium in the upper atmosphere are suggested to be caused by a rapid convective and / or di GLYPH<11> usive up-mixing of warm gases from deeper atmospheric layers (e.g. Saumon et al. 2000, 2006, 2007; Leggett et al. 2007) in cold Brown Dwarfs (e.g. Gliese 229B, 570D) and giant gas planets. A similar process is discussed to a GLYPH<11> ect the abundance determination in the solar system for Jupiter, Saturn, Uranus, and Neptune (Visscher & Moses 2011). Another possibility may be the impact of cosmic rays on the chemistry of the upper atmospheres of Brown Dwarfs. Various groups (Zahnle et al. 2009a,b; Line et al. 2010; Moses et al. 2011; Venot et al. 2012) study the chemical kinetics under the influence of vertical mixing and photodissociation. All networks consider molecules formed of the elements H, C, O and N (Line et al. 2010 does not include nitrogen). The number of species and reaction di GLYPH<11> er between the chemical networks: Zahnle et al. (2009a) (used in Miller-Ricci Kempten et al. 2012) consider 58 species (561 kinetic reactions incl. 33 photo-chemical) and their network is accurate to C2H n . The species C4H and C4H2 are also included in their network. Line et al. (2010) consider 32 species (299 kinetic reactions incl. 41 photo-chemical), accurate to C2H2. Moses et al. (2011) consider 90 species ( GLYPH<24> 1600 reactions), and incorporate hydrocarbons up to the size of benzine. Venot et al. (2012) consider 46 C / Ospecies and 42 N-containing species ( > 2000 reactions incl. 34 photo-chemical reactions), accurate to C2H n . All these complex networks prescribe the atmospheric temperature-pressure profile and treat the eddy di GLYPH<11> usion coe GLYPH<14> cient Keddy as a parameter. The pressure-dependence of Keddy is roughly determined by Line et al. (2010) and Moses et al. (2011). Moses et al. (2011) obtains Keddy from the circulation models of Showman et al. (2009). Zahnle et al. (2009a) treat the gas temperature as isothermal, and Moses et al. (2011) and Venot et al. (2012) calculate ( T ; p ) profiles from radiative transfer (Fortney et al. 2006, 2010) and hydrostatic equilibrium (Showman et al. 2009). None of these models has a comprehensive treatment of grain formation or the deposition of gas-phase species onto grains. The deviation from chemical equilibrium (or a steady state solution of the kinetic models) increases with increasing mixing e GLYPH<14> ciency which is a generally unknown parameter for planetary objects outside the solar system (Miller-Ricci Kempten et al. 2012). The deviations also tend to become more significant at lower pressures. The non-equilibrium steady-state and thermochemical equilibrium abundances agree to within an order of magnitude when p gas & 1bar ( = 10 GLYPH<0> 6 dyn / cm 2 ) (see Moses et al. 2011, their Fig. 8.) The inner boundary is an additional free parameter the impact of which is demonstrated by Venot et al. (2012, their Fig. 1). They also point out uncertainties in the NH3 and HCN abundances based on the use of di GLYPH<11> erent networks. Our ansatz considers the collisional dominated inner part of an atmosphere where kinetic gas-phase modelling is not required, but which composes the inner boundary for kinetic gas-phase rate networks. In this part of the atmosphere, the local thermodynamic conditions are well constrained by atmosphere simulations that combine radiative transfer and cloud formation (Sect. 2.1 for more details). We consider the part of the atmosphere where the formation of dust clouds influences the local gas-phase chemistry, the local temperature and the density. The cloud formation causes a depletion of those elements which take part in the condensation process (e.g. Fe, Mg, O; Fig. 2) resulting in reduced abundances of respective molecules. Once the cloud particles have formed, they represent a very strong local opacity source absorbing in the optical and reemitting isotropically in the infrared. The consequence is a backwarming e GLYPH<11> ect which causes a local increase of the gas temperature below the cloud layer (Fig. 1). Given that collisional gas-phase processes dominate in the atmospheric part of interest, we apply a chemical equilibrium routine that allows us to provide first estimates of the abundances of carbon-bearing macro-molecules and small PAHs. This approach allows us to study the gas-phase abundances at the inner boundary of future kinetic considerations, and to look at species not presently included in most of the current networks (e.g. C6H6). Furthermore, we show the influence of element abundances that are inhomogeneously depleted by the formation of dust clouds causing, for exam- ple, a considerable shift in the local carbon-to-oxygen (C / O) ratio (Fig. 3). We discuss vertical mixing that transports gas to higher, hence, cooler atmospheric regions in comparison to the chemical relaxation timescale, and we assess the influence of uncertainties in rate coe GLYPH<14> cient data on the quenching height. The influence of cosmic rays on the upper atmosphere will be addressed in a forthcoming paper.", "pages": [ 1, 2 ] }, { "title": "2 METHOD", "content": "We investigate the abundances of complex carbon-binding molecules in dust-forming, oxygen-rich atmospheres of Brown Dwarfs and giant gas planets by utilising chemical equilibrium calculations in combination with results from model atmosphere simulations (see Sect. 2.1). We are particularly interested in studying the e GLYPH<11> ect of a changing C / O ratio on the remaining gas-phase chemistry as caused by element depletion during cloud formation.", "pages": [ 2 ] }, { "title": "2.1 Model atmosphere with cloud formation", "content": "We utilise results from the D rift -P hoenix (Dehn 2007; Helling et al. 2008; Witte et al. 2009) grid of model atmosphere simulation which solves the classical 1D model atmosphere problem (radiative transfer, mixing length theory, hydrostatic equilibrium, gasphase chemistry; P hoenix ; Hauschildt & Baron 1999) coupled to a cloud formation model (nucleation, surface growth and evaporation, gravitational settling, convective replenishment, element conservation; D rift ; Woitke & Helling 2003, 2004; Helling & Woitke 2006). Each of the model atmospheres is determined by the e GLYPH<11> ective temperature (Te GLYPH<11> [K]), the surface gravity (log(g) with g in [cm / s 2 ]), and a set of element abundances which have been chosen to be solar. These element abundances will be altered where dust forms as demonstrated in Fig. 2. The metallicity may be used as an additional parameter, and can be varied by homogeneously increase or decrease all elements to mimic a sub- or supersolar element abundance set. Additional input quantities are absorption coe GLYPH<14> cients for all atomic, molecular and dust opacity species considered. The cloud's opacity is calculated applying Mie and e GLYPH<11> ective medium theory. For more details on Drift-Phoenix , refer to Witte et al. (2009). Providing details on the dust clouds, such as height-dependent grain sizes, and the height-dependent composition of the mixedmaterial cloud particles, the model atmosphere code calculates atmospheric properties, like the local convective velocity, and the spectral energy distribution, etc. The relevant output quantities which we use for the present study are the temperature-pressure (Tgas [K], pgas [dyn / cm 2 ]) structure and the height-dependent element abundances GLYPH<15> i (Figs. 1, 2). The local temperature is the result of the radiative transfer solution and the local gas pressure of the hydrostatic equilibrium. The element abundances are the result of the element conservation equations that include the chance of elements by dust formation and evaporation.", "pages": [ 2 ] }, { "title": "2.2 Chemical equilibrium calculation", "content": "In Local Thermodynamic Equilibrium (LTE), at a given temperature, gas pressure, and for a elemental composition, the chemical abundances depend on the thermodynamic properties of the species through their pressure equilibrium constants, K p (T). For an example dielemental molecule being formed from gaseous constituent atoms, K p (T) is a function of temperature only, and is given by the law of mass action (e.g. Tsuji 1973; Gail & Sedlmayr 1986), where p (AaBb), p (A) and p (B) are the partial pressures of the molecule A a B b in LTE and constituent atoms A and B, respectively. The temperature dependence of K p (T) can be fitted with a 4 th order polynomial where GLYPH<18> is the reciprocal temperature equal to 5040 / T (e.g. Tsuji 1973). Tabulated fitting parameters for the complex carbon-bearing molecules are from Cherchne GLYPH<11> & Barker (1992). Each atom, molecule and ion is represented by the law of mass action, as well as satisfying element conservation and charge conservation. The partial pressure of the molecules, ions and atoms can then be converted into a number density, nA aBb , by use of the ideal gas law The equations are solved simultaneously for all gas-phase species. The atmospheric profile of the local gas temperature, gas pressure, and element abundances are prescribed (D rift -P hoenix model atmosphere results; Dehn (2007); Witte et al. (2009)) and the chemical equilibrium is evaluated for each atmospheric layer.", "pages": [ 3 ] }, { "title": "2.3 Chemical kinetic approach through quenching kinetics", "content": "In the deep atmospheric layers, at high temperatures and pressures, chemical equilibrium can prevail if reaction kinetics operate faster than convective mixing. This is when the time-scale for a species to reach thermochemical equilibrium, t chem, is less than the timescale for atmospheric mixing, t mix (Sect. 2.3.3 for more details). We consider vertical mixing as the only cause of non-equilibrium gasphase chemistry here. We consider non-irradiated objects only, and hence, can neglect the e GLYPH<11> ect of photo-chemistry for the time being.", "pages": [ 3 ] }, { "title": "2.3.1 Chemical relaxation time scale", "content": "Consider, for example, a gas-phase species, A, which is formed and destroyed solely by the gas-phase reaction: The change in the number density of A, [A] [cm GLYPH<0> 3 ], is: where kf ( p ; T ) is the forward reaction rate coe GLYPH<14> cient, and kr ( p ; T ) is the reverse reaction rate coe GLYPH<14> cient. Consider a given pressure and temperature, ( p 0 ; T 0), where ( d [A]0 = dt ) p 0 ; T 0 = 0. The densities [A]0,[B]0, etc. are the equilibrium densities for ( p 0 ; T 0). We now quickly 1 transport A in a large gas parcel to a new pressure and temperature, ( p 1 ; T 1), with new equilibrium densities, [A]1,[B]1, etc., where ( d [A]1 = dt ) p 1 ; T 1 = 0. If [A]0 GLYPH<29> [A]1, then the time-scale, t chem, for A to go from [A]0 ! [A]1 can be expressed as (Prinn & Barshay 1977): The condition, [A]0 GLYPH<29> [A]1, requires that: If this were not the case, then the formation rate for A would be nearly equal or greater than the destruction rate for A, and at steady state, we would find [A]0 . [A]1, violating our required condition. In the case where [A]0 GLYPH<29> [A]1, the reverse reaction rate can be neglected, and: With this we consider that species A is destroyed by N reactions, each involving a species B i , governed by a rate constants ki . In this case, Eq. 4 is generalised as If a particular kj [B j ]0 is much larger than any of the other ki [B i ]0, then Eq. 5 simplifies to The dominating destruction rate depends both on the rate coe GLYPH<14> -cient, kj , and on the number density of species B, [B j ]0 in the gas parcel. The chemical kinetic conversion timescale, t chem(A) [s], for a given gas-phase species A, is the time for relaxation towards an equilibrium state. It is defined here as the time it takes for the number density, [A] [cm GLYPH<0> 3 ], to reach the equilibrium value. We describe our method for finding the dominant destruction reactions for select hydrocarbons in Section 3.2.", "pages": [ 3, 4 ] }, { "title": "2.3.2 Mixing time scale", "content": "Large-scale convection is the transport of gases with the mean bulk flow. In contrast, di GLYPH<11> usion refers to the transport of gases along a negative concentration gradient by the action of random motions. In the radiative zone, the vertical transport timescale by eddy diffusivity was suggested to be (e.g. Saumon et al. 2006; Moses et al. 1 2011). where Hp( z ) is the local pressure scale height. The coe GLYPH<14> cient for eddy di GLYPH<11> usion K eddy was taken to be 10 4 and 10 8 cm 2 s GLYPH<0> 1 , a set of reasonable values for substellar atmospheres (Saumon et al. 2006, 2007). A comparison with Zahnle et al. (2009a) and Moses et al. (2011) show that K eddy = 10 8 cm 2 s GLYPH<0> 1 is on the low end of the values used in kinetic models. However, Miller-Ricci Kempten et al. (2012) studies mixing e GLYPH<14> ciencies down to K eddy = 10 6 cm 2 s GLYPH<0> 1 for GJ1214b. Increasing the value of K eddy increases the mixing which can yield larger departures from chemical equilibrium as shown by e.g Zahnle et al. (2009a). The convective zone is situated in deeper atmospheric layers (the convective velocity is , 0), however, Woitke & Helling (2004) developed the idea of convective overshooting. Driven by their momentum, the parcels of gas are able to ascend the atmosphere beyond the Schwarzschild boundary into the radiative zone. We compare the chemical kinetic conversion timescales ( t chem; coloured lines in Figs. 12, 13) for the hydrocarbon species to the vertical convective overshooting timescales (thin black lines in Figs. 12, 13) of each atmosphere (see Woitke & Helling 2004, their Eq. 9) and to the eddy di GLYPH<11> usion timescales (Eq. 7, thick black lines in Figs. 12, 13). The convective mixing involved in the Drift-Phoenix models does mainly impact chemistrywise the cloud forming part of the atmosphere and decreases exponentially in the upper atmosphere. The di GLYPH<11> usive mixing as applied in most of the complex chemical network evaluations sustains its e GLYPH<14> ciency throughout the whole atmosphere, moving slower in the inner atmosphereand faster in the higher atmosphere.", "pages": [ 4 ] }, { "title": "2.3.3 Quenching level", "content": "Deep in the atmosphere, the species in a gas parcel reach equilibrium with the surrounding gas faster than the time it takes for the gas parcel to reach the upper atmosphere. In the outer cooler atmosphere, however, energy barriers can become significant and vertical transport can dominate over chemical processes ( t chem > t mix). As a result, departures from chemical equilibrium can potentially be observed for some molecules (e.g. CH4 / CO, Saumon et al. 2006). The abundance of a molecular constituent may become 'quenched' at a value called the 'quench level', when t chem = t mix (Prinn & Barshay 1977; Saumon et al. 2006; Visscher & Moses 2011; Moses et al. 2011). Above that level, at lower gas temperatures, the chemical reactions are frozen by vertical mixing, i.e. the forward chemical timescale is significantly slower than the gasdynamic timescales involved. The intersection between the mixing timescale and the reaction timescale of a chemical species marks the point inside the atmosphere where the equilibrium number density of that species is 'frozen in', and from where on it remains roughly constant.", "pages": [ 4 ] }, { "title": "3.1 Equilibrium gas-phase chemistry calculation", "content": "A combination of 199 gas-phase molecules (including 33 complex carbon-bearing molecules), 16 atoms, and various ionic species were used under the assumption of LTE. This is an extension of the gas-phase chemistry routine used so far in our dust cloud formation according to Helling et al. (1996). The data for the large carbon-bearing molecules considered are taken from Cherchne GLYPH<11> & Barker (1992) and they are grouped according to their structure as follows: large N-bearing species HC x N, complex hydrocarbons C n H2 n GLYPH<6> 2, C2H2 n , CH-bearing radical C x H, CH x , and C x . The Grevesse et al. (2007) solar composition is used for calculating the gas-phase chemistry outside the metal depleted cloud layers and before cloud formation. No solid particles were included in the chemical equilibrium calculations but their presence influences the gas phase by the reduced element abundances due to cloud formation and the cloud opacity impact on the radiation field, both accounted for in the D rift -phoenix model simulations (Sect 2.1). We utilize D rift -P hoenix model atmosphere (Tgas, pgas, GLYPH<15> i,) structures as input for our chemistry calculations.", "pages": [ 4, 5 ] }, { "title": "3.2 Reaction kinetics and rate determining steps", "content": "Various gas-phase rate networks are applied in the literature to study non-equilibrium gas-phase abundances in the upper lowpressure planetary atmospheres for irradiated objects (Zahnle et al. 2009a,b; Line et al. 2010; Moses et al. 2011; Venot et al. 2012; Miller-Ricci Kempten et al. 2012; Kopparapu et al. 2012). Com- plete chemical networks provide the ideal basis for determining quenching heights. The standard approach is to first model the atmosphere of a given object using a full rate network, and then to analyse the results of this network. Sensitivity analysis of networks can find rate-determining steps (Moses et al. 2011), and can even guide construction of a simplified network that includes only the dominant reactions (Carrasco et al. 2008). A comparison of the results for these networks is generally di GLYPH<14> cult because the rate networks di GLYPH<11> er in the number of rates. Therefore, the completeness of reaction paths, and often the reaction coe GLYPH<14> cient (e.g. Venot et al. 2012) and mixing parameter (e.g. Miller-Ricci Kempten et al. 2012) are not well constrained. A further challenge for rate network calculations is the choice of the inner boundary and the initial values for their kinetic equations. Thus far, we have followed a chemical equilibrium approach to gain a first insight of how much element depletion by dust formation would support the existence of large carbon molecules in the collisional dominated part of the atmosphere which has welldetermined thermodynamic properties (see Sect 2.1) compared to the diluted photo-chemically drive exosphere. To assess potential non-equilibrium e GLYPH<11> ects on macroscopic carbon-binding molecules in the gas-phase due to quenching, we study time-scale for only C2H2, C2H3 and C2H6, for which we can provide the equilibrium values. We also investigate the possible e GLYPH<11> ects of varying model atmosphere parameters (e.g. the surface gravity log(g)) in order to uncover potential di GLYPH<11> erences between the extended atmosphere of a planet, and the much more compact atmosphere of Brown Dwarfs. To investigate which molecule is driven out of equilibrium, one ideally uses an entire rate network because minor species might become unexpectedly important. However, it may also be possible to identify the dominant rate determining step which then would allow us an e GLYPH<14> cient assessment by using Eq. 6. This approach has been followed e.g. by Saumon et al. (2006) and was addressed also in Moses et al. (2011), and it has the advantage of flexibility. It further allows us to evaluate uncertainties in material quantities efficiently (Sect. 6.2). Carrasco et al. (2008) demonstrate how to produce a skeletal chemistry for Titan with the help of reference mass spectrum from INMS measurements of the Cassini Spacecraft. Their research is driven by the understanding that not all 700 reactions are needed to describe the Cassini observations. Their criterion is the reproduction of a certain reference observable within a certain uncertainty by a reduced set of reactions, starting from the full network. We will explore the quenching heights of C2H2, C2H3, and C2H6 using a simplified method: We consider only the immediate destruction reactions for C2H2, C2H3, and C2H6 in order to calculate the value of t chem using Eq. 6. We search for the dominant destruction reaction over the entire (Tgas, pgas) profile. Three criteria guide our search for the dominant immediate destruction reaction. For a given reaction involving species A with abundance [A]0, with a destruction reaction frequency GLYPH<23> f [s GLYPH<0> 1 ] 2 , and reverse reaction with a frequency GLYPH<23> r [s GLYPH<0> 1 ] 2 , there may be a range of (Tgas, pgas) where: 2 The reaction frequency is defined as GLYPH<23> GLYPH<17> k 2[ M ] for combustion reactions, GLYPH<23> GLYPH<17> k 2[B] for two-body reactions involving the species B, and GLYPH<23> GLYPH<17> k 3[B][ M ] for three-body reactions involving B. In each case, k 2 [cm 3 s GLYPH<0> 1 ] and k 3 [cm 6 s GLYPH<0> 1 ] are measured rate coe GLYPH<14> cients and [ M ] [cm GLYPH<0> 3 ] is the number density of the third body. (iii) GLYPH<23> f is much greater than the measured destruction frequency for other destruction reactions. If these three criteria are met for a range of our (Tgas, pgas) profile, then we consider that reaction to be the dominant destruction reaction for that range. We applied these three criteria in the case of C2H2, C2H3, and C2H6. We obtained the rate coefficients for the destruction reactions by searching the NIST Kinetics Database (http: // kinetics.nist.gov / ), the UMIST Database for Astrochemistry, the reaction list from Moses et al. (2005, www.lpi.usra.edu / science / moses / reaction list.pdf) and from (Venot et al. 2012), available on the KIDA database (Wakelam et al. 2012, http: // kida.obs.u-bordeaux1.fr). Reverse reactions were obtained from NIST, where available. When NIST did not list the reverse reactions, the rate coe GLYPH<14> cients were calculated from the assumption of microscopic reversibility, employing the method outlined in Visscher & Moses (2011). For each of the dozens of reactions, we calculated the destruction reaction frequency, and the reverse (formation) reaction frequency. The reactions that met all three criteria above are listed along with their rate coe GLYPH<14> cients in Table 1. The rate coe GLYPH<14> cients for the dominant destruction reactions, k , are given by the Arrhenius equation: Where A , n and Ea are parameters taken from the sources mentioned above. Units of k are s GLYPH<0> 1 , cm 3 s GLYPH<0> 1 and cm 6 s GLYPH<0> 1 for first-, second- and third-order reactions, respectively. Table 1 contains the dominant destruction reactions for C2H2, C2H3 and C2H6. Multiple destruction rates are listed because di GLYPH<11> erent destruction reactions dominate at di GLYPH<11> erent pressures. The listed reactions may di GLYPH<11> er for di GLYPH<11> erent atmospheres of di GLYPH<11> erent objects or for an atmosphere with di GLYPH<11> erent element abundances, e.g. for a N2 dominated atmosphere which is the case also for full rate networks. This method cannot account for the nonlinear nature of the full kinetics treatment, but no current chemical kinetics model is able to account for the deposition of gas-phase species onto dust grains as done in the D rift -P hoenix atmosphere models used here. The purpose of our approach is to constrain quenching heights for select small hydrocarbons to indicate potentially arising nonequilibrium e GLYPH<11> ects starting from our equilibrium abundances of carbon-binding macro-molecules. Since our intent is not to incorporate any non-equilibrium processes other than atmospheric mixing, this simplistic time-scale comparison seems capable of determining whether a given species is quenched, and what range of heights at which it is likely to be quenched. Although solving the series of nonlinear coupled di GLYPH<11> erential equations describing the full chemical kinetics is in general the most complete approach, in this case it would risk amplifying uncertainties (Wakelam et al. 2005) by using large computing resources. A detailed discussion of the reasons for these uncertainties and their e GLYPH<11> ect on our investigation is presented in Sec. 5. A considerable uncertainty is hidden in the designation of the reactions in Table 1 due to uncertainties in laboratory and theoretical investigations. Errors in the treatment of reaction kinetics (incorrect assumptions about the rate-determining step or rate coe GLYPH<14> -cient) are common, and may lead to uncertainties in the conclusions regarding disequilibrium abundances. This has been demonstrated by Venot et al. (2012) for the competing abundances of NH3 and HCN. This problem is known in the wider astronomical community and has lead to comparative studies of PDR chemical networks (e.g. Rollig et al 2007; PDR - photo-dissociation regions). Our list of reactions in Table 1 is constructed based on the best available measured and calculated rate coe GLYPH<14> cients. As this knowledge improves, the list of reactions might change, and so would the quenching height, independent of the approach chosen. Similar risks are present when analyzing the chemistry using a comprehensive chemical network. To our knowledge, the reactions in Table 1 are the most e GLYPH<14> cient destruction routes for the species we consider.", "pages": [ 5, 6 ] }, { "title": "4 CHEMICAL EQUILIBRIUM RESULTS", "content": "In what follows, we present our results of the composition of the gas of the deeper atmospheric layers, i.e. below the uppermost at- mosphere which could be a GLYPH<11> ected by photochemistry. We are interested in how cloud formation may indirectly impact the occurrence or the increase of the number density of complex carbon molecules in oxygen-rich environments. Such an indirect influence results from an inhomogeneous depletion of elements due to the formation of cloud particles, and from the feedback on the local temperaturepressure structure due to the large opacity of a dust cloud. We study the e GLYPH<11> ect of the atmosphere model parameter (Te GLYPH<11> , log(g), metallicity), and investigate a case for an artificially increased carbon over oxygen abundance. We present the calculations of the gas-phase abundances for a given atmospheric structure. The local temperature, pressure and element abundances are a result of the D rift -P hoenix atmospheric simulations (see Sect. 2.1). In Sect. 4.1 the local gas-phase element abundances are discussed with respect to the element depletion due to cloud formation. These element abundances are input quantities for our chemical equilibrium routine. Their impact on the gasphase compositions is demonstrated in Sect. 4.2 for an example of a giant-gas planet model atmosphere (Te GLYPH<11> = 1500K, log(g) = 3.0, initial solar element abundances). Section 4.3 shows which carbonbinding macro-molecules, small PAHs and HCN molecules could be expected in chemical equilibrium in the dense, cloud forming part of a Brown Dwarf or giant gas planet's atmosphere. Both dif- fer by their gravitational surface acceleration (log(g)), which causes the giant gas planet to have a much larger pressure scale height in the atmosphere, hence, to be less dense than a Brown Dwarf atmosphere. Section 4.5 tests how our results change if the C / O ratio is increased to larger than one.", "pages": [ 6, 7, 8 ] }, { "title": "4.1 Element abundances", "content": "The cloud formation as part of the D rift -P hoenix atmosphere simulations a GLYPH<11> ects the metal elements O, Ti, Mg, Si, Fe, Al which are depleted by the amount needed to form the cloud particles. This results a metal-depleted gas-phase. This becomes apparent in Fig. 2: the metal elements decrease as dust forms. Deeper inside the atmosphere, after evaporation of the dust grains, the metal elements are released back into the gas-phase. The relative depletion is independent on the initial element abundances as a comparison between the upper (solar) and the lower (subsolar) panel of Fig. 2 shows. The depletion of the oxygen causes the carbon-to-oxygen ratio to increase as relatively more carbon is available than without dust formation. Figure 3 demonstrates further that the change in C / O from the solar value to GLYPH<24> 0 : 7 is independent of the stellar parameters Te GLYPH<11> and log(g). Only the atmospheric gas temperature range concerned becomes wider with increasing gravity, hence for Brown Dwarfs, slightly more oxygen is available at the cloud base compared to a gas giant's atmosphere. The drop below the initial value (solar or [M / H] = GLYPH<0> 3 : 0) of the C / O ratio, is caused by a rise in the number fraction of oxygen at the cloud base following the gravitational settling and complete evaporation of dust particles containing oxygen.", "pages": [ 8 ] }, { "title": "4.2 Reference calculations", "content": "The elemental abundances determine how much of a given element is available to form molecules or ions. We have chosen one example model to demonstrate how di GLYPH<11> erent the molecular abundances are for a dust-depleted (blue) and a non-depleted (grey) element composition for an oxygen-rich planetary atmosphere of Te GLYPH<11> = 1500K, log(g) = 3.0 and (initial) solar metallicity (Figure 4). Oxygen-bearing, dust forming molecules can be considerably less abundant inside the cloud layers which coincides with the deviation of the elements from their initial values, because the dust formation processes consumes the local elements. This is shown in Fig. 2 where the cloud layer spans a temperature range from Tgas = 100 : : : GLYPH<24> 1600K for the low-metallicity giant gas planet atmosphere (bottom panel). As the metal elements are depleted by dust formation, they become unavailable to the formation of typical oxygen-rich gasphase molecules (e.g. SiO, FeO, MgOH). The Ti-bearing gases are the least abundant; the Fe-, Al-, Si-, and Mg-bearing gases have concentrations log( n y = n gas) GLYPH<25> 10 GLYPH<0> 5 : : : GLYPH<25> 10 GLYPH<0> 15 , hence they decrease by orders of magnitude inside the cloud region. We observe further that molecules such as, e.g. NH3, which are not directly involved in dust formation can change their abundance if dust forms. This is largely due to the e GLYPH<11> ect of dust formation on the gas-phase metallicity, and its e GLYPH<11> ect on the atmospheric temperature structure. Figure 4 demonstrates the major reservoir for oxygen in sub- stellar oxygen-rich atmospheres are gaseous H2O and CO. CH4 is the most abundant hydrocarbon (Fig. 5). Its concentration ( GLYPH<25> 10 GLYPH<0> 5 -10 GLYPH<0> 10 ) is comparable to the concentration of the typical oxygenbinding molecular species encountered in substellar atmospheres. We refer for a more complete plots regarding to oxygen-rich gasphase abundances to e.g. Lodders & Fegley (2002).", "pages": [ 8 ] }, { "title": "4.3 Hydrocarbons & PAH's", "content": "Carbon and hydrocarbon macro-molecules are not included in the dust cloud formation considered in the D rift -P hoenix model at- mospheres, nor in any of the other model atmosphere simulations for substellar objects. But dust formation has a strong impact on the oxygen abundance. Because of the large abundance of oxygen compared to iron, magnesium etc, the change seems small at a first glance, but the resulting C / O ratio is considerable (Sect. 4.1). The question is: how many carbon-bearing macromolecules would we expect in the collisionally dominated chemistry of a denser atmospheric environment, and how would this change if an oxygendepleting process like cloud formation occurred? Our calculations will serve as an inner boundary for more complex, kinetic models that are not yet able to resolve all possible reaction paths, and that address the outer atmospheric regions only (Zahnle et al. 2009, Line et al. 2010, Moses et al. 2011, Venot et al. 2012). We find that inside the cloud layer where oxygen is depleted, the carbon-bearing molecules, including the CN-complex, are more abundant than outside the cloud layer. For example, C2H6 increases by 3 orders of magnitude in concentration, C4H4 by 4 orders of magnitude and CH4 by 1 : 5 order of magnitude (Figs. 5, 6; Te GLYPH<11> = 1500K, log(g) = 3.0 and solar). CO, on the other hand, does not vary visibly (Fig. 4). The e GLYPH<11> ect is smaller for the CN-molecules (Fig. 7). All number densities rise, with the exception of C4H, with local gas temperature as the gas density increase inside the atmosphere. A higher density allows more molecules to form as the atmospheric number density increases. The smaller member of the hydrocarbon groups is always the most abundant. However, C6H6 (the first aromatic ring , or PAH) becomes one of the most abundant hydrocarbons in the deeper atmospheric layers. Its number density is comparable to the one of TiO or to hydrocarbons bearing 2 or 3 carbon atoms (Fig. 4). The results may appear surprising, but the atmospheric environments that we investigate here are considerably more dense than for example the atmosphere of an AGB star by which the study of carbon-molecules was inspired. The PAHs, other than C6H6, do not form in significant quantities. The little 'dip' observed in each curve is due to the drop of the C / O ratio at GLYPH<24> 1550K (model dependent). It is attributed to a sudden small rise in the number fraction of oxygen upon elemental oxygen replenishment of the cloud base following the gravitational settling and complete evaporation of dust particles containing oxygen. Varying T e GLYPH<11> , log(g) and [M / H]: We study the influence of Te GLYPH<11> , log(g) and [M / H] on the molecular concentrations. An increase of the e GLYPH<11> ective temperature to Te GLYPH<11> = 2000K causes decreasing number densities of the macro-molecules considered here. This is a consequence of a lower gas pressure for a given temperature in the atmosphere. It must be noted that all carbon-bearing molecular abundances remain greater inside the cloud layers than outside, even for increasing Te GLYPH<11> . Carbon-bearing species are able to form in higher abundances inside the cloud layers for an increasing surface gravity, i.e. inside Brown Dwarfs' atmospheres. This is a consequence of the higher gas pressure throughout their atmosphere compared to a gas giant's atmosphere. Increasing the surface gravity increases the partial pressures of the gas-phase molecules and hence their number densities. The pronounced jump in the number densities at GLYPH<24> 2150K coincides with the (T, p )-structure of the model. It is a feedback that results from dust formation causing a backwarming e GLYPH<11> ect. In spite of this increase in gas pressure, hydrocarbons with 6 or 4 car- bon atoms are still very rare ( < 10 GLYPH<0> 20 in concentration). Benzene is again an exception in the deeper atmospheric layers, where it reaches a concentration of 10 GLYPH<0> 9 for the very low-metallicity cases, which is comparable to the oxygen-bearing molecules' concentration shown in Fig. 4. Note, that low-metallicity atmospheres are considerably denser that solar metallicity atmospheres as demonstrated by Fig. 1. The metallicity is another parameter that determines the structure of a (model) atmosphere, but it is not very well constrained as our results in Sect. 4.1 suggest. The metallicity parameter is often introduced because detailed knowledge of individually varying element abundances is only very scarcely available. Works by Burgasser and co-workershow that metallicity needs to be considered as a parameter also for Brown Dwarfs and giant gas planets. By decreasing the metallicity, [M / H], atomic hydrogen becomes even more abundant relative to heavier elements. The concentrations of the di GLYPH<11> erent H-bearing species throughout the atmosphere increase compared to the solar composition case. Molecules with a higher number of H-atoms (e.g. CH4 and C2H6) remain in higher concentration inside the cloud layer, although by a much smaller fraction than in the solar-metallicity models studied. This small surplus in the concentration of some hydrocarbons leads to a lowering of the abundances of the other carbon-bearing molecules and compensates for the disappearance of hydrogen-saturated carbon-containing species.", "pages": [ 8, 10 ] }, { "title": "4.4 The carbon fraction locked up in carbonaceous molecules", "content": "In substellar atmospheres, most of the hydrogen is locked in H2, and most of the carbon is in CO or CH4. In this section, we calculate the carbon fraction locked in some hydrocarbon and cyanopolyyne molecules to test for the presence of alternative dominant equilibrium forms of carbon under the temperature and pressure conditions considered in this work following the work by Helling et al. (1996). To calculate the fraction of carbon, by mass, locked into each carbon-bearing molecule, we used where NC is the number of carbon atoms in molecule y , m H is the mass in grams of a carbon atom, [ y ] is the number density of molecule y [cm GLYPH<0> 3 ] and GLYPH<26> C is the total carbon mass density [g cm GLYPH<0> 3 ] in the gas-phase. Figure 8 depicts the fraction F y of carbon atoms bound in species y along the temperature profile of the cloud layer of a giant gas planet with Te GLYPH<11> = 1500K, log(g) = 3.0 and solar element abundances. It is remarkable that, in the outer layers, C4H traps most of the available atomic carbon, before dropping exponentially. From 500K onwards, CO, CO2 and CH4 establish themselves as the dominant carbon-bearing molecules throughout the cloud layer. Influence of T e GLYPH<11> , log(g) and [M / H]: When the e GLYPH<11> ective temperature is increased to 2000K (lower left panel in Fig. 8), F y generally diminishes since the molecular number densities have decreased. FC 4 H substantially decreases and substitutes CO and CO2 as the main carbon-bearing species in the very outer layers of the atmosphere. In a Brown Dwarf (log(g) = 5 : 0), CH4 is substituted as a more important carbon-bearing molecule than CO2. In a low metallicity environment ([M / H] = -3.0), CO and CO2 are overtaken by CH4. This e GLYPH<11> ect is more pronounced in high surface gravity objects. Due to the favourable conditions to the formation of H-bearing molecules, C2H6 is now a more important atmospheric reservoir for carbon than C2H3 and C3H3 throughout the entire cloud layer.", "pages": [ 10, 11 ] }, { "title": "4.5 Carbon-rich substellar atmospheres", "content": "In extrapolation of our results so far, we show how the gasphase composition would change if more carbon becomes available in an example giant gas planet atmosphere. We use the same D rift -P hoenix model atmosphere structures as before (Te GLYPH<11> = 1500K, log(g) = 3.0 and solar element abundances), but now we change the carbon element abundance such that more carbon than oxygen is available, hence C / O = 1.1 in our equilibrium chemistry routine (Figure 9). No feedback onto the radiative transfer is taken into account. In comparison to an oxygen-rich atmosphere (C / O < 1), the hydrocarbons and the PAHs are significantly more abundant if C / O > 1 even only moderately. For example, the number densities of CN and C2H2 have increased significantly by 5 and 10 orders of magnitude, respectively (compare Sect. 4.3). C2H2 is now as abundant as the typical molecules encountered in oxygen-rich environments (compare Fig. 4). C2H2 now dominates over H2O throughout the entire atmosphere while CH4 dominates over H2O until T = 3000K. Figure 9 shows further that carbon-monoxide is the most dominant C-bearing species in the model atmosphere used for this study. H2O and NH3 are of comparable abundance. The importance of the larger amongst the PAHs has increased considerably by increasing the carbon such that C / O > 1 (lower panel, Fig. 9 and Fig. 10). Helling et al. (1996) predicted the formation of PAHs with large concentrations in the layers of dynamical carbon-rich stellar atmospheres when T 6 850K (for higher e GLYPH<11> ective temperature and smaller surface gravity than the models studied in the present work). In our study, the PAHs are important between a gas temperature of 1000K and 1500K (Fig. 9). We recovered the same molecular concentration for C6H2, C2H2 and HCN as Helling et al. (1996) throughout the atmosphere; whereas CH4 and HC3N are higher in concentration in our model giant gas planet.", "pages": [ 11, 12 ] }, { "title": "5 DEPARTURE FROM CHEMICAL EQUILIBRIUM BY CONVECTIVE MIXING", "content": "So far, we have studied how abundant carbon-binding macromolecules can be in local chemical equilibrium in brown dwarf and giant gas planet atmospheres. Our results suggest that C2H6 becomes gradually more important than C2H2 in low-metallicity atmospheres as relatively less carbon is available. We now study possible deviations from the thermochemical equilibrium values by a simple time-scale comparison. The major carbon-binding molecules in a solar metallicity gas, CO and CH4, are rather small. Both, however, have been shown to be a GLYPH<11> ected when vertical mixing processes are faster than their destruction kinetics. According to Moses et al. (2011) and Prinn & Barshay (1977) transport time scale arguments can be used to predict the abundance of CH4 at its quenching point in the atmosphere. Next, we examine the potential influence of the transportinduced quenching on the hydrocarbon chemistry. Above the thermochemical regime in the deep atmosphere, where equilibrium is maintained via rapid reaction kinetics, a quenched regime may exist for some species, where rapid atmospheric transport and slow reaction kinetics drive constituents out of equilibrium. As a result, the abundance of molecules can be di GLYPH<11> erent from their equilibrium value at the same height in the atmosphere. We investigate three pre-selected hydrocarbon species, C2H2 , C2H6 and C2H3, which among the atmospheric hydrocarbon inventory, become increasingly important in the deeper atmospheric layers. There is good reason to think that CO and CH4 are quenched, and their quenching height may a GLYPH<11> ect the dominant destruction pathway for C2H x species. This may happen in two ways. Either (1) CO or CH4 directly destroys C2H x , or (2) CO or CH4 is involved in the formation or destruction of the species that destroy C2H x . We are capable of applying our first-order approximation of t chem to explore (1). Considering only the direct destruction of C2H x is a simplification that involves profound uncertainties. The full e GLYPH<11> ect of CH4 and CO cannot be accounted for under such a simplification, and these may impact the abundances of various species that destroy C2H x such that other destruction routes may dominate. This would be the case, however, even if CH4 and CO were not quenched; we only consider direct destruction routes for these C2H x species. The reactions that destroy C2H2, C2H3 and C2H6 that involve CO or CH4 and meet criteria (i) - (iii) in Section 3.2 are: with rate coe GLYPH<14> cients taken from Tsang & Hampson (1986). There are no known reactions involving C2H6 and CO / CH4. In order to find the maximum direct e GLYPH<11> ect of CO and CH4 quenching on relaxation time-scales for C2H2, C2H3 and C2H6, we consider the rates of these reactions with CO and CH4 quenched at the highest pressures considered in our D rift -P hoenix model atmospheres. It turns out that Reactions (11),(12) are both much slower than the destruction of C2H3 by H2, and so the quenching of CO and CH4 does not directly a GLYPH<11> ect the quenching height for C2H3. The relaxation time-scale t chem is shorter for both destruction reactions for C2H2 in Table 1 than for Reaction 10, even assuming the maximum possible abundance of CO due to quenching. Reaction (13) is endothermic, with a barrier of GLYPH<24> 43000 K, but the reverse reaction is severely impeded in the upper atmosphere by the depletion of HCO. If CO is not quenched, then the reverse reaction, HCO + C2H5 dominates throughout the atmosphere. If, however, CO is quenched at the highest pressure we consider for our model atmospheres, then Reaction (13) dominates in the log g = 3, solar metallicity case when p gas . 10 GLYPH<0> 5 bar. Nevertheless, the relaxation time-scale for this reaction is much larger than any of the dynamical time-scales, and so this reaction does not help to determine the quenching height of C2H6. The time-scale comparison for Reactions (10)-(12) are plotted in Fig. 11. The reaction pathways studied here are approximated to be rate-determining steps, since no data could be found available with full reaction schemes of hydrocarbons in substellar environments. Each molecular interconversion, such as CO / CH4 and N2 / NH3, are usually a full reaction scheme that consists of many reactions including the rate-determining step. The applicability of reaction schemes are also dependent on the atmospheric structure studied, and the uncertainties on the kinetic rate coe GLYPH<14> cients make the field of reaction kinetics very challenging as is clear from the wide range of time-scales for single reactions in Figs. 12,13. For three pre-selected hydrocarbon species, C2H2, C2H6 and C2H3, we find the following: C2H2: Figure 12 compares the chemical relaxation time-scales for C2H2 to the convective mixing time-scale and turbulent di GLYPH<11> usion time-scale. In the solar-composition giant gas planet model, the quench level of C2H2 can occur at atmospheric pressures as high as p gas GLYPH<25> 3 GLYPH<2> 10 GLYPH<0> 3 bar down to p gas GLYPH<25> 10 GLYPH<0> 5 bar. The intersection of t mix and t chem occurs at di GLYPH<11> erent pressures for all three models. When log g = 5 (solar metallicity), quenching occurs somewhere in the range 10 GLYPH<0> 3 bar < p gas < 3 GLYPH<2> 10 GLYPH<0> 2 bar. In the low metallicity case (log g = 3, [M / H] = GLYPH<0> 3), quenching occurs within the range 10 GLYPH<0> 4 bar < p gas < 3 GLYPH<2> 10 GLYPH<0> 2 bar. Transport-induced quenching has been extensively studied also by Moses et al. (2011) in the atmospheres of hot Jupiters. A dominant interconversion scheme for C2H2 ! CH4 is proposed (Moses et al. 2011, their Eq. 9), of which the rate-determining step is included in Table 1, along with the combustion reaction for C2H2. Moses et al. (2011) found that disequilibrium chemistry enhances the abundances of acetylene (C2H2) in the atmospheres of hot Jupiter exoplanets. C2H3: The quenching-study approach fails for species that have no quench point, i.e. no disequilibrium number density of the species can be estimated for above the quench level, only a comparison between the mixing and chemical timescales can be made. For example, the destruction of C2H3 through reaction with molecular hydrogen is unquenchable. This can be easily determined from Fig. 11, since the relaxation time-scale for C2H3 is orders of magnitude lower than the fastest dynamical time-scales considered for our model atmospheres. It results that vertical mixing cannot freeze out C2H3 destruction, and one cannot use our simple approximation of the quenching kinetics in this case to draw conclusions on the disequilibrium abundance of C2H3. C2H6: Figure 13 shows our results for the quenching of C2H6 according to our alternative method. The quench level candidates are intersection points between the eddy di GLYPH<11> usion timescales and t chem for the reaction: For C2H6, quenching may occur at pressures as low as 10 GLYPH<0> 3 bar. By increasing the eddy di GLYPH<11> usion, the eddy di GLYPH<11> usivity mixing time decreases. Consequently, the intersection of t mix ; eddy and t chem occurs at a higher range of pressures yielding a higher non-equilibrium number density of acetylene and C2H6 over a large extent of the atmosphere (compare thick black line in Figs. 12, 13). Despite the uncertainties in the rate coe GLYPH<14> cient for Reaction 15, a lower limit to the quenching height can be established.", "pages": [ 12, 13, 14 ] }, { "title": "6 DISCUSSION", "content": "Our exploration of the e GLYPH<11> ects that dust formation has on gas-phase metallicity and as a result on the equilibrium chemistry has yielded some surprising results. Although only a GLYPH<11> ecting the C / O by bringing it from 0 : 5 to 0 : 7, dust formation results in the depletion of metals like silicon and titanium by several orders of magnitude, and has had an orders-of-magnitude impact on the chemistry. Also remarkable is the stability of benzine in the deep atmosphere. We note particularly the high abundance of C4H at low temperatures and the impact of various published rate coe GLYPH<14> cients on predictions for the quenching heights and time-scales, both of which we discuss below.", "pages": [ 14 ] }, { "title": "6.1 C 4 Habundance", "content": "Our results suggest high abundances of C4H which is somewhat surprising as C4H is a radical and should therefor be destroyed easily. We checked the equilibrium constants and none of them behaves strangely when plotted, nor are they used outside that tested temperature interval. We cannot find any obvious errors with our calculation with respect to any of the molecules considered here, including C4H. Surprisingly high abundances of C4H have been observed in the interstellar medium (e.g. Pety et al. 2005), in cometary ice (Geiss et al. 1999), and are believed to play an important role in the complex carbon chemistry within Titan's atmosphere (Berteloite et al. 2008). It is believed that the high amount of C4H cannot be accounted for by gas-phase thermochemistry, but is formed as a product of PAH destruction, within the cometary ice itself, or via photodissocation (Leonori et al. 2008). It is interesting that our thermochemical equilibrium contains high amounts of C4H, without having accounted for any of these possible sources. The condensation of heavy elements in the gas onto grains, and the impact on the metallicity, may partially explain the high abundances of C4H observed in comets and the interstellar medium, and inferred to be present in the atmosphere of Titan.", "pages": [ 14, 15 ] }, { "title": "6.2 Uncertainties due to di GLYPH<11> erences in rate coe GLYPH<14> cient", "content": "Given the various rate networks employed in the literature, we assess the impact of the di GLYPH<11> erent material data on the quenching point. For this, we utilise C2H6 quenching. The uncertainties in the rate coe GLYPH<14> cient for Reaction (15) (Table 1) span an order of magnitude or more amongst the di GLYPH<11> erent rate coe GLYPH<14> cient data sources. For example, two values for k 0 [cm 3 s GLYPH<0> 1 ] for the C2H6 combustion reaction given in Baulch et al. (1992) have identical values of n and Ea , but values of A = 7 : 5 GLYPH<2> 10 GLYPH<0> 20 cm 3 s GLYPH<0> 1 and 4 : 5 GLYPH<2> 10 4 cm 3 s GLYPH<0> 1 , a di GLYPH<11> erence of about 24 orders of magnitude. A more typical disagreement would be between the k 0 values for this same reaction between Baulch et al. (1992) and Warnatz (1984), which at 800 K is about an order of magnitude, although these two rate coe GLYPH<14> cients come into much better agreement at high temperatures. Concerning the destruction of C2H2 by the three-body interaction with hydrogen, we examined in detail published rate coe GLYPH<14> -cients from reviews (Baulch et al. 1992; Tsang & Hampson 1986), experiment (Hoyermann et al. 1968) and theory (Benson & Haugen 1967). For the combustion of C2H2, we examined the review of rate coe GLYPH<14> cients from Dur'an et al. (1989), as well as the experimentally determined rate coe GLYPH<14> cients from Thraen et al. (1982) and Palmer & Dormish (1964) and the rate coe GLYPH<14> cients determined theoretically by Benson (1989). Finally, for C2H6 combustion into 2CH3, we considered the reviewed rates from Baulch et al. (1992); Warnatz (1984), as well as experimental (Oehlschlaeger et al. 2005; Izod et al. 1971) and theoretical calculations (Kiefer et al. 2005) of these rate coe GLYPH<14> cients. The range of values for the rate coe GLYPH<14> cients over the pressure range is incredible, and its impact on the chemical time-scales spans 15 orders of magnitude, as can be seen in Figs. 12 and 13. Three-body rate coe GLYPH<14> cients are very di GLYPH<14> cult to constrain from high temperature experiments, and theoretical work is therefore also fairly unconstrained. These large uncertainties pose a problem not only for our work, but even more so for the non-linear chemical kinetics models applied to these atmospheres. Improved experimental and theoretical determinations of these rate constants are essential to progress beyond the first order approximation employed in this paper. This leads in the case of C2H6 that there is no definite upper limit to the quenching pressure. Figure 13 depicts the results for the di GLYPH<11> erent rate coe GLYPH<14> cients in di GLYPH<11> erent lines styles which shows that it is possible that t chem > t di GLYPH<11> for the entire range of pressures considered in the D rift -P hoenix model atmospheres. The reason for this is the uncertainty of the values of the rate coe GLYPH<14> cients alone. It will be important for future chemical kinetics modelling to more carefully explore the e GLYPH<14> ciency of Reaction (15). In the case of C2H2, the termolecular destruction with atomic hydrogen has a relatively small uncertainty, and if it were the only dominant reaction for destroying C2H2, its quenching height would likewise be well-constrained. The rate coe GLYPH<14> cient for C2H2 combustion is far less accurate, but it still provides a range of quenching heights; the uncertainties do not overwhelm our analysis. The fact that C2H3 reacts with molecular hydrogen means that it should not have a quenching height at all. We note again that the data uncertainties do also apply to fullnetwork considerations. Every quenching height given in the literature will therefore change if the material data change. Hence, quenching heights should in general be rather given as a limit GLYPH<6> uncertainty.", "pages": [ 15 ] }, { "title": "7 CONCLUSION", "content": "It must be acknowledged that small hydrocarbon molecules are able to form in an oxygen-rich environment such as the atmosphere of Brown Dwarfs and giant gas planets. These molecules do not form in very significant concentrations in comparison to carbon-rich atmospheres; nonetheless, an increased surface gravity and / or decreasing metallicity combined with a greater C / O ratio inside the dust clouds improve the chance of PAH formation. A decrease of the oxygen abundance caused by oxygen-depletion due to cloud formation does support the appearance of complex carbon-binding molecules. These results contradict the general belief that hydrocarbon equilibrium chemistry is not expected in the atmospheres of Brown Dwarfs and giant gas planets where the C / Oratio is less than unity. It must be noted that hydrocarbon chemistry in irradiated giant planets, through non-equilibrium photochemistry, is theoretically predicted and observed (Zahnle et al. 2009). The formation of hydrocarbons in hot Jupiters with temperatures below 1000K is driven by the photodissociation of methane; the products - C2H2, C2H4 and C2H6 - further polymerise to build complex PAHs and hydrocarbon aerosols, called soots, which are thought to be involved in the prebiotic evolutionary processes towards the emergence of amino acids (Tielens 2008). For wavelengths at which the dust cloud is transparent, the deep atmospheric layers can be observed. Due to the relatively significant number densities of benzene (C6H6), the vinyl radical (C2H3) and acetylene (C2H2) predicted in our work in this region, one might expect a signature in the absorption lines. A recent work by Fortney (2012) discussed the possibility of the formation of carbon-rich giant planets in disks where the 'condensation of solids can lead to non-stellar C / O ratios in nebula gases', in accordance with the idea used in the present work. Furthermore, Fortney (2012) raises the question of the detection of carbon-rich Brown Dwarfs that may have been eluded so far (2MASS and SDSS) simply because the spectral appearance of a Brown Dwarf with refractory clouds that remove oxygen from the gas-phase will be di GLYPH<11> erent: di GLYPH<11> erent molecules will influence the opacities and thus, the absorption lines in the atmosphere. The present work is a step forward in determining the chemical species whose opacities may yield to spectra that appear distinctly di GLYPH<11> erent from objects with no oxygen-depleted refractory clouds. An interesting process to consider would be the transport of the deeper-layer hydrocarbons upward into the cloud layer. Additionally, one could imagine hydrocarbons sticking on the surface of dust grains, producing dark soot grains. This new piece of chemistry could lower the albedo of a gas giant or a Brown Dwarf, by rendering its spectral appearance 'darker'. A recent work by Tian et al. (2012) studied the formation mechanism of PAH molecules in interstellar and circumstellar environments by looking at reactions of acetylene over silicate particles like forsterite (MgSiO4), a particularly abundant dust particle in the clouds of brown dwarfs. Their experiments lead to the production of gas-phase PAHs such anthracene, naphthalene, phenanthrene and pyrene.", "pages": [ 15, 16 ] }, { "title": "8 ACKNOWLEDGEMENTS", "content": "We highlight financial support of the European Community under the FP7 by an ERC starting grant. The computer support at the School of Physics & Astronomy in St Andrews is highly acknowledged. Most literature serach has been performed using ADS. Our local computer support is highly acknowledged.", "pages": [ 16 ] }, { "title": "REFERENCES", "content": "Verlag, NY Witte S., Helling C., Hauschildt P. H., 2009, A&A, 506, 1367 Woitke P., Helling C., 2003, A&A, 399, 297 Woitke P., Helling C., 2004, A&A, 414, 335 Zahnle K., Marley M. S., Fortney J. J., 2009a, ArXiv e-prints Zahnle K., Marley M. S., Freedman R. S., Lodders K., Fortney J. J., 2009b, ApJ, 701, L20", "pages": [ 17 ] } ]

[ { "title": "ABSTRACT", "content": "In long-baseline interferometry, bandwidth smearing of an extended source occurs at finite bandwidth when its different components produce interference packets that only partially overlap. In this case, traditional model fitting or image reconstruction using standard formulas and tools lead to biased results. In this paper, we propose and implement a method to overcome this effect by calculating analytically a corrective term for the conventional interferometric observables: the visibility amplitude and closure phase. For that purpose, we model the interferogram taking into account the finite bandwidth and the instrumental differential phase. We obtain generic expressions for the visibility and closure phase in the case of temporally-modulated interferograms, either processed using Fourier analysis or with the ABCD method. The expressions can be used to fit arbitrary models to the data. We then apply our results to the search and characterisation of stellar companions with PIONIER at the Very Large Telescope Interferometer, assessing the bias on observables and model-fitted parameters of a binary star. Finally, we consider the role of the atmosphere, first with an analytic model to identify the main contributions to bias and, secondly, by confirming the model with a numerical simulation of the atmospheric turbulence. In addition to the analytic expressions, the main results of our study are: (i) the chromatic dispersion in the beam transport in the instrument has a strong impact on the closure phase and introduces additional biases even at separations where smearing is not expected to play an important role; (ii) the atmospheric turbulence introduces additional biases when smearing is present, but the impact is important only at very low spectral resolution; (iii) the bias on the observables strongly depends on the recombination scheme and data processing; (iv) the goodness of model fits is improved by modelling a Gaussian bandpass as long as the smearing is moderate. Key words: Instrumentation: interferometers - Atmospheric effects - Methods: data analysis - Methods: analytic", "pages": [ 1 ] }, { "title": "R. Lachaume 1 , 2 &J.-P. Berger 3", "content": "3 November 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "ference fringes are formed and their contrast and shift will be used to retrieve partial or total information on the complex visibilities. Long-baseline interferometry is an observational technique used from the optical (Michelson 1920) to the radio domain (Ryle & Vonberg 1946; Pawsey et al. 1946) that allows to overcome the resolution limit of single-dish telescopes, as ultimately set by diffraction. To achieve such a goal an ideal interferometer measures the complex degree of coherence and relates this so-called complex visibility to the object intensity distribution through a Fourier transform (van Cittert 1934; Zernike 1938). Practically speaking, inter- There are numerous sources of error and biases that have to be evaluated and as much as possible corrected in order to provide a proper estimation of the interferometric observables. Among them, bandwidth smearing occurs in finite bandwidth for objects spanning an extended field of view. The interferogram corresponding to a point-like source has a coherence length of the order of Rλ where R is the spectral resolution and λ the central wavelength. For two points of an extended source separated by a distance θ along the projected baseline length B corresponding to two telescopes of 1 1 the array, individual fringe packets are shifted with respect to each other by an optical path difference θ/B . When the OPD shift θ/B becomes of the order of, or greater than, the fringe packet width, i.e. when θ ≈ Rλ/B , the fringe packets of these points do not overlap correctly and bandwidth smearing of the interferogram occurs (see bottom left panel of Fig. 1). In other words, one can consider that the coherence length of an interferogram Rλ corresponds to an angular extension on the sky θ ≈ Rλ/B : it is called the interferometric field of view. Objects composed of multiple incoherent sources, either discrete or continuous, are affected by the smearing when their extent becomes of the order of the interferometric field of view. interferogram envelope; 3) the smeared regime where the separation is not anymore a small fraction and interferometric estimators are altered; 4) the 'double packet' regime where two fringes packets are well separated. While this effect has been known for decades (Thompson 1973), it cannot be remedied by calibration as other biases. This was analysed in a review by Bridle & Schwab (1989) in the radioastronomy context, in which the observer had no other choice but to define, somewhat heuristically, the best compromise between observing performance and limiting the bandwidth smearing. However, modern radio techniques, using a posteriori software recombination, can overcome the problem in many situations by using several phase centres, around which smearing does not occur. In the optical and the infrared, software recombination is not technically feasible and bandwidth smearing must be dealt with. Zhao et al. (2007) recommend to limit the field of view θ to 1 / 5 of the theoretical value of the interferometric field of view i.e. θ ≈ Rλ/ (5 B ) to remain in the resolved regime. For an interferometer working in the near-IR with 100 m baselines, this corresponds to 5-10 milliarcseconds of separation when a wide-band filter ( λ/ ∆ λ ∼ 5 ) is used without spectral resolution. The main leverage to increase the interferometric field of view is adapting the spectral resolution or the baseline length. However, it comes very often at a prohibitive sensitivity cost (spectral resolution) or a loss of spatial resolution (baseline length). In this paper, we present the first analytic calculation of the bandwidth smearing effect on the two main optical interferometric observables, namely the squared visibility and closure phase. We restricted the calculation to temporally encoded interferograms, including the so-called Fourier mode (a full scan of the fringe packet) and the temporal ABCD (a 4-point scan of a central fringe), which are among the most popular optical schemes. Fourier mode has been or is being used at COAST (Baldwin et al. 1994), IOTA with IONIC (Berger et al. 2003) and FLUOR (Coudé du Foresto et al. 1998), CHARA with FLUOR (Coudé du Foresto et al. 2003) and CLIMB (ten Brummelaar et al. 2012), VLTI with VINCI (Kervella et al. 2000), PIONIER (Berger et al. 2010; Le Bouquin et al. 2011), and MIDI (Leinert et al. 2003). Temporal ABCD is the choice at PTI (Colavita et al. 1999) and the Keck Interferometer (Ragland et al. 2012). It should be stressed that a similar line of reasoning can be used with very little adaptation to the 8-point time-encoded interferograms of NPOI (Johnston et al. 2006), and, with more efforts, to spatially encoded interferograms such as in VLTI/AMBER (Petrov et al. 2007) and static ABCD systems such as VLTI/PRIMA (Delplancke 2008). The derived formulae can be applied to correct any squared visibility and closure phase analytic formula describing the object angular intensity distribution. We apply this corrective technique to the study of binary stellar systems. Indeed optical long baseline interferometry is a unique tool to study the astrometry of close binary systems with milli-arcsecond accuracy to provide direct means to measure accurate masses. Moreover very recently several attempts at searching for substellar companions (Absil et al. 2011; Zhao et al. 2011b) are pushing the technique down to dynamical ranges where no adverse effects can be neglected. Since most studies forgo bandwidth smearing correction without assessing the biases that may arise from such approximation, we felt a proper treatment had become mandatory and would be useful in the future. For practical purposes we used the PIONIER instrument characteristics to provide an application of that work. PIONIER is currently being used at the Very Large Telescope Interferometer (VLTI, Haguenauer et al. 2012) to combine four beams in the H band ( 1 . 5 to 1 . 8 µ m ). Sect. 2 gives the analytic expression of the observables in the absence of atmospheric turbulence for an instrument working in fringe-scanning mode. Section 3 is an application of these formulas to a binary star, which allows us to analyse the bias that smearing produces on the interferometeric observables and the modelfitted parameters of the binary. We also show there how simulated fringes of PIONIER are much better fitted with the smeared model than with the standard expression. Finally, section 4 studies the impact of atmospheric turbulence on the observables, indicating that a moderate spectral resolution is enough to alleviate most of its effects.", "pages": [ 1, 2, 3 ] }, { "title": "2 MODELLING THE BANDWIDTH SMEARING: TURBULENCE-FREE CASE", "content": "In order to introduce the basic concepts of the data processing for fringe-scanning mode instruments, we remind the reader here how observables are derived in monochromatic light. Ignoring the atmosphere and instrumental signatures, the interferogram of a binary on baseline ij can be written as where N 1 and N 2 are the fluxes of each component, δ is the OPD between the arms of the interferometer, and α ij = (2 πσ B ij · θ ) is proportional to the binary separation θ , the projected baseline B ij , and wavenumber σ . It is convenient to use the coherent flux, a complex quantity representing the interferogram, from which the continuum N 1 + N 2 is removed and the negative frequencies are filtered out. In practice, one can take the Fourier transform of the interferogram, remove all frequencies but a small interval centred on the frequency of the fringes, and take the inverse Fourier transform. The coherent flux can be written as The square visibility amplitude is obtained by dividing the power contained in the coherent flux by that in the continuum:", "pages": [ 3 ] }, { "title": "fringe packet", "content": "where < x > δ means the average of variable x over the OPD. In practice, the power may be computed using the Fourier transform of the coherent flux, which is strictly equivalent (Parseval's identity). When a triplet of telescopes ijk is used, the closure phase is used to obtain partial information on the phase because it is independent of atmospheric turbulence. It is the argument of the bispectrum given by: where δ ij , δ jk , and δ ki are the time-modulated OPDs on the three baselines, meeting the closure relation δ ij + δ jk + δ ki = 0 . (Eq. 4 gives a compact, generic expression for the bispectrum in the same way Le Bouquin & Absil (2012) did for the specific case of highcontrast binaries.) The goal of this section is to describe the coherent flux, squared visibility, and closure phase of time encoded interferograms processed by means of Fourier analysis, when observing a source of arbitrary geometry in finite bandwidth. In other words, we seek to generalise Eqs. (2, 3, & 4) and provide ready-to-use formulas to fit object models to smeared data. For the sake of simplicity we use a discrete formalism valid for a collection of point-like sources. The results presented here are easily generalised to systems of resolved, compact sources (Appendix A1) and to any system with our summations over a finite number of point-like sources replaced by integrations on the plane of the sky. The most frequently used notations and symbols used in this section are given in Table 1.", "pages": [ 4 ] }, { "title": "2.1 Interferogram", "content": "We consider an interferometer with stations i , j , etc. separated by a baseline B ij operating in a spectral channel centred on wavenumber σ 0 . In the following developments we shall use σ , the wavenumber, and ξ = σ -σ 0 as 'reduced' wavenumber. Without losing generality, we assume that we observe an object made of several point sources o , p , etc. with positions θ o , θ p , etc. in the plane of the sky and spectra n glyph[star] o ( σ ) , n glyph[star] p ( σ ) , etc. The interferometer measures the complex coherent flux of the electromagnetic field by forming dispersed fringes on a detector. In our case, fringes are obtained by a temporal modulation of the optical path difference (OPD) δ around an ideal position x ij o . This position is related to the angular position of the source in the sky θ o through the relation x ij o = B ij · θ o . Each of the point sources contributes to a quasi-monochromatic interferogram per instrument spectral channel. Once the incoherent photometric contribution has been removed from the two telescopes and the negative frequencies have been filtered out in Fourier space, the complex coherent flux of one source reads: where n ij o ( ξ ) is the 'instrumental' coherent flux density primarily due to the wavelength-dependent instrumental effects, but also to some extent to the spectrum of the source. We can define this coherent flux density as: where: We assume that these instrumental signatures do not depend on the OPD position in the interferogram, which is a good approximation in fringe-scanning mode, since the OPD modulation is obtained through a few micrometres of air or vacuum, with negligible dispersion. In other words, we assume that the instrumental differential phase is a static term that is not impacted by the movement of the differential delay lines. However, this is usually not true for spatially dispersed fringes (see Tatulli & LeBouquin 2006, for a generic expression for the fringes), so that our approach needs adaptation to instruments like AMBER (Petrov et al. 2007). It is now possible to describe the coherent flux for an arbitrary number of sources and across a wider spectral bandpass: For practical purposes we use the Fourier transform substitute Eq. (5) into Eq. (7), and obtain where α ij o = 2 πσ 0 x ij o . In the following, we will use the coherent flux expression (Eq. 9) to compute the most commonly used interferometric observables i.e. the square visibility and the closure phase. In practice, n ij o is not known a priori. However, it can be inferred from fringes obtained on an internal lamp. The coherent flux of the lamp fringes yield n ij lamp (see Eq. 9). If both the spectrum of the source n glyph[star] o and that of lamp n lamp are known, n ij o = n ij lamp t ij int ( n glyph[star] o /n lamp ) (see Eq. 6) where t ij int is the transmission of the interferometer before the calibration lamp. The amplitude of the VLTI transmission is a smooth function of wavelength that can be considered constant. Its phase results from dispersive elements in the optical path. The optical elements of the VLTI before PIONIER are all in reflection and the most dispersive ones (the M9 dichroics) have been designed to display the least differential dispersion, so that the dispersion is dominated by the air in the non evacuated delay line. In the rest of this paper, we have considered near-zenithal observations for which the interferometric delay is small so that the air dispersion could be ignored as Appendix A5 shows. While the presence of dispersion in non zenithal observations has a significant impact on the amount of smearing, it neither changes its order of magnitude nor the general conclusions of this paper. When the atmospheric dispersion must be tackled, it can be done either explicitly (Appendix A5 explains how) or implicitly by letting the parameters of Sect. 2.2 free in model fits, as Zhao et al. (2007) do for the spectral resolution. As an illustration, we show in the left panels of Fig. 2 the spectral coherence transmission n ij lamp (amplitude and phase) that we measured on the internal source of PIONIER using three spectral channels across the H band on three baselines. The right panels correspond to the coherent flux of the fringes M ij lamp (amplitude and phase).", "pages": [ 4, 5 ] }, { "title": "2.2 Instrumental spectral response", "content": "In this paper, after providing generic formulas using Fourier formalism, we will also give closed form expressions for direct use. To do so, we need an analytic description of the instrumental transmission ( t i ) and differential phase ( ψ ij ). PIONIER's instrumental coherent flux density is obtained on a calibration lamp (Fig. 2, left panels). It displays a near-quadratic behaviour of the differential phase and a spectral transmission intermediate between top-hat and Gaussian functions. We therefore describe the instrumental differential phase as The linear term β ij in the instrumental differential phase ψ ij ( ξ ) translates into a fringe packet shift of β ij / 2 πσ 0 with respect to the nominal zero OPD (see Fig. 1, bottom right panel). It is called group delay. In a single-spectral channel interferometer it is possible to zero it by means of fringe tracking. When several spectral channels are observed at the same time, it is no longer possible to do so in all channels simultaneously. For instance, if a central spectral channel is centred at zero OPD, adjacent channels may be shifted with respect to it if there is a differential behaviour of the dispersive elements (such as waveguides, dichroics, or air whose refractive index depend on wavelength) in the beam paths before the recombiner. In the bottom panels of Fig. 2 (baseline 1-3), the central spectral channel is approximately centred at zero OPD (the solid line on the right panel shows the envelope of the fringe packet, i.e. the amplitude of the coherent flux) with a slope of the phase averaging to ≈ 0 (same line of the left panel). The adjacent channels feature some shift (dashed lines on the right panels) and non-zero phase slope (same lines on the left). Appendix A5 gives a further description of the group delay and its correction through fringetracking. The quadratic term in the instrumental differential phase D ij has a less visible impact on the fringe packet. We will give results both for Gaussian and top-hat transmissions of FWHM ∆ σ :", "pages": [ 5 ] }, { "title": "2.3 Square visibility amplitude", "content": "The square visibility amplitude is obtained from the coherent flux using: where N ij is a normalisation factor that relates to the total flux of the target ( ∝ N 2 ) and x ∗ stands for the complex conjugate of x . In the first line of the previous equation, we substitute Eq. (9) and expand the product into a double sum to find: (14) Using the change of variables δ → u = δ + x ij o , a correlation of Fourier transforms is recognised and simplified into the Fourier transform of a product. Thus, The bandwidth smearing is contained in ˜ n ij o n ji p . It can be made clearer by introducing the complex smearing where α is an angular variable that is linked to the OPD by the relation α = 2 πσ 0 δ . It is convenient to use the amplitude and phase of the smearing: Γ ij op = | ג ij op | is the contrast loss due to smearing and γ ij op = arg ג ij op is a phase shift induced by it. We also define the flux product equivalent-the equivalent to N o N p in the monochromatic case-as With these definitions, we can rearrange the square visibility amplitude: These results are independent of the instrumental phase ψ ij . If Γ ij op = 1 and γ ij op = 0 (no smearing) this formula is equivalent to the monochromatic case (Eq. 3 in the case of a binary). In practice, model-fitting of square visibility amplitudes by multiple stellar systems uses Eqs. (16, 17, & 18). Knowledge of n ij o , needed in Eqs. (16 & 17), can be inferred from fringes obtained on a calibration lamp (or a calibrator) if the spectra of both lamp and source o are known, as we discussed in Sect. 2.1. When the different sources share the same spectrum, i.e. n glyph[star] o ( ξ ) ∝ n glyph[star] p ( ξ ) , we may express the visibility as a function of the individual fluxes N o and the total flux N . In Eq. 18, we then use the flux products in lieu of the flux products equivalents, i.e. N ij op = V ins N o N p and N ij = N 2 , where is the 'instrumental' square visibility amplitude. Note that V ins also depends on the spectral profile. It only disappears in the calibration if the calibrator has the same spectral profile as the source. In the cases of the Gaussian and top hat transmissions with FWHM ∆ σ around central wavelength σ 0 and a constant contrast loss C ij in the spectral channel, the smearing is purely real ( γ = 0 ) and where R = σ 0 / ∆ σ is the spectral resolution. For small enough baselines, we have shown in Appendix A3 that an exponential formula can be used by properly choosing the value of R . On real data, R will need to be set to a value that differs from the spectral resolution in order to account from the departure from Gaussian profile and the wavelength dependence of the contrast. In practice, a model fit of smeared data may leave it as a free parameter. If high precision is needed, the asymmetry of the spectral band and the slope of C ij give a non zero γ . Cubic developments for the smearing terms Γ and γ are given in Appendix A3.", "pages": [ 5, 6 ] }, { "title": "2.4 Closure phase", "content": "A triple correlation or its Fourier transform, the bispectrum, or an equivalent method, is generally used to determine the closure phase (Lohmann et al. 1983; Roddier 1986). The determination of the closure phase in direct space uses the phase of the bispectrum, given by: where t is time in the case of linear OPD variations. By substitution of Eq. (7) into Eq. (21) and writing δ ij ( t ) = ˙ δ ij t It follows from Eqs. (9 & 22) and closure relation ˙ δ ij + ˙ δ jk + ˙ δ ki = 0 that Using the change of variables t → u = x ij o /δ ij + t , a triple crosscorrelation of Fourier transforms can be recognised and expressed as the two-dimensional Fourier transform of the triple product The bispectrum therefore reads The bandwidth smearing is contained in ˜ ˜ n ijk opq . In order to make it clearer we need to introduce several terms. The triple flux product equivalent (corresponding to N o N p N q in the monochromatic case) is given by the 'instrumental' closure phase by and the smearing by The 'instrumental' closure phase is a flux-weighted mean over the spectral channel and thus also depends on the spectrum of the source. The triple flux product equivalent can be simplified to the triple flux product ( N ijq opq ∝ N o N p N q ) when the sources have the same spectrum, i.e. n glyph[star] o ( ξ ) ∝ n glyph[star] p ( ξ ) . Note that the instrumental closure phase cancels out in the calibration only if the sources o , p , q and the calibrator all share the same spectrum. With these notations, the bispectrum reads If ג ijk opq = 1 (no smearing) and ψ ijk opq = 0 (no bandwidth-related differential phase), the formula is equivalent to the monochromatic case (Eq. 4 for a binary). In practice, Eqs. (25, 27, 28, 29, & 30) allow us to to model fit multiple stellar systems to smeared interferometric data. The knowledge of n ij o needed in Eq. (25) can be inferred from calibration fringes obtained on an internal lamp (or a calibrator) as discussed in Sect. 2.1. This modelling can be further simplified using an analytic description of the bandpass. In that case, Eqs. (30 & 31) can be used for the model fit of closure phases. In our cases of top-hat and Gaussian transmission of FWHM ∆ σ , with a linear instrumental phase, we reorder baselines so that ˙ δ ki has the largest absolute value, and we can assume it negative without losing generality. Then, the smearing can be simplified to In the equations above, the 'group delay closure' is expressed as The group delay closure is the consequence of the incorrect centering of the three fringe packets on the three baselines of the telescope triplets. Because of this de-centering, the centres of these packets are not scanned at the same time. In order to yield a usable closure phase, there should still be an overlap in the time intervals when the high contrast part of the packets are scanned. It means that the individual group delays β ij , β jk , and β ki , and thus the group delay closure, should be of the order of a few times the spectral resolution or less ( β ijk glyph[lessorsimilar] 2 π R ). Since this overlap in time depends on the relative scanning speeds along the baselines, the group delay closure depends on ˙ δ ij , ˙ δ jk , and ˙ δ ki . In our analytic approach to the spectral transmission, the instrumental closure phase reduces to a constant term, independent of the sources Appendix A2 explains how to use the Gaussian formula if the the quadratic chromatic dispersion term D ij is non zero.", "pages": [ 6, 7 ] }, { "title": "3.1 Bias on the interferometric observables", "content": "The first impact of the smearing is a tapering of the peak-to-peak amplitude of the oscillation of the visibility with baseline, hour angle, or spectral channel, due to the smearing amplitude Γ . The second impact only concerns the closure phase in multi-channel observations. It originates from the imperfect alignment of the fringe packets on baseline triplets, as measured by β ijk . In order to make these influences clearer, we give in Fig. 3 the interferometric observables of a binary with a high flux ratio 0.6, whose characteristics are given in Table 2. Square visibility amplitude. Figure 3(a), top panel, shows the theoretical smearing of the visibility amplitude of a binary as a function of reduced separation θB/λ (in mas · hm · µ m -1 ) for three different spectral resolutions ( ≈ 7 , 18 , 42 ) corresponding to the observing modes available on PIONIER at the VLTI. The lower panel of the figure displays the error on the square visibility occurring from not taking smearing into account, as a function of separation and spectral resolution. The result is easily generalised to binaries of different flux ratios, as the relative error on the visibility ∆ | V 2 | / | V 2 | remains unchanged. Closure phase. Figure 3(b), top panel, shows the theoretical closure phase of a binary for three different spectral resolutions ( ≈ 7 , 18 , 42 ) corresponding to the observing modes available on PIONIER at the VLTI. It can be seen at small separations (510 mas · hm · µ m -1 ) that the intermediate spectral resolution ( ≈ 18 ) shows more smearing than expected for these separations, in particular more than the broad-band ≈ 7 observing mode. The reason lies in the dispersive elements in the light beams of the interferometer and instrument that decentre fringe packets more in some spectral channels than in others, thus making it impossible to centre all fringes packets at the same time. (see the imperfect centering of some spectral channels of PIONIER in Fig. 2 and a description of the group-delay tracking in Appendix A5). This effect is not seen in the broad band, where the single fringe packet of each baseline can be centred with a fringe tracker, thus eliminating the groupdelay. This low-separation smearing approximately scales linearly with separation, as fβ ijk θ/ R 2 , where f is the flux ratio of the binary, θ the separation, and β ijk the group-delay closure (This can be obtained analytically by linearising Eq. 31b and normalising by the bispectrum of a point-source calibrator.) At larger separations g] [de k j i j [p] ( glyph[greaterorsimilar] 10mas · hm · µ m -1 in Fig. 3(b)), the closure phase is impacted by a combination of the tapering of the oscillation of the visibility (a purely spectral resolution effect, as seen in the visibility in Fig. 3(a)) and the instrumental phase, the impact is relatively complex, and we can only recommend to use Eq. (31b) to model it. As an illustration, Fig. B1 of Appendix B compares the closure phase of the three spectral channels of PIONIER for a given configuration of the interferometer, and it is quite clear the the behaviour radically changes with channel and telescope triplet. The lower panels displays the error on the closure phase occurring from not taking smearing into account, as a function of separation and spectral resolution. The figure shows a sharp discontinuity at resolution R = 8 where the transition occurs from a single spectral channel (where the single fringe packet of each baseline is positioned at zero OPD by an ideal fringe-tracker) to spectrally dispersed fringes (with the fringe packets of each baseline that do not align well because they are shifted with respect with each other by the instrumental phase). Even for moderately resolved sources, percent precision requires a good enough spectral resolution ( R glyph[greaterorsimilar] 40 or more), adequate modelling of bandwidth smearing, or a good fringe-tracking on a single spectral channel at moderate spectral resolutions ( R glyph[greaterorsimilar] 10 ).", "pages": [ 7, 8, 9 ] }, { "title": "3.2 Retrieving binary parameters", "content": "We assess here the bias on the binary parameters that smearing produces. In order to model the data as realistically as possible we build synthetic binary fringes corresponding to a typical scenario: near-zenith object observed in a sequence of three sets of fringes separated by one hour using a large telescope quadruplet at VLTI (see Fig. 4 for u , v coverage). They are obtained from calibration fringes obtained by PIONIER on an internal calibration lamp, which can be considered as a point source observation for our purpose. Then, we feed these synthetic data to the PIONIER data reduction software and get visibility amplitudes and closure phases. They are calibrated using simulated fringes of a point-source calibrator. They are then fit with a binary model to derive the parameters of the binary. In a first step, the model is that of an unsmeared binary (Eqs. 3 & 4), then we use the smeared model of Sect. 2 with Gaussian bandpass (Eq. 20b & 31b). Additional transmission ef- fects of the VLTI from the telescope up to the internal calibration lamp, positioned after the delay lines, have been ignored: the nearzenith observations we consider here are dominated by PIONIER's instrumental effects (as we discuss in Sect. 2.1). For non zenithal observations, where the interferometric delay in the delay lines is several tens of metres, the air dispersion in the delay lines becomes a factor of the same order of PIONIER's instrumental phase and can be modelled using Appendix A5. In our analysis, the separations in right ascension and declination are varied from -30 to 30 mas or approximately 10 times the angular resolution the interferometer and the magnitude differences from 0.1 to 3.3 (flux ratios from 0.05 and 0.95). For each point triplet of parameters, the difference between the fitted values and the input gives us the bias on the binary position and magnitude difference. The reduced chi square was determined assuming a 2% accuracy on visibilities and 20 mrad on closure phases typical of single-mode instrument performances on bright objects (like PIONIER). Figure 5 shows the absolute values of the errors and reduced chi square at each separation and position angle at the given magnitude difference of 0.55 (flux ratio of 0.6). In Figure 6, we consider possible biases and give the median value of the error with its confidence intervals for a given binary separation, considering all the position angles and flux ratios at that separation. Smearing-free binary model. A binary model with the classical expression for the visibility amplitude and closure phase (Eqs. 3 & 4) is fitted to synthetic PIONIER data with the three-channel spectral resolution. The left panel of Fig. 5 displays from top to bottom the absolute value of the error on the secondary's position, the absolute value of the error on the magnitude difference, and the reduced chi square for errors of 2% and 20 mrad on individual measurements of square visibility amplitudes and closure phases respectively. We checked that the results for other flux ratios are similar. The errors (with median value and confidence intervals) for the parameters are given in Fig. 6 (left panel) as a function of separation when the flux ratio is allowed to vary between detectable limits (0.05 to 0.95). The median value of the error indicates a bias, if it is non zero and consistently of one sign. The main impact of the smearing is a degradation of the goodness of fit at all separations, followed by errors on the flux ratio and separation at moderate separations, and a clear bias of both observables at larger separations. In our models, the secondary is dimmer than the input of the simulation more often than not and the separation tends to be smaller more often than not. (For instance, the confidence intervals on the errors of Fig. 6 show that the error on the separation is approximately 5 times more likely to be negative than positive at a separation of 30 mas.) The apparent dimming of the secondary is easily explained by the tapering of the fringe contrast that occurs due to smearing. The bias on separation is independent of smearing as we will see later on. Even at moderate separations (5-10 mas) the reduced chi square is around 3. However, the errors on the flux ratio and positions become significant (50 µ as and 20 mmag) only at higher separations ( glyph[greaterorsimilar] 15 mas), as Fig. 5. At first sight, it seems to contradict the trend of Sect. 3.1. In that section, we have found a significant smearing of the closure phase at small separations, as a result of the imperfect centering of fringe packets in an observation with multiple spectral channels. We easily reconcile these findings by noting that, as an average over the spectral band, the group delay is zero, i.e. both ends of the bands have group delays of same magnitude but opposite signs; thus their respective impacts on the observables approximately cancel out in the fit. The deviation of the individual spectral channels from the average over the band still explains the larger chi square. (Fig. B1 in Appendix B shows how the closure phases are impacted differently for the three spectral channels of PIONIER in low resolution mode.) Smeared binary model. We performed similar fits to synthetic smeared fringes of a binary by using the Gaussian formulas for the smearing (see Sect. 2). The absolute values of the errors on the position and flux ratio are given for a binary with a flux ratio of 0.6 in the right panel of Fig 5. The errors on the position and magnitude difference, and the quality of the fit are given in the right panel Fig. 6 for a wide range of flux ratios. In Fig. 6, the median value of the error indicates a bias if it is non zero and consistently of one sign. Taking the smearing into account eliminates most of the errors and bias on the flux ratio. It also largely improves the quality 300", "pages": [ 9, 10, 11 ] }, { "title": "Smeared fringes Smeared binary model", "content": "of the fit, with a reduced chi square of 3 found at significant separations ( glyph[greaterorsimilar] 15 mas) in most cases. The errors on the separation are improved at all separations but the bias remains at larger separations. We have found that the bias is related to the uncertainty on the effective wavelength of the interferometer, which varies by ≈ 0 . 1 % across baselines on PIONIER; this phenomenon is independent of our adequate modelling of the smearing. It is difficult to calibrate in the first place, because a deviation of the piezo scan speed from its nominal value has exactly the same observable consequence. (We note that including a proper spectral calibration in the instrument would solve for this problem.) At 30 mas of separation, a 0.1% bias translates into 30 µ as, which is what we indeed find: the solid lines in the top panels of Fig. 6 show this bias both in the monochromatic model and the smeared one. At specific binary parameters, seen as high error values islands on Fig. 5, the discrepancy originates from the difference between the smeared visibility and the Gaussian model: This happens close to smearing-induced phase jumps (see Fig. B1 of Appendix B for a comparison between Gaussian smearing and simulated values). High contrast binaries do not feature these phase jumps and are not impacted. For precision work of high to moderate flux ratio binaries, we strongly recommend to discard closure phases close to predicted jumps.", "pages": [ 11 ] }, { "title": "4 MODELLING THE ATMOSPHERE", "content": "The estimators of the interferometric observables have been chosen to be mostly immune to atmospheric biases in the typical interferometric regime of a moderately resolved source, i.e. when bandwidth smearing can be ignored. In this section, we investigate possible biases when bandwidth smearing becomes significant, as Zhao et al. (2007) did for IOTA's closure phase estimator. For temporal scanning, it is possible to write the differential piston-the variable differential phase induced by the atmosphere-as a function of OPD since time and OPD are linked (see for instance Colavita 1999). The jittered coherent flux can be expressed as a function of the ideal coherent flux where glyph[pi1] ij is the atmospheric differential piston on baseline ij . The exponential term is the contrast loss due to piston variation during the integration, of the order of one millisecond for one OPD step of a temporal scan. It bears the assumption that the spectral envelope of the fringes does not have features as sharp as the fringe frequency and that the integration during one OPD step is fast enough (of the order of a millisecond in practice) to allow for a linearisation of piston.", "pages": [ 11, 12 ] }, { "title": "4.1 Orders of magnitude", "content": "An analytic approach to the atmospheric turbulence can be taken, using the assumption that scanning is fast enough for the piston to vary linearly during a sub-second scan, i.e. glyph[pi1] ij = glyph[pi1] ij 0 + glyph[pi1] ij 1 δ ij , where glyph[pi1] ij 0 is the group-delay tracking error and glyph[pi1] ij 1 a rate of piston variation during scan. glyph[pi1] 0 and glyph[pi1] 1 are random variables when statistics over a large number of scans are derived. Using this approach, the coherent flux is: This approach can be used to determine the orders of magnitude of the atmospheric effects. Visibility. The piston variation term 1 + glyph[pi1] ij 1 comes as a product of the OPD variable in Eq. (35), so we recognise it as a scaling factor. glyph[pi1] ij 0 is a mere shift of the central OPD and has no impact-the square visibility does not depend on centering. Therefore, we can link the jittered visibility to the ideal case: The impact of atmospheric jitter is independent of the geometry of the source and, thus, smearing. For all separations it can be calibrated out if science target and calibrators are observed with similar atmospheric conditions. Closure phase. The group-delay tracking term glyph[pi1] ij 0 can be seen as a fringe shift that adds to the predicted fringe position β ij → β ij +2 πσ 0 glyph[pi1] ij 0 and the linear variation of the piston can be seen as a scanning velocity change ˙ δ ij → ˙ δ ij (1 + glyph[pi1] ij 1 ) . With these substitutions, the formulas of Sect. 2.4 can be used directly to determine the jittered closure phase. As we have seen, the predominant impact of the bandwidth smearing on the closure phase is the fringe decentering β ij , so we expect the group-delay tracking errors to be the main source of bias.", "pages": [ 12 ] }, { "title": "4.2 Numerical modelling", "content": "In the high frequency regime the pistons at the different stations can be considered as uncorrelated when the baselines are larger than the outer scale of turbulence L 0 (Kellerer & Tokovinin 2007). With a median value L 0 = 22 m at Paranal (Martin et al. 2000) baselines of the medium and large telescope quadruplets used with PIONIER normally fulfil the criterium. At other sites, for smaller baselines, or under relatively uncommon atmospheric conditions at Paranal, the pistons can be correlated. This correlation decreases the amount of atmospheric jitter for given coherence time and seeing, which in turns tends to decrease the bias on the interferometric observables. Therefore, we model the random piston glyph[pi1] i ( t ) using its spectral density where A and B are constants and Φ i ( ν ) is chosen randomly for each sampled temporal frequency ν . For Kolmogorov turbulence, the fast scan ( glyph[lessmuch] 1 s) regime has B = 17 / 6 (Conan et al. 1995) but there is experimental evidence (di Folco et al. 2003) that the slope is not as steep at VLTI, with simulations by Absil et al. (2006) explaining it in terms of the piston induced at high frequency by the adaptive optics (imperfect) correction ('bimorph piston', see Vérinaud & Cassaing 2001) and wavefront errors produced by the injection into single-mode waveguides ('coupled piston', see Ruilier & Cassaing 2001). Linfield et al. (1999) have also measured a deviation from the Kolmogorov behaviour at PTI. We used B = 2 , which experimentally reproduces well the accordion features of temporal scans obtained under below average atmospheric conditions (see Fig. 7). We normalise A to match the group-delay tracking rms in the differential piston glyph[pi1] ij = glyph[pi1] j -glyph[pi1] i . By substituting in Eq. 35, we perform a numerical integration of Eqs. (13 & 22) and obtain the jittered visibility amplitude and closure phase.", "pages": [ 12 ] }, { "title": "4.3 Bias on the observables", "content": "As we have seen in Sect. 4.1 there is little bias of the atmosphere on the square visibility amplitude and we could confirm it numerically. However, the bias can be substantial on the closure phase. Figure 8 displays in its top panel the bias on the closure phase of our test-case binary as a function of separation, for the three spectral resolutions R = 7 , 18, 42 corresponding to PIONIER's modes. For each separation, baseline, and spectral resolution considered in the simulation, 100 random scans with a remaining scatter of the fringe tracking of 6 λ (typical value by average conditions) have been generated. The closure phase on the telescope triplet is the average closure phase of the scans. To better identify the biases, the closure phase of a jitter-free observation has been subtracted from the results. In the lower panel of the figure, the bias on the phase · · is given in the separation-spectral resolution plane. As one can see, the impact of the atmosphere is very important at low resolution but quickly vanishes for R glyph[greaterorsimilar] 20 . For three spectral channels across a typical IR band, the error on the phase is at most a few degrees or less.", "pages": [ 12, 13 ] }, { "title": "5.1 Impact of the instrument and visibility estimator", "content": "As already discussed by Perrin & Ridgway (2005), the square visibility amplitude is impacted differently for different estimators that otherwise would be equivalent in the absence of smearing. Not only is the amount of smearing different but the behaviour can be changed. Because it is a popular recombination method and it illustrates this argument, we have given the formulas for the smeared complex visibility of a time-modulated ABCD recombiner in Appendix A4. In Sect. 2, we have seen that the square visibility amplitude is not impacted by the fringe centering in full scans processed by Fourier analysis : in Eq. (16), smearing is independent of absolute source position-only on source distances α ij o -α ij p -and group delay β ij . Conversely, the ABCD visibility estimator shows explicit dependence on α ij o and β ij (see for instance Eq. A14b), and this propagates to the square visibility estimator. Also, we have clearly put in evidence that instrumental features such as the OPD modulation scheme (ABCD or Fourier mode, stroke speeds on the different baselines) or the chromatic dispersion have a strong impact on the closure phase. In particular, the smearing behaviour of the closure phase of PIONIER (Fig. B1) shows different trends on different triplets or different spectral channels: on one hand, different telescope triplets are impacted differently because of the different OPD modulations; on the other hand, different spectral channels of the same triplet behave in different manners, as a consequence of different chromatic signatures. While the square visibility amplitude did not show a strong dependence on instrumental signature for full scans processed by Fourier analysis (Sect. 2), this is not necessarily the case. For instance, a timemodulated ABCD method displays impact for both visibility and phase (see Eq. A14b in Appendix A4). We therefore stress that each data reduction pipeline and each instrument require their own modelling of the smearing. In this paper, we have provided a generic formalism which can be used as is for VLTI/PIONIER and probably with little adaptation to other instruments that temporally scan most of the fringe packet.", "pages": [ 13 ] }, { "title": "5.2 When only part of the fringe packet is sensed", "content": "Also, our developments make the implicit assumption that most of the flux of the fringe packet is measured, i.e. that the OPD range is significantly larger than the FWHM of the fringe envelope. Actually, our developments still hold if the centres of the fringe packets originating from the different parts of the source are scanned but the extremities of the fringe packet are cropped, providing that the cropping is not too aggressive. In the case of PIONIER, the partial cropping on some baselines does not prevent a good agreement between simulated fringed and our analytic development, as Fig. B1 shows. However, it is clearly not the case in the ABCD method when a fringe-tracker locks the recombiner on the 'central' fringe (e.g Shao & Staelin 1980). While the smearing can be derived theoretically for this method (see Appendix A4), its magnitude will depend on the location of the fringe (i.e the OPD) onto which the fringe tracker locks. In the aforementionned Appendix it is shown that the visibility depends on the position of a source which in turns depends on the value of the group delay β ij (see Eq. A14). For relatively compact objects, the fringe tracker locks onto the brighter fringe or a local zero of the group delay and possible biases are calibrated out when observing an (almost) point-like calibrator un- er similar conditions. When a source is smeared, the fringe tracker does not necessarily behave in the same manner on source and calibrator, since there is no longer an obvious location of a central fringe (e.g. in the extreme case of a double fringe packet, it may lock on either packet). Therefore, it is quite likely that instruments sensing the central fringe of sources more resolved than a few beam widths (i.e. a few times the resolution power of the interferometer) will lead to altered measurements, unless (a) a high spectral resolution is used ( R glyph[greatermuch] β ij in Eq. A14) or (b) the fringe tracking scheme can be modelled with enough detail to know on which part of a given smeared fringe packet it locks. In particular, instruments that target high accuracy astrometry with the ABCD method like GRAVITY (Eisenhauer et al. 2011) and PRIMA (Delplancke 2008) will require that both the tracking reference and the science target are not very resolved.", "pages": [ 13, 14 ] }, { "title": "5.3 Image reconstruction", "content": "Our approach clearly targets parametric analysis, by providing formulas to model fit interferometric data by systems of compact sources. Image reconstruction however, usually relies on the Fourier relation between visibility and image, a relation which is broken in finite bandwidth. Thus, image reconstruction is made difficult as Bridle & Schwab (1989) already noted in radio interferometry.", "pages": [ 14 ] }, { "title": "5.4 Dealing with bandwidth smearing in practice", "content": "The angle of attack of radio astronomers to limit bandwidth smearing (see e.g Bridle & Schwab (1989)), is to restrict its effects either by increasing the spectral resolution to optimise the interferometric field of view or centering the phase tracking delay optimally to reduce the radial spread. Optical interferometry users do not have necessarily such a flexibility. One of the important differences between the wavelength regimes is that, in the optical, because the arrays have many fewer telescopes, most of the users do not actually reconstruct images but rather model directly the interferometric observables. This has been done to an extreme level of precision where visibilities are measured to a fraction of percent (e.g. Absil et al. 2008) and closure phases to a fraction of a degree (see e.g Zhao et al. 2011a). The particularly large impact of the smearing, even for moderately resolved sources, undermine the idea that the parameters for a large number of objects might be derived effortlessly using the traditional techniques. It therefore appears reasonable to adopt a two step strategy to deal with bandwidth smearing first by limiting the static instrumental smearing by design and secondly by operating the instrument under conditions that allow a proper modelling of the induced biases . Limiting the instrumental smearing. We have seen that the 'group delay closure' is the major contributor to a static smearing effect in the closure phase for instruments that operate in Fourier mode; it depends on the group delays and the OPD modulation scheme. The scanning speed scheme can be chosen so as to minimise the average group delay closures. For the temporal ABCD, visibility amplitudes and closures phases are directly impacted by the group delay, and this mitigation can longer be used. Since the group delay is mostly produced by a static chromatic dispersion in the instrument (waveguides, optical elements), an integrated approach to differential dispersion and birefringence compensation can be attempted as discussed in (Lazareff et al. 2012). Solutions exist that can provide guided or free space optics instrument with dispersion compensation (Vergnole et al. 2005). Correcting the air dispersion in the delay lines in real time may prove more difficult to implement than static correction of the dispersion in the optical elements, so that evacuated delay lines are probably part of the solution for larger baseline lengths ( glyph[greatermuch] 100 m) and at shorter wavelengths where the air dispersion is larger. Modelling the biases. We have shown that bandwidth smearing can be modelled provided that, a moderate spectral resolution is used (the first obvious step) and the estimators of the observables are properly calculated. In very low spectral resolution or in fullband ( R ∼ 5 ) observations atmospheric effects must also be decently constrained. For the latter, initial studies (e.g. Linfield et al. 1999; di Folco et al. 2003) have shown the correlation between atmospheric turbulence and low frequency statistics of piston but these are not necessarily well adapted to the sub second exposure (e.g. Absil et al. 2006). Dedicated further characterisation of piston statistics vs. monitored atmospheric properties would be needed. In summary, the ultimate tool to obtain a smeared source's properties will simulate the instrumental visibility numerically taking the instrumental signatures, in particular a dedicated spectral calibration, and the atmosphere into account.", "pages": [ 14 ] }, { "title": "5.5 Concluding remarks", "content": "Optical interferometry is increasingly used for precise measurements of high flux ratios and/or separation. Application of this precision techniques range from the detection of hot dust components around debris-disc host stars or the search for direct detection of hot Jupiters to the accurate astrometry of binary systems in search of precise mass determination. We have focused our work on a rarely studied effect that can alter significantly these astrophysical measurements, the so-called the bandwidth smearing. This bias-inducing phenomenon arises from the wavelength-dependence in the characteristics of the instrument, the atmosphere, and the source. We have modelled its impact by analysing its influence on the instrumental fringe contrast and determined how it alters the visibility amplitudes and closure phases. The magnitude of this effect will depend, for a given instrument, on the spectral resolution and the extension of the observed field of view and in some cases on the atmospheric piston. We have demonstrated analytically how to calibrate for this degradation in the context of popular temporal fringe scanning instruments and applied this analysis to the specific case of binary systems by computing the error or biases induced on the separation vector and flux ratio. We have further discussed 'real-life' constraints such as the influence of the atmospheric piston, the use of different fringe encoding schemes or the imperfections of the fringe tracking quality. We believe that the current analysis can be used with little effort to correct for potential bandwidth smearing biases in almost any astrophysical case.", "pages": [ 14 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We would like to thank an anonymous referee and Chris Haniff who helped us to improve this paper. This research has made use of NASA's Astrophysics Data System, the free softwares maxima, Yorick, and python. It has been supported by Comité Mixto ESOChile and Basal-CATA (PFB-06/2007).", "pages": [ 14 ] }, { "title": "REFERENCES", "content": "Absil, O., den Hartog, R., Gondoin, P., et al. 2006, A&A, 448, 787 Absil, O., di Folco, E., Mérand, A., et al. 2008, A&A, 487, 1041 Absil, O., Le Bouquin, J.-B., Berger, J.-P., et al. 2011, A&A, 535, A68 Baldwin, J. E., Boysen, R. C., Cox, G. C., et al. 1994, in Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series, Vol. 2200, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. J. B. 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A., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8445, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series van Cittert, P. 1934, Physica, 1, 201 Vergnole, S., Kotani, T., Perrin, G., Delage, L., & Reynaud, F. 2005, Optics Communications, 251, 115 Vérinaud, C. & Cassaing, F. 2001, A&A, 365, 314 Zernike, F. 1938, Physica, 5, 785 Zhao, M., Monnier, J. D., Che, X., et al. 2011a, PASP, 123, 964 Zhao, M., Monnier, J. D., Che, X., et al. 2011b, PASP, 123, 964 Zhao, M., Monnier, J. D., Torres, G., et al. 2007, ApJ, 659, 626", "pages": [ 15 ] }, { "title": "A1 System of compact, resolved sources", "content": "We consider sources indexed by o with with complex visibilities (amplitude and phase) given for baseline ij by when taken with respect to their nominal centre. The interferograms of the compact sources are not individually impacted by bandwidth smearing, so it is straightforward to see that the developments are similar to those of Sect. 2, with n o substituted to by n o ν ij o in the complex coherent flux (Eq. 9). Under the assumption that the individual visibilities are approximately constant over the bandwidth, formulas for the visibility, differential phase, and closure phase can be derived from Eqs. (18 & 30) using the following substitutions: with V ij o and φ ij o the mean visibility amplitude and phase over the bandwidth.", "pages": [ 15 ] }, { "title": "A2 Chromatic dispersion", "content": "In this section we provide analytic formulas for the smearing of interferometric observables when the quadratic dispersion term D ij cannot be neglected in the instrumental phase ψ ij (Eq. 10). It is only valid for the Gaussian bandpass. In that case, two quadratic terms appear in the exponential of n ij o (Eq. 6 with Eqs. 10 & 11 substituted in). One is the bandpass (real-valued) and the other the dispersion (imaginary). They can be gathered by using a 'complex' resolution R r ij instead of R in the Γ g g Reduced binary separation [mas · hm · m m - ] equations, where From there calculations are very similar (involving affine transforms of factor r ij ) and results of Sect. 2.4 can be used provided that the following substitutions are made: in Eqs. (30, 31b & 32).", "pages": [ 15, 16 ] }, { "title": "A3 Approximation of small smearing", "content": "For small enough baselines, the Taylor development of the complex exponential in the Fourier transforms can be used. We find that the smearing of the visibility can be linked to the moments of band pass using With these definitions the amplitude of the smearing is a second-order term of baseline while the phase-shift has first and third order terms: For sources with the same spectrum as the calibration lamp, X ij op cancels out because of the spectral calibration. We conveniently chose to define the Gaussian-equivalent spectral resolution as so that the formula of the Gaussian bandpass (Eq. 20b) may be applied for small separations. The Taylor developments of our two test-cases, top-hat and Gaussian bandpass, are This is not in agreement with Zhao et al. (2007) who noticed a difference by a factor of 1 . 5 2 in the coefficient between both bandpasses. They may have assumed that the bandpass was given by the power spectral density instead of the fringe envelope-there is a square involved, which leads to a factor 1.4 for the Gaussian FWHM. If we do the same, we then find results consistent with theirs. In Figure A1 we show the error budget resulting from approximations in the treatment of the smearing. While the source's spectrum must be taken into account to achieve a good precision, we find that in most cases a quadratic or exponential approximation for the smearing amplitude Γ is accurate enough. The source's spectrum may be implicitly taken into account by fitting the free parameter R to the data.", "pages": [ 16 ] }, { "title": "A4 Smearing for the time-modulated ABCD method", "content": "The visibility amplitude and closure phase are deduced from the complex visibilities. For a white fringe centred at zero OPD and a sampling at OPD points -3 λ/ 4 , -λ/ 4 , λ/ 4 , 3 λ/ 4 for A, B, C, and D respectively, the non-normalised complex visibility is given by where A , B , C and D stand for the flux in the interferogram, i.e. the real part of M ij ( δ ) . We deal here with the OPD modulation that probes the interferogram at OPDs of exactly -3 λ/ 4 , -λ/ 4 , λ/ 4 , 3 λ/ 4 , which can be achieved in practice by a time-modulation of the OPD in steps (see Fig. A2). For the other OPD modulation scheme, the variation is linear with time and the interferogram is integrated over an OPD interval of width π/ 2 . Our formulas are still a very good approximation for this scheme as long as there is some spectral resolution ( R glyph[greaterorsimilar] 5 ). We substitute Eq. 9 into A, B, C, and D to write where the smearing terms read In the case of a linear instrumental phase (Eq. 10 with no dispersion D ij ) and top-hat or Gaussian transmission (assumed identical for all objects), the smearing amplitude is the smearing phase reads and N ij o is proportional to the flux of the source. Note that the visibility, in particular the amplitude, is impacted by the instrumental group delay β ij in contrary to the full scans processed with Fourier analysis. The reason is that the ABCD only scans a small part of the interferogram where the smearing decreases the fringe contrast. As a consequence, the visibility amplitude of spectrally dispersed fringes are biased by a chromatic instrument because the fringe-tracking is not able to zero the group 100 delay in all channels simultaneously (see Appendix A5). Also, since the atmospheric differential piston introduces a random group delay (see Sect. 4.1), the ABCD visibility amplitudes are sensitive to the atmospheric turbulence unless the fringe tracking is close to ideal. The workaround in both cases is to increase the spectral resolution so that the fringe contrast remains approximately constant over the OPD excursion due to the atmosphere and the instrumental group delay, typically several tens of microns.", "pages": [ 16, 17 ] }, { "title": "A5 Group delay: fringe tracking & atmospheric dispersion", "content": "We consider a state of the interferometer, atmosphere, and recombiner in which the 'instrumental' differential phase is ψ ij ∗ ( σ ) . If the fringe-tracker introduces an OPD δ , the instrumental phase becomes In group-delay tracking around wavenumber σ 0 so that, in the end, the group delay at reference wavenumber becomes zero. If several channels are present, the group delay in channel c with central wavenumber σ c is Figure A3 shows how fringe tracking can cancel the group delay (and instrumental phase slope) at a reference channel but not at adjacent ones. This is due to the non linearity of the instrumental phase as the second derivative in Eq. (A19) shows. In the particular case of differential air dispersion in the delay lines, the additional differential phase is where ∆ ij is the interferometric delay and r ( σ ) is the refractive index of air minus 1 at wavenumber σ . Using Eqs. (A16, A17) we write interferometer transmission as The corresponding group delay (Eq. A19) has been computed for the different infrared bands and its typical value is reported in Table A1. In the H band, the atmospheric group delay becomes of the order of PIONIER's internal group delay ( ∼ 10 µ m, see Fig. 2) for interferometric delays of about 50 metres. Near-zenith observations with 100 metres baselines are therefore more impacted by PIONIER's instrumental phase.", "pages": [ 17, 18 ] }, { "title": "APPENDIX B: VALIDITY OF THE GAUSSIAN APPROXIMATION", "content": "When the amount of smearing becomes significant, the Gaussian model and actual smearing differ significantly. Figure B1 gives a comparison between the simulated and modelled closure as a function of separation; the model assumes a binary with a West-East separation observed in the test PIONIER configuration of Sect. 3. It is worth noting that the antisymmetry of the closure phase (with respect to orientation of the binary) is broken by the instrumental signatures; also the different spectral channels display different behaviours. ∆ α mas", "pages": [ 18, 19 ] } ]

[ { "title": "ABSTRACT", "content": "By combining optical and near-IR observations from the Hubble Space Telescope with near-IR photometry from the Spitzer Space Telescope it is possible to measure the rest-frame UV-optical colours of galaxies at z = 4 -8. The UV - optical spectral energy distribution of star formation dominated galaxies is the result of several different factors. These include the joint distribution of stellar masses, ages, and metallicities (solely responsible for the pure stellar spectral energy distribution), and the subsequent reprocessing by dust and gas in the interstellar medium. Using a large cosmological hydrodynamical simulation ( MassiveBlack-II ) we investigate the predicted spectral energy distributions of galaxies at high-redshift with a particular emphasis on assessing the potential contribution of nebular emission. We find that the average (median) pure stellar UV-optical colour correlates with both luminosity and redshift such that galaxies at lower-redshift and higher-luminosity are typically redder. Assuming the escape fraction of ionising photons is close to zero, the effect of nebular emission is to redden the UV-optical 1500 -V w colour by, on average, 0 . 4 mag at z = 8 declining to 0 . 25 mag at z = 4. Young and low-metallicity stellar populations, which typically have bluer pure stellar UV-optical colours, produce larger ionising luminosities and are thus more strongly affected by the reddening effects of nebular emission. This causes the distribution of 1500 -V w colours to narrow and the trends with luminosity and redshift to weaken. The strong effect of nebular emission leaves observed-frame colours critically sensitive to the redshift of the source. For example, increasing the redshift by 0 . 1 can result in observed frame colours changing by up to ∼ 0 . 6. These predictions reinforce the need to include nebular emission when modelling the spectral energy distributions of galaxies at high-redshift and also highlight the difficultly in interpreting the observed colours of individual galaxies without precise redshift information.", "pages": [ 1 ] }, { "title": "Theoretical predictions for the effect of nebular emission on the broad band photometry of high-redshift galaxies", "content": "Stephen M. Wilkins , William Coulton , Joseph Caruana , Rupert Croft Tiziana Di Matteo , Nishikanta Khandai , Yu Feng , Andrew Bunker , Holly Elbert 1 , 2 glyph[star] 2 2 , 3 2 , 4 , 2 , 4 4 , 5 4 2 2 26 August 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The availability of deep Hubble Space Telescope surveys utilising the Advanced Camera for Surveys (ACS) and more recently Wide Field Camera 3 (WFC3) means it is now possible to routinely identify galaxies at very-high redshift, with large ( > 50) samples identified to z ≈ 8 (e.g. Oesch et al. 2010a, Bouwens et al. 2010a, Bunker et al. 2010, Wilkins et al. 2010, Finkelstein et al. 2010, Wilkins et al. 2011a, Lorezoni et al. 2011, Bouwens et al. 2011, Lorenzoni et al. 2013, McLure et al. 2013, Schenker et al. 2013) and a few candidates now identified at z > 10 (e.g. Oesch et al. 2012a, Bouwens et al 2012b, Coe et al. 2012, Oesch et al. 2013, Ellis et al. 2013). While Hubble ACS and WFC3 observations (which probe the rest-frame UV continuum at z > 3) alone allow us to learn a great deal about high-redshift galaxies, including the UV luminosity function (e.g. Oesch et al. 2010a, Bouwens et al. 2010a, Wilkins et al. 2011a, Lorezoni et al. 2011, Bouwens et al. 2011, Lorenzoni et al. 2013, McLure et al. 2013, Schenker et al. 2013), the UV continuum slope (e.g. Stanway et al. 2005, Bunker et al. 2010, Bouwens et al. 2010b, Wilkins et al. 2011b, Wilkins et al. 2013a, Bouwens et al. 2013), and UV morphologies (e.g. Oesch et al. 2010b), by combining them with Spitzer Infrared Array Camera (IRAC) photometry it is possible to also probe the rest-frame optical emission. This is extremely difficult (given the lower sensitivity of the IRAC observations) for all but the brightest individual objects. However, by stacking large samples of galaxies together it becomes possible to robustly probe the average SEDs of even the faintest galaxies (e.g. Eyles et al. 2005, Labb'e et al. 2010, Gonz'alez et al. 2011, Labb'e et al. 2012). The UV/optical spectral energy distribution of a star formation dominated galaxy are affected by a complex mixture of different factors including the joint distribution of stellar masses, ages, and metallicities, dust, and nebular emission, many of which are closely coupled. The large number of effects makes it difficult to ab initio interpret observations, especially at high-redshift, where typically only broadband photometry is available, in the context of any of these individual quantities. In this paper we use a state-of-the-art cosmological hydrodynamical simulation of structure formation ( MassiveBlack -II) to investigate the UV-optical colours of high-redshift galaxies and in particular the effect of nebular emission thereupon. This paper is organised as follows: in Section 2 we discuss in turn the various factors affecting the rest-frame UV/optical colours of high-redshift galaxies. In Section 3 we present predictions from our large cosmological hydrodynamic simulation MassiveBlack -II. In Section 4 we describe how strong nebular emission makes robust estimates of galaxy stellar masses difficult. Finally, in Section 5 we present our conclusions. Magnitudes are calculated using the AB system (Oke & Gunn 1983). Throughout this work we assume a Salpeter (1955) stellar initial mass function (IMF), i.e.: ξ ( m ) = d N/ d m ∝ m -2 . 35 .", "pages": [ 1, 2 ] }, { "title": "1.1 Filters used to probe the UV-optical SEDs", "content": "Throughout this work we make use of several Hubble and Spitzer filters, including: Hubble /ACS ( B f 435 w , V f 606 w , i f 775 w , and z f 850 lp ), Hubble /WFC3 ( Y f 105 w , J f 125 w , and H f 160 w ), and Spitzer /IRAC ([3.6] and [4.5]) filters. We also introduce four rest-frame bandpasses (1500 1 , B 2 , V w 3 , and R 4 ) 5 . These rest-frame bandpasses allow us to consistently compare the properties of galaxies at different redshifts. The simple shape of these filters is chosen for convenience and the transmission profiles of all these filters are shown in Figure 1. Figure 2 shows the observed frame SED of a star forming galaxy at z ∈ { 5 . 0 , 5 . 9 , 6 . 9 , 8 . 0 } in relation to this filter set.", "pages": [ 2 ] }, { "title": "2 FACTORS AFFECTING THE UV-OPTICAL COLOURS OF STAR FORMING GALAXIES", "content": "The observed spectral energy distributions (SEDs) of galaxies are formed from the intrinsic stellar and AGN SEDs with reprocessing by dust and gas (both in the local ISM and IGM). The intrinsic SED of a stellar population (i.e. the pure stellar SED) is determined by the joint distribution of stellar masses, ages, and metallicities. To demonstrate the effect of various changes to the star formation and metal enrichment histories we utilise the Pegase.2 (Fioc & RoccaVolmerange 1997,1999) stellar population synthesis (SPS) model. We first, in § 2.1-2.2, describe how the rest-frame intrinsic pure stellar UV-optical 1500 -V w and optical B -R colours are affected by the properties of the stellar population (distribution of masses, ages, and metallicities). In § 2.3 we extend this to include the effect of dust and critically in § 2.4 we discuss the effect of nebular emission. It is also important to stress that the predicted SED, for a given IMF, star formation history, and metal enrichment history, is also sensitive to the choice of SPS model. In Appendix 5 we investigate how changing the SPS model affects the predicted UV-optical colours.", "pages": [ 2 ] }, { "title": "2.1 Distribution of stellar mass and ages", "content": "The SEDs of individual stars vary strongly with both stellar mass and evolutionary stage (and therefore age). As such the intrinsic SED of a composite stellar population is predominantly determined by the joint distribution of stellar masses and ages. This, in turn, is determined by both the initial mass function (IMF) and the star formation history (SFH).", "pages": [ 2 ] }, { "title": "2.1.1 Star formation history", "content": "In general, stellar populations with protracted star formation histories will contain a higher proportion of low-mass stars (as many of the original high-mass stars will have evolved off the main-sequence) causing the SED of the population to redden relative to a younger population. The sensitivity of the UV/optical colours to the recent star formation history can be seen in Figure 3; this shows the sensitivity of both the rest-frame 1500 -V w (UV-optical) and B -R (optical) colours to the duration of previous (constant) star formation. Increasing the duration of previous star formation from 10 → 1000 Myr causes the pure stellar 1500 -V w colour to redden by ∼ 1 . 4 mag. The effect on the optical B -R colour is more subtle, with the colour increasing by only ∼ 0 . 2 mag as the previous duration of star formation is increased from 10 → 1000 Myr. Assuming an increasing SFH, which is likely to be more representative of high-redshift star forming galaxies (see for example the predictions of Finlator et al. 2011), will suppress the evolution of the 1500 -V w colour as the SED is remains dominated by the most massive stars. In contrast if we instead consider the colour evolution of an instantaneous burst over the same time period (10 → 1000 Myr) the 1500 -V w and B -R colours redden by ∼ 3 . 5 mag and ∼ 0 . 5 mag respectively.", "pages": [ 2 ] }, { "title": "2.1.2 Initial mass function", "content": "Changing the choice of initial mass function will also effect the distribution of stellar masses. As such it can potentially have a significant affect on the colours of a stellar population. Changes to the low-mass ( < 0 . 5 M glyph[circledot] ) end of the IMF, have only a small effect on the shape of the SED as these stars contribute only a small fraction of the total luminosity (especially in the UV/optical) of actively star forming galaxies. On the other hand, changes to the high-mass end, will affect the mass-distribution of luminous massive stars. The high-mass IMF can be most simply parameterised as a power law, i.e. ξ ( m> 0 . 5 M glyph[circledot] ) = d N/ d m ∝ m α 2 , where α 2 is the high-mass slope (which for the Salpeter 1955 IMF would be α 2 = -2 . 35). Increasing α 2 increases the relative proportion of very-high mass stars resulting in a bluer 1500 -V w colour, as can be seen in Figure 4. Changing α 2 from -2 . 35 to -1 . 5 results in the pure stellar colour decreasing by ∼ 0 . 2 mag.", "pages": [ 4 ] }, { "title": "2.2 Metal enrichment history", "content": "Stars of similar mass and age with lower metallicities generally have higher effective-temperatures and thus bluer UV/optical colours (see also Wilkins et al. 2012, Wilkins et al. 2013a). A stellar population with a similar star for- mation history, but lower metallicity, will also then tend to have bluer UV-optical colours. This can be seen in Figure 3, where the UV/optical colours (1500 -V w , B -R ) are shown for two metallicities ( Z ∈ { 0 . 02 , 0 . 0004 } ). While (assuming the same duration of previous star formation) the high metallicity population is always redder, the difference is sensitive to the previous star formation duration (and thus the distribution of stellar masses and ages).", "pages": [ 4 ] }, { "title": "2.3 Dust", "content": "Throughout the UV to NIR dust acts to preferentially absorb light at shorter wavelengths. One effect of dust is then to cause the observed colour to redden relative to the intrinsic colour, i.e. ( m a -m b ) obs = ( m a -m b ) int + E ( B -V )( k a -k b ) where λ a < λ b , E ( B -V ) ≥ 0 . 0 and k a > k b . The extent of the reddening due to dust is then a product of the attenuation/reddening curve k ( λ ) and a measure of the total attenuation (often expressed by the colour excess E ( B -V )). Assuming an SMC-like curve (Pei et al. 1992), which is favoured at high-redshift by recent observations (e.g. Oesch et al. 2012b), an optical attenuation of A V = 0 . 4 mag (which corresponds to A 1500 ≈ 1 . 76 mag) will redden the 1500 -V w and B -R colours by ≈ 1 . 36 and ≈ 0 . 15 respectively (assuming no distinction in the effect of dust between nebular and stellar emission). If instead we assume a Calzetti et al. (2000) starburst curve (and again assume no distinction in the effect of dust between nebular and stellar emission) the same V -band attenuation (i.e. A V = 0 . 4 mag, which corresponds to A 1500 ≈ 1 . 03 mag) reddens the 1500 -V w colour by ≈ 0 . 63 mag and the B -R colour by only ≈ 0 . 12 mag. However, Calzetti et al. (2000) found that the nebular emission of intensely star forming galaxies is more strongly affected by dust than the stellar emission (at the same wavelength) with a the relationship between the colour excess of the nebular and stellar emission of E ( B -V ) stellar = (0 . 44 ± 0 . 03) × E ( B -V ) nebular . A significant consequence of assuming the Calzetti et al. (2000) is that there is no longer a unique mapping between the intrinsic and observed colours for a given attenuation. While in this work we are more concerned with effect of nebular emission on the intrinsic photometry it is however worth noting that the similar consequences of dust attenuation, the star formation history and metallicity make interpreting the observed broad-band colours of stellar populations in the context of these quantities extremely challenging.", "pages": [ 4, 5 ] }, { "title": "2.4 Nebular emission", "content": "Ionising radiation, which is produced predominantly by hot, young ( < 10 Myr), massive stars ( > 30 M glyph[circledot] ) 6 , is potentially reprocessed by gas in the ISM into nebular (line and continuum) emission. At high-redshift ( z > 4), galaxy formation models (see Section 3) suggest that virtually all galaxies continue to actively form, or have recently formed, stars 7 . Assuming the the escape fraction of ionising photons ( X f ) is small these galaxies are likely to contain strong nebular line emission. To include the effect of nebular emission on our predictions we use the number of ionising photons predicted (by Pegase.2 ) to determine the fluxes in the Hydrogen recombination lines. Fluxes in non -Hydrogen lines are determined using the metallicity dependent conversions of Anders & Fritze et al. (2003) 8 . Throughout this analysis we assume that the escape fraction ( X f ) is zero. This assumption allows us, when combined with the pure stellar colours, to explore the full range of potential colours.", "pages": [ 5 ] }, { "title": "2.4.1 Effect on rest-frame UV-optical colours", "content": "Due to the number of strong emission lines (e.g. O[II], H β , O[III], and H α ) in the rest-frame optical, unless the bandpass used to probe the UV encompasses Lymanα , the effect of nebular emission will be to redden the UV-optical colour relative to that of the pure stellar colour. This can be seen in Figure 3 where both the pure stellar and the stellar with nebular emission colours are shown (as function of previous duration of star formation). For a 100 Myr duration of previous constant star formation and Z = 0 . 02, the effect of including nebular emission is to redden the 1500 -V w colour by ∼ 0 . 3 (see Figure 3). However, the relative effect of nebular emission changes with both metallicity and previous star formation duration. For protracted ( > 1000 Myr) constant star formation, the effect decreases to ∼ 0 . 1 mag (if there has been no star formation for > 10 Myr the contribution of nebular emission will also fall to virtually zero). This variation is predominantly due to the sensitivity to the ratio of ionising photon flux to the optical flux which is itself sensitive to the (joint) distribution of stellar masses, ages, and metallicities. The variation with metallicity is in part also due to the metallicity dependent line ratios. The inclusion of nebular emission can (for actively star forming populations) also dramatically modify the trend with the high-mass slope of the IMF compared to the pure stellar case ( § 2.1.2), as can be seen in Figure 4. When nebular emission is included the trend between the 1500 -V w colour and high-mass slope reverses: stellar populations with shallower IMFs (high-mass biased) have redder 1500 -V w colours. This is again a result of the increased proportion of massive hot stars which produce large amounts of ionising radiation.", "pages": [ 5 ] }, { "title": "2.4.2 Redshift sensitivity and effect on observed colours", "content": "The effect of nebular emission on observed frame colours is critically sensitive to both the filter transmission curve and the redshift of the source. Small changes in redshift can leave strong emission lines in adjacent bands, dramatically affecting the observed colour. This can be seen in Figure 5, where the X -[3.6] (where X ∈ { z f 850 lp , Y f 105 w , J f 125 w , H f 160 w } ) and [3.6] -[4.5] colours (both from the pure stellar SED and the stellar+nebular) of a young star forming stellar population (100Myr continuous star formation) are shown as a function of redshift (Figure 2 also shows the spectral energy distribution of a stellar population forming stars for 100 Myr along with the transmission functions of the various observed frame filters). For example, at z = 5 -5 . 3 there are no strong emission lines within the IRAC [3.6] bandpass, thus leaving the X -[3.6] colour virtually unchanged relative to that for a pure stellar SED (unless the X filter encompasses Lymanα ). In contrast at z < 5 and z > 5 . 3 the IRAC [3.6] filter includes strong emission lines (O[III] and H β , or H α respectively). This results in an extremely strong sensitivity to the redshift. An increase in redshift of 0 . 1 (i.e. z = 5 . 0 → 5 . 1) can decrease the X -[3.6] colour by 0 . 5 9 while correspondingly increasing the [3.6] -[4.5] colour by 0 . 7. A similar situation occurs at z ≈ 7 as the [OIII] and H β lines move out of the [3.6] band into the [4.5] band. The [3.6] -[4.5] colour (which probes the rest-frame optical at z = 4 -8) also experiences significant variation as a function of redshift, as shown in the lower-panel of Figure 5. Specifically, both z = 5 and z = 6 . 9 are approximately rising inflexion points, again caused by the shifting locations of the various strong emission lines. This highlights that interpreting the observed colours of galaxies with strong emission lines is extremely challenging without precise knowledge of the redshift (or redshift distribution).", "pages": [ 5 ] }, { "title": "3 PREDICTIONS FROM GALAXY FORMATION SIMULATIONS", "content": "The preceding analysis demonstrated that the observed frame optical/NIR colours of high-redshift galaxies can be extremely sensitive to nebular emission. By using a galaxy formation model to predict both the star formation and metal enrichment histories we can predict the stellar spectral energy distributions (SEDs) and nebular (line and continuum) emission and thus the intrinsic observed colours.", "pages": [ 6 ] }, { "title": "3.1 MassiveBlack -II", "content": "We make use of a state-of-the-art cosmological hydrodynamic simulation of structure formation: MassiveBlack -II (for a more detailed description of this simulation see: Khandai et al. in-prep ). The MassiveBlack -II simulation is performed using the cosmological TreePM-Smooth Particle Hydrodynamics (SPH) code P-Gadget , a hybrid version of the parallel code Gadget2 (Springel 2005) tailored to run on the new generation of Petaflop scale supercomputers. MassiveBlack -II includes N par = 2 × 1792 3 ≈ 11 . 5 billion particles in a volume of 10 6 Mpc 3 /h 3 (100 Mpc /h on a side) and includes not only gravity and hydrodynamics but also additional physics for star formation (Springel & Hernquist 2003), metal enrichment, black holes and associated feedback processes (Di Matteo et al. 2008, Di Matteo et al. 2012).", "pages": [ 6 ] }, { "title": "3.1.1 Properties of galaxies in the simulation", "content": "A detailed overview of the properties of galaxies (galaxy stellar mass functions, luminosity functions etc.) in the simulation is presented in Khandai et al. in-prep ). Nevertheless, it is useful to present predictions for the properties which directly influence the UV-optical colours of galaxies, i.e. the star formation and metal enrichment histories. Instead of presenting the full star formation and metal enrichment histories, in Figure 6 we show the median massweighted stellar age in bins of intrinsic UV luminosity and in Figure 7 the median mass-weighted stellar metallicity, in both cases for a range of redshifts ( z ∈ { 5 , 6 , 7 , 8 , 9 , 10 } ). The redshift trends can be seen more clearly in Figure 8 where the evolution of median mass-weighted stellar age and metallicity for galaxies with -20 . 5 < M 1500 < -18 . 5 are shown.", "pages": [ 6 ] }, { "title": "3.1.2 Stellar spectral energy distributions", "content": "The stellar spectral energy distributions (SEDs) of galaxies are generated by combining the SEDs of individual star particles taking account of their metallicity and age using the stellar population synthesis (SPS) model Pegase.2 and assuming a Salpeter (1955) IMF. As noted previously, the SED predicted from a given star formation and metal enrichment history, is sensitive to the choice of SPS model. In Appendix we consider the effect of alternative SPS models on the spectral energy distribution of galaxies predicted by the simulation.", "pages": [ 6, 7 ] }, { "title": "3.1.3 Nebular emission and dust attenuation", "content": "Nebular emission is included in the simulated SEDs using the same prescription described in § 2.4. We take the ionising flux predicted from the stellar SED and determine the fluxes in the various hydrogen recombination lines assuming the escape fraction is zero. We then use the calibrations of Anders & Fritze et al. (2003) to determine the fluxes in the non-hydrogen lines. We also use Pegase.2 to predict the contribution of nebular continuum emission (though this is typically very small except in extreme cases). For the results presented here we do not include dust attenuation concentrating solely on the intrinsic photometry.", "pages": [ 7 ] }, { "title": "3.2 Stellar colours", "content": "The top panel of Figure 9 shows the median rest-frame pure stellar 1500 -V w colour as a function of the intrinsic UV luminosity ( M 1500 , int ) for galaxies at z ∈ { 5 , 6 , 7 , 8 , 9 , 10 } . The 1500 -V w colour is correlated, albeit weakly, with the UV luminosity with the colour reddening by ∼ 0 . 15 as M 1500 , int = -18 → -20 (at z = 7). The 1500 -V w colour is also strongly correlated with redshift as can be seen more clearly in Figure 10 where the median pure stellar colour (of galaxies with -20 . 5 < M 1500 < -18 . 5) is shown as a function of redshift. For example, the median 1500 -V w colour increases by ∼ 0 . 5 from z = 8 → 5. The distribution of pure stellar colours for galaxies with -20 . 5 < M 1500 < -18 . 5 is shown in Figure 11. The 16 th -84 th percentile range is 0 . 15 mag though this increases slightly to lower redshift. The correlation with redshift is driven by the variation in both the average star formation and metal enrichment his- tories of galaxies (as shown in Figure 8) while the correlation with luminosity is driven predominantly by the variation in the metal enrichment history.", "pages": [ 7, 8 ] }, { "title": "3.3 The effect of nebular emission", "content": "As noted in § 2.4 the effect of nebular emission will, be adding additional flux in the V w -band, have the result of reddening the rest-frame 1500 -V w colour relative to the pure stellar colour. This is shown for our simulated galaxies in Figures 9 and 10. As can be seen in Figures 9 and 10 the effect of adding nebular emission is to flatten the correlation between the 1500 -V w colour and the UV luminosity and redshift. The median 1500 -V w colour including nebular emission only reddens by ∼ 0 . 05 from M 1500 , int = -18 →-20 (at z = 7). This is because the relative strength of nebular emission is inversely correlated with both stellar metallicity and age. Those galaxies with bluer stellar colours, which are indicative of more recent star formation or lower metallicity (which are more common at higher redshift and lower luminosity) will then have stronger nebular emission and consequently will be reddened by nebular emission more than those with redder stellar colours. This has the effect of diminishing the correlation of the 1500 -V w colour with redshift and luminosity. The median 1500 -V w colour including nebular emission only reddens by ∼ 0 . 1 from M 1500 , int = -18 →-20 and ∼ 0 . 3 from z = 8 → 5 (c.f. 0 . 1 and 0 . 3 for the pure stellar case respectively).", "pages": [ 8 ] }, { "title": "3.3.1 The distribution of colours", "content": "Because galaxies with bluer pure stellar galaxies typically have lower metallicities and/or ages (and consequently stronger nebular emission) the inclusion of nebular emission reduces the scatter in the 1500 -V w colour (as measured by the 16 th -84 th percentile range), as can be seen in Figure 11. At z = 8 this reduces the scatter (as measured by the 16 th -84 th percentile range) by almost a factor of × 2 while at lower redshift it is less important.", "pages": [ 8 ] }, { "title": "3.3.2 The distribution equivalent widths", "content": "One alternative measure of the relative strength of nebular emission is the distribution of the equivalent widths of the prominent H α and [OIII] λ 5007 emission lines. These are shown in Figure 12 and follow a similar patten to the distribution of colour increments shown in Figure 11.", "pages": [ 8 ] }, { "title": "3.3.3 The effect on observed colours", "content": "As noted in § 2.4, the effect of nebular emission on observedframe colours is strongly sensitive to both the choice of filters and the redshift of the source. Even a small change in redshift can have a dramatic effect on the observed colour. Figure 13 shows the average simulated (for galaxies with -20 . 5 < M 1500 < -18 . 5) observed frame X -[3.6] and [3.6] -[4.5] colours for redshift ranges centred on the median redshift of observed drop-out samples (see Labbe et al. 2010, Gonzalez et al. 2011, Labbe et al. 2012): V f 606 w -band drop out: 〈 z 〉 = 5 . 0, i f 775 w : 〈 z 〉 = 5 . 9, z f 850 lp : 〈 z 〉 = 6 . 9, and Y f 105 w : 〈 z 〉 = 8 . 0. In each case the X filter is chosen (from the available Hubble ACS and WFC3 filters) such that it approximately probes the rest-frame UV continuum at 1500 ˚ A. Therefore we use: z ≈ 5 → z f 850 lp , z ≈ 6 → Y f 105 w , z ≈ 7 → J f 125 w , and z ≈ 8 → H f 160 w . The trends seen in Figure 13 closely reflect those demonstrated in § 2.4 (and seen in Figure 5). We see at z ≈ 5 and z ≈ 7 the observed colours are particularly sensitive to the redshift, with changes in redshift of 0 . 1 changing the average observed colours by up to 0 . 3 mag. Stark et al. (2012, hereafter S12) studied the observed frame [3 . 6] -[4 . 5] colours of a spectroscopically confirmed sample of galaxies at 3 < z < 7. S12 compare the [3 . 6] -[4 . 5] colour distribution at 3 . 8 < z < 5 . 0 (where H α emission, if present, will contaminate the [3 . 6] filter) with those at 3 . 1 < z < 3 . 6 (where there are expected to be no strong nebular emission lines in either the [3 . 6] or [4 . 5] filters) finding the [3 . 6] -[4 . 5] colour at 3 . 8 < z < 5 . 0 is 0 . 33 mag bluer than at 3 . 1 < z < 3 . 6. This is interpreted as being due to the presence of nebular line emission. This closely matches our prediction at 4 . 5 < z < 4 . 9 where we find the effect of nebular emission results in the observed [3 . 6] -[4 . 5] colour being 0 . 25 mag bluer than the pure stellar colour. S12 also attempt to determine the increase in the [3 . 6] flux due to nebular emission at 3 . 8 < z < 5 . 0 by comparing the observed [3 . 6] flux with that expected from the best fit stellar continuum models, and find an increase in the [3 . 6] flux ([3 . 6] neb -[3 . 6] stellar = -0 . 27 mag) consistent with our predictions at 4 . 5 < z < 4 . 9 ([3 . 6] neb -[3 . 6] stellar = -0 . 29 mag).", "pages": [ 8, 9 ] }, { "title": "4 THE EFFECT ON STELLAR MASS ESTIMATES", "content": "We have seen that the effect of nebular emission is to typically redden the observed colour probing the rest-frame UVoptical relative to the pure stellar colour. For individual galaxies at z ≈ 7 this can be as large as +0 . 6 mag with the average at z = 6 . 8 being +0 . 4 mag for J f 125 w -[3.6]. The effect is also strongly dependent on the redshift and choice of observed filters with changes in the redshift of as little as ± 0 . 1 is able to change the observed colours by > 0 . 3 mag. Because accurate estimates of the stellar mass-to-light ratio require a measurement of the shape rest-frame UVoptical SED (e.g. Wilkins et al. 2013b) the effect of nebular emission can introduce a large bias, particularly where the redshift is not known accurately. The exact size of the effect of nebular emission on stellar mass estimates however depends on various factors including the method used to measure the mass-to-light ratio, the redshift. As a simple illustration we consider the case where the stellar mass-to-light ratio is measured using a single colour (e.g. Taylor et al. 2012, Wilkins et al. 2013). Figure 14 shows the relationship between the observed frame J f 125 w -[3.6] colours and J f 125 w -band stellar mass-to-light ratios of galaxies at z = 6 . 8 and z = 7 . 0. In both cases the J f 125 w -[3.6] colour is correlated with the mass-to-light ratio (though the correlation is much stronger at z = 7 . 0 where the effect of nebular emission is smaller than at z = 6 . 8), however, for the same observed colour the average mass-to-light ratio at z = 7 . 0 is ∼ 0 . 25 dex (1 . 8 × ) larger. That is, were the redshift erroneously assumed to be at z = 7 . 0 instead of the true redshift of z = 6 . 8 the stellar mass-to-light ratio would be overestimated by around ∼ 0 . 25 dex.", "pages": [ 9, 10 ] }, { "title": "5 CONCLUSIONS", "content": "We have explored the evolution of the rest-frame UV/optical (and observed frame near-IR) colours of high-redshift galaxies ( z = 5 -10) predicted by our large cosmological hydrodynamical simulation MassiveBlack -II. We find that the median rest-frame pure stellar UVoptical (1500 -V w ) colour is correlated with both luminosity and redshift. The 1500 -V w colour reddens by ∼ 0 . 2 as the luminosity increases from M 1500 = -18 to -20 (at z = 7) and by ∼ 0 . 5 from z = 8 to z = 5. In both cases, this reflects the trend of increasing ages and metallicities with luminosity and to lower redshift. However, when nebular emission is included, these correlations weaken. The 1500 -V w colour reddens by only ∼ 0 . 3 mag as z = 8 → 5 and ∼ 0 . 1 mag as M 1500 = -18 → -20. This occurs because galaxies with very blue stellar colours (indicative of galaxies with recent star formation or low metallicity) typically have stronger nebular emission causing their colours to redden by a greater relative amount. The effect of nebular emission on observed frame colours is very sensitive to both the choice of filters and redshift. For example, at z = 7 . 0, nebular emission only reddens the observed J f 125 w -[3.6] colour by, on average, ∼ 0 . 1 mag while at z = 6 . 8 it reddens the J f 125 w -[3.6] colour by ∼ 0 . 45 mag (i.e. a difference of ∼ 0 . 35 mag). Similarly, at z = 7 . 1, nebular line emission causes the [3.6] -[4.5] to redden by ∼ 0 . 2 mag, while at z = 6 . 8 it causes [3.6] -[4.5] colour to shift blue-ward by ∼ 0 . 4 mag (a net difference of ∼ 0 . 6 mag). This strong sensitivity of observed colours to the redshift makes interpreting the colours of individual objects extremely difficult unless precise redshifts are known. Indeed, if the stellar mass-to-light ratio were inferred from the J f 125 w -[3.6] colour alone a difference of ± 0 . 35 mag (as expected between z = 6 . 8 and 7 . 0) would roughly double While the general trends we observe hold true irrespective of the choice of initial mass function and stellar population synthesis model both these factors can strongly affect the predicted colours. For example, utilising the Maraston et al. (2005) model yields stellar colours between 0 . 1 -0 . 3 mag redder than the Pegase.2 model (which is assumed throughout this work).", "pages": [ 10, 11 ] }, { "title": "Acknowledgements", "content": "We would like to thank the anonymous referee for their useful comments and suggestions that we feel have greatly improved this manuscript. SMW and AB acknowledge support from the Science and Technology Facilities Council. WRC acknowledges support from an Institute of Physics/Nuffield Foundation funded summer internship at the University of Oxford. RACC thanks the Leverhulme Trust for their award of a Visiting Professorship at the University of Oxford. The simulations were run on the Cray XT5 supercomputer Kraken at the National Institute for Computational Sciences. This research has been funded by the National Science Foundation (NSF) PetaApps program, OCI-0749212 and by NSF AST-1009781.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "Anders, P., & Fritze-v. Alvensleben, U. 2003, A&A, 401, 1063 Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2010b, ApJ, 709, L133 This paper has been typeset from a T E X/ L A T E Xfile prepared by the author.", "pages": [ 11 ] }, { "title": "APPENDIX A: THE CHOICE OF STELLAR POPULATION SYNTHESIS MODEL", "content": "A key ingredient in our analysis is the transformation of the simulated star formation and metal enrichment history into a spectral energy distribution through the use of stellar population synthesis (SPS) modelling. Stellar population synthesis models work by combining stellar tracks, which model different stellar evolution phases, with spectral libraries, which empirically or theoretically relate the spectral output to individual stars based on their mass, age, and chemical composition. By combining these with a choice of initial mass function and initial chemical composition it then becomes possible to model the spectral energy distributions (SEDs) of simple stellar populations. However, due to differences in the treatment of stellar evolution and the utilised spectral libraries, SPS models produce varying results for the spectral energy distributions of stellar populations (see, for example, Conroy & Gunn 2010). An example of some of these differences can be seen in Figure A1 where we show the SEDs of simple stellar populations with ages between 10 Myr and 10 Gyr predicted assuming three popular SPS models: Pegase.2 ; BC03: Bruzual & Charlot (2003); M05: Maraston et al. (2005). For the youngest ages considered (10 and 100 Myr) there are significant differences between the models. For older populations ( > 1 Gyr) the models are relatively consistent over the rest-frame UV-optical. In the nearIR however the M05 model predicts an excess of flux at 1 Gyr, attributed to a more detailed treatment of the TPAGB stage. While this produces significant enhancement of the near-IR flux it will have little effect on this work due to our focus on rest-frame UV-optical colours and very-high redshift galaxies. The variation between different SPS models, and its relevance to this work, can be seen more clearly in Figure A2 where we show the evolution of the rest-frame stellar 1500 -V w as a function of age for a simple stellar population (with Z = 0 . 02). While the BC03 and Pegase.2 models yield a similar colour evolution the M05 model predicts significantly redder colours at ages 10 -500 Myr. This variation between different models will then leaves the colours predicted by our analysis of the MassiveBlack -II simulation sensitive to the choice of model. In Figure A3 we show the predicted rest-frame stellar 1500 -V w colour assuming various SPS models (c.f. Figure 10); the lower panel of this figure shows the difference between the alternative models considered and the Pegase.2 model utilised throughout this work. While the use of the BC03 model yields colours similar ( < | 0 . 1 | mag difference) to those assuming the Pegase.2 (default) model the M05 model yields colours which are typically between 0 . 1 -0 . 2 mag redder. At the most extreme, use of the M05 model at z = 10 yields colours ≈ 0 . 3 mag redder than using Pegase.2 .", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "We use hydrodynamical simulations from the OWLS project to investigate the dependence of the physical properties of galaxy populations at redshift 2 on the assumed star formation law, the equation of state imposed on the unresolved interstellar medium, the stellar initial mass function, the reionization history, and the assumed cosmology. This work complements that of Paper I, where we studied the e ff ects of varying models for galactic winds driven by star formation and AGN. The normalisation of the matter power spectrum strongly a ff ects the galaxy mass function, but has a relatively small e ff ect on the physical properties of galaxies residing in haloes of a fixed mass. Reionization suppresses the stellar masses and gas fractions of low-mass galaxies, but by z = 2 the results are insensitive to the timing of reionization. The stellar initial mass function mainly determines the physical properties of galaxies through its e ff ect on the e ffi ciency of the feedback, while changes in the recycled mass and metal fractions play a smaller role. If we use a recipe for star formation that reproduces the observed star formation law independently of the assumed equation of state of the unresolved ISM, then the latter is unimportant. The star formation law, i.e. the gas consumption time scale as a function of surface density, determines the mass of dense, star-forming gas in galaxies, but a ff ects neither the star formation rate nor the stellar mass. This can be understood in terms of self-regulation: the gas fraction adjusts until the outflow rate balances the inflow rate. Key words: cosmology: theory - galaxies: formation - galaxies: evolution - galaxies: fundamental parameters - methods: numerical", "pages": [ 1 ] }, { "title": "Physical properties of simulated galaxy populations at z = 2 -II. E ff ects of cosmology, reionization and ISM physics", "content": "Marcel R. Haas 1 , 2 , 3 /star , Joop Schaye 2 , C. M. Booth 4 , 5 , 2 , Claudio Dalla Vecchia 6 , Volker Springel 7 , 8 , Tom Theuns 9 , 10 and Robert P. C. Wiersma 2 † 1 Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Rd., Piscataway, NJ 08854, USA 2 Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA, Leiden, The Netherlands Submitted to MNRAS", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Understanding the star formation (SF) history of the Universe represents one of the most fundamental pieces of the galaxy formation puzzle. There exists significant uncertainty as to which physical processes control the rate of star-formation in galaxies of a given mass. At the coarsest level, the rate at which gas enters galaxies is controlled, at high redshift, by the rate of growth of the dark matter halo, while at low redshift it also depends sensitively on the rate at which gas can cool into galaxies (e.g. White & Rees 1978; Hernquist & Springel 2003; Choi & Nagamine 2010; Schaye et al. 2010; van de Voort et al. 2011). However, matching observational constraints on galaxy masses, ages and star formation rates (SFRs) is much more complex than this simple picture would suggest, and although many attempts have been made to model the formation of a galaxy population using complex hydrodynamical simulations and semi-analytic models, there exists no clear consensus on what combination of physical processes are required to explain the distribution of galaxies seen in the local Universe. Di ffi culties in the simulations begin where gas becomes dense enough to form stars. This interstellar medium (ISM) gas has an extremely complex structure. Amongst other processes, magnetic fields, turbulence, cosmic rays and radiative transfer may all play some role in determining the rate at which stars form (e.g. McKee & Ostriker 2007). For this reason, most cosmological simulations treat the gas in the ISM with very simple 'subgrid' prescriptions. Additionally, the limited numerical resolution of the simulations prevents a detailed modelling of SF, and they need to rely on empirical laws (e.g. Kennicutt 1998). However, recent observations (Kennicutt et al. 2007; Bigiel et al. 2008) show that the star formation rate surface density is a function of the molecular hydrogen density, or the surface density of cold gas (Schaye 2004; Krumholz et al. 2011; Glover & Clark 2012). Simulations with su ffi cient resolution to resolve gas with temperatures /lessmuch 10 4 K can include a more detailed treatment of the ISM (e.g. Gnedin et al. 2009; Ceverino & Klypin 2009; Agertz et al. 2009; Christensen et al. 2012), but are currently limited to a very small number of galaxies. Furthermore, the stellar initial mass function (IMF) assumed in the simulations is not predicted from first principles, but is imposed. Observational determinations of the stellar IMF are very challenging outside the solar neighbourhood so the local IMF is usually assumed to be universal and is applied to all galaxies at all redshifts. In addition to uncertainties related to processes that take place inside galaxies, the ultra-violet (UV) background produced by the first generation of galaxies, ionizes the Universe and bathes gas in ionizing radiation. This ionizing UV background can then strongly suppress the infall of gas into low-mass haloes (e.g. Quinn et al. 1996; Okamoto et al. 2008; Pawlik & Schaye 2009; Hambrick et al. 2011). It is therefore important to assess, independently, how each of these uncertain physical processes a ff ects the properties of galaxies formed in cosmological simulations. In Haas et al. (2013, hereafter Paper I) of this series we use cosmological hydrodynamical simulations from the OverWhelmingly Large Simulations (OWLS; Schaye et al. 2010) to investigate the e ff ects of cooling and feedback on the galaxy population at z = 2. In this companion paper we turn our attention to other physical processes, namely cosmology, reionization, the treatment of the high density gas, the star formation law and the stellar initial mass function. This work (together with Paper I) complements that of Schaye et al. (2010), where the cosmic SF histories predicted by the OWLS simulations were analysed. In Paper I we describe the behaviour of the simulations in detail, and in particular we focus there on the e ff ects of radiative cooling and energetic feedback from star formation and AGN. One of the main conclusion from Paper I is that the SF is self-regulated by the interplay between gas cooling onto galaxies and the feedback that o ff sets this gas accretion. The cooling rate onto galaxies is, for a fixed halo mass and redshift, mainly determined by the radiative cooling rate, which itself depends on chemical composition. We found that for many integrated galaxy properties the uncertainties in models for cooling and feedback give rise to a very large spread in galaxy physical properties (see Fig. 2). In this paper we discuss the importance of cosmology, reionization, the prescription for the ISM, the SF law and the stellar initial mass function. We will show that uncertainties in these processes are of secondary importance for the total amount of stars formed and for the SFR of simulated galaxies. However, we will also show that some of the processes discussed in this paper are important for the amount of mass residing in the star forming ISM of galaxies, which we will argue can be explained in terms of self-regulated star formation. The structure of this paper is follows: in Sec. 2 we briefly describe the simulations used in this study and the numerical techniques we employ. In Sec. 3 we describe how galaxy properties depend upon the physics included in the simulation. In this paper we consider the e ff ects of: cosmology (Sec 3.1), reionization (Sec. 3.2), the e ff ective equation of state of the ISM (Sec. 3.3), the star formation law (Sec. 3.4), and the stellar IMF (Sec. 3.5). Finally, in Sec. 4 we summarize our findings and conclude.", "pages": [ 1, 2 ] }, { "title": "2 NUMERICALTECHNIQUES", "content": "The simulations comprising the OWLS project are described fully in Schaye et al. (2010). Here we briefly summarize the reference simulation, with a focus on the physical prescriptions relevant for this paper. This simulation will be referenced throughout this paper as ' REF '. All simulations are performed with an extended version of the N-Body Tree / SPH code G adget 3 (last described in Springel 2005) in periodic boxes of 25 co-moving h -1 Mpc and contain 512 3 dark matter and the same number of baryonic particles (which can be either collisionless 'stars' or collisional 'gas' particles). The particle mass of the simulations we use here is 8 . 68 × 10 6 M /circledot for dark matter and 1 . 85 × 10 6 M /circledot for baryons (initially, the baryonic particle masses change in the course of the simulation due to mass transfer from star particles to gas particles). The gravitational softening length is initially fixed in co-moving coordinates at 1 / 25 the mean inter-particle spacing (1.95 co-moving h -1 kpc). Below z = 2 . 91 the softening is fixed in proper units, at 0.5 h -1 kpc. We provide tests showing convergence of our results with respect to simulation box size and the particle number in the appendix of Paper I. The cosmology assumed in the reference simulation is summarized in Table 1 and is deduced from the WMAP 3 year results (Spergel et al. 2007). The results are largely consistent with the more recent WMAP7 results (Komatsu et al. 2011), the most notable di ff erences are in σ 8, which is 2.3 σ lower in WMAP3 than in WMAP7 and in the Hubble parameter which is 1 σ lower in the WMAP3 than in the WMAP7 data. The primordial helium mass fraction is set to 0.248. As the subgrid model variation is the main power of the OWLS suite, we will now describe the parameters and subgrid models used in the reference simulation, before varying them in later sections. Radiative cooling and heating are calculated element-by-element by explicitly following the 11 elements H, He, C, N, O, Ne, Mg, Si, S, Ca and Fe in the presence of the Cosmic Microwave Background and the Haardt & Madau (2001) model for the UV / X-ray background radiation from quasars and galaxies, as described in Wiersma et al. (2009a). The timed release of these elements by stars is followed as described by Wiersma et al. (2009b). The gas is assumed to be optically thin and in photo-ionization equilibrium. The simulations model hydrogen reionization by switching on the Haardt & Madau (2001) background at z = 9. Helium reionization is modelled by heating the gas by an extra amount of 2 eV per atom. This heating takes place at z = 3.5, with the heating spread in redshift with a Gaussian filter with σ ( z ) = 0 . 5. As shown by Wiersma et al. (2009b), the reionization prescription used in these simulations roughly matches the temperature history of the intergalactic medium (IGM) inferred from observations by Schaye et al. (2000). In the centres of haloes the density and pressure are so high, that the gas is expected to be in multiple phases, with cold and dense molecular clouds embedded in a warmer, more tenuous gas. This multi-phase structure is not resolved by our simulations (and the simulations lack the physics to describe these phases), so we im- pose a polytropic e ff ective equation of state for particles with densities n H > 10 -1 cm -3 . These particles are also assumed to be star forming, as this is the density required to form a molecular phase (Schaye 2004). We set the pressure of these particles to P ∝ ρ γ e ff , where γ e ff is the polytropic index and ρ is the physical proper mass density of the gas. In order to prevent spurious fragmentation due to a lack of numerical resolution we set γ e ff = 4 / 3, as then the ratio of the Jeans length to the SPH kernel and the Jeans mass are independent of density (Schaye & Dalla Vecchia 2008). The normalization of the polytropic equation of state is such that for atomic gas with primordial composition, the energy per unit mass corresponds to 10 4 K, namely ( P / k = 1 . 08 × 10 3 K cm -3 for n H = 10 -1 cm -3 ). The implementation of SF is stochastic, as described in Schaye & Dalla Vecchia (2008), with a pressure-dependent SFR, obtained from the assumption of local hydrostatic equilibrium and the observed Kennicutt-Schmidt law (Kennicutt 1998). Energetic feedback from star formation is implemented kinetically. On average, we give 2 of the SPH neighbours of each newly formed star particle a 'kick' such that the total energy in the outflow corresponds to roughly 40% of the energy available from supernovae (SNe) of type II (including Ib,c). η = 2 is the mass loading factor. For our assumed Chabrier (2003) IMF this corresponds total wind velocity of 600 km s 1 . See Dalla Vecchia & Schaye (2008) for more details on the kinetic implementation of SN feedback. Haloes are identified using a Friends-of-Friends algorithm, as described in Sec. 2.2 of Paper I and we only show results over mass ranges where the simulation achieves numerical convergence (Appendix A of Paper I). This corresponds to haloes that contain at least 100 star particles when considering properties as a function of stellar mass and a minimum of 2000 dark matter particles when we plot properties against total halo mass.", "pages": [ 2, 3, 4 ] }, { "title": "3 ISOLATING THE EFFECTS OF THE INPUT PHYSICS", "content": "In this section we discuss each of the variations to the input physics in turn. Table 2 summarizes the simulations used in this paper and indicates in which subsection they are discussed. Bold-face values indicate departures from the reference model. In the subsequent sections we will discuss the sensitivity of galaxy properties to cosmology (Section 3.1), reionization (Section 3.2), the polytropic equation of state of the ISM (Section 3.3), the assumed SF law (Section 3.4) and the IMF (Section 3.5). Agraphical representation of the gas density of a z = 2 galaxy formed in the di ff erent simulations (except ' MILL ', see below) is shown in Fig. 1. The galaxy resides in a halo of total mass ∼ 10 12 . 5 M /circledot . It was first identified in the ' REF ' simulation, where its position (defined as the centre of mass of all particles within 10% of the virial radius) was determined. The line of sight is along the z-axis, which is almost perfectly aligned with the angular momentum vector of the gas within 10% of the virial radius (cos( φ ) = 0 . 994). For the other simulations the image is centered on the same position, illustrating the remarkable similarity in the positions and orientations of the galaxies. The ' MILL ' simulation was run with di ff erent cosmological parameters, resulting in a di ff erent distribution of galaxies over the volume. This model is therefore not plotted in Fig. 1. The physics variations discussed in this paper do not lead to large di ff erences in the total stellar content of galaxies. Nevertheless, we see in Fig. 1 that there are significant di ff erences in galaxy morphologies, although a gaseous disk forms in all cases. These di ff erences will be discussed throughout the remainder of this section. Fig. 2 of paper I, as well as Figs. 2 through 6 show, as a function of total halo mass the nine di ff erent galaxy properties we consider in this paper: medians of stellar mass fraction ( f star = M star / M tot, panel A), SFR (panel B), baryon fraction ( f baryon = M baryon / M tot, panel C), fraction of star-forming gas ( f ISM = M ISM / M tot, panel D) and gas mass fraction ( f gas, halo = ( M gas, total -M ISM) / M tot, panel E). Then, as a function of stellar mass: medians of the molecular gas mass in the ISM (panel F), specific SFR (sSFR = SFR / M ∗ , panel G), the inverse of the gas consumption timescale (SFR / M ISM, panel H) and galaxy number density (the galaxy stellar mass function, panel I). In panels F, G and I we compare to observations of, respectively, Genzel et al. (2010), Daddi et al. (2007) and Marchesini et al. (2009), as described in detail in Paper I. The black line in all panels is the ' REF ' simulation. The physics variations presented in Paper I strongly influence the amount of stars formed, the star formation rate and the gas and baryon fractions in the halo. As discussed below, the physics vari- ions described in this paper mainly influence gas consumption time scales and ISM mass fractions.", "pages": [ 4 ] }, { "title": "3.1 Cosmology", "content": "To investigate the dependence of the galaxy properties on cosmology, and to facilitate comparisons to earlier work, we investigate the e ff ects of changing the cosmological parameters from the WMAP 3-year results (Spergel et al. 2007) used in the ' REF ' simulation to the so-called 'concordance' or WMAP year 1 cosmology that was used in many previous studies including the Millennium Simulation (Springel et al. 2005). We will refer to this set of cosmological parameters as the 'Millennium cosmology' and denote the model assuming this cosmology ' MILL '. The main di ff erences between the WMAP3and Millennium cosmologies are firstly σ 8, which is 0.74 in WMAP3 and 0.9 in the Millennium cosmology and secondly, the universal baryon fraction Ω b, which is 0.0418 in WMAP3 and 0.045 in the Millennium cosmology. Other parameter values are summarized in Table 1. The WMAP3 cosmological parameters are largely consistent with the most recent 7-year results of WMAP (Komatsu et al. 2011), although in WMAP7 the value of σ 8 is 2.3 σ higher (0.8) and the Hubble parameter is 1 σ in WMAP7 than in WMAP3. Increasing σ 8 and Ω b leads to the formation of many more stars than in the ' REF ' simulation. In order to roughly match the global SFR in the ' MILL ' simulation, it is therefore necessary to double the amount of SN energy injected into the ISM per unit mass. The ' MILL ' simulation therefore uses a mass loading of η = 4 for SN-driven winds, rather than the η = 2 used in the reference model. To isolate the e ff ect of cosmology, we therefore compare ' MILL ' to a simulation that uses the WMAP 3 cosmology, but the same SN feedback model as ' MILL '. We term this simulation ' WML4 '. In Fig. 2 the e ff ect of the cosmology can thus be isolated by comparing the blue dashed and red dotted curves, which correspond to the WMAP3 (' WML4 ') and 'Millennium' (' MILL ') cosmological parameters (as indicated in Table 1), respectively. The main e ff ect of σ 8 is to set the time-scale for structure formation. With a higher σ 8 structures of a given mass form earlier (e.g. Peebles 1993). The concentration of a DM halo at a given mass is set by formation time. Indeed, Du ff y et al. (2008) showed that dark matter halo concentrations are significantly lower in the WMAP5 cosmology ( σ 8 = 0 . 77) than in the WMAP1 cosmology ( σ 8 = 0 . 90). Galaxy SFRs at a given halo mass could thus be influenced by the value of σ 8 through its e ff ect on the halo gravitational potential. Comparing ' MILL ' to ' WML4 ' in Fig. 2 the total stellar mass (panel A), the SFR (panel B) and the ISM mass (panel D) of high-mass haloes are all slightly higher for the WMAP1 cosmology than for the WMAP3 cosmology, due to the higher central densities in the WMAP1 cosmology at a given halo mass. Comparing these results to the global SFR (Schaye et al. 2010), we note that cosmology has a much larger e ff ect on the global SFR than on individual objects. This reflects the e ff ect that the cosmological parameters have on the halo mass function. A higher σ 8 leads to more haloes of a given mass, and the larger number of haloes results in a much higher global SFR density. The primary e ff ect of cosmology on the galaxy stellar mass function (panel I) is through its e ff ect on the number density of haloes at a given mass, i.e. the halo mass function, see also Crain et al. (2009). At low masses, the M tot -f star (panel A) and the M ∗ -M ISM (panel F) relations di ff er in ' MILL ' and ' WML4 ' by ∼ 0 . 1 dex). The stellar mass function at the low-mass end changes by up to 0.2 dex, because haloes of a given mass are more common . in the simulation with the higher values of σ 8. Panel I also demonstrates that a higher σ 8 also leads to the formation of more massive galaxies ( M star goes up to ∼ 10 12 M /circledot for the ' MILL ' simulation). We do not see an exponential cut-o ff in the predicted stellar mass function. This is because (as demonstrated by the gas consumption time-scale in panel H and the star forming gas mass in panel F), gas is being consumed rapidly in massive haloes where SN feedback is ine ff ective (see Paper I for more discussion on this point). Also, 'radio-mode' AGN feedback, which is not considered here, is often assumed to cause this drop.", "pages": [ 4, 6 ] }, { "title": "3.2 Reionization", "content": "The reionization of the Universe by quasars and galaxies has a profound e ff ect on the temperature of the IGM and may impact the galaxy formation process (e.g. Okamoto et al. 2008). Reionization is modelled in the simulations by turning on a UV + X-ray background from galaxies and quasars (Haardt & Madau 2001). All gas in the simulations is assumed to be optically thin to this radiation, which is assumed to be uniform and isotropic. In the ' REF ' simulation, the redshift of reionization is set to z r = 9. As shown in Wiersma et al. (2009a), the main e ff ect of the UV background is to heat low-density, cold gas to T ∼ 10 4 K. This leads to gas in haloes with virial temperatures /lessorsimilar 10 4 K ( M tot /lessorsimilar 10 8 M /circledot ) being evaporated from their host haloes, and so e ff ectively imposes a minimum halo mass in which star-formation can take place (e.g. Quinn et al. 1996; Okamoto et al. 2008; Pawlik & Schaye 2009; Hambrick et al. 2011). This mass scale corresponds to only ∼ 10 particles in our simulations. Hence, our simulations do not probe the regime where photo-heating is expected to be most important and they may underestimate the e ff ect for somewhat higher masses because the e ff ect of photo-heating on their progenitors is not resolved. To isolate the e ff ects of reionization on the galaxy population, we compare the ' REF ' simulation to a simulation without reionization (' NOREION ') and to two models in which the redshift of reionization is changed to z r = 6 or z r = 12 from its default value z r = 9, which are denoted ' REIONZ06 ' and' REIONZ12 ' respectively. As described in Sec. 2, we match the observed temperature evolution of the IGM by modeling helium reionization as an additional heat input of 2 eV per atom around z = 3 . 5. The simulation ' NOHeHEAT ' neglects this extra heating. Inspection of Fig. 3 (all panels) reveals that by z = 2 all knowledge of the redshift of reionization has been washed out of the resolved galaxy population, see also Schaye et al. (2010); Wiersma et al. (2011). Comparing ' NOREION ' to the rest of the simulations, we see that the e ff ects of neglecting the heating due to reionization are to increase the stellar masses (panel A), star forming gas masses (panel D) and gas masses (panels C and E), and hence the SFRs (panels B and G) of low-mass objects ( M tot /lessorsimilar 10 11 M /circledot ). Gas that should have been removed from these objects by the extra energy it receives from the photo-ionizing background is allowed to remain. This e ff ect is also visible in the stellar mass function (panel I), where at low masses ( M star /lessorsimilar 10 9 M /circledot ), a given stellar mass corresponds to a lower halo mass, making these objects more abundant. At high halo masses ( M tot /greaterorsimilar 10 11 M /circledot ), neglecting reionization results in slightly lower halo gas fractions because more of the gas is in the ISM and in stars. We can examine the e ff ect of helium reionization by comparing ' REF ' (solid, black curve) to ' NOHeHEAT ' (dashed, purple curve). From all panels we can immediately see that, consistent with Schaye et al. (2010), the extra heat input due to helium reionization has no e ff ect on the properties of resolved haloes and galaxies. All massive objects have already formed a very significant proportion of their mass by the time that helium reionization takes place ( z = 3 . 5) and this heat input has a negligible e ff ect at gas densities typical of haloes. The unimportance of helium reionization holds for all of the properties of galaxies we investigate in this work. The extra heat input to the IGM from helium reionization is therefore only important for controlling the temperature of the low-density gas of the IGM (Theuns et al. 2002) and has no effect on the properties of galaxies formed in the simulations, at least for M ∗ > 10 8 M /circledot .", "pages": [ 6 ] }, { "title": "3.3 The polytropic equation of state for high-density gas", "content": "Large volume cosmological simulations lack both the resolution and the physics to model the multiphase ISM. We therefore impose an e ff ective equation of state (EoS) for all gas particles with densities higher than n H = 0 . 1 cm -3 . The EoS we use is polytropic, P ∝ ρ γ e ff and in the ' REF ' simulation γ e ff = 4 / 3. This value of γ e ff ensures that both the Jeans mass and the ratio of the Jeans length and the kernel of the SPH particles are independent of density (Schaye & Dalla Vecchia 2008), provided that they are resolved at n H = 0 . 1 cm -3 , preventing spurious numerical fragmentation due to a lack of resolution. In order to isolate the e ff ect of the EoS, we compare ' REF ' to two simulations in which the slope of the EoS changed to be either isothermal, with γ e ff = 1 (' EOS1p0 '), or adiabatic, with γ e ff = 5 / 3 (' EOS1p67 '). Changing the slope of the EoS has a significant e ff ect on the visual appearance of the galaxy disk. Comparing ' EOS1p0 ' and ' EOS1p67 ' in Fig. 1, we can immediately see that using a steeper EoS pressurizes the gas more strongly, resulting in a smoother gas distribution. Despite markedly changing the visual appearance of the disks, it is clear from Fig. 4 that the galaxy stellar properties (panels A and I) as well as the total amount of baryons associated with the galaxies (panels C, D, E and F) and the SFRs (panels B and G) are virtually una ff ected by the structure of the ISM on small scales. The only noticeable (although very weak) di ff erence between the simulations is that in the very most massive galaxies, steepening the EoS leads to slightly more e ffi cient SF at a given halo mass. This is because, at a given density, pressures (and hence star-formation e ffi ciencies) are higher for a steeper EoS. It is striking that the properties of the galaxy are almost entirely insensitive to changes in the polytropic index (at least in the range γ e ff = 1 -5 / 3). This is a consequence of our use of the prescription for SF of Schaye & Dalla Vecchia (2008). In this prescription the observed Kennicutt-Schmidt surface density law is analytically converted into a pressure law under the assumption of vertical, local hydrostatic equilibrium. As demonstrated by Schaye & Dalla Vecchia (2008), this enables the same recipe to reproduce the observed SF law regardless of the equation of state, and without tuning any parameters. Because the pressure profile is determined by the gravitational potential in hydrostatic equilibrium, which is generally not badly violated, the SF we predict is the same if the potential is the same, irrespective of the assumed equation of state (which does, however, change the thickness of the gas disk). Note that, as far as we know, all other groups use Schmidt-type SF laws, i.e. volume density SF laws, rather than pressure laws. In that case the SFR is expected to depend on the assumed equation of state, even if the gravitational potential and the pressure profile remain unchanged. Because the observed SF law that the simulations are calibrated to match is a surface density rather than a volume density law, the calibration would have to be repeated if the equation of state is changed. This is, however, generally not done, which means that the simulations can no longer be expected to reproduce the observed SF law when γ e ff is changed.", "pages": [ 7, 8, 9 ] }, { "title": "3.4 The star formation law", "content": "Because large-volume cosmological simulations have neither the resolution nor the physics to model SF, the standard approach is to implement a simple, stochastic volume density SF law and to calibrate it to match the observed Kennicutt-Schmidt surface density law, where n = 1 . 4 and A = 1 . 151 × 10 -4 M /circledot yr -1 kpc -2 and the law steepens below a gas surface density Σ g ∼ 10 M /circledot pc -2 (Kennicutt 1998). In our simulations we do something similar, except that we implement SF as a pressure law (see eqns 12, 14 and 18 in Schaye & Dalla Vecchia 2008), which has the advantage that no tuning is required, since we can convert the observed surface density law in a pressure law under the assumption of vertical hydrostatic equilibrium (Schaye & Dalla Vecchia 2008). As we have seen in the previous section, another advantage of this approach is that the predicted SFRs become independent of the assumed equation of state of the star forming gas (see also Schaye & Dalla Vecchia 2008; Schaye et al. 2010). Following Schaye & Dalla Vecchia (2008), the SFR of a star-forming gas particle of mass mg as a function of the pressure ( P ) is given by where γ = 5 / 3 is the ratio of the specific heats (not to be confused with the e ff ective equation of state described in the previous section), and the parameters A and n are those of the KS-law in Eq. 1. In order to isolate the e ff ect of changing the SF law, we compare the ' REF ' simulation to three simulations in which the parameters of the Kennicutt-Schmidt are varied. Simulation ' SFAMPLx3 ' increases the normalization of the Kennicutt-Schmidt law, A , by a factor of three, which implies that for a gas particle at a given pressure, the SFR is a factor 3 higher in SFAMPLx3 ' than in ' REF '. In simulation ' SFSLOPE1p75 ', the power-law slope of the KS-law is increased from n = 1 . 4 to 1 . 75. The normalisation of the Kennicutt-Schmidt law is chosen such that the SFR surface density is the same as ' REF ' at Σ g = 1 M /circledot / pc 2 . As this is below the SF threshold, this KS-law forms stars more e ffi ciently than the ' REF ' simulation at all densities. The third variation on the ' REF ' model, termed ' SFTHRESHZ ' is a simulation in which the threshold density for SF, n ∗ H , depends on the gas metallicity, n ∗ H ∝ Z -0 . 64 , such that the threshold density is equal to the reference simulation's SF threshold at a metallicity of 0 . 1 Z /circledot ( n H = 0 . 1 cm -3 ) and there is a maximum allowed critical density of n ∗ H = 10 cm -3 . This change mimics the metallicity dependence of the critical surface density for the formation of a cold, molecular phase predicted by Schaye (2004). Panel (H) of Fig. 5 shows that, as expected, the gas consumption time scale is significantly (about a factor of 3) shorter for ' SFAMPLx3 ' and ' SFSLOPE1p75 ' than for ' REF '. Model ' SFTHRESHZ ' predicts nearly the same gas consumption time scale as ' REF ', although SF is slightly less e ffi cient in ' SFTHRESHZ ' for all but the highest stellar masses. Intriguingly, the much shorter gas consumption time scale in models ' SFAMPLx3 ' and ' SFSLOPE1p75 ' has virtually no e ff ect on the galaxy stellar mass (panels A and I) or its SFR (panels B and G). This agrees with Schaye et al. (2010), who found that all these simulations predict nearly the same cosmic SF histories. This, initially surprising, result can be understood by considering the distribution of gas inside haloes. Panels (C) and (E) demonstrate that the total baryon fraction inside haloes does not depend on the assumed SF law, but that the amount of ISM gas, i.e. the amount of star forming gas, is very sensitive to the same parameters (panels D and F). Indeed, the response of the galaxies to a change in the gas consumption time scale, is to decrease the amount of star forming gas by a similar factor (compare panels D and H). This supports the explanation put forward by Schaye et al. (2010) that star formation is self-regulated by the interplay between the available fuel supply and feedback processes : The feedback accompanying SF regulates the amount of gas condensing onto the ISM, and hence SF itself, so that outflow roughly balances inflow (see also Bouch'e et al. 2010; Dav'e et al. 2012). As the outflow rate depends on the SFR rather than on the gas consumption time scale, the SFR, and hence the stellar mass, remain the same if the SF law is changed. To produce the same SFR with a di ff erent SF law, the galaxy will have to adjust the amount of star forming gas. Naively one would expect that the metallicity-dependent SF threshold should have a large e ff ect on the amount of stars formed in a simulation (zero metallicity gas has a SF threshold 100 times higher than in the ' REF ' simulation), but Fig. 5 shows that this is not the case: the ' SFTHRESHZ ' (green, dashed) curves closely track the ' REF ' (black, solid) curved in all panels. This is another manifestation of the self-regulation described above. If the gas in a galaxy is insu ffi ciently dense to form stars, then there are no processes to prevent gas from condensing into the galaxy and increasing the gas density until SF becomes possible. At this point the galaxy continues to form stars until the rate at which they inject energy into the galaxy is su ffi cient for the resulting galactic wind to balance the rate of gas inflow. Seemingly contrary to our results, e.g. Gnedin & Kravtsov (2011) found that a metallicity dependent cut-o ff is important for the SF properties of galaxies. The dependency on metallicity in those studies is, however, stronger than the one assumed here and they focus on low-mass galaxies. Hopkins et al. (2011) performed high-resolution galaxy simulations in which they varied the SF laws in a similar manner to that investigated in this section. Because of their much higher resolution ( M star ∼ 10 2 -10 3 M /circledot ), which enables them to use a much higher SF threshold ( n H = 100 cm -3 ), their feedback operates on a much smaller spatial scale. Because star particles form in the high-density peaks, their feedback is acting on much higher-density gas than it is in our simulations. Hopkins et al. (2011) find, in common with the results presented here, that the galaxy SFR is independent of the parameters of their SF law. However, in apparent contrast to our results, they find little dependence of the amount of gas in the ISM on these same parameters. This apparent discrepancy can be understood by noting that in the simulations of Hopkins et al. (2011) there is a strong dependence of the high-density tail of the gas density distribution function on the SF law parameters. They find that there is more gas at very high densities when the SF e ffi ciency is lowered. The di ff erence, therefore, results from the scale at which the feedback is imposed on the gas. For their simulations this occurs at much higher densities, and the feedback therefore only regulates the high-density end of the gas distribution - the gas where stars are actually forming. Our lower resolution simulations impose the feedback on larger scales and at densities comparable to our SF threshold, which coincides with our definition of the ISM (except for ' SFTHRESHZ '). Therefore, in both cases feedback from SF acts to regulate the amount of star-forming gas. As we do not a priori know on what scales feedback regulates SF in real galaxies, it is not clear which gas densities are sensitive to the SF law.", "pages": [ 9, 10, 11 ] }, { "title": "3.5 The stellar initial mass function", "content": "The assumed stellar IMF can potentially have a large e ff ect on the galaxy population. The IMF determines the ratio of low- to highmass stars. This, in turn, changes the integrated colours of stellar populations, their chemical yields and the number of SNe per unit stellar mass. In this section we compare ' REF ', which assumes a Chabrier (2003) IMF to a simulation that assumes a Salpeter (1955) IMF (' IMFSALP '). Above 1 M /circledot the two IMFs have similar shapes, but whereas the Salpeter IMF is a pure power law, the Chabrier IMF includes a log-normal decrease at the low-mass end, resulting in much lower mass-to-light ratios. Both IMFs are integrated over the same mass range (0 . 1 -100 M /circledot ). We increase the normalization of the SF law by a factor of 1.65 in the ' IMFSALP ' simulation to take into account the di ff erence in the number of ionizing photons produced per unit stellar mass (the observed SF law is based on the flux from massive stars combined with an assumed IMF). We do not, however, change the feedback parameters (wind velocity and mass loading are the same as in the reference model; for the e ff ect of changing those parameters, see Paper I). Thus, because in a Salpeter IMF there are fewer high mass stars per unit stellar mass formed but we use the same feedback parameters, the SNe in ' IMFSALP ' inject 40% · 1 . 65 ≈ 66% of the total available SN energy into their surroundings (i.e. ∼ 34% is assumed to be lost radiatively). Fig. 6 demonstrates that the total stellar masses of the galaxies (panel A), as well as the total amount of baryons in the halo (panels Cand E), are not strongly a ff ected by the choice of IMF. Inspection of Fig. 1 shows that the extent and morphology of the gaseous disk is also not strongly a ff ected by the IMF. However, at a given halo or stellar mass, SFRs (panel B) are somewhat lower in the ' IMFSALP ' simulation. This is because the Salpeter IMF produces less metals and returns less mass to the ISM than the Chabrier IMF. This decreases the available fuel supply both by keeping more mass locked up in stars and by decreasing the e ff ect of metal-line cooling (see also Paper I). At a given mass, the amount of star-forming gas is significantly lower in ' IMFSALP ' than in ' REF ' (panels D and F). This occurs because the higher SF e ffi ciency implied by the change in the SF law means that for a Salpeter IMF more stars are formed at a given gas density (see panel H). As discussed in Sec. 3.4 this means that the stellar population is injecting more energy into its surroundings at a given gas density (recall that the wind parameters are the same in the two simulations, only the SF e ffi ciency has changed), and this allows the galaxy to self-regulate at lower gas densities.", "pages": [ 11 ] }, { "title": "4 CONCLUSIONS", "content": "We have analysed a large set of high-resolution cosmological simulations from the OWLS project (Schaye et al. 2010). We focused on the baryonic properties of (friends-of-friends) haloes at redshift 2, while varying parameters in the sub-grid models for reionization, the pressure of the unresolved multiphase ISM, star formation (SF), the stellar initial mass function, as well as the cosmology. This paper complements Paper I, which focuses on variations in the energetic feedback from star formation and the e ff ects of metal-line cooling and feedback from AGN. Our main conclusions can be summarised as follows: There are inconsistencies between observations and our simulations. On the one hand, the simulated sSFRs at fixed stellar mass are lower than observed, and the relation between the two is steeper than observed in the observed range, which is in the mass range where SN feedback in the simulations in this paper is ine ff ective. In the regime where SN feedback is e ffi cient, the slope of the relation is inverted (rising sSFR with stellar mass). The stellar mass function falls within the observational errors for almost all observed masses, except possibly the very lowest masses, where the simulations likely overproduce the number of galaxies. Weconclude, also on the basis of the results presented in Paper I, that the integrated physical properties of galaxies are mainly determined by the e ffi ciency of the feedback from star formation and AGN, while the e ffi ciency of star formation (i.e. the star formation law) determines the gas fractions. To make further progress, it is therefore most important to improve our understanding of galactic winds.", "pages": [ 11, 12 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "The authors thank the anonymous referee for a helpful report. We are grateful to the members of the OWLS collaboration for useful discussions about, and comments on, this work. The simulations presented here were run on Stella, the LOFAR BlueGene / L system in Groningen, on the Cosmology Machine at the Institute for Computational Cosmology in Durham (which is part of the DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS, and Durham University) as part of the Virgo Consortium research programme, and on Darwin in Cambridge. This work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from the Netherlands Organization for Scientific Research (NWO), also through a VIDI grant. The research leading to these results has re- ceived funding from the European Research Council under the European Union's Seventh Framework Programme (FP7 / 2007-2013) / ERC Grant agreement 278594-GasAroundGalaxies and from the Marie Curie Training Network CosmoComp (PITN-GA-2009238356). VS acknowledges support through SFB 881, The Milky Way System, of the DFG.", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "It is not understood whether long-lived star clusters possess a continuous range of sizes and masses (and hence densities), or if rather, they should be considered as distinct types with different origins. Utilizing the Hubble Space Telescope (HST) to measure sizes, and long exposures on the Keck 10m telescope to obtain distances, we have discovered the first confirmed star clusters that lie within a previously claimed size-luminosity gap dubbed the 'avoidance zone' by Hwang et al (2011). The existence of these star clusters extends the range of sizes, masses and densities for star clusters, and argues against current formation models that predict well-defined size-mass relationships (such as stripped nuclei, giant globular clusters or merged star clusters). The red colours of these gap objects suggests that they are not a new class of object but are related to Faint Fuzzies observed in nearby lenticular galaxies. We also report a number of low luminosity UCDs with sizes of up to 50 pc. Future, statistically complete, studies will be encouraged now that it is known that star clusters possess a continuous range of structural properties. Key words: galaxies: formation - galaxies: star clusters - globular clusters: general", "pages": [ 1 ] }, { "title": "Duncan A. Forbes, 1 glyph[star] , Vincenzo Pota 1 , Christopher Usher 1 Jay Strader 2 , Aaron J. Romanowsky, 3 , 4 , Jean P. Brodie 4 , Jacob A. Arnold 4 , Lee R. Spitler 5 , 6", "content": "1 Centre for Astrophysics and Supercomputing, Swinburne University, Hawthorn, VIC 3122, Australia 2 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA 6 Australian Astronomical Observatory, PO Box 915, North Ryde, NSW 1670, Australia 30 March 2022", "pages": [ 1 ] }, { "title": "1 INTRODUCTORY REMARKS", "content": "Old, compact star clusters have traditionally been classified into several types. These include globular clusters (GCs) first discovered in 1665 by Abraham Ihle (as noted by Schultz 1866). They are compact (having projected half-light sizes R h of ∼ 3 pc) and span a wide range of mass. All large galaxies, including our own Milky Way, host a system of GCs. In the last decade, several new types of star cluster containing an old stellar population have been identified. Deep imaging of the nearby lenticular galaxy NGC 1023 by the Hubble Space Telescope and spectroscopic follow-up using the 10m Keck I telescope revealed a population of low luminosity GC-like objects with large sizes ( ∼ 10 pc) dubbed Faint Fuzzies (FFs) by Larsen & Brodie (2000). Objects with similar sizes and luminosities were discovered around M31 by Huxor et al. (2005) and named Extended Clusters (ECs). Similar extended objects have been identified in galaxies ranging from dwarfs to giant ellipticals (e.g. Peng et al. (2006), Georgiev et al. (2009)), and may be related to the Palomar-type GCs found in the outer halo of the Milky Way. Searches beyond the Local Group have revealed an additional population of star clusters called Ultra Compact Dwarfs (UCDs; Drinkwater et al. 2000). These spherical collections of stars were first thought to be very compact dwarf galaxies but they also resemble extended (R h > 10 pc) GCs, some two magnitudes brighter than EC/FFs. The origin of these various star clusters (GCs, EC/FFs and UCDs) and their relationship to each other is the subject of debate (e.g. Forbes & Kroupa (2011); Willman & Strader (2012)). The size and luminosity distribution of star clusters was summarised recently by Brodie et al. (2011) (to download database see: http://sages.ucolick.org/downloads/sizetable.txt). They included all types of known star cluster with old ( glyph[greaterorequalslant] 5 Gyr) stellar ages. They also restricted their sample to objects with confirmed distances. This is important if one is exploring size and luminosity trends, but this has not always been the case in the literature. From size and luminosity, the projected surface and volume densities can also be derived. In Figure 1 we show the fundamental parameters of size and luminosity from this state-of-the-art compilation for long-lived star clusters. The figure shows a U-shaped distribution. The high luminosity, extended star clusters are generally referred to as UCDs, the base of the U-shape is occupied by compact GCs and the low luminosity, extended size regime is associated with ECs and FFs. Two extreme Milky Way GCs are highlighted in the figure; NGC 2419 (the largest Galactic GC, which lies in the region near low luminosity UCDs) and ω Cen (the most luminous Galactic GC). The figure shows that star clusters with V band magnitudes M V brighter than -10 and projected half-light radii R h greater than 5 pc are very rare, if not completely absent, in the Local Group of galaxies which is dominated by the Milky Way and Andromeda. Only a few objects beyond the Local Group are known with M V fainter than -8.5. This corresponds to an apparent magnitude limit of V < 22.5 at the distance of the Virgo cluster (a typical limiting magnitude for spectroscopic studies on 8m class telescopes). The exception is the deep HST and Keck telescope observations of FFs in NGC 1023 by Larsen & Brodie (2000). The figure also highlights the lack of very compact, very luminous objects, i.e. those with ultra high densities. It has been argued by Hopkins et al. (2010) that feedback from massive stars sets an upper density limit, beyond which star clusters do not form. However, perhaps the most interesting feature of Figure 1 is the deficiency of objects around M V ≈ -9 and R h glyph[greaterorequalslant] 7 pc, i.e. sizes and luminosities intermediate between EC/FFs and UCDs. This gap in the size-luminosity distribution has been called the star cluster 'avoidance zone' by Hwang et al. (2011). Such a gap could be due to physical processes or to an observational selection effect. A real gap would imply that EC/FFs are physically distinct from low luminosity UCDs and hence are formed by different mechanisms that have inherent upper and lower mass limits respectively. Continuity across the gap might suggest that one family of star cluster has a wider range of properties than previously known or that a new type of star cluster exists. Here we briefly present the recession velocities, and hence physical sizes and luminosities for extended (R h > 5pc) star clusters around three elliptical galaxies. In particular, we investigate whether these star clusters occupy the 'avoidance zone' seen in Figure 1 or not.", "pages": [ 1, 2 ] }, { "title": "2 THE DATA", "content": "To identify potential star clusters in the 'avoidance zone' the candidates need to be resolved in order to measure their sizes. This is best achieved with the superior spatial resolution of the Hubble Space Telescope (HST). A small number of nearby elliptical galaxies have been imaged by HST in two filters (required for colour selection) and over half a dozen pointings (needed to identify a large number of candidate star clusters associated with each galaxy). In particular, half-light sizes have been measured from g and z band HST/ACS images for candidate star clusters in NGC 4278 by Usher et al. (2013, in prep.) and NGC 4649 by Strader et al. (2012). In both the Usher et al. and Strader et al. works, objects were selected on the basis of having colours that matched those expected of candidate star clusters. Sizes were then determined using the ISHAPE software and visual inspection to remove obvious background galaxies. For NGC 4697 a similar procedure was used. The galaxies are located at distances of 15.6 Mpc (NGC 4278), 17.3 Mpc (NGC 4649) and 11.4 Mpc (NGC 4697). At these distances HST can resolve sizes as small as 1-2 pc. After selecting resolved star cluster candidates (with GC-like colours) around these three galaxies, we designed several multiobject slit masks for the DEIMOS instrument on the 10m Keck II telescope. Typical exposures of 2 hrs, in 0.8-1.2 arcsec seeing conditions during the nights of 2013 January 11-12, were obtained. The resulting spectra were reduced using standard procedures and radial velocities measured, e.g. following the method of Pota et al. (2013). For each galaxy we confirmed several tens of GCs, with sizes of ∼ 3 pc, to have velocities consistent with that of their host galaxy. A small number of background galaxies, with significantly higher velocities, were confirmed in each mask. Their magnitudes, colors and angular sizes of the background galaxies are provided in the Appendix. Here we focus on the confirmed objects with sizes greater than 5 pc. Table 1 lists their magnitudes, colours, average half-light radii from the g and z bands and apparent V band magnitudes from the transformation: 0 . 753 × ( g -z ) -0 . 108 + z (based on a large sample of GCs from Usher et al. 2013, in prep.). IDs for the objects come from Usher et al. (2013, in prep.), Strader et al. (2012) and this work for NGC 4278, 4649 and 4697 respectively.", "pages": [ 2 ] }, { "title": "3 FILLING THE GAP", "content": "In Figure 2 we again show the data points from Brodie et al. (2011) and now include all the confirmed star clusters in NGC 4278, 4649 and 4697. Our main finding is that old star clusters do indeed occupy the 'avoidance zone' gap. The avoidance zone is therefore simply the result of a selection bias in previous works which were unable to reach low enough surface brightness levels beyond the Local Group. Here we confirm that long-lived star clusters cover a wide and continuous range of sizes and luminosities (and hence densities). A clue to the nature of the extended size star clusters comes from their instrinsic colours. In Figure 2 objects have been coded by their colour, i.e. red or blue for a colour separation at (g-z) = 1.1, which corresponds to a metallicity of [Fe/H] ∼ -1. We find that the high luminosity star clusters tend to be blue (or metal-poor) and the low luminosity ones red (metal-rich). Focusing on the gap itself, the clusters are mostly red in colour indicating that they are metal-rich. This suggests that they are more closely related to the lower lu- nosity FFs found in NGC 1023 by Larsen & Brodie (2000) and the Diffuse Star Clusters (DSCs) of Peng et al. (2006) which are metal-rich and red in colour. These objects are typically associated with the disks of lenticular galaxies that reveal signs of a past interaction. Burkert, Brodie & Larsen (2005) suggest that FFs form in metal-rich disks as the result of an interaction and subsequent starburst. Goudfrooij (2012) has argued that the intermediate-aged diffuse star clusters in the merger remnant NGC 1316 may evolve to resemble FFs after the continued disruption by tidal shocks. Although all three host galaxies studied here are classified as ellipticals, we note that NGC 4278 contains a large HI ring (Raimond et al. 1981) that is perhaps a remnant of a past interaction, NGC 4649 reveals strong rotation in its outer region as might expected after a major merger (Hwang et al. 2008) and NGC 4697 is highly flattened (E6) and so may be a mis-classified S0 (Dejonghe et al. 1996). We have also confirmed the existence of several other interesting objects. They include a number of blue low luminosity UCDs, similar to those found originally by Strader et al. (2011) and listed in their table 9. Two of these have sizes and luminosities very similar to the Milky Way GC NGC 2419, the largest known GC in the Milky Way. Like other massive GCs in the Milky Way, NGC 2419 contains multiple stellar populations, e.g. Cohen & Kirby (2012), which are traditionally associated with galaxies (Forbes & Kroupa (2011); Willman & Strader (2012)). Indeed Cohen & Kirby (2012) have suggested that NGC 2419 is not in fact a GC but the remnant nucleus of a stripped dwarf galaxy. If it was once part of a dark matter dominated dwarf galaxy, that dark matter appears to have been largely stripped away as none is detected today in its outer regions (Conroy, Loeb & Spergel (2011); Ibata et al. (2013)). We also confirm an object (acs580) around NGC 4697 with a similar luminosity to the red FFs but with a larger size (26 pc). This object has the lowest surface density of any confirmed old star cluster beyond the Local Group. Finally, we note that one star cluster (D68) has a luminosity of M V = -10.8, similar to that of ω Cen (the most massive GC or remnant nucleus in the Milky Way) but with a half light radius some six times larger at 47 pc and hence a lower surface density by a factor of ∼ 35. In Figure 3 we show several examples of our Keck spectra for selected star clusters. The examples include: acs112 which has a size and luminosity similar to that of the Milky Way GC NGC 2419; D68, the largest star cluster confirmed in this work; A51, a red star cluster located in the 'avoidance zone' and acs580 a large faint fuzzy (FF) analogue around NGC 4697. We remind the reader that it becomes increasingly difficult with decreasing brightness and increasing size to confirm low density star clusters (the upper left hand side of Figure 2) and so a reduction in the number of star clusters in that region of the figure is probably due to current observational limitations. Future deep surveys may rectify this, finding that region is well-populated.", "pages": [ 2, 3, 4 ] }, { "title": "4 CONCLUDING REMARKS", "content": "A number of theories have been put forward to explain the origin of the different types of extended star cluster with corresponding predictions for their structural properties. For example, if UCDs are simply giant GCs (Murray (2009)) or the remnant nuclei of stripped dwarf galaxies (Bekki et al. (2001)) then a well-defined size-luminosity trend of near constant density is predicted. In the merging star cluster simulations of Bekki et al. (2004) the resulting UCDs are also predicted to have a well-defined size-luminosity relationship. Although a distinct size-luminosity relation may exist for more luminous objects (such as compact ellipticals), for luminosities fainter than M V = -13.5 we find a continuous range in size and luminosity for old star clusters. With the introduction of an external tidal field, and exploring a larger range of masses, the simulations of Bruns et al. (2011) produced merged star clusters with a large range of size and luminosity. However, their work indicated an upper limit to the maximum size that increased with star cluster mass. This is not generally seen in our data. Individual star clusters were assumed to follow a distinct initial size-mass relationship in the simulations of Gieles et al. (2010) but the effects of stellar evolution, binaries and two-body relaxation over time resulted in their old clusters having large (R h ∼ 10 pc) sizes. Tidal effects would tend to reduce this size further. While matching some aspects of our data, this model has difficulty reproducing the largest (R h > 10 pc) star clusters. In summary, we find a continuity of structural properties across a gap in size and luminosity called the 'avoidance zone'. The red colours of these gap objects suggests that they are not a new class of object but are related to the Faint Fuzzies observed in nearby lenticular galaxies. We also report a number of low luminosity UCDs with sizes of up to 50 pc. No single model for the formation of extended star clusters can currently reproduce the diversity of structural properties now observed for old star clusters.", "pages": [ 4 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The analysis pipeline used to reduce the DEIMOS data was developed at UC Berkeley with support from NSF grant AST-0071048. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. DF thanks the ARC for support via DP130100388. JB acknowledges support from NSF grant AST-1109878. We thank the referee for several useful suggestions that have improved the paper.", "pages": [ 4 ] }, { "title": "REFERENCES", "content": "Bekki K., Couch W. J., Drinkwater M. J., Gregg M. D., 2001, ApJL, 557, L39 Bekki K., Couch W. J., Drinkwater M. J., Shioya Y., 2004, ApJL, 610, L13 Brodie J. P., Romanowsky A. J., Strader J., Forbes D. A., 2011, AJ, 142, 199 Bruns R. C., Kroupa P., Fellhauer M., Metz M., Assmann P., 2011, A&A, 529, A138 Burkert A., Brodie J., Larsen S., 2005, ApJ, 628, 231 Cohen J. G., Kirby E. N., 2012, ApJ, 760, 86 Conroy C., Loeb A., Spergel D. N., 2011, ApJ, 741, 72 Dejonghe H., de Bruyne V., Vauterin P., Zeilinger W., 1996, A&A, 306, 363 Drinkwater M., Jones J., Gregg M., Phillipps S., 2000, PASA, 17, 227 Forbes D., Kroupa P., 2011, PASA, 28, 77 Goudfrooij P., 2012, ApJ, 750, 140 Huxor A. P., Tanvir N. R., Irwin M. J., Ibata R., Collett J. L., Ferguson A. M. N., Bridges T., Lewis G. F., 2005, MNRAS, 360, 1007 Ibata R., Nipoti C., Sollima A., Bellazzini M., Chapman S. C., Dalessandro E., 2013, MNRAS, 428, 3648 Larsen S., Brodie J., 2000, AJ, 120, 2938 Murray N., 2009, ApJ, 691, 946 Peng E. W. et al., 2006, ApJ, 639, 838 Pota V. et al., 2013, MNRAS, 428, 389 Schultz E., 1866, AN, 67, 1 Strader J. et al., 2011, ApJS, 197, 33 Strader J. et al., 2012, ApJ, 760, 87 Willman B., Strader J., 2012, AJ, 144, 76", "pages": [ 4, 5 ] }, { "title": "APPENDIX A: BACKGROUND OBJECTS", "content": "We caution that some researchers have attempted to explore the size-luminosity distribution of star clusters without having a confirmed distance to each object. This is a dangerous practice and can lead to incorrect conclusions. For example, some have been tempted to explore mean size trends with luminosity and to make subsequent comparisons with theoretical predictions. In Table A1 we list the objects which have similar apparent sizes and magnitudes to our confirmed objects but our spectroscopic redshifts indicate that they are actually distant background galaxies. The columns are ID, z magnitude and error, (g-z) colour and error, halflight radius and error.", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "Current surveys are underway to utilize gravitational lensing by galaxy clusters with Einstein radii > 35 '' in the search for the highest redshift galaxies. Associated supernova from the epoch of reionization would have their fluxes boosted above the detection threshold, extending their duration of visibility. We predict that the James Webb Space Telescope (JWST) will be able to discover lensed core-collapse supernovae at redshifts exceeding z = 7-8.", "pages": [ 1 ] }, { "title": "Tony Pan 1 , Abraham Loeb 1", "content": "1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 14 September 2021", "pages": [ 1 ] }, { "title": "Key words:", "content": "supernovae: general - gravitational lensing - galaxies: clusters: general - early Universe", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Clusters of galaxies act as gravitational lenses, focusing light-rays from sources behind them and magnifying their images. As this effect enables observers to probe higher redshifts than ever probed before, surveys are being conducted with the Hubble Space Telescope (HST) to obtain deep images of the sky through massive galaxy clusters. One such ongoing program is the Cluster Lensing and Supernova survey (CLASH), which is imaging 25 clusters each to a depth of 20 orbits (Postman et al. 2012). The 5 clusters selected for this program have large Einstein radii of 35 '' to 55 '' , maximizing their potential for discovering ultra-high redshift galaxies. Indeed, three candidate galaxies at redshifts z ≈ 9-10 and another candidate galaxy at z ≈ 11 have already been found in the CLASH fields (Bouwens et al. 2012; Coe et al. 2013). Similarly, the planned HST Frontier Fields 1 program will target 6 strong lensing galaxy clusters to reveal yet higher redshifts galaxies. The James Webb Space Telescope (JWST), the successor to HST scheduled for launch in 2018, is likely to have analogous observational programs with comparable integration times on a similar number of lensing clusters. Although the CLASH survey does aim to detect Type Ia supernova (SN) out to redshifts of z ∼ 2 . 5, the current HST cluster observations are unlikely to detect gravitationally lensed SN from the epoch of reionization at z > 6. Indeed, transient science was not identified as a science priority for the Frontier Fields program, which will not revisit the same field twice. The greater sensitivity of JWST and its optimization for observations in the infrared could potentially allow it to find lensed supernova from the cosmic dawn in these same cluster fields. In this Letter , we estimate the cosmic star formation rate during the epoch of reionization by requiring that enough Pop II stars were formed to ionize the universe. Using model spectral time series for Type II SN, as well as a simple isothermal sphere model for lensing, we calculate in § 2-5 the required magnification and duration of detectability of such SN at z > 6 for different JWST bands and integration times. Combining the above, we derive the snapshot rate, i.e. the expected number of gravitationally lensed core collapse SNe detected in the field-of-view of JWST around these high magnification clusters.", "pages": [ 1 ] }, { "title": "2 STAR FORMATION & SUPERNOVA RATE", "content": "We infer the volumetric supernova rate R SN ( z ) as a function of redshift by relating it to the cosmic star formation rate density (SFRD) ˙ ρ /star ( z ): where we use a Salpeter initial mass function (IMF), ψ ( M ) ∝ M -2 . 35 , and include a factor of 0 . 7 in the mass integral to account for the shallower slope at M /lessorsimilar 0 . 5 M /circledot in a realistic IMF (Fukugita, Hogan & Peebles 1998). For the stellar mass range between M min = 8 M /circledot and M max = 40 M /circledot appropriate for optically-luminous core-collapse supernova, the conversion coefficient between the star formation rate and the supernova rate is η SN ∼ 0 . 0097 M -1 /circledot . We require that enough massive stars were formed by the end of reionization so as to produce sufficient ionizing UVradiation to ionize the intergalactic medium by z end = 6. This follows the approach used in Pan, Kasen & Loeb (2012), albeit with different parameters to bring our estimates closer to other inferences in literature, as detailed below. The star formation rate during reionization peaks at late times, when metals expelled from a prior generation of star formation enriched the interstellar gas, so we assume that early Pop II stars ( Z = 0 . 02 Z /circledot ) with a present-day IMF dominated the ionizing photon budget. Using the stellar ionizing fluxes of Schaerer (2002), we find the average number of ionizing photons produced per baryon incorporated into a Pop II star was ¯ η γ = 5761. Thus, the mass in stars per comoving volume ρ /star ( z ) should satisfy where C is the number of ionizing photons necessary to ionize each baryon after accounting for recombinations, ρ b is the cosmic baryon density, and f esc is the average escape fraction of ionizing photons from their host galaxies into the intergalactic medium. Also, we can relate the mass in stars per volume ρ /star ( z ) to the mass in virialized halos per volume via a star formation efficiency f /star : where we use the Sheth-Tormen mass function of halos for dn/dM (Sheth & Tormen 1999), and M min ∼ 10 8 M /circledot is the minimum halo mass with atomic hydrogen cooling. The cosmological parameters, such as the matter and baryon densities Ω M , Ω b , were taken from Planck Collaboration et al. (2013). Assuming f /star is constant, we can calibrate f /star via equations (2), (3), and then evaluate ρ /star ( z ) at any redshift. The star formation rate is simply, dρ /star ( z ) /dt . Figure 1 shows our estimated SFRD, with C = 3 and f esc = 0 . 2, resulting in a SFRD ≈ 2 × 10 -2 M /circledot yr -1 Mpc -3 (comoving) between redshifts of z = 6 to 8. This corresponds to volumetric rates of approximately 2 × 10 -4 yr -1 Mpc -3 for core-collapse supernova. Our simple SFRD model and the resulting SN rates linearly scale with C and f -1 esc , so the JWST snapshot rates calculated later can be easily scaled for different parameter choices of the SFRD.", "pages": [ 1, 2 ] }, { "title": "3 LIGHT CURVES", "content": "We adopt the spectral time series of a Type II plateau SN from a red giant progenitor with an initial mass 15 M /circledot , computed by Kasen & Woosley (2009) using a code that solves the full multi-wavelength time-dependent radiative transfer problem. We plot the SN light curves in the observer frame for the best possible HST and JWST filters in Figure 2. Note that Type II SN are diverse transients with peak luminosities that can vary by more than an order of magnitude, and the relationship between the progenitor mass and the brightness of the supernova is uncertain; we adopt a single characteristic model to represent all core collapse SNe for the sake of simplicity. Type IIP SNe are the most common events, and the model light curves and spectra used here agree very well with observed SNe of average luminosities. We verified that HST is incapable in practice of detecting a core-collapse SNe from the epoch of reionization. The sensitivity of the HST 1.6 µ m filter is only a factor of 2 worse than the JWST F444W filter, but its overwhelming drawback is its waveband, which can only probe the SN restframe UV flux at z /greaterorequalslant 4. Although the JWST F356W is more sensitive, the F444W band will be optimal for detecting the highest redshift SN that gravitational lensing could provide. Figure 3 shows the magnification necessary to detect Type II supernova at high redshifts for different integration times. Even with a 10 5 s exposure, a large magnification factor of µ /greaterorequalslant 10 will be necessary for detecting Type IIP SNe at z > 10 with JWST.", "pages": [ 2 ] }, { "title": "4 LENSING MAGNIFICATION", "content": "For simplicity, we adopt a singular isothermal sphere (SIS) model for the mass distribution of the lensing cluster, within which the magnification properties are uniquely specified by the Einstein radius θ E (Schneider, Ehlers & Falco 1992). We denote the angular separations of the source and the image from the center axis of the lens as β and θ , respectively. If the source lies within the Einstein radius β < θ E , two images are created at locations θ ± = β ± θ E , with magnifications µ ± = 1 ± θ E /β . Note that µ -has negative magnification, that is, the image is flipped compared to the source. If the source lies outside the Einstein radius β > θ E , there is only one image at θ = θ + with magnification 1 < µ + < 2. We conservatively consider only the higher-magnification image at θ + , for which the source angle β = θ E / ( µ -1). Then, the differential source volume (comoving) of magnified events as a function of magnification and redshift is: where the differential comoving distance dD C ( z ) is with E ( z ) ≈ √ Ω M (1 + z ) 3 +Ω Λ , and the differential source area is Here D A ( z ) is the angular diameter distance, and the extra (1+ z ) 2 is to adjust the area to comoving units. In Figure 4, we plot the source volume for a range of Einstein radii typical of high-magnification clusters. Given core-collapse SN rates of ∼ 10 -3 yr -1 Mpc -3 , capturing SN with high mag- fications within source volumes < 10 2 Mpc 3 is unlikely. Hence, we expect most lensed supernova detected to have their fluxes moderately boosted with µ /lessorsimilar 5; the benefit of lensing is to probe somewhat deeper redshifts, and to greatly extend the duration of visibility. Also, since high-redshift observations are background-limited, for a target signal-tonoise ratio, the limiting flux is proportional to t -1 / 2 , so even a modest magnification of µ ∼ 3 can reduce the required integration time by an order-of-magnitude. This volume limitation of lensing also justifies our focus on core-collapse SNe, which have the highest volumetric rates. Although Type Ia SNe are brighter, their volumetric rate is a factor of 4 smaller than the core collapse rate at z ≈ 7 (Pan, Kasen & Loeb 2012), with the difference drastically increasing with redshift due to the long delay times needed between star formation and explosion for some Type Ia events (Maoz, Mannucci & Brandt 2012). Pair-instability SNe from Pop III stars have volumetric rates at least two orders of magnitude lower.", "pages": [ 2, 3, 4 ] }, { "title": "5 SNAPSHOT RATE", "content": "The snapshot 'rate' is the total number of events observed at a limiting flux within a given field (not per unit time). The differential snapshot rate can be calculated from equations (1) and (4) via where t ( F ν , µ, z ) is the rest-frame duration over which an event with magnification µ will be brighter than the limiting flux F ν at redshift z , for the observation wavelength ν under consideration. We find t ( F ν , µ, z ) using our spectral timeseries for the Type IIP SN model described in § 3. As we care about the apparent SN rate for observers, there is an implicit factor of (1+ z ) -1 in front of the intrinsic volumetric supernova rate R SN ( z ), but that cancels with a (1+ z ) factor for t ( F ν , µ, z ) due to cosmic time dilation. In Figure 5, we plot the expected snapshot rate of magnified core-collapse SN detected by JWST above target redshifts, calculated by integrating equation (7) over µ and partially over z . Since NIRCam has two modules each with a 2.2 × 2.2 arcmin 2 field-of-view, we limit the source area in equation (4) to images that lie within this field-of-view. We find that a 10 5 s JWST snapshot with the F444W filter is expected to detect ∼ 1 magnified core collapse SN at z > 7 around each cluster, and ∼ 0 . 1 SNe at z > 8. Using ∼ 5 clusters with θ E /greaterorequalslant 35 '' , the prospects for detecting a few non-superluminous SNe at high redshifts via lensing are high. If the other ∼ 20 galaxy clusters in the CLASH survey with smaller Einstein radii of θ E ∼ 15 '' -30 '' are also included, the expected number of gravitationally lensed highz SNe detected should double.", "pages": [ 4 ] }, { "title": "6 DISCUSSION", "content": "At z > 6, the observed duration of gravitationally-lensed core-collapse SNe can reach /greaterorsimilar 1 year, lending their detection to a search strategy of taking images separated by ∼ 0 . 5 -1 year, and looking for flux differences between consecutive snapshots. Ideally, the cluster survey should cover most of Redshift the critical curve area, and not just known locations of magnified images of highz galaxies, as the lensed SN may appear in currently 'dark' critical curve areas, and serve as a flag for its fainter host galaxy. The spectral energy distribution of Type II SNe is sufficiently different from blackbody to allow for photometric redshift determination, however, typing the SNe accurately may require time-consuming spectroscopy. Our quantitative results improve upon previous calculations of the frequency of lensed SNe. For example, Marri, Ferrara & Pozzetti (2000) first explored the effects of gravitational lensing on highz Type II SNe by intervening cosmological mass for different cosmologies, but the predicted detection rates were unrealistically high because of optimistic assumptions about JWST capabilities. Gunnarsson & Goobar (2003) explored the lensing by massive clusters of distant Type Ia and Type II SNe observed at wavelengths of 0.81.25 microns, but found the discovery rate tapered off at z ∼ 3. Also, gravitational lensing is not required per se to detect Type II SNe from the epoch of reionization. A moderate JWST blank-field survey can obtain similar snapshot rates, albeit trading off the highest redshift events for more lower redshift ones compared to a lensing survey. For example, Mesinger, Johnson & Haiman (2006) found that a 10 5 s exposure with JWST can detect 4-24 SNe per field at z > 5, although the assumed SFRD ∼ 0 . 1 M /circledot yr -1 Mpc -3 was an order-of-magnitude higher than our estimates here, and the current specifications for JWST NIRCam filter sensitivities are now ∼ 3 times worse than the values assumed at that time. For more luminous SNe, Whalen et al. (2012) found that core collapse SNe from Pop III progenitors in the earliest galaxies could be visible with the deepest JWST surveys (reaching M AB = 32) even at z > 10, as these SNe are bluer and almost an order-of-magnitude brighter than the average Type II SNe considered in this paper. Whalen et al. (2013) also found that superluminous Type IIn SNe powered by circumstellar interactions from Pop III stars could be visible out to z ∼ 20. Truly massive Pop III stars with masses M /greaterorsimilar 200 M /circledot can also die as extremely bright pairinstability supernova, detectable with JWST at z > 15 (Pan, Kasen & Loeb 2012; Hummel et al. 2012); indeed, the current record for the highest-redshift supernova ever observed is likely a pair-instability or pulsational pair-instability event at z = 3 . 90 (Cooke et al. 2012). However, the small volumetric density of Pop III stars makes it unlikely that these events will be strongly lensed. Finally, there is growing evidence of a prompt population of Type Ia SNe, so their volumetric rates during the later stages of reionization may not be negligible. With the fiducial SFRD model in this Letter , we estimate /greaterorsimilar 1 gravitationally lensed Type Ia SNe could be discovered at z > 7 in the snapshots across the ∼ 5 high-magnification clusters at any given time. At lower redshifts, the measured core collapse SN rate is a factor of ∼ 2 lower than that predicted from the cosmic star formation rate (Horiuchi et al. 2011); the most likely explanation is that some SN are dim, whether intrinsically faint or due to dust obscuration. This will reduce our predicted snapshot rate. However, we ignored the contribution of multiple lensing images in our analysis. Due to the gravitational lens time delay, which could be ∼ 1 -100 years for strong lensing around the clusters of interest (Coe et al. 2013), multiple images arriving at different times can increase the expected snapshot detection rate of separate SN within the same field-of-view. Although our SIS lens model can produce a maximum of only 2 magnified images, substructure and ellipticity in actual galaxy clusters will likely increase both the number of images and their magnifications.", "pages": [ 4, 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are grateful to Dan Kasen for providing the spectral time series data for the Type IIP SN model used in this letter. TP was supported by the Hertz Foundation and the National Science Foundation via a graduate research fellowship. This work was supported in part by NSF grant AST-0907890 and NASA grants NNX08AL43G and NNA09DB30A.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Bouwens R. et al., 2012, ArXiv e-prints Coe D. et al., 2013, ApJ, 762, 32 Cooke J. et al., 2012, Nature, 491, 228 Ellis R. S. et al., 2013, ApJ, 763, L7 Fukugita M., Hogan C. J., Peebles P. J. E., 1998, ApJ, 503, 518 Gunnarsson C., Goobar A., 2003, A&A, 405, 859 Horiuchi S., Beacom J. F., Kochanek C. S., Prieto J. L., Stanek K. 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[ { "title": "ABSTRACT", "content": "We present results from long cadence Kepler observations covering 97 . 6 days of the newly discovered eclipsing cataclysmic variable KIS J192748.53+444724.5/KIC 8625249. We detect deep eclipses of the accretion disk by the donor star every 3.97 hours. Additionally, the Kepler observations also cover a full outburst for this cataclysmic variable, making KIS J192748.53+444724.5 the second known eclipsing cataclysmic variable system in the Kepler field of view. We show how in quiescence a significant component associated to the hot-spot is visible preceding the eclipse, and that this component is swamped by the brightness increase during the outburst, potentially associated with the accretion disk. Furthermore we present evidence for accretion disk radius changes during the outburst by analysing the out-of-eclipse light levels and eclipse depth through each orbital cycle. We show how these parameters are linearly correlated in quiescence, and discuss how their evolution during the outburst is suggesting disk radius changes and/or radial temperature gradient variations in the disk. Key words: accretion, accretion discs - binaries: close - stars: individual: KIS J192748.53+444724.5, KIC 8625249 - cataclysmic variables", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Cataclysmic variables (CVs) are interacting close binary systems where a late-type star transfers material to a white dwarf (WD) companion via Roche lobe overflow. With a system orbital period in the range of hours, the transferred material from the secondary star forms an accretion disk surrounding the WD. As angular momentum is transported outwards in the disk, material will approach the inner-most regions close to the WD in the absence of strong magnetic fields, and eventually accrete onto the compact object. Eclipsing CVs are particularly useful not only because the system parameters can be recovered (such as the masses of the two stars), but also because modelling the eclipses allows us to study in great detail the physics of the accretion disk (see Horne 1985; Feline et al. 2004). In total, 208 eclipsing systems are known (Ritter & Kolb 2003 version 7.12). In this Letter we report on the discovery of an eclipsing dwarf-nova type CV within the Kepler field-of-view: KIC 8625249/KIS J192748.53+444724.5 (hereafter KIS J1927). This is the sec- ond know eclipsing CV in the Kepler field, after V447 Lyr (Ramsay et al. 2012). KIS J1927 was first discovered as a CV by Scaringi et al. (2013) via spectroscopic follow-up of sources from the Kepler -INT Survey (Greiss et al. 2012) displaying both H α and blue colour excess. Soon after the discovery, the Kepler satellite began monitoring this object ( Kepler passband magnitude of Kp = 18 . 4) with a timing resolution of 29 . 4 minutes (long cadence mode). Only 7 CVs have been observed and studied with Kepler (Still et al. 2010; Cannizzo et al. 2010; Wood et al. 2011; Cannizzo et al. 2012; Barclay et al. 2012; Scaringi et al. 2012b,a; Ramsay et al. 2012; Osaki & Kato 2012; Kato & Maehara 2013; Osaki & Kato 2013; Kato & Osaki 2013), but an additional 8 have recently been monitored. In Section 2 we introduce the Kepler satellite and the available observations, and present the orbital lightcurve of KIS J1927. In section 3 we provide an ephemeris for the system, whilst in section 4 we discuss the folded lightcurve on the orbital period. Section 5 discusses how the Kepler photometry provides evidence for accretion disk radius changes and/or radial temperature gradient variations during the observed outburst of KIS J1927, and places our observations in the context of previous work. Our conclusions are drawn in section 6, and prospects for future Kepler observations of KIS J1927 are discussed.", "pages": [ 1, 2 ] }, { "title": "2 OBSERVATIONS", "content": "The Kepler mission's primary science objective is to discover Earth-sized planets in the habitable zone of Sunlike stars (Borucki et al. 2010; Haas et al. 2010; Koch et al. 2010). The spacecraft is in an Earth-trailing orbit allowing it to continuously monitor the same field-of-view (FOV). The shutterless photometer (with a response function covering the wavelength range 4000 -9000 ˚ A) has a 116 deg 2 FOV and makes use of 6.02 second integrations (plus an additional 0 . 52 seconds for CCD readout). Only pixels containing preselected targets are saved due to telemetry bandwidth and onboard memory constraints. Up to 170,000 targets can be observed in long cadence (LC) mode, where 270 integrations are summed onboard the spacecraft for an effective 29.4 minute exposure, and up to 512 targets can be observed in short cadence (SC) mode, where nine integrations are summed for an effective 58.8 second exposure. Gaps in the photometric lightcurves are the result of quarterly data downlinks, as well as Kepler occasionally entering anomalous safe modes. During such events no data are recorded and for a few days following these events the data are always correlated due to the spacecraft not being in thermal equilibrium. Further details of artefacts within Kepler light curves can be found in the Kepler Data Release Notes 20 (Thompson et al. 2013). Here, we make no attempt to correct these artefacts, but simply remove them from the light curve. The data for KIS J1927 discussed in this Letter is that of the first quarter with available and reduced observations (Quarter 15: 04 October 2012 - 06 January 2013) obtained in LC mode. KIS J1927 resides in a crowded field with a number of close neighbours identified, including the Kp = 17 . 4 object KIC 8625243. The archived Simple Aperture Photometry is based upon the summation of two collected pixels with CCD module 13.3 coordinates (271,859) and (272,859). We expect these two pixels to be contaminated by the Pointspread-function (PSF) wings of near neighbours, and the correction for contamination in the archived Pre-search Conditioning Data is overly simplified, resulting in the eclipse depths to be underestimated in the archived Simple Aperture Photometry. To rectify this situation we extract new photometry from the archived target pixels using PSF photometry. The PSF model was downloaded from the Mikulski Archive for Space Telescopes (MAST 1 ). A more precise PSF distribution was obtained by interpolation over the position of KIS J1927 and this model was fit to the target pixels, at each photometric time stamp. A fit adopting three significant sources within the target mask proved sufficient to reduce the residuals to an acceptable level (Pearson's χ 2 = 173 for 26 degrees of freedom). The resulting photometric time series for the target star is provided in Fig. 1. A typical fit to the pixels collected from a single time stamp is provided in Fig. 2. The median eclipse depth (relative to the out-of-eclipse light) within the Simple Aperture Photometry was 90 . 3 electrons/second, whereas the the median eclipse depth within the PSF photometry is 161.2 electrons/second. The lightcurve in Fig. 1 shows a large amplitude outburst starting shortly after the beginning of the observations, similar to the outbursts observed in dwarfnova type CVs (see Still et al. 2010; Cannizzo et al. 2010, 2012; Wood et al. 2011; Kato & Maehara 2013). The Fourier transform of the quiescent interval (BJD > 2456226 days) is shown in Fig. 2, where the orbital period is clearly visi- ble, as well as higher harmonics due to the non-sinusoidal lightcurve shape through each orbit. Also alias periods are present, which are due to the close proximity of integer multiples of the sampling frequency to the system orbital period. The most notable aspect of the lightcurve are the eclipses occurring every 3.97 hours as the binary inclination angle is high enough to cause the donor star to eclipse the accreting white dwarf and/or its associated accretion disk and hot spot.", "pages": [ 2, 3 ] }, { "title": "3 ECLIPSE TIMINGS", "content": "We estimated the arrival times of the eclipse minima by fitting a spline function independently to each orbital cycle, and determining the time of minimum flux within the fitted curve. In order to obtain the eclipse ephemeris we then fitted a linear curve to the observed mid-eclipse times for every observed cycle. The accuracy of the ephemeris is then assumed to be the small scatter around the fit, which is mainly caused by the aliasing of the sampling frequency to the orbital period. The mid-eclipse ephemeris is then: where N is the cycle number. The 1 σ uncertainty for the parameters are given in parentheses for the last digits.", "pages": [ 3 ] }, { "title": "4 FOLDED LIGHTCURVE", "content": "Fig. 4 shows the lightcurve folded on the orbital period of 3.97 hours during the quiescent interval (BJD > 2456226 days). The median eclipse depth relative to the out-of-eclipse flux is 34%. However, this value is most-likely underestimated as a result of the data cadence. Future observations with a faster cadence may better resolve each orbital cycle and reveal the eclipses to be deeper than what is observed here. The increase in flux at the orbital phase just preceding the eclipse ( φ ≈ 0 . 8 -0 . 9, where φ is the orbital phase) can be explained by the hot-spot (where the accretion stream from the donor star impacts the outer-edges of the accretion disk) being observed nearly face on (Wood et al. 1986). It is also possible that the hot-spot itself is also being eclipsed by the donor star. This would explain the observed flux descent after the eclipse ( φ ≈ 0 . 1 -0 . 3) as a continuation of the hot-spot emission, and would imply that the maximum hot-spot brightness occurs during the eclipse. Assuming we can derive the mass ratio of the system, we can place constraints on the system inclination. To do this we use the semi-empirical donor tracks of Knigge et al. (2011) to infer the mass of the secondary star of M 2 = 0 . 32 M /circledot from the observed orbital period. Since the mass ratio q = M 2 /M 1 has to be smaller than 2 / 3 for stable mass transfer to occur (Warner 2003), and since the mass of the primary can at most be the Chandrasekhar limit (1 . 4 M /circledot ), we infer a mass ratio range of 0 . 23 < q < 0 . 67. We note however that the mass ratio might be larger than 0.35 for this system as no superhumps are detected during outburst (although this might be caused by the low sampling rate), but we employ a larger conservative range. By fitting a Gaussian function to the folded eclipse profile (between 0 . 85 < φ < 1 . 1) we also infer a full width at half maximum of the eclipse of ∆ φ = 0 . 095. We then employ the method of Horne (1985) to deduce an inclination of i > 80 o .", "pages": [ 3 ] }, { "title": "5 VARYING ACCRETION DISK RADIUS", "content": "Fig. 1 shows the Kepler lightcurve of KIS J1927 as a function of orbital cycle, with insets displaying zoomed portions of the lightcurve during outburst and quiescence. It is clear from Fig. 1 that there are significant changes in the eclipse profiles as the system switches between outburst and quiescence. Most notably, the increase in flux at φ ≈ 0 . 8 -0 . 9 associated with the hot-spot is much less pronounced during the outburst. This is potentially suggesting that during the outburst, the optical light is dominated by the accretion disk and the bright spot makes a significantly smaller contribution to the total optical flux as compared to quiescence, and/or that the hot-spot emitting region has changed from a small, compact, region to a larger structure over the disk. Additionally, the eclipse depth relative to the out-of-eclipse light changes from 34% in quiescence to 58% in outburst. The Kepler observations of KIS J1927 displayed an increase in brightness of 2 . 5 magnitudes during the outburst. Fig. 5 shows the relation between the out-of-eclipse brightness and the eclipse depth. Studies on how these variations are correlated in eclipsing CVs have been presented by Groot et al. (1998) and Walker (1963). These studies found that during quiescence the out-of-eclipse light is linearly correlated with the eclipse depth, but also that deviations are observed from this correlation, namely the 'Walker branch' (as these excursions were first noted by Walker 1963) and the 'Shallow branch' (naming after Fig. 2 of Groot et al. 1998). The explanation for the linear correlation between the out-of-eclipse light and the eclipse depth is simple geometry, where the same part of the disk is eclipsed during every orbital cycle, assuming the radial temperature gradient of the disk remains constant. If the accretion disk brightens by say one magnitude (but does not increase in size), then the eclipse depth will also increase by the same amount. In principle, during each cycle, the secondary star may totally eclipse the accretion disk, and the correlation line has been named the 'Line of Totality'. However a total eclipse of the accretion disk is not a necessity to produce a linear correlation between the out-of-eclipse light and the eclipse depth: all that is required is that the same fraction of the disk be eclipsed during each orbital cycle, and we thus rename this line the 'Line of maximal eclipse'. We have performed a similar analysis as that presented by Groot et al. (1998). For each orbital cycle we measured the out-of-eclipse light and eclipse depth as discussed in section 4, and show the out-of-eclipse light as a function of eclipse depth for KIS J1927 in Fig. 5. The 'Line of maximal eclipse' has been derived from fitting a linear relation to observations in quiescence (BJD > 2456226 days) by keeping the gradient fixed to unity and only allowing the intercept to vary, as expected from the 'Line of maximal eclipse' (see top-left inset of Fig. 5). To test whether the assumption of unity gradient is consistent with the data we performed a two-parameter fit, allowing both the gradient and intercept to vary. We assigned counting (Poisson) errors to the out-ofeclipse flux axis, whilst we summed in quadrature those for the eclipse depth, taking into account the covariance term as the two parameters are not independent. We recover a gradient of 1 . 07 ± 0 . 14, consistent with unity. We thus fix the gradient to unity and obtain an out-of-eclipse flux intercept is 299 . 1 ± 2 . 7 electrons/second, which, in the case of a total eclipse, would represent the brightness of the secondary star of Kp = 18 . 9. The line represents the maximal fractional eclipse depth allowed by the system, which occurs during quiescence when the largest relative disk fraction is eclipsed. The outburst rise is marked with blue squares, whilst the decline with red circles, and the thick line joins consecutive orbital cycles. Several apparent 'branches' are also present in Fig. 5, however these seem to be correlated with the aliasing pattern, causing the eclipse depth to be systematically shallower at regular intervals. However, from Fig. 5 it can be seen how the observed tracks immediately deviate from the 'Line of maximal eclipse' at the outburst onset. During the outburst evolution, a decrease in the radial temperature gradient (say from T ≈ R 0 to T ≈ R -3 / 4 ) is expected. This will cause observations to lie above the 'Line of maximal eclipse' in the case where the inner-most edges of the disk are being eclipsed during each cycle. The only way to then explain our observations is by increasing the radial size of the disk during the outburst. If the inner-most edges of the disk are not being eclipsed, then an increase in the radial temperature gradient could explain our observations. However we note that CV accretion disks are theoretically expected to expand in radius, and the temperature gradient is theoretically expected to decrease, during outbursts (Frank et al. 2002; Anderson 1988). Furthermore, many CVs display this expansion in their observational properties (e.g. U Gem: Smak 1984 and Z Cha: O'Donoghue 1986).", "pages": [ 4 ] }, { "title": "6 CONCLUSION", "content": "Wehave reported on long cadence Kepler observations of the eclipsing CV KIS J1927. The system has an orbital period of 3.97 hours, and displayed a ≈ 2 . 5 magnitude, ≈ 10 daylong, outburst during the 97.6 day-long observation. The quiescent folded lightcurve displays a significant contribution from the hot spot, which is swamped during the outburst, potentially by the bright accretion disk. We have also reported on evidence for accretion disk radius changes during the outburst of this system by studying the out-of-eclipse light levels versus the eclipse depth during the outburst. We find that the eclipse depth and out-of-eclipse light levels are linearly correlated in quiescence, and that this correlation is offset during outburst. This result is suggesting that the accretion disk is increasing in radius and/or temperature during the outburst. Similar implications on the other eclipsing CV in the Kepler field (V447 Lyr, Ramsay et al. 2012) have also been deduced, as well as previous work using ground based observations (Groot et al. 1998; Rutten et al. 1992). The main drawback from our analysis has been the lightcurve cadence, which coarsely samples the eclipses. Short-cadence Kepler observations of this object, with a sampling timescale of 58.5 seconds, would potentially allow to model the lightcurve via eclipse mapping, and to separate out each individual contribution of the lightcurve for every orbital cycle (accretion disk, hot spot, white dwarf and secondary star). Furthermore, eclipse mapping on short cadence data will allow us to locate the physical origin of flickering in CVs (Baptista & Bortoletto 2004; Scaringi et al. 2012b) with unprecedented precision. Additionally, eclipse mapping of this system using short cadence data will potentially allow us to track the disk evolution during the outburst, and to infer both radius as well as temperature changes during the rise and fall of the outburst. If additional observations by Kepler become viable after investigation of further 2- or 3- reaction wheel operations, KIS J1927 is a valuable potential target for short cadence observations.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "This research has made use of NASA's Astrophysics Data System Bibliographic Services. S.S. acknowledges funding from the FWO Pegasus Marie Curie Fellowship program. Additionally, S.S. acknowledges the use of the astronomy & astrophysics package for Matlab (Ofek in prep.).", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Anderson N., 1988, ApJ, 325, 266 Baptista R., Bortoletto A., 2004, AJ, 128, 411 Barclay T., Still M., Jenkins J. M., Howell S. B., Roetten- bacher R. M., 2012, MNRAS, 422, 1219 Borucki W. J., Koch D., Basri G., Batalha N., Brown T., Caldwell D., et al., 2010, Science, 327, 977 Cannizzo J. K., Smale A. P., Wood M. A., Still M. D., Howell S. B., 2012, ApJ, 747, 117 Cannizzo J. K., Still M. D., Howell S. B., et al., 2010, ApJ, 725, 1393 Feline W. J., Dhillon V. S., Marsh T. R., Brinkworth C. S., 2004, MNRAS, 355, 1 Horne K., 1985, MNRAS, 213, 129 Kato T., Osaki Y., 2013, 1305.5636 Knigge C., Baraffe I., Patterson J., 2011, ApJS, 194, 28 O'Donoghue D., 1986, MNRAS, 220, 23P Osaki Y., Kato T., 2012, 1212.1516 -, 2013, 1305.5877 Ramsay G., Cannizzo J. K., Howell S. B., Wood M. A., Still M., Barclay T., Smale A., 2012, MNRAS, 425, 1479 Ritter H., Kolb U., 2003, A&A, 404, 301 Smak J., 1984, Acta Astron., 34, 93 Warner B., 2003, Cataclysmic Variable Stars. Cataclysmic Variable Stars, by Brian Warner, pp. 592. ISBN 052154209X. Cambridge, UK: Cambridge University Press, September 2003.", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "Using ∼ 1,700,000 target- and sky-fiber spectra from the Sloan Digital Sky Survey (SDSS), we have carried out a systematic search for Galactic planetary nebulae (PNe) via detections of the [O iii ] λλ 4959, 5007 lines. Thanks to the excellent sensitivity of the SDSS spectroscopic surveys, this is by far the deepest search for PNe ever taken, reaching a surface brightness of the [O iii ] λ 5007 line down to about 29.0 magnitude arcsec -2 . The search leads to the recovery of 13 previously known PNe in the Northern and Southern Galactic Caps. In total, 44 new planetary nebula (PN) candidates are identified, including 7 candidates of multiple detections and 37 candidates of single detection. The 7 candidates of multiple detections are all extremely large (between 21 ' and 154 ' ) and faint, located mostly in the low Galactic latitude region and with a kinematics similar to disk stars. After checking their images in H α and other bands, three of them are probably HII regions, one is probably associated with a new supernova remnant, another one is possibly a true PN, and the remaining two could be either PNe or supernova remnants. Based on sky positions and kinematics, 7 candidates of single detection probably belong to the halo population. If confirmed, they will increase the number of known PNe in the Galactic halo significantly. All the newly identified PN candidates are very faint, with a surface brightness of the [O iii ] λ 5007 line between 27.0 - 30.0 magnitude arcsec -2 , very challenging to be discovered with previously employed techniques (e.g. slitless spectroscopy, narrow-band imaging), and thus may greatly increase the number of 'missing' faint PNe. Our results demonstrate the power of large scale fiber spectroscopy in hunting for ultra-faint PNe and other types of emission line nebulae. Combining the large spectral databases provided by the SDSS and other on-going projects (e.g. the LAMOST Galactic surveys), it is possible to build a statistically meaningful sample of ultra-faint, large, evolved PNe, thus improving the census of Galactic PNe. Key words: ( ISM: ) planetary nebulae: general - techniques: spectroscopic", "pages": [ 1 ] }, { "title": "H. B. Yuan 1 glyph[star] † & X. W. Liu 1 , 2", "content": "1 Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He Yuan Road 5, Hai Dian District, Beijing 100871, China 2 Department of Astronomy, Peking University, Yi He Yuan Road 5, Hai Dian District, Beijing 100871, China Received:", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Planetary nebulae (PNe) represent the last stages of evolution for low- and intermediate-mass stars of masses less than about 8 solar masses. The PN phase begins when the central contracting white dwarf reaches an effective temperature of above 30,000 K and starts to ionize the gaseous envelope ejected during the previous Asymptotic Giant Branch (AGB) phase. The PN phase is short, lasting for just tens of thousands years maximally. PNe enrich the interstellar medium (ISM) with dust grains, helium, carbon, nitrogen, oxygen and other products of nucleosynthesis, and are vital probes of the stellar nucleosynthesis processes, abundance gradients and chemical evolution of galaxies. In addition, PNe play a key role in studying the physics and time-scales of mass-loss and stellar evolution for low- and intermediatemass stars (Iben 1995). PNe may also be progenitors of Type Ia supernovae that return large amounts of iron to the ISM. New PNe have been continually discovered via various surveys (Parker et al. 2012). The Strasbourg-ESO Catalog of Galactic Planetary Nebulae (SECGPN) compiled by Acker et al. (1994, 1996) contains ∼ 1,500 true, probable or possible PNe. Kohoutek (2001) lists ∼ 1,500 objects classified as Galactic PNe known up to the end of 1999, as well as a number of possible pre-PNe and post-PNe. Using the the Anglo-Australian Observatory UK Schmidt Telescope (AAO/UKST) SuperCOSMOS H α Survey (SHS; Parker et al. 2005), Parker et al. (2006) and Miszalski et al. (2008) present ∼ 1,200 newly discovered true, possible or likely Galactic PNe in the Macquarie/AAO/Strasbourg H-Alpha Planetary Nebula Catalogue (MASH) and its supplement (MASH-II). From the Isaac Newton Telescope (INT) Photometric H α Survey (IPHAS; Drew et al. 2005), Viironen et al. (2009a, b) discover several PN candidates. Jacoby et al. (2010) describe a technique to search for additional unknown PNe via visually inspecting the existing data collections of the digital sky surveys (DSS) such as the POSS-I and POSSII surveys, and find tens of new PNe. Up to now, there are in total less than 2,850 true or probable PNe known in the Milky Way (Miszalski et al. 2012). However, the number is still a small fraction of the total predicted by any PN population model. For example, the stellar population synthesis models of Moe & De Marco (2006) predict a total number of 46,000 ± 13,000 PNe with radii glyph[lessorequalslant] 0.9 pc in the Milky Way if all low- and intermediate-mass stars of 1 - 8 solar mass will go through a PN phase. The number drops to ∼ 6,600 if all PNe have to form via the channel of close binaries (through the common envelope phase) (De Marco & Moe 2005). Observationally, based on a solar neighborhood ( glyph[lessorequalslant] 2 kpc) sample of about 200 PNe, Frew (2008) estimates a total Galactic population of 24,000 ± 4,000 PNe of radii glyph[lessorequalslant] 1.5 pc, or 13,000 ± 2,000 PNe of radii glyph[lessorequalslant] 0.9 pc. A variety of multi-waveband observations ranging from the radio up to the X-ray have been utilized to detect PNe. Essentially, all the efforts have been based on the technique of wide field, interference filter or objective prism slitless spectroscopic imaging surveys, primarily searching for emission in the light of the [O iii ] λ 5007 line and/or H α . However, it is a very challenging task to to find all PNe in the Galaxy directly. Firstly, since most PNe are distributed in the Galactic plane where dust extinction is high, a large fraction of them are unobservable at optical wavelengths where PNe are most luminous. Secondly, when only imaging data are available, it is difficult to differentiate PNe from other diffuse emission line nebulae such as HII regions, supernova remnants, nova shells, symbiotic nebulae and distant emission line galaxies. The Sloan Digital Sky Survey (SDSS; York et al. 2000) provides uniform and contiguous imaging photometry for about one-third of the sky in the u, g, r, i and z bands, as well as over two million high quality low resolution optical spectra of stars, galaxies and quasars. The latest SDSS Data Release 9 (DR9; Ahn et al. 2012) delivers spectra for about 1.46 million galaxies, 0.23 million quasars and 0.67 million stars. The SDSS observations mainly target high Galactic latitude regions of the Northern and Southern Galactic Caps. If by chance a PN (or an extended emission line nebula of other types) falls in the sightline of some SDSS targets (including sky background targets by sky fibers), lines emitted by the nebula, such as the [O iii ] λλ 4959, 5007, typically the strongest for a medium to high excitation nebula, will appear in the spectra of those targets. Then by systematically searching and measuring the [O iii ] λλ 4959, 5007 emission lines of Galactic nebular origins in those millions of SDSS spectra, one may expect to discover and study previously unknown PNe or other types of extended emission line nebula in the Galaxy. This spectroscopic approach has the fol- ng advantages: a) Compared to previously commonly used techniques, such as narrow-band imaging and slitless objective prism imaging spectroscopy, the sky background is much reduced in slit or fiber spectroscopy. Thus the latter is much more sensitive to faint nebular emission and capable of detecting large (evolved and/or nearby) PNe of very low surface brightness; b) With spectral information available over a wide wavelength coverage, it is much easier to differentiate different types of emission line nebulae; and c) With a contiguous and uniform coverage over a huge sky area such as that provided by the SDSS, it is possible to construct a statistically meaningful sample to improve the census of Galactic PNe. In this paper, we present results of a systematic search for PNe in the SDSS spectroscopic database. The paper is organized as following. In Section 2, we introduce the data and method used to search for PNe. The results are presented in Section 3 and discussed in Section 4. A brief summary then follows in Section 5.", "pages": [ 1, 2 ] }, { "title": "2.1 Data", "content": "We used the SDSS DR7 (Abazajian et al. 2009) spectroscopic database in the current work. The release contains over 1.6 million low-resolution ( R ∼ 1 , 800) spectra targeting approximately 930,000 galaxies, 120,000 quasars, 460,000 stars and 97,000 blank sky positions. Most targets are within a large contiguous area over 7,350 deg 2 in the Northern Galactic Cap, with the remaining ones from a number of stripes in the Southern Galactic Cap and a few stripes across the Galactic plane targeted by the program of the Sloan Extension for Galactic Understanding and Exploration (SEGUE; Yanny et al. 2009). Spectroscopic observations of the SDSS are usually undertaken in non-photometric conditions when the imaging camera is not in use. At least three fifteen-minute exposures are taken until the cumulative mean signal-to-noise ratio (SNR) per pixel exceeds 4 for a fiducial fiber magnitude of g = 20.2 and i = 19.9. For faint SEGUE plates, a total exposure time of about 1.5 hours is required. A total number of 640 spectra are collected simultaneously covering 3,800 9,200 ˚ A, at a spectral resolution of ∼ 1,800. The large aperture size of the SDSS telescope (2.5 meter) and the long exposures ( glyph[greaterorequalslant] 45 minutes) make the SDSS spectra extremely sensitive to narrow nebular emission lines. Note that each SDSS fiber samples a sky area of angular diameter of 3 '' and the SDSS wavelength scale is based on vacuum wavelengths.", "pages": [ 2 ] }, { "title": "2.2 Method", "content": "For each spectrum from the SDSS DR7 spectroscopic database, we have performed Gaussian fitting to the [O iii ] λλ 4959, 5007 lines around their rest wavelengths. The two lines are fitted independently. To reduce the degrees of freedom of fit, we have adopted a fixed line width of FWHM 2.82 ˚ A and assumed a flat continuum in the vicinity of both lines. When a positive signal is detected, the SNR of the detection is then computed by dividing the peak value of the fitted Gaussian by the standard deviation of a segment spectrum of a wavelength span of 40 ˚ A around the rest wavelength of the line. Note that the SNRs thus calculated are lower limits of the true values. In cases of very strong signals, the SNRs would be greatly under-estimated. To minimize spurious detections, we require that: a) SNRs of the [O iii ] λ 5007 line should be higher than 3; b) The difference of radial velocities deduced from the two [O iii ] λλ 4959, 5007 lines must be smaller than 60 km s -1 ; c) The intensity ratio of the two lines F 5007 / F 4959 must fall between 2 and 4, bracketing the intrinsic ratio of 2.98 (Mathis & Liu 1999; Storey & Zeippen 2000). The [O iii ] λ 5007 line emission from Galactic PNe may be contaminated by emission lines (e.g. the [O ii ] λλ 3726, 3729 lines, the Balmer lines and the [O iii ] λλ 4959, 5007 lines) of target galaxies and quasars. Such cases are also carefully avoided by visual check of the SDSS images and spectra to remove nearby galaxies whose [O iii ] emission lines are regarded as Galactic emission and distant galaxies and quasars that have an emission line shifted to the wavelength of the [O iii ] λ 5007 line. Faint background emission line galaxies at a given redshift, such as [O ii ] emission line galaxies at redshift about 0.35 and Ly α emitters at redshift about 3.1, are important contaminations for PNe surveys beyond the Milky Way (e.g. Gerhard 2006 and references therein). To further avoid such contaminations, we have excluded a few PN candidates whose [O iii ] λ 5007 line widths are larger than 4.23 ˚ A(1.5 times the expected line width). These measures have lessened the extragalactic contamination in our work. The radial velocities of the PN candidates (see Fig. 14 in Section 4) also suggest their Galactic origins, as extragalactic origins would result a uniform radial velocity distribution. In addition, we find a few cases that some spectral features of a normal galaxy at a given redshift, such as the 4661 ˚ A feature at redshift around 0.074 and the 4819 ˚ A feature at redshift about 0.039, can mimic the [O iii ] λ 5007 emission. Such cases are also excluded. Detection of the [O iii ] λλ 4959, 5007 lines is a good indicatation but not sufficient to identify a PN. The PN candidates found in this work may be contaminated by reflection nebulae, diffuse ionized gas, HII regions, supernova remnants, emission line stars and nova shells (Frew & Parker 2010). Imaging data, particularly narrow-band images, can provide important clues to classify the PN candidates. Therefore, both the SDSS spectra and available imaging data are used to help classify the nature of the PN candidates in this work.", "pages": [ 2, 3 ] }, { "title": "3 RESULTS", "content": "After applying the above criteria, a total number of 160 spectra with reliable detections of the Galactic [O iii ] λλ 4959, 5007 emission lines are selected. The spectral sequence number, SDSS spectral ID, the type, redshift and equatorial and Galactic coordinates of the target, the surface brightness and radial velocity of the detected Galactic [O iii ] λ 5007 line of each spectrum are listed in Tab. 1. Note the redshifts here indicate the SDSS targets and are given by the SDSS pipeline. While the radial velocities are associated the PNe or PN candidates and measured from the [O iii ] λ 5007 emission lines in this work. The SDSS spectra and colour-composite thumbnails are displayed in Figs. A1 and A2 in the Appendix, respectively. Based on the coordinates and radial velocities, they are divided into 58 groups, with each group representing a PN (candidate). After cross-correlating with existing PN catalogs including SECGPN, MASH, MASHII, Kohoutek et al. (2001) and the IPHAS PN catalog, the 58 groups are further divided into 3 categories: previously known PNe or other types of emission line nebulae recovered in this work (Spectral sequence number SEQ 1 - 37), PN candidates of multiple [O iii ] λ 5007 detections (SEQ 38 - 123) and PN candidates of single detection (SEQ 124 - 160), with each category having 14, 7 and 37 members, respectively. Their Galactic distribution is shown in Fig. 1. As listed in Tab. 1, the 160 SDSS spectroscopic targets include 59 galaxies (GAL), 28 blank skies (SKY), 16 blue horizontal-branch stars (BHB), 12 F-turnoff stars (FTO), 7 quasars (QSO), 9 white dwarfs (WD), 7 X-ray sources from the ROSAT All-Sky Survey (ROS; Voges et al. 1999), 7 G-dwarfs (GD), 3 hot stars (Hot), plus a few other types of object. Note that the target types are assigned by the SDSS. The fraction of spectra with detectable Galactic [O iii ] line emission are 0.007, 0.004, 0.006 and 0.016 per cent for target type GAL, QSO, GD and FTO, respectively. The fractions increase to 0.022, 0.025 and 0.022 per cent for target type ROS, BHB and SKY, and further to 0.087 and 0.12 for Hot and WD, respectively. Such increases are caused by the facts that it is easier to detect the [O iii ] λλ 4959, 5007 emission on spectra of blue stars or blank skies and some of the WDs and hot stars are exactly the central stars of the PNe.", "pages": [ 3 ] }, { "title": "3.1 Previously known PNe", "content": "Thirteen previously known PNe (IC 4593, NGC 6210, A 39, H4-1, K1-16, BE UMa, NGC3587, A28 JnEr1, HDW7, A31, EBG6 and LoTr5) are recovered in this work. They are all cataloged in the SECGPN catalog except for LoTr 5, which is from Kohoutek et al. (2001). Their standard PNG identifications, names, angular sizes and heliocentric radial velocities if available are listed in Tab. 1. Note that the faint low-excitation nebula PHL 932 in the SECGPN catalog, which in fact is a small HII region around a subdwarf B star (Frew et al. 2010), is also recovered. This group of PNe range widely in sizes, from compact ones of a few arcsec to very large ones of about 1,000 '' . The radial velocities measured in this work agree well with those in the literature. Four of the known PNe are only detected in one SDSS spectrum, by marginally detected signals only for 1 of them (corresponding to a [O iii ] λ 5007 line surface brightness S 5007 about 28.0 magnitude arcsec -2 ). The results demonstrate the reliability and sensitivity of the current method. SDSS true-color ( g, r, i ) images centered at the positions of the 13 PNe are displayed in Fig. 2. They are clearly visible on those images except for BEUMa (G144.8+65.8), A28 (G158.8+37.1), EBG6 (G221.5+46.3) and LoTr5 (G339.9+88.4). For the latter 4 PNe, the measured S 5007 values are mostly around 27.0 magnitude arcsec -2 . However, they are all visible in the SDSS g -band images after rebinning (10 '' per pixel). Representative spectra for each of the recovered PNe and the spatial distribution of SDSS fibers leading to their discovery are shown in the first 13 panels of Figs. 3, 4, respectively. Note that the halo PN H 4-1 was selected as a quasar candidate by the SDSS due to its very blue colors. Its 'broad' [O iii ] λ 5007 and H α emission lines are due to saturation. The outer haloes of PNe IC4593, NGC6210 and NGC3587 are also recovered, and fainter and larger than previous findings in the first two cases. IC 4593 has an bright core of 10 '' in diameter and an extended irregular outer halo of 130 '' × 120 '' (Corradi et al. 1997). Its faint halo is recovered in this work but at a larger distance of 108 '' and of S 5007 = 28.13 magnitude arcsec -2 . NGC6210 has a bright core of 12 '' × 8 '' in size, which is surrounded by a faint halolike structure with a diameter about 20 '' (e.g. Pottasch et al. 2009). In this work we find that NGC 6210 has a much larger and fainter halo extending to a few arcmin away and of S 5007 ∼ 27.0 magnitude arcsec -2 . NGC3587, also named as M97 or Owl Nebula, is a very bright spherical PN with a diameter about 2.8 ' in the SDSS image. But it also has a surrounding halo (e.g. Kwitter, Chu & Downes 1991; Hajian et al. 1997), which is well detected at 2.6 ' away in this work. Some known PNe and PN candidates in the SDSS footprint are missed in our search. For example, 6 PNe (G013.3+32.7, G061.9+41.3, G064.6+48.2, G208.5+33.2, G238.0+34.8 and G303.6+40.0) and 4 PN candidates (G221+45, G095+38, G275+72 and G315+59) from the SECGPN catalog and 4 objects (G144.8+65.8, G003.3+66.1, G085+52 and G052.7+50.7) from Kohoutek (2001) of Galactic latitude larger than 20 · are missed. We have investigated these cases. The 6 PNe are clearly visible in the SDSS images but missed simply because there are no SDSS spectroscopic targets surrounding them. For the remaining 8 objects, they are missed because there are no SDSS spectroscopic observations, the SNRs of the [O iii ] λ 5007 detections are too low or they are possibly false PNe. The most oxygen-deficient halo PN TS 01 (Stasi'nska et al. 2010) was observed by the SDSS. However, due to its unusually low oxygen abundance and consequently extremely weak [O iii ] lines relative to H β , we failed to detect its [O iii ] λ 4959 line and missed it in the search. The result suggests that our method can effectively find true PNe if there are SDSS fibers pointing to them, but may miss unusual halo PNe such as TS 01.", "pages": [ 3, 8 ] }, { "title": "3.2 PN candidates of multiple detections", "content": "Seven PN candidates of multiple (2 - 39) detections are identified based on the spatial positions and radial velocities. Each of them is assigned a PNG identification according to its median Galactic coordinates. The '?' in the PNG identifications indicates they are PN candidates which need confirmation. Their minimum sizes and average radial velocities are also estimated and given in Tab. 1. Their Galactic distribution is shown in Fig. 1. Six candidates are located in the low Galactic latitude region ( | b | glyph[lessorequalslant] 30 · ) and discovered by the SEGUE plates. All the candidates have extremely large sizes, extending from 21 ' to 154 ' . Since the typical values of S 5007 for these candidates are around 28.0 magnitude arcsec -2 , they are invisible on Fig. A2. Representative spectra of each newly discovered PN candidate and the spatial distribution of SDSS fibers leading to its discovery are shown in the last 7 panels of Figs. 3, 4, respectively. Based on Fig. 4 alone, we can not rule out the small possibility that the first 4 PN candidates, namely PN?G055.9 -3.9, PN? G070.8+10.4, PN? G108.9+10.7 and PN?G117.1 -26.3, might be multiple PNe lying closely in the field. These PN candidates may be contaminated by diffuse ionized gas, HII regions, supernova remnants and so on (Frew & Parker 2010). It's possible to distinguish PNe from HII regions and supernova remnants based on line flux ratios (e.g. Kniazev et al. 2008; Sabin et al. 2013). However, due to the faintness of the candidates and the H α and H β lines may be strongly affected by stellar absorptions, it's very difficult to classify the candidates based on line flux ratios. Each candidate has spectra that show very weak or non-detections of the H α , H β and [N ii ] λλ 6548, 6584 lines relative to the [O iii ] λλ 4959, 5007 lines, indicating that they are PN candidates. However, some candidates have spectra that show good detections of the [N ii ] λλ 6548, 6584 lines (e.g. G055.9 -3.9 and G070.8+10.4) and the [S ii ] λλ 6716, 6731 lines (G108.9+10.7, G126.8 -15.5 and G202.0+19.8), indicating that they could also be HII regions or supernova remnants.", "pages": [ 8 ] }, { "title": "3.3 Imaging analysis of PN candidates of multiple detections", "content": "To further investigate the nature of these PN candidates, We have checked their images from the Virginia Tech SpectralLine Survey (VTSS; Dennison, Simonetti & Topasna 1998), the Southern H-Alpha Sky Survey Atlas (SHASSA; Gaustad et al. 2001), the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) and the DSS. Both VTSS and SHASSA have modest angular resolution (96 '' per pixel for VTSS and 48 '' per pixel for SHASS) and are very deep (down to ∼ 1 Rayleigh at H α ), thus very suitable to examine the large and faint PN candidates in this work. To increase the SNR, the DSS and WISE images are rebinned to 10 '' per pixel. The results are shown in Figs. 5-10. Fig. 5 displays the VTSS continuum-corrected H α and WISE W 4 images of the PN candidate G055.9 -3.9. The circles indicate the positions where the [O iii ] λλ 4959, 5007 lines are detected. The candidate is invisible in the DSSII blue plate image. It's shown that PN? G055.9 -3.9 is not a PN but part of an HII region. Fig. 6 displays the VTSS continuum-corrected H α image of the PN candidate G070.8+10.4. This candidate is probably not a PN but an ionized H α filament associated with an HII region. PN? G055.9 -3.9 and PN?G070.8+10.4 are not PNe but HII regions, consistent with the fact that both candidates show good detections of the [N ii ] λλ 6548, 6584, the [S ii ] λλ 6716, 6731, and relatively strong H α and H β to the [O iii ] λλ 4959, 5007 lines. Fig. 7 displays the VTSS continuum-corrected H α , WISE W 4 and DSS-II blue plate images of the PN candidate G108.9+10.7. In the bottomleft corner of the VTSS image, there is probably a new supernova remnant (SNR? G107.1+9.0) based on its spherical morphology and filamentary structure. PN? G108.9+10.7 is probably not a PN but associated with this supernova remnant, consistent with the facts that it shows relatively strong [S ii ] λλ 6716, 6731 emission lines and it's discovered in the spectra of ROS targets. Fig. 8 displays the VTSS continuumcorrected H α and DSS-II blue plate images of the PN candidate G117.1 -26.3. The detected [O iii ] emission is probably from H α filaments of the diffuse ISM, as seen in the DSS-II image. Therefore, this candidate is not a PN. Fig. 9 shows the VTSS continuum-corrected H α , WISE W 4 and DSS-II blue plate images of the PN candidate G126.8 -15.5. Based on the VTSS and WISE images, this candidate is probably a true PN. However, it also shows strong [S ii ] λλ 6716, 6731 emission lines and could be a new supernova remnant. For PN candidate G202.0+19.8, the VTSS image is not available. It's invisible in either the WISE W 4 or DSS-II blue plate images. Although it is covered by the Wisconsin H-Alpha Mapper Survey (WHAM; Haffner et al. 2003), the spatial resolution of WHAM data is too low (about 1 · ). Its arc-like morphology and the detections of strong [S ii ] λλ 6716, 6731 emission lines suggest it's a supernova remnant. But, it could also be a normal PN that has experienced an interaction with the ISM. Deep narrow-band imaging and spectroscopic observations are needed to confirm the nature of PN? G126.8 -15.5 and PN?G202.0+19.8. Fig. 10 shows the SHASSA continuum-corrected H α , WISE W 4 and DSS-II blue plate images of the PN candidate G247.7+47.8. The candidate is seen in all the three images and possibly a true PN, consistent with the fact that all its spectra show very weak or non-detections of the H α , H β and [N ii ] λλ 6548, 6584 lines. In summary, PN?G055.9 -3.9, PN? G070.8+10.4 and PN?G117.1 -26.3 are probably not PNe but HII regions, PN?G108.9+10.7 is probably associated with a new supernova remnant, PN? G126.8 -15.5 and PN?G202.0+19.8 could be either PNe or supernova remnants, and PN?247.7+47.8. is a possible PN.", "pages": [ 8, 9 ] }, { "title": "3.4 PN candidates of single detection", "content": "Thirty-seven PN candidates of single detection are found. Compared to the PN candidates of multiple detections, most PN candidates of single detection are distributed in the Northern and Southern Galactic Caps, as shown in Fig. 1. Therefore, the possibilities of contamination by HII regions and supernova remnants are relatively low. Some of them have large radial velocities, suggesting that they are probably halo PNe (see Section 4.3). As the PN candidates of multiple detections, the candidates of single detection have S 5007 around 28.0 magnitude arcsec -2 , thus invisible on Fig. A2 either. Eleven candidates have either SHASS (SEQ 136, 144, 148, 149, 152, 153, 160) or VTSS (SEQ 126, 129, 133, 159) H α images, as shown in Fig. 11. Based on the images, 3 candidates (SEQ 126, 129, 159) are probably HII regions, 4 candidates (SEQ 144, 148, 152, 160) seem to be around very diffuse HII re- 2 gions, 1 candidate (SEQ 133) may be associated with the supernova remnant candidate SNR? G107.1+9.0, 1 candidate (SEQ 153) may be a true PN, and 2 candidates (SEQ 136, 149) show nil H α emission. The results indicate that a significant fraction of PN candidates may be (diffuse) HII regions. We also checked the rebinned (about 10 '' per pixel) SDSS g -band images of the candidates. One candidate (SEQ 127) is clearly visible, as shown in Fig. 12, suggesting that it is probably a spherical PN of a radius of about 3.0 ' . PG1204+543, a hot subdwarf O star (Green et al. 1986), is probably its ionizing star.", "pages": [ 9, 11, 14 ] }, { "title": "4.1 Most highly evolved PNe", "content": "The evolved, large PNe play a critical role in studying the transition from PN to white dwarf (e.g. Napiwotzki 1995), the PN-ISM interaction on a range of spatial scales (Tweedy & Kwitter 1994) and calibrating the distances of the diverse population of local PNe (Ciardullo et al. 1999; Frew & Parker 2006; Frew 2008), therefore meriting detailed study. However, such PNe are inherently of low surface brightness and difficult to detect, especially in the Galactic plane where interstellar extinction is large. As mentioned earlier, the method in this work is mostly sensitive to large and faint PNe, thus very suitable to find most highly evolved PNe. We have found 7 PN candidates of multiple detections in this work. Based on their spectra and images in H α and other bands, three of them are probably HII regions, one is probably associated with a new supernova remnant, another one is probably a true PN, and the remaining two could be either PNe or supernova remnants. They all exhibit extremely low surface brightness and large sizes, suggesting that they are highly evolved if they are true PNe. Acker et al. (2012) reported the discovery of a possible PN candidate (Ou4) of the largest angular extent ever found then that extends about 72 ' . We have found some PN candidates that are of similar sizes or even larger. Note that the surface brightness of Ou4 is highest in the [O iii ] λ 5007 emission line. It takes a pre-PN 32,600 years to expand to a radius of 0.5 pc at a typical expansion velocity of 30 km/s. Such a PN has an angular extent of 1 · at a distance of 57 pc. Thus, we infer that these PN candidates are very old and local ( glyph[lessorequalslant] ∼ 50 pc) if their PN nature are confirmed. They have radial velocities consistent with disk population (as to be shown in Fig. 14), suggesting that they are descendants of local disk stars.", "pages": [ 14, 15, 16 ] }, { "title": "4.2 A population of faint PNe", "content": "multiply-detected and singly-detected PN (candidate) samples, respectively. Note the S 5007 of the halo PN H 4-1 is 18.14, out of the x-range of this figure. There is not a continuum of S 5007 from known Halo PNe (e.g. H 4-1) to the strongly clustered faint candidates found in this study. It is probably because that the technique in this work is very biased to large, evolved and faint PNe. The recovery of H 41 in this work is lucky, because it is observed as a quasar candidate. Given its small size about 10 '' and the sampling density of SDSS fibers about 100 per sqr.deg., the proba- ty of having such a small PN observed by the SDSS by chance is very tiny, about 0.1 per cent. Thanks to the extremely high sensitivity of the SDSS spectra in detecting narrow and strong [O iii ] λλ 4959, 5007 lines from Galactic PNe, we reach PNe of S 5007 as faint as 29.0 - 30.0 magnitude arcsec -2 , much fainter than most previously known PNe. Note that there are a few measurements for the previously known PNe reaching down to S 5007 ∼ 28.0 magnitude arcsec -2 . But these measurements are for their fainter outer haloes that are firstly discovered in this work. For an extended source of uniform surface brightness, its surface brightness doesn't depend on its distance if interstellar extinction is not taken into account. Thus, very faint PNe mean that they are very old, large and highly evolved or they are intrinsically fainter than others. Deep imaging and spectroscopic observations are needed to explore the possibilities. All the newly identified PN candidates are very faint, very challenging to be discovered with previously employed techniques (e.g. slitless spectroscopy, narrow-band imaging), and thus may greatly increase the number of 'missing' faint PNe.", "pages": [ 16, 17 ] }, { "title": "4.3 Halo PNe", "content": "Halo PNe are descendants of stars formed in the early history of the Galaxy. They are important tracers to study the evolution of metal-poor stars and the early physical and chemical conditions of the Galaxy. Halo PNe are mainly characterized by their large height above the Galactic plane, peculiar velocity compared to the Galactic rotation curve of the disk stars and their low metallicity. Currently, very few halo PNe have been identified. There are only 14 PNe from the SECGPN catalog regarded as halo members based on their location and kinematics, including H 4-1 recovered in this work. The SDSS legacy survey concentrates on the Northern Galactic Cap, thus is a very suitable database to search for halo PNe. To identify possible halo PNe, we plot radial velocities of the SDSS stars and PNe (candidates) as a function of Galactic longitude in Fig. 14. The stars and PNe (candidates) are marked by dots and crosses, respectively. The black dots indicate 30,000 randomly selected metal-rich disk stars of [Fe/H] glyph[greaterorequalslant] -0 . 5 and the cyan ones indicate 5,000 randomly selected metal-poor halo stars of [Fe/H] glyph[lessorequalslant] -1 . 5. The stars have radial velocity errors smaller than 4.0 km s -1 . Here the stellar parameters and their errors are from the SEGUE Stellar Parameter Pipeline (SSPP, Lee et al. 2008a, b; Allende Prieto et al. 2008; Lee et al. 2011; Smolinski et al. 2011). The disk and halo stars are clearly separated in the figure. According to their kinematics, the PNe (candidates) are divided into disk population and halo population, as indicated by red and blue crosses, respectively. In total, 8 halo PNe (candidates) are found and marked in Tab. 1, including", "pages": [ 17, 18 ] }, { "title": "4.4 Total number of Galactic PNe", "content": "To estimate the total number of Galactic PNe, a widely used method is based on the identification of a complete sample of PNe within a local volume and then extrapolating that PN density (usually relative to either mass or luminosity) to the entire Milky Way (e.g. Ishida & Weinberger 1987; Phillips 2002; Frew 2008). Such method requires knowing distances to the local sample. However, accurate PN distances are very difficult to obtain, resulting uncertainties in the estimated total number of Galactic PNe of a factor of 2 - 10. Consequently, this method yields total counts that have a wide spread in values - from 13,000 (Frew 2008) to 140,000 (Ishida & Weinberger 1987). With sophisticated modeling of Galactic PN population (e.g. their luminosity and size distributions) and the sampling effects of the SDSS spectroscopic surveys, it is possible to obtain a reliable estimate of the total number of PNe in the Galaxy without knowing distances of PNe. We leave such an exploration to a future paper. Compared to the SDSS DR7, the SDSS DR9 has increased the number of spectra significantly thanks to the projects SEGUE-II and Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013). In addition, the other on-going and up-coming large scale spectral survey projects such as LAMOST (Cui et al. 2012; Zhao et al. 2012; Liu et al. 2013) and HERMES (Freeman 2010) will provide supplementary data-sets for finding PNe (and other types of emission line nebulae) and improving their Galactic census. The search limits can be further increased by using the template subtraction technique, which has been used to detect the diffuse interstellar bands in the SDSS and LAMOST stellar spectra (Yuan & Liu 2012; Yuan et al. 2013). The results of searching for new PNe in the SDSS DR9 and LAMOST datasets will be presented in another work.", "pages": [ 18, 19 ] }, { "title": "5 SUMMARY", "content": "We have carried out a systematic search for Galactic PNe by detecting the [O iii ] λλ 4959, 5007 lines in ∼ 1,700,000 spectra from the SDSS DR7. Thanks to the excellent sensitivity of the SDSS spectroscopic surveys, this is by far the deepest search for PNe ever taken, reaching a surface brightness of the [O iii ] λ 5007 line S 5007 down to about 29.0 magnitude arcsec -2 . A number of interesting results are found: and NGC3587 are also recovered, and much larger and fainter than previous findings in the first two cases. Acknowledgments We would like to thank the referee for his/her valuable comments, which helped improve the quality of the paper significantly. This work made use of the SDSS and SIMBAD databases. This research made use of Montage, funded by the National Aeronautics and Space Administration's Earth Science Technology Office, Computational Technnologies Project, under Cooperative Agreement Number NCC5-626 between NASA and the California Institute of Technology. The code is maintained by the NASA/IPAC Infrared Science Archive. This research made use of the Virginia Tech Spectral-Line Survey (VTSS) and the Southern H-Alpha Sky Survey Atlas (SHASSA), which are supported by the National Science Foundation. This work is supported by the Natural Science Foundation of China (No. 10933001)", "pages": [ 19, 20 ] }, { "title": "REFERENCES", "content": "Abazajian, K. N., Adelman-McCarthy, J. K., Agueros, M. A., et al. 2009, ApJS, 182, 543 in press Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, APPENDIX A: A COLLECTION OF THE SDSS SPECTRA AND IMAGES OF THE TARGETS IN TAB.1", "pages": [ 20, 21 ] } ]

[ { "title": "ABSTRACT", "content": "We combine two Gaussianization techniques - Wavelet Non-Linear Wiener Filter (WNLWF) and density reconstruction - to quantify the recovery of Fisher information that is lost in the gravitational collapse. We compute a displacement fields, in analogy with the Zel'dovich approximation, and apply a Wavelet Non-Linear Wiener Filter that decomposes the reconstructed density fields into a Gaussian and a non-Gaussian component. From a series of 200 realizations of N -body simulations, we compute the recovery performance for density fields obtained with both dark matter particles and haloes. We find that the height of the Fisher information trans-linear plateau is increased by more than an order of magnitude at k > 1 . 0 h Mpc -1 for particles, whereas either technique alone offers an individual recovery boost of only a factor of three to five. We conclude that these two techniques work in a symbiosis, as their combined performance is stronger than the sum of their individual contribution. When applied to the halo catalogues, we find that the reconstruction has only a weak effect on the recovery of Fisher Information, while the non-linear wavelet filter boosts the information by about a factor of five. We also observe that non-Gaussian Poisson noise saturates the Fisher information, and that shot noise subtracted measurements exhibit a milder information recovery. Key words: cosmology: theory-dark matter-large scale structure of universemethods: statistical", "pages": [ 1 ] }, { "title": "Optimizing the Recovery of Fisher Information in the Dark Matter Power Spectrum", "content": "Joachim Harnois-D'eraps /star 1 , 2 , Hao-Ran Yu 3 , 1 , Tong-Jie Zhang 3 , 4 and Ue-Li Pen 1", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Understanding the nature of dark energy has been identified internationally as one of the main goal of modern cosmology (Albrecht et al. 2006), and many dedicated experiments attempt to constrain its equation of state: LSST 1 (LSST Science Collaborations et al. 2009), EUCLID 2 (Beaulieu et al. 2010), JDEM 3 (Gehrels 2010), CHIME 4 (Peterson et al. 2006), SKA 5 (Schilizzi 2007; Dewdney et al. 2009), BOSS 6 (Schlegel et al. 2009) and Pan-STARRS 7 . One of the favored technique involves a detection of the Baryonic Acoustic Oscillations (BAO) sig- nal (Seo & Eisenstein 2003, 2005; Eisenstein et al. 2006; Seo & Eisenstein 2007), which has successfully constrained the dark energy parameter in current galaxy surveys (Eisenstein et al. 2005; Tegmark et al. 2006; Percival et al. 2007; Blake et al. 2011). The analyses are based on a detection of the BAO wiggles in the matter power spectrum, which act as a standard ruler and allow one to map the cosmic expansion. With the new and upcoming generation of dark energy experiments, the precision at which we will be able to measure the cosmological parameters is expected to drop at the sub-percent level, therefore it is essential to understand and suppress every sources of systematic uncertainty. In a BAO analysis, one of the main challenge is to extract an optimal and unbiased observed power spectrum, along with its uncertainty; the latter propagates directly on the dark energy parameters with Fisher matrices (Fisher 1935; Tegmark et al. 1997). This task is difficult for a number of reasons. yses sit at the transition between the linear and the nonlinear regime, at least for the redshift at which current galaxy surveys are sensitive, hence the underlying uncertainty on the matter power spectrum is affected by the non-linear dynamics. These effectively couples the phases of different Fourier modes (Zhang et al. 2003) and the Gaussian description of the density fields has been observed to fail (Meiksin & White 1999; Rimes & Hamilton 2005, 2006; Neyrinck et al. 2006; Neyrinck & Szapudi 2007). For an estimate of the BAO dilation scale to be robust, one must therefore include in the analysis the full non-linear covariance of the power spectrum. Although results from Takahashi et al. (2011) seem to suggest that non-Gaussianities had no real effect on the final results, it was recently shown that this was only true if the original power spectrum was measured in an unbiased and optimal way, which is rarely the case (Ngan et al. 2011). Otherwise, the discrepancy on the constraining power is at the percent level. One of the way to reduce the impact of the non-linear dynamics is to transform the observed field into something that is more linear. Over the last few years, many 'Gaussianization' techniques have been developed, which all attempt to undo the phase coupling between Fourier modes. The number of degrees of freedom - i.e. uncoupled phases - can be simply quantified by the Fisher information, and recovering parts of this erased information can lead to improvements by factors of a few on cosmological parameters. For example, a density reconstruction algorithm (Eisenstein et al. 2007; Noh et al. 2009; Padmanabhan et al. 2009), based on the Zel'dovich approximation, has been shown to reduce by a factor of two the constraints on the BAO dilation scale (Eisenstein et al. 2007; Ngan et al. 2011). This technique was applied on the SDSS data recently (Padmanabhan et al. 2012) to improve the BAO detection, with small modifications to the algorithm such as to correct for the survey selection function and redshift space distortions. As discussed therein, an important issue is that two main mechanism are reducing our ability to measure the BAO ring accurately: 1) a large coherent ∼ 50 Mpc infall of the galaxies on to overdensities, which tends to widen the BAO peak, and 2) local non-linear effects, including non-linear noise, which also erase the smallest BAO wiggles. Reconstruction addresses the first of these mechanisms, and its is important to know whether something can be done about the second, after reconstruction has been applied. Wavelet Non-linear Wiener Filters (hereafter WNLWF, or just wavelet filter) were used to decompose dark matter density fields (Zhang et al. 2011) and weak gravitational lensing κ -fields (Yu et al. 2012) into Gaussian and nonGaussian parts, such as to condense in the latter most of the collapsed structure; the Gaussian part was then shown to contain several times more Fisher information than the original field. Other methods include log-normal transforms (Seo et al. 2011), Cox-Box (Joachimi et al. 2011), running N -body simulation backwards (Goldberg & Spergel 2000), or direct Gaussianization of the one-point probability function (Yu et al. 2011), just to name a few. This technique seems perfectly suited to address the issue of non-Gaussian noise described above. Our focus, in this paper, is to discuss how two of these techniques can be used in conjunction to maximize the re- covery of Fisher information. Not all combinations of Gaussianization techniques are winning, however. It was recently shown (Yu et al. 2012) that WNLWF and log-transforms are not combining in an advantageous way. On one hand, if the log-transform is applied onto a Gaussianized field, the prior on the density field is no longer valid, and the logtransform maps the density into something even less Gaussian. On the other hand, it was shown that the log-transform is less effective than WNLWF alone at recovering Fisher information, at least on small scales. Applying the filter after the log-transform does not improve the situation, since the Gaussian/non-Gaussian decomposition is less effective. In other words, the Fisher information that the log-transform could not extract is not recovered by WNLWF, and we are better off with the WNLWF alone. It seems, however, that this unfortunate interaction is not a constant across all combinations. In this paper, we discuss how non-linear Wiener filters, constructed in Wavelet space, can improve the results of a density reconstruction algorithm, which takes the density back in time using linear perturbation theory. Our first result is that these two techniques work well together, in the sense that the final Fisher information recovery is larger than the two techniques stand alone. We first obtain these results with particle catalogues extracted from N -body simulations, and extend our techniques to halo catalogues, which provide a sampling of the underlying matter field that is much closer to actual observations. The structure of the paper is as follows: in Section 2, we briefly review the theoretical background of the density reconstruction and WNLWF, and review how we extract the density power spectra, their covariance matrices, and the Fisher information. We discuss our results in Section 3 and conclude in Section 4.", "pages": [ 1, 2 ] }, { "title": "2.1 Numerical Simulations", "content": "Our sample of 200 N -body simulations are generated with CUBEP3M , an enhanced version of PMFAST (Merz et al. 2005) that solves Poisson equation with sub-grid resolution, thanks to the p 3 m calculation. Each run evolves 512 3 particles on a 1024 3 grid, and is computed on a single IBM node of the Tightly Coupled System on SciNet (Loken et al. 2010) with Ω M = 0 . 279, Ω Λ = 0 . 721, σ 8 = 0 . 815, n s = 0 . 96 and h = 0 . 701. We assumed a flat universe, and started the simulations at z i = 50. Each simulation has a side of 322 . 36 h -1 Mpc, and we output the particle catalogue at z = 0 . 054. We search for haloes with a spherical over-density algorithm (Cole & Lacey 1996) executed at run time, which sorts the local grid density maxima in descending order of peak height, then loops over the cells surrounding the peak center and accumulates the mass until the integrated density drops under the collapse threshold of 178, and finally empties the contributing grid cells before continuing with the next candidate, ensuring that each particle contributes to a single halo. Halo candidates must consist of at least one hundred particles, ensuring the haloes are large and collapsed objects. The center-of-mass of each halo is calculated and used as position, as opposed to its peak location, even though both quantities differ by a small amount. We mention here that algorithms of this kind have the unfortunate consequence to create an exclusion region around each halo candidate, thus effectively reducing the resolution at which the halo distributions are reliable. Each field contains about 88 , 000 haloes, for a density of 2 . 6 × 10 -3 h 3 Mpc -3 . For comparison, this is about eight times larger than the density of the BOSS density (Schlegel et al. 2009).", "pages": [ 2, 3 ] }, { "title": "2.2 Density reconstruction algorithm", "content": "We use a density reconstruction algorithm that is based on the linear theory prediction, first found by Zel'dovich (Zel'Dovich 1970), that couples the density field δ ( q , t 0 ) to the displacement field s ( q ) via In the above expression, q is the grid, or Lagrangian, coordinate, and the displacement field is obtained in Fourier space as where F ( k ) = exp[ -( kR ) 2 / 2] is a smoothing function suppressing features smaller than R = 10 h -1 Mpc. Particles at Eulerian coordinate x are displaced from their grid positions following with D ( t ) the linear growth factor. These calculations are commonly used for the generation of initial conditions in N -body simulations, and are accurate as long as the smallest scales probed are still on linear regime at the starting redshift. In the case where the particles - or haloes - to be displaced are not at t 0 , one must subtract from the result the displacement field from the grid location (see Noh et al. (2009) for a detailed explanation of this technique).", "pages": [ 3 ] }, { "title": "2.3 Wavelet non-linear Wiener filter (WNLWF)", "content": "In this subsection we briefly review the WNLWF algorithm, and direct the reader to Zhang et al. (2011); Yu et al. (2012) for more details. We consider in this paper the Daubechies-4 (Daubechies 1992) discrete wavelet transform (DWT), which contains certain families of scaling functions φ and difference functions (or wavelet functions) ψ . The density fields are expanded into combinations of these orthogonal bases, and weighted by scaling function coefficients (SFCs) and wavelet function coefficients (WFCs). In our WNLWF algorithm, we deal with only WFCs, each of which characterizes the amplitude of the perturbation on a certain wavelength and at a certain locations. In the three dimensional case, the properties of each perturbation depend on three scale indices ( j 1 , j 2 , j 3 ) - controlling the scales of the wavelet Daubechies-4 functions - and three location indexes ( l 1 , l 2 , l 3 ) - controlling their translations. Specifically, on a given dimension, the grid scale corresponding to a specified dilation is L/ 2 j ( L = 1024 in our case), and the spatial location is determined by lL/ 2 j < x < ( l +1) / 2 j . After the wavelet transform, all SFCs and WFCs are stored in a 3-dimensional field, preserving the grid resolution (see Fang & Thews (1998); Press et al. (1992) for more details). Our non-linear Wiener filter (NLWF) construction strategy relies on the fact that in wavelet space, the nonGaussianities are clearly characterized in the probability distribution function (PDF) of the WFCs ˜ /epsilon1 j 1 ,j 2 ,j 3 ; l 1 ,l 2 ,l 3 . We thus construct our filter by splitting the wavelet transform of the original density, which we label D , into a Gaussian ( G ) and a non-Gaussian ( N ) contribution. Since wavelet transforms are linear operations, this Gaussian/non-Gaussian decomposition happens also in real space when we wavelet transform back the contributions. Namely, we can write in wavelet space and real space respectively, where the original density ( d ) is expressed as the sum over a Gaussian contribution ( d G ) and a non-Gaussianized contribution ( d NG ). Our goal is thus to design a filter that concentrates most of the collapsed structure in d NG , and thus produces d G that are closer to linear theory. The NLWF acts on individual wavelet modes , which are defined as combinations (not permutations) of all WFCs having the same three scale indices ( j 1 , j 2 , j 3 ). For each wavelet mode, the NLWF is determined completely by the PDF f ( x ) of the corresponding WFCs, which is constructed by looping over the other three indices ( l 1 , l 2 , l 3 ). We then fit this PDF with the analytical function presented in Yu et al. (2012) (equation (15)): and extract the two parameters α and s . These are actually dependent on the second and the fourth central moment of the PDF f ( x ) m 2 and m 4 , hence we measure the moments first, then extract α and s via: ∣ ∣ We then loop back over all spatial indices ( l 1 , l 2 , l 3 ) of this wavelet mode and decompose each WFC into two components: which are functions of x only. Note that the final filter function depends only on s , which corresponds to the full width at half maximum of the Gaussian NLWF function w G . It characterizes the extent of the departure from a Gaussian PDF: the greater the s , the smaller departure is from Gaussian statistics. In comparison, α parameterizes the central deviation of the PDF: σ central = α -1 2 . The same decomposition is performed on the reconstructed density fields and on those obtained from the halo catalogues. In this paper, we do not make use of the information contained in the non-Gaussian component and simply discard it, although it serves as a powerful probe of small scale structures and could help identifying haloes in a (Gaussian) noisy environment.", "pages": [ 3 ] }, { "title": "2.4 Information recovery", "content": "The calculation of uncertainty about dark energy cosmological parameters is based on a propagation of the uncertainty about the matter power spectrum. In this process, the number of degrees of freedom - i.e. the Fisher information contained in the field is directly related to the constraining power. In this section, we review how the Fisher information about the amplitude of the matter power spectrum is calculated from simulated dark matter particles and haloes. The power spectrum P ( k ) of a density contrast δ ( x ) is calculated in a standard way: where the angle brackets refer to an average over our 200 different realizations and over the solid angle. The uncertainty about the power spectrum is estimated from a covariance matrix C is defined as where N is the number of realizations and 〈 P ( k ) 〉 is the mean angular power spectrum over all realizations. The crosscorrelation coefficient matrix is somehow more convenient to plot since it has higher contrasts, and is defined as: The diagonal is normalized to one, and each element represents the degree of correlation between the scales ( k, k ' ). In the Gaussian approximation, the density is completely characterized by the power spectrum. Namely, Consequently, ρ G is identical to the identity matrix. As expected from the theory of structure formation, the nonlinear collapse of the matter density field tends to couple the Fourier modes, which are otherwise independent, starting from the smallest scales and progressing towards larger scales with time. This coupling is responsible for highly correlated region in ρ , and can be understood in terms of higherorder corrections, including the bispectrum, the trispectrum, etc. The Fisher information measures the number of independent Fourier modes in a density field up to a resolution scale k max . We see from equation(12) that dividing the covariance by the (square of the) power spectrum is proportional to the number of independent measurements N ( k ). Therefore, the normalized covariance is defined as: Then, the number of degrees of freedom up to a scale k max is otained by inverting the corresponding sub-sample of the normalized matrix, then summing over all the elements: The inversion of the covariance matrix involved in the calculation of the Fisher information amplifies the noise, hence such measurements typically requires a very strong convergence on the forward matrix. This can otherwise lead to biases of a few percent on derived quantities like the BAO dilation scale (Ngan et al. 2011). Generally, a forward matrix that is closer to diagonal contains more Fisher information; the theoretical maximum corresponds to the Gaussian case, where all modes are independent. Any non-vanishing off-diagonal element reduces the information.", "pages": [ 4 ] }, { "title": "3 RESULTS", "content": "In this section, we describe and quantify our ability at recovering Fisher information with our two Gaussianization techniques, for density fields measured with simulated particles and haloes. We recall that the later is much closer to actual observations, since galaxies trace highly collapsed structures.", "pages": [ 4 ] }, { "title": "3.1 Density fields", "content": "To illustrate the effect of different Gaussianization techniques, we present in Fig. 1 the projections through a thickness of 50 cells of a given realization after density reconstruction alone, after WNLWF alone, and with both techniques applied. We observe that the reconstruction reduces the size of each halo, as expected from this algorithm: particles attempt to travel out of the gravitational potential. WNLWF has a slightly different visual effect on the density: it removes most of the smallest structure perturbations, leaving behind the larger ones. As discussed in Pen (1999); Zhang et al. (2011), the geometry of the Cartesian wavelet leaves behind a grid patterns, which only affects the smallest scales of the power spectrum and has no impact on the scales we are interested in. The combination of both techniques is presented in the middle right panel, and visually presents the least amount of collapsed structures. We also show the non-Gaussian part of the wavelet filter with and without reconstruction in the bottom panels. The largest peaks and sharpest structures are indeed filtered out. Fig. 2 shows the power spectrum of the dark matter particles and haloes before and after the Gaussianization techniques. We first observe that the measurement form the original particle field agrees at the few percent level with the non-linear predictions obtained from CAMB (Lewis et al. 2000) up to k ∼ 2 . 0 h Mpc -1 , which sets the resolution scale of our power spectrum measurements. As expected from WNLWF, the Gaussian component of WNLWF preserves the power on linear scales - up to k ∼ 0 . 1 h Mpc -1 - while signals from trans-linear and non-linear scales are mostly transferred to the non-Gaussian contribution of the WNLWF decomposition, which explains the drop in power. At the same time, we note that the power spectrum after WNLWF actually traces quite well the linear predictions, to within a factor of two, at all scales. The density reconstruction algorithm also has a significant impact on the shape of the power spectrum, as particles are pumped out of the gravitational potential. As a result, small scale power is directly transferred to larger scales, as seen in the figure. Interestingly, the turning point between these two effects is also locate close to k ∼ 0 . 1 h Mpc -1 . This common feature to both Gaussianization technique is Mpc] -1 Distance [h Mpc] -1 Distance [h 300 250 200 150 100 50 0 300 250 200 150 100 50 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 explained by the fact that both attempt to pump back information from scales in the trans-linear regime, whose lowerk limit indeed corresponds to this mode. When looking at the halo measurements, we observe that the original and reconstructed power spectra are dominated by shot noise at scales smaller than k ∼ 1 . 0 h Mpc -1 . This noise, however, is strongly suppressed by the wavelet filter. In practice, a common way to deal with Poisson noise is to compute the cross power spectrum between two populations randomly selected out of the original catalogue. The shot noise is eliminated in this operation, and the signals left behind are stronger on large and intermediate scales. Small scales are typically anti-correlated due to the 'halo exclusion' effect, which is a result of our halo-finder that collapses all the structure to a single point, and leaves the surrounding region empty. Although precise and robust, this procedure is hard to apply to all the cases under study in this paper, since the density reconstruction algorithm looses efficiency if we under-resolve the small scale structures. Instead, we use another common approach which consists in subtracting from the measured power spectrum an estimate of the shot noise, defined as P shot = V 3 /N halos . To illustrate this, we show in Fig. 3 a comparison between halo power spectrum from the full catalogue, the cross spectrum, and the shot-noise subtracted power. We observe that the two shot noise subtraction techniques agree up to k = 0 . 3 h Mpc -1 , beyond which the P ( k ) -P shot ( k ) approach looses power in comparison; by k = 1 . 0 h Mpc -1 , it is smaller by a factor of 2.4. This difference between the two shot noise subtraction techniques means that our approach is not optimal, and that the results on the smallest scales are not as robust as one would wish.", "pages": [ 4, 5, 6 ] }, { "title": "3.2 Covariance matrices", "content": "The two Gaussianization techniques that are discussed in this paper both attempt to bring cosmological information, or degrees of freedom, back to the power spectrum. Consequently, the covariance matrices of the Gaussianized fields will be more diagonal. The top left panel of Fig. 4 shows the cross-correlation coefficient matrix of the original particle fields in the upper triangle, and the wavelet filtered ones on the lower triangle. To ease the comparison between the figures, we show in the main figure of each panel the positive components only, and present in the insets the negative entries. There is a mild anti-correlation (less than 10 per cent) in some matrix elements of the original fields, which comes from residual noise in the largest scale. This is a mild effect that has very little impact, hence we do not attempt to correct for it. The top right panel shows, on the upper triangle, the results after a density reconstruction has been applied, then, on the lower triangle, the measurements after both technique have been executed. The off-diagonal elements of the covariance matrix are reduced by 20-30 per cent by both Gaussianization techniques. We see that those two techniques combine well and reduce to a minimal value the correlation between the modes. The bottom panels of Fig. 4 show the same measurements, when carried on halo fields. The wavelet filter produces a band of negative elements, correlating the k > 1 . 0 h Mpc -1 - Poisson dominated - with all scales. This anti-correlation does not carry any physics about the signal in it, hence these scales should be left out or carefully interpreted in future analyses. Most off-diagonal elements are about 30 per cent less correlated than in the unfiltered matrix, showing that wavelet filtering is also very efficient on halo fields. We also observe that the density reconstruction algorithm has very little impact on the correlation of the halo measurements. This is caused by the fact that haloes are non-overlapping by construction, hence the region of exclusion prevents an accurate construction of the gravitational potential 8 . This is a strong limit of the technique, as real galaxy data behave much more like halos than particles. However, higher multiplicity in the galaxy population of many haloes is likely to improve the construction of the gravitational potential, hence we can expect a modest gain. When we combined both technique, most of the diagonalization comes from wavelet filter. This can be seen visually by comparing the lower triangles of both bottom panels.", "pages": [ 6 ] }, { "title": "3.3 Fisher information", "content": "When we extract the Fisher information from the covariance matrices presented above, we expect the original particle fields to exhibit the global shape first measured in Rimes & Hamilton (2005). Namely, the information should follow the Gaussian predictions on large scales, then reach 1 a trans-linear plateau where the gain is very mild as one increases the resolution of the survey, then hit a second rise on scales smaller than about 1 . 0 h Mpc -1 . We first see from Fig. 5 that we are able to recover those results, plus those of Ngan et al. (2011), which showed that the density reconstruction algorithm can raise the height of the trans-linear plateau by a factor of a few. We also recover the results from Zhang et al. (2011) and obtain a similar gain with the wavelet non-linear Wiener filtering technique. As mentioned in the introduction, it was shown by Yu et al. (2012) that different Gaussianization techniques do not always combine well. In the current case, however, we observe that on all scales, the Fisher information from the combined techniques are larger than the sum of the two separate contributions. For k > 0 . 6 h Mpc -1 , notably, we are able to extract more than ten times the Fisher information of the original particles fields, whereas individual techniques offer a recovery of about a factor of four. This symbiosis effect grows larger as one goes to smaller scales. When considering the halo fields, we observe in Fig. 6 that the density reconstruction technique, taken alone, has little impact on the recovery of information, due to a poor modeling of the gravitational potential. In contrast, wavelet filtering recovers five times more information by the time we have reached k = 1 . 0 h Mpc -1 , before shot noise subtraction. The Poisson noise is a non-Gaussian effect, which also saturates the Fisher information. A hard limit one can think of is the following: the number of degrees of freedom can not exceed the number of objects in our fields of view. We therefore plot the (non-Gaussian) Poisson noise limit as a flat line corresponding to the halo number density, and observe that the original halo Fisher information approach but never exceed that limit. Wavelet fields, however, reduces the Poisson noise significantly, hence allows the information to reach higher values. We also see that shot noise subtracted Fisher information curves show a lower information recovery, which means that the number density needs to be high enough in order to maximize the recovery. As mentioned in section 2.1, the halo density is about eight times larger than current spectroscopic surveys. Next generation experiments and current photometric redshift surveys have a much larger number counts, hence the corresponding Poisson noise limit will be much higher.", "pages": [ 6, 7, 8 ] }, { "title": "4 DISCUSSION AND CONCLUSION", "content": "This paper explores the recovery of Fisher information with the combined use of two Gaussianization techniques, and show that wavelet non-linear Wiener filtering and density reconstruction can extract an order of magnitude more information than the original fields. We also reproduce the calculations on halo catalogues and find that 1) the density reconstruction has only a mild impact on its own, due to a poor modelling of the gravitational potential, 2) wavelet filter recovers about five times more information by k = 1 . 0 h Mpc -1 , and 3) the combined techniques recovers about three times more Fisher information than in the original fields by k > 0 . 7 h Mpc -1 , even after shot noise subtraction. Interestingly, we find that in both matter tracers, the recovery of the combination of the two Gaussianization techniques is larger than the sum of the individual contributions. k [ h/ Mpc]", "pages": [ 8, 9 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "This work was supported by the National Science Foundation of China (Grants No. 11173006), the Ministry of Science and Technology National Basic Science program (project 973) under grant No. 2012CB821804, and the Fundamental Research Funds for the Central Universities. Computations were performed on the TCS supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. UP and JHD would like to acknowledge NSERC for their financial support.", "pages": [ 9 ] }, { "title": "References", "content": "Albrecht A., Bernstein G., Cahn R., Freedman W. 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[ { "title": "ABSTRACT", "content": "Utilizing the observational data of 55,318 star-forming galaxies (SFGs) selected from the catalog of MPA-JHU emission-line measurements for the SDSS DR8, we investigate the galaxy downsizing effect of their O and N enrichments, and the nitrogen production mechanism in them. We show the redshift evolution of O and N abundances and specific star formation rates for different galaxy mass ranges, demonstrating the galaxy downsizing effect caused by less massive progenitors of less massive galaxies. The O and N abundances do not remain constant for different galaxy mass ranges, and the enrichment (and hence star formation) decreases with increasing galaxy stellar mass. We find evidence of the O enrichment for galaxies with stellar masses M ∗ > 10 11 . 0 (in units of M /circledot ), i.e. ∆(log(O / H)) ∼ 0 . 10 and ∆(log(N / H)) ∼ 0 . 28 from redshift 0.023 to 0.30. Based on the evolutionary schematic model of N/O ratios in Coziol et al., who proposed the scheme that the production of nitrogen is the consequence of a sequence of bursts in SFGs, we conclude that the nitrogen production is dominated by the intermediate-mass stars, which dominate the secondary synthesis in SFGs. However, for galaxies with M ∗ > 10 10 . 35 we find evidence of enhanced N/O abundance ratios, which are significantly above the secondary synthesis line. This suggests that outflows of massive stars, which deplete oxygen efficiently, are more important in massive galaxies. Finally we find an excellent linear relation between M ∗ and log(N/O), indicating that the N/O abundance ratio is a good indicator of the stellar mass in a SFG and may be used as a standard candle for studying cosmology, if confirmed with further studies. Subject headings: galaxies: abundances - galaxies: starburst - galaxies: statistics", "pages": [ 1 ] }, { "title": "Evolution of oxygen and nitrogen abundances and nitrogen production mechanism in massive star-forming galaxies", "content": "Yu-Zhong Wu 1 and Shuang-Nan Zhang 1 , 2", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The metallicity of a galaxy is a crucial parameter for understanding its formation and evolution. The element abundances of the interstellar medium (ISM) are obtained by tracers of the chemical compositions of stars and gas within a galaxy. Optical emission lines from H ii regions have long been regarded as the principal tools of gas-phase chemical diagnostics in galaxies (Aller 1942; Peimbert 1975; Pagel 1986; de Robertis 1987; Liang et al. 2006). Since estimating metallicities needs theoretical models, empirical calibrations or a combination of both (Kewley & Dopita 2002; Kewley & Ellison 2008), we have different methods to obtain metallicities. Assuming a classical H ii region model, the ratio of the auroral line [O iii ] λ 4363 to a lower excitation line such as [O iii ] λ 5007, can be used to determine the electron temperature of the gas, which is then converted into the metallicity of the gas (Osterbrock 1989). The method of using the observed line ratios to infer directly the electron temperatures and to estimate metallicities in galaxies is known as 'direct T e method' (Pagel et al. 1992; Skillman & Kennicutt 1993). This method is generally regarded as the most accurate abundance measurement for estimating metallicities in galaxies. It, however, has two disadvantages. In most instances, [O iii ] λ 4363 line is too weak to be detected. It is well known that in metal-rich galaxies, the electron temperature decreases (as the cooling is via metal lines) and the auroral lines eventually become too faint to be measured, when the metallicity increases (Yin et al. 2007). In addition, in low-metallicity galaxies, the oxygen abundance is usually underestimated by the [O iii ] λ 4363 diagnostic in low-metallicity galaxies (Kobulnicky et al. 1999). Due to the above reasons, photoionization models are used instead for estimating abundances of high metallicity star-forming galaxies (SFGs). The most wide and common method used is the R23 method proposed by Pagel et al. (1979) and Alloin et al. (1979); the oxygen indicator R 23 =([O ii ] λλ 3727 , 3729+ [O iii ] λλ 4959 , 5007) / H β suggested by Pagel et al. (1979) is widely accepted and used. Moreover, the relation between the line intensities of strong oxygen lines and the oxygen abundance has been calibrated by photoionization models (e.g., Edmunds & Pagel 1984; McCall et al. 1985; Dopita & Evans 1986; Kobulnicky et al. 1999; Kewley & Dopita 2002). However, it has one problem that the relationship between R 23 and 12 + log(O / H) is double valued, which shows the transition between the upper metal-rich branch and the lower metal-poor branch occurring near 12+log(O / H) ∼ 8 . 4 (Liang et al. 2006). With the releases of catalogues of several large spectral surveys, especially the Sloan Digital Sky Survey (SDSS) that has released a large number of spectral data, the number of good-quality spectra of emission-line galaxies has increased dramatically (York et al. 2000). These open a new era of utilizing the large survey spectra to study the evolution of O and N abundances in galaxies. Using the line flux measurements of SDSS spectra, Thuan et al. (2010) not only found the evolution of O and N abundances in galaxies with different stellar masses, but also found evidence for galaxy downsizing that metal enrichment shifts from higher-mass galaxies at early cosmic times to lower-mass ones at later epochs (Cowie et al. 1996). In the last decade, the evolution of the mass-metallicity relation of galaxies with redshift has been investigated by several groups (Lilly et al. 2003; Savaglio et al. 2005; Erb et al. 2006; Cowie & Barger 2008; Maiolino et al. 2008; Lamareille et al. 2009; Lara-L'opez et al. 2009). In these studies, they used different methods to obtain O abundances and found the O abundance change of SFGs, with ∆(log(O / H)) ∼ 0 . 3 or lower. In addition, Thuan et al. (2010) have paid attention to the redshift evolution of N abundances in galaxies and have shown two advantages for studying the chemical evolution of galaxies. Firstly, at 12+(log(O / H)) /greaterorsimilar 8 . 3, nitrogen abundance change with redshift has a larger amplitude than that of oxygen. Then, compared with oxygen production, the nitrogen production has a time delay (Maeder 1992; van den Hoek & Groenewegen 1997; Pagel 1997), and it can give some limits for the chemical evolution of galaxies. Using a large sample of galaxies in the Great Observatories Origins Deep Survey-North (GOODS-N), Cowie & Barger (2008) found that star formation ceases in most massive galaxies (with stellar masses M ∗ > 10 11 in units of M /circledot ) at z < 1 . 5. Thuan et al. (2010) also found ∆(log(O / H)) = 0 for massive galaxies with M ∗ > 10 11 . Although the evolution of O and N abundances of galaxies with redshift has been widely studied, we still cannot fully understand the downsizing effect (i.e., some disputes for the origin of the downsizing; Poggianti et al. 2004; Bundy et al. 2006). Moreover, some studies seem to show no signs of the evolution of O abundance with redshift in most massive galaxies ( M ∗ > 10 11 ). Therefore, we utilize the MPA-JHU DR8 release of spectral measurements to investigate these issues. In this work, we first present the galaxy downsizing effect; we then show evidence of the O enrichment for galaxies with M ∗ > 10 11 . 0 . On the basis of the evolutionary schematic model of N/O ratios, we can clarify the nitrogen production mechanisms in SFGs. For the line fluxes, we use the following notations throughout this paper: R 2 = I [O II] λ 3727+ λ 3729 /I H β , R 3 = I [O III] λ 4959+ λ 5007 /I H β , N 2 = I [N II] λ 6548+ λ 6584 /I H β , R 23 = R 2 + R 3 .", "pages": [ 2, 3 ] }, { "title": "2. THE SAMPLE and DATA", "content": "The catalog of MPA-JHU emission-line measurements for the SDSS Data Release 8 (DR8) is chosen as our primary sample. The measurements are available for 1,843,200 different spectra. Compared to the previous DR4 release used for a similar study (Wu, Zhao, & Meng 2011), this represents a significant extension in size and a number of improvements in the data. The MPA-JHU DR8 release of spectral measurements provides many parameters, such as the redshift, galaxy stellar mass, and various emission line measurements. These data are available online at the following address: http://www.sdss3.org/dr8/spectro/galspec.php. We initially select 71,888 objects from the above sample with the criterion of S/N > 5 for the redshift, H β λ 4861, H α λ 6563, [O ii ] λ 3727, [O ii ] λ 3729, [N ii ] λ 6584, [O iii ] λ 5007, [O i ] λ 6300, [S ii ] λ 6717, and [S ii ] λ 6731 lines. In addition, we have excluded these galaxies for their FLAG keywords of zero and -9999 . 0 in SFR and sSFR. Due to that 3800 ˚ A -9300 ˚ A is the wavelength range of the SDSS spectra, the range of redshifts in our nearby galaxies can be determined. If the [O ii ] λλ 3727 , 3729 emission lines are not out of the observed range, then the lower limit of redshifts in our nearby galaxies is z ≈ 0 . 023. As shown in Fig. 1, a tiny fraction of the objects show anomalously large [O ii ] λ 3727/[O ii ] λ 3729 ratios; we thus exclude 30 objects with [O ii ] λ 3727/[O ii ] λ 3729 > 10. If the [S ii ] λλ 6717 , 6731 emission lines are not out of the observed range, then the upper limit of redshifts in our nearby galaxies is z ≈ 0 . 33 (Pilyugin & Thuan 2011). The SFG sample presented in Fig. 2 is based on the BPT diagram (Baldwin et al. 1981; Veilleux & Osterbrock 1987), and the separation line between AGNs and SFGs proposed by Kauffmann et al. (2003a) is the criterion of the SFG sample. We select all the objects which reside in the lower left corner of the Kauffmann et al. (2003a) line as our SFG sample, and we exclude the objects with redshift z > 0 . 30. Therefore, we obtain the galaxy sample with 55,318 objects. In addition to the redshift of each galaxy from the release of the MPA-JHU DR8 catalog, the release also provides the stellar masses of all galaxies based on the fits to the photometry following the philosophy of Kauffmann et al. (2003a) and Salim et al. (2007); the stellar masses are FITS binary tables with keys MEDIAN. The galaxy stellar masses come all from the MPA-JHU DR8 catalog in this paper. In addition, the specific star formation rate (sSFR), taken as the median of the PDF of the sSFR for each SFG, is also added for completeness.", "pages": [ 5 ] }, { "title": "3. Metallicity calibrations", "content": "In this section, we estimate the oxygen and nitrogen abundances utilizing both emissionline fluxes and recent calibrations. The oxygen abundances of the sample galaxies are estimated using the R 23 method (Pilyugin et al. 2006; Pilyugin et al. 2010; Zahid et al. 2012), We adopt the calibration of Tremonti et al. (2004), where x is log R 23 . The log(N/O) abundance ratios of these SFGs are estimated by using the algorithm given by Thurston et al. (1996), where t II is the [N ii ] temperature (in units of 10 4 K), and x is log R 23 . Based on a five-level atom calculation, Pagel et al. (1992) given a convenient formula: Here, we take N 2 = 1 . 33 [N ii ] λ 6584 instead of the standard N 2 =[N ii ] λ 6548+[N ii ] λ 6584, because the measurements of the [N ii ] λ 6548 line are less reliable than those of the [N ii ] λ 6584 line (Thuan et al. 2010). To derive N/O, we make the following assumption, With regard to the accuracy of this assumption, some discussions were given by Vila-Costas & Edmunds (1993), and Thurston et al. (1996) suggested that this assumption is fairly accurate. The above calibration and procedure of obtaining the O abundance and the N/O abundance ratio together for SFGs have been widely used in literature, e.g., Liang et al. (2006), Mallery et al. (2007), and Lara-L'opez et al. 2009. The total N abundance can then be obtained from", "pages": [ 6 ] }, { "title": "4. RESULTS AND DISCUSSION", "content": "In this section, we firstly present the galaxy downsizing effect, i.e., enrichment ceases in higher-mass galaxies at earlier times and shifts to lower-mass galaxies at later epochs (Pilyugin & Thuan 2011); we show that both enrichment rates (i.e. slopes in Figs 3 -6) and sSFRs (in Fig. 7) decrease between redshift 0.3 and 0.023, as well as that abundances at z ∼ 0 . 023 increase, but sSFRs decrease with the galaxy stellar mass growth. Then we demonstrate the O and N enrichments for M ∗ > 10 11 . 0 . Finally we discuss the relation of the nitrogen productions between intermediate-mass stars and massive stars using the model proposed by Coziol et al. (1999) in SFGs.", "pages": [ 7 ] }, { "title": "4.1. The Galaxy Downsizing Effect", "content": "In Fig. 3, we show the redshift evolution of O and N abundances for different galaxy mass ranges; their contours are presented in Fig. 4. In each panel, the red and black points represent O and N abundances of SFGs at different redshifts, respectively, and the best least-squares fits are shown by the solid lines. The upper panel shows that the N and O abundance variations with redshift in galaxies with M ∗ < 10 9 . 5 . This indicates that these galaxies have not reached a high astration level some 3 Gyr ago and are experiencing intense star formation in their evolution processes. This is consistent with the result of Pilyugin & Thuan (2011). From Fig. 3 and Table 1, we can see that the slope for O abundance increases from -1 . 08 to -0 . 33 when the stellar mass increases from M ∗ < 10 9 . 5 to M ∗ < 10 11 . 0 , as seen visually in Fig. 4. This indicates that the O enrichment (or star formation) decreases with the galaxy stellar mass growth (see Table 1). This demonstrates the galaxy downsizing effect, consistent with Pilyugin & Thuan (2011) who noted that O enrichment seems to decrease slowly with increasing galaxy stellar mass. In addition, the O and N abundances do not remain constant between different galaxy mass ranges, and O abundance at z ∼ 0 . 023 increases from 8.76 to 9.12 with increasing galaxy stellar mass. This is inconsistent with the result of Pilyugin & Thuan (2011), who noted that O abundances are always ∼ 8 . 5 for different stellar mass galaxies at z ∼ 0 . 023. Moreover, the galaxies with the same stellar mass range present significantly larger N enrichment than O enrichment (see Table 1), which confirms the result of Thuan et al. (2010). It is also noticeable that the difference between 12+log(O/H) and 12+log(N/H) decreases with the galaxy mass, indicating that the O enrichment lags behind the N enrichment as galaxies grow, which will be extensively discussed in section 4.4. In Fig. 5, we show the redshift evolution of N/O abundance ratios for different galaxy Note : Col.(1): Galaxy mass. Col.(2): the sample size of SFGs. Cols.(3)-(5), (6)-(8), and (9)(11): For each sample of the log(N/O), 12+log(N/H), and 12+log(O/H). 'a' are the values of log(N/O), 12+log(N/H), and 12+log(O/H) at z ∼ 0 . 023, respectively. 'b' are slopes (enrichment rates) of log(N/O), 12+log(N/H), and 12+log(O/H) between z ∼ 0 . 3 and z ∼ 0 . 023, respectively. 'c' are ∆(log(N / O)), ∆(12 + log(N / H)) , and ∆(12 + log(O / H)) between z ∼ 0 . 3 and z ∼ 0 . 023 , respectively. mass ranges; their contours are presented in Fig. 6. In the upper panel of Fig. 5, the sample size of galaxies with M ∗ < 10 9 . 5 is 17226, and the N/O abundance ratio at z ∼ 0 . 023 is -1 . 11. In the middle and lower panels of Fig. 5, the sample sizes of galaxies with M ∗ of 10 9 . 5 -10 10 . 5 and > 10 10 . 5 are 34100 and 3992, respectively, and the N and O abundances at z ∼ 0 . 023 are -0 . 74 and -0 . 36, respectively. These indicates that the N/O abundance ratio increases with increasing galaxy stellar mass. This is in good agreement with the relation between N/O and the stellar mass of P'erez-Montero & Contini (2009). Fig. 7 shows the redshift evolution of the specific star formation rate (sSFR) for different galaxy mass ranges. In each panel, the black points represent sSFRs of SFGs at different redshifts, and the best least-squares fits are shown by the solid (red) lines. From Fig. 7, we find that the slopes for sSFRs decrease from 6.39 to 4.39 and then to 3.58, and the intercepts for sSFRs at z ∼ 0 . 023 decrease from -9.70 to -10.00 and then to -10.11 with galaxy stellar mass from M ∗ < 10 9 . 5 to 10 9 . 5 -10 10 . 5 and then to > 10 10 . 5 . This is consistent with Fig. 3 of Pilyugin et al. (2013). Clearly in this low redshift range the sSFRs of less massive galaxies are on average always higher than that of more massive galaxies, explaining the galaxy downsizing effect shown in Fig. 3 or 4. In other words, less massive galaxies are less massive because of their less massive progenitors, despite of their higher sSFRs.", "pages": [ 7, 8, 11 ] }, { "title": "4.2. The O and N Enrichments for Galaxies with M ∗ > 10 11 . 0", "content": "In this section, we firstly present the evidence of the O and N enrichments for galaxies with M ∗ > 10 11 . 0 . Then we show the redshift evolution of N/O abundance ratios for galaxies with M ∗ > 10 11 . 0 . Finally, we discuss the O and N enrichments for these most massive galaxies. In Fig. 8, we show the O and N enrichments and the redshift evolution of N/O abundance ratios for galaxies with M ∗ > 10 11 . 0 . The sample size of galaxies with M ∗ > 10 11 . 0 is 284. The upper panel shows evidence of the N and O enrichments. The slopes are -0 . 33 and -1 . 01 for O and N abundances, respectively, and the N/O abundance ratios at z ∼ 0 . 023 are 9 . 10 and 8 . 77, respectively. The lower panel shows the redshift evolution of N/O abundance ratios. The slope is -0 . 67 for N/O abundance ratios. In Fig. 13 of Thuan et al. (2010), the O and N enrichments were shown for galaxies with M ∗ of 10 10 . 0 -10 10 . 3 ; however, the O and N enrichments were not seen for galaxies with M ∗ of 10 11 . 2 -10 11 . 5 . Employing data of the VIMOS VLT Deep Survey, Lamareille et al. (2009) obtained the mass-metallicity relation of SFGs and have found that the galaxies of 10 10 . 2 show a larger O enrichment than those of 10 9 . 4 . In addition, Lara-L'opez et al. (2009) have studied the O abundance of relatively massive (log( M ∗ ) /greaterorequalslant 10 . 5) SFGs from SDSS/DR5 at different redshift intervals from 0.4 to 0.04. They found an oxygen enrichment ∆(log(O / H)) ∼ 0 . 1 from redshift 0.4 to 0. From Fig. 9 and Table 1, we can see clearly ∆(log(O / H)) ∼ 0 . 10 and ∆(log(N / H)) ∼ 0 . 28 from redshift 0.023 to 0.30. Comparing with the above results, we may safely conclude that the O and N are enriched for galaxies with M ∗ > 10 11 . 0 .", "pages": [ 11, 13 ] }, { "title": "4.3. The N Production in Star Forming Galaxies", "content": "In this section, we first introduce a schematic demonstration of productions of primary and secondary nitrogen in a sequence of bursts (Coziol et al. 1999). Then we show that the nitrogen production of intermediate-mass stars is larger than that of massive stars in SFGs. Based on the evolutionary schematic model of N/O ratios in Garnett (1990), Coziol et al. (1999) proposed a scheme that the production of nitrogen is the consequence of a sequence of bursts in SFGs. This means that the N abundance and the metallicity increase in SFGs, because these galaxies experience a couple of star formation processes in several cycles. During a cycle, the 12 + log(O / H) increases due to the evolution of massive stars, while the N/O ratio decreases (Garnett 1990; Olofsson 1995; Coziol et al. 1999). After ∼ 0 . 4 Gyrs of massive star active onset, the intermediate-mass stars contribute most nitrogen, and the N/O ratio increases sharply, while the 12 + log(O / H) does not increase. Utilizing the SDSS DR7 data, Torres-Papaqui et al. (2012) found that the intensity of the bursts seems to result in the chemical differences between the nitrogen-poor and nitrogen-rich SFGs, which is well explained by the sequence of bursts model (Coziol et al. 1999). In addition, the SDSS DR8 data are selected as the SFG sample, and Wu & Zhang (in preparation) found that the metallicity increase on the BPT diagram can be explained by several star formation processes. These suggest that a sequence of bursts model may help to understand the chemical evolution of SFGs. In Fig. 9, we show the relations of log(N / O) versus 12 + log(O / H) and 12 + log(N / H), respectively. Generally, the N/O ratio will be constant for primary nucleosynthesis, while this ratio will be a linear correlation for secondary nucleosynthesis. The combination of both primary and secondary nucleosyntheses gives rise to a nonlinear relation (Mallery et al. 2007). During the first cycle, the evolution of massive stars contribute not only the 12 + log(O / H) increase but also the N/O ratio decrease in the left panel; in the meantime, this process also produces some nitrogen (the right panel in Fig. 9). With the evolution of these galaxies, the massive stars start dying off, and the intermediate-mass stars open large scale nitrogen production (the right panel in Fig. 9), but the oxygen production ceases (the left panel in Fig.9). In the right panel, it can be seen that the nitrogen production of the intermediate-mass stars is larger than that of the massive stars. Therefore, the nitrogen production is dominated by the intermediate-mass stars in SFGs during the first cycle. The second cycle begins with an increase in the 12+log(O / H) and a decreases in the N/O ratio, and this cycle will follow the first cycle regardless of that the star formation process has a lower or higher amplitude (intensity) than that of the first cycle. Actually, some models of successive bursts assume that the bursts will have decreasing intensities (Gerola, Seiden, & Schulman 1980; Krugel & Tutukov 1993; Marconi, Matteuci, & Tosi 1994; Koeppen, Theis, & Hensler 1995; Coziol et al. 1999). However, our result that the nitrogen production is dominated by the intermediate-mass stars in SFGs will not be changed, because the nitrogen production processes of intermediate-mass and massive stars are the same as those of the first cycle. Regarding relative high-metallicities (12 + log(O / H) /greaterorsimilar 8 . 3) objects, the N/O ratio increases significantly with the O abundance growth. The trend seems to originate from the nitrogen production with metallicity-dependence in both massive and intermediate-mass stars (e.g., Vila-Costas & Edmunds 1993; Pilyugin et al. 2003). Therefore, the nitrogen is generally a secondary element in the metallicity range (Vila-Costas & Edmunds 1993; Henry, Edmunds, & Koppen 2000; van Zee & Haynes 2006; L'opez-S'anchez & Esteban 2010). Since primary synthesis is commonly assumed (but not definitely) to confine in intermediate-mass stars, while secondary synthesis could occur in stars of all masses (Renini & Voli 1981; Vila-Costas & Edmunds 1993), our result that the nitrogen production is dominated by the intermediate-mass stars is reasonable. During the main sequence (MS), intermediate mass stars burn hydrogen through the CNO cycle. The conversion of almost all central C 12 into N 14 is the consequence of the CNO cycle (Smiljanic et al. 2006). In addition, intermediate-mass stars seem most likely to contribute, during the asymptotic giant branch, the CNO-processed material (Cannon et al. 1998). In term of initial mass function (IMF), equations 1 and 2 in Kroupa (2001) show that a mean stellar mass 〈 m 〉 =0.36 M /circledot for stars with 0.01 ≤ m ≤ 50 M /circledot , and 5.7 % 'intermediatemass (IM) stars' (1.0-8.0 M /circledot ) contribute 34 % mass, and 0.37 % 'O' stars ( > 8.0 M /circledot ) contribute 17 % mass (Kroupa 2001). Therefore intermediate-mass stars can contribute twice as much mass as massive star in SFGs. This is consistent intermediate-mass stars dominate the nitrogen production in SFGs.", "pages": [ 13, 14, 17 ] }, { "title": "4.4. Enhanced N/O ratio in massive galaxies", "content": "However, most of the galaxies in our sample are located significantly above the secondary synthesis line, which has been proposed to dominate the total synthesis of N/O. It is evident that the last arrows that marks the nitrogen production by the intermediate-mass stars in both panels point to the direction of the observed N/O enhancement. This means that no O abundance enhancement is needed for the following star-forming processes. This is inconsistent with Coziol et al. (1999), who suggested that the sum of the vectors of alternating oxygen (by massive stars) and nitrogen (by intermediate-mass stars) converges to the secondary synthesis line. Due to the existence of the outflow/winds, the separation in Fig. 9 may originate from the galactic wind driven by the starburst (Strickland, Ponman, & Stevens 1997; Moran, Lehnert, & Helfand 1999). The outflow will preferentially deplete the oxygen, and nitrogen will remain at its ISM value at the time of the starburst/wind (Lehnert & Heckman 1996; Heckman 2003; Torres-Papaqui et al. 2012). Therefore, the outflow effect may result in the separation of the metallicities of the galaxies from the secondary + primary chemical evolution model of Vila-Costas & Edmunds (1993). In order to understand the above discrepancy, in Figs 10 and 11 we show their contours and central values (with '+' sign) for different galaxy mass and redshift ranges. Each contour includes 68.3% of the total galaxies for this subsample; all subsamples have almost an equal size in each Figure. Fig. 10 shows clearly that larger mass galaxies show more significant deviations of N/O abundance ratio above the secondary synthesis line. This is in good agreement with the relation between N/O abundance ratio and O abundance with the stellar mass of Amor'ın, P'erez-Montero, & V'ılchez (2010). Fig. 11 shows that higher redshift galaxies have statistically larger N/O abundance ratios than lower redshift ones. Using a good indicator of the stellar age, Kauffmann et al. (2003b) displayed a correlation between the D n (4000) index and stellar mass; this relation indicates that massive galaxies seem to be older (Tremonti et al. 2004). This means that higher redshift and more massive galaxies have started their nitrogen enrichment earlier; this is another manifestation of the downsizing effect. Since the N abundance enrichment process is not accompanied with any significant oxygen enrichment, we conclude that the outflows of massive stars, which deplete oxygen efficiently, are more significant in massive galaxies. In Table 2, we list the sample size, mean and median values of M ∗ , log N O , 12 + log N H and 12 + log O H of these galaxies in each subsample shown in Figs 10 and 11; in each case, the mean and median values are approximately the same. In Fig 12, we show the relations between the median values of M ∗ and log N O , 12 + log N H and 12 + log O H ' respectively; a linear fit to each relation is also shown as the dashed line. Clearly a good linear relation exists between log M ∗ and log N O , i.e., suggesting that N/O abundance ratio is a statistically excellent indicator of the mean/median stellar mass of a sample of SFG. Since M ∗ is derived from the total stellar lights of a galaxy, a cosmological model (luminosity distance) must be assumed when calculating M ∗ from the multi-band photometric data. The ability of predicting reliably M ∗ with Equation (7) from the observed N/O abundance ratio, which is cosmological model independent, suggests that the relation in Equation (7) can be used a standard candle to study cosmology. The parameters (slope and intercept) in Equation (7) are now determined with a given cosmological model in the catalog, thus cannot be used directly to study alternative cosmological models. However, we can use the observed SNe Ia, which are excellent standard candles, to calibrate the relation in Equation (7), in a model-independent way similar to that the method of calibrating several luminosity relations of gamma-ray bursts (Liang et al. 2008, 2010). Since the relationship in equation (7) is derived using only the median values of N/O abundance ratio, its statistical robustness should be carefully studied in the future with, e.g. the method of Kelly (2007). Finally, this relationship needs to be tested in a broader redshift range before it is applied for cosmological studies. However further more detailed discussion on this subject is beyond the scope of this present work and will be presented elsewhere. With the observational data of 55,318 SFGs selected from the catalog of MPA-JHU emission-line measurements for the SDSS DR8, we find evidence of the galaxy downsizing effect, the O and N enrichments for galaxies with stellar masses larger than 10 11 . 0 , and the nitrogen production dominated by the intermediate-mass stars in SFGs. We summarize our main results below. Note : Col.(1): M ∗ or redshift (z) ranges. Col.(2): the subsample size of SFGs. Cols.(3)-(4), (5)-(6), (7)-(8), and (9)-(10): 'mean' and 'median' are the mean and median values of M ∗ or z, log(N/O), 12+log(N/H), and 12+log(O/H), respectively, in each subsample. YZWthanks Yanchun Liang for valuable discussions. The anonymous referee is thanked for many constructive comments and suggestions, which allowed us to improve the paper significantly. SNZ acknowledges partial funding support by 973 Program of China under grant 2009CB824800, by the National Natural Science Foundation of China under grant Nos. 11133002 and 10725313, and by the Qianren start-up grant 292012312D1117210.", "pages": [ 17, 18, 19, 20, 22 ] }, { "title": "REFERENCES", "content": "Aller L. H. 1942, ApJ, 95, 52 Alloin D., Collin-Souffrin S., Joly M., Vigroux L. 1979, A&A, 78, 200 Amor'ın R. O., P'erez-Montero E., V'ılchez J. M. 2010, ApJ, 715, L128 Baldwin J. A., Phillips M. M., Terlevich, R., 1981, PASP, 93, 5 Bundy K. et al. 2006, ApJ, 651, 120 Cannon R. D., Croke B. F. W., Bell R. A., Hesser J. E., Stathakis R. A., 1998, MNRAS, 298, 601 Cowie L. L., Songaila A., Hu E. M., Cohen J. G. 1996, AJ, 112, 839 Cowie L. L., Barger A. J. 2008, ApJ, 686, 72 Coziol R., Reyes R. E. C., Consid'ere S., Davoust E., Contini T., 1999, A&A, 345, 733 de Robertis M. M. 1987, ApJ, 316, 597; Dopita M. A., Evans I. N. 1986, ApJ, 307, 431 Edmunds M. G., Pagel B. E. 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[ { "title": "ABSTRACT", "content": "We present nebular spectra of the nearby Type IIb supernova (SN) 2011dh taken between 201 and 678days after core collapse. At these late times, SN 2011dh exhibits strong emission lines including a broad and persistent H α feature. New models of the nebular spectra confirm that the progenitor of SN 2011dh was a low-mass giant ( M ≈ 13-15M /circledot ) that ejected ∼ 0.07 M /circledot of 56 Ni and ∼ 0.27 M /circledot of oxygen at the time of explosion, consistent with the recent disappearance of a candidate yellow supergiant progenitor. We show that light from the SN location is dominated by the fading SN at very late times ( ∼ 2yr) and not, for example, by a binary companion or a background source. We present evidence for interaction between the expanding SN blastwave and a circumstellar medium at late times and show that the SN is likely powered by positron deposition /greaterorsimilar 1yr after explosion. We also examine the geometry of the ejecta and show that the nebular line profiles of SN 2011dh indicate a roughly spherical explosion with aspherical components or clumps. Key words: supernovae: general - supernovae: individual: SN 2011dh - techniques: spectroscopic", "pages": [ 1 ] }, { "title": "Nebular Spectroscopy of the Nearby Type IIb Supernova 2011dh", "content": "Isaac Shivvers, 1 † Paolo Mazzali, 2 , 3 Jeffrey M. Silverman, 4 , 5 J'anos Boty'anszki, 6 S. Bradley Cenko, 1 , 7 Alexei V. Filippenko, 1 Daniel Kasen, 1 , 6 , 8 Schuyler D. Van Dyk, 9 Kelsey I. Clubb 1 1 Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA 9 Spitzer Science Center/Caltech, Mailcode 220-6, Pasadena, CA 91125 Accepted to MNRAS; 2013 September 25. † Email: [email protected]", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Type IIb supernovae (SNe; Woosley et al. 1987; Filippenko 1988) are a relatively rare class of core-collapse supernova (SN), constituting only ∼ 7% of all SNe (Li et al. 2011). Like other SNe II, they show strong hydrogen features in their early-time spectra, yet within only a few weeks after core collapse the H fades and the spectra of SNe IIb most closely resemble those of stripped-envelope SNe Ib (for a review of the spectral classification of SNe, see Filippenko 1997). SNe IIb therefore represent a transitional class of core-collapse SNe with only partially stripped envelopes. Exactly what process removes most (but not all) of their hydrogen envelope is still an open question, though interaction with a binary companion increasingly appears to be the most likely answer. Thus far, there have been only a handful of nearby and intensely studied SNe IIb, including SN 2008ax ( ∼ 9.6 Mpc; e.g., Chornock et al. 2011), SN 2001ig ( ∼ 11.5 Mpc; e.g., Silverman et al. 2009), SN 2003bg ( ∼ 21.7 Mpc; e.g., Hamuy et al. 2009; Mazzali et al. 2009), and SN 1993J ( ∼ 3.69 Mpc; e.g., Filippenko, Matheson & Ho 1993; Matheson et al. 2000). SN 2011dh in M51 ( ∼ 8.05 Mpc; see § 2.3 below) has become another nearby and very well-observed example of this unusual class of SN. In early June 2011, SN 2011dh (also known as PTF11eon) was independently discovered within ∼ 1 day of core collapse by several amateur astronomers (Griga et al. 2011) and the Palomar Transient Factory collaboration (PTF; Rau et al. 2009; Law et al. 2009; Arcavi et al. 2011). The SN is apparent in an image taken by A. Riou of France on May 31.893 (UT dates are used throughout), while a PTF image taken May 31.275 does not detect a source down to a 3 σ limiting magnitude of m g = 21 . 44 (Arcavi et al. 2011). These observations most likely bracket the time of explosion, and for this paper we assume an explosion date of May 31.5. After discovery, a spectrum was promptly obtained by Silverman, Filippenko & Cenko (2011), and a possible progenitor star was first identified in archival Hubble Space Telescope (HST) images by Li & Filippenko (2011). Maund et al. (2011) and Van Dyk et al. (2011) confirmed the identification of the likely progenitor star in HST images through ground-based adaptive optics imaging of the SN, measuring a spatial coincidence of the HST source and the SN to within 23 and 7 mas, respectively. Both reported that the source in the HST images has a spectral energy distribution consistent with a single star: a yellow (mid-F) supergiant with an extended envelope ( R ≈ 200 R /circledot ), a temperature of ∼ 6000 K, and a mass of 13-18 M /circledot . However, Van Dyk et al. (2011) expressed doubt that the yellow supergiant (YSG) is the true progenitor of SN 2011dh, instead preferring a scenario with a faint and compact progenitor as a binary companion to the YSG. This was largely motivated by the results of Arcavi et al. (2011), who favored a compact (10 11 cm) binary companion based on the rapidity of the shock breakout and the relatively cool early photospheric temperature. Soderberg et al. (2012) supported this interpretation with radio and X-ray observations, estimating the progenitor size to be ∼ 10 11 cm through modeling of the cooling envelope. In this compact-star scenario, the progenitor of SN 2011dh was theorised to be a faint Wolf-Rayet star with a zero-age main sequence mass /greaterorsimilar 25 M /circledot and a history of mass loss through vigorous winds. Bersten et al. (2012) disagreed; their hydrodynamical models suggested that an extended progenitor was required to produce the early-time light curve, at odds with the analytic relation used by Arcavi et al. (2011), originally from Rabinak & Waxman (2011). Bersten et al. (2012) found that a progenitor with a zero-age main sequence mass of 12-15 M /circledot and a radius ∼ 200 R /circledot was consistent with the early-time light curve and photospheric temperature, and showed that any model with a zero-age main sequence mass /greaterorsimilar 25 M /circledot (i.e., a Wolf-Rayet star) was strongly disfavoured. Benvenuto, Bersten & Nomoto (2012) presented a model of a possible binary progenitor scenario with a ∼ 16 M /circledot YSG primary star losing material to a much fainter ∼ 10 M /circledot companion (undetectable in the pre-explosion HST images). In addition, Murphy et al. (2011) argued that the mass of the SN 2011dh progenitor must be either 13 +2 -1 M /circledot or > 29 M /circledot , based upon an analysis of the star-formation history of the SN's environment. Star formation in the vicinity of the SN overwhelmingly occurred in two discrete bursts at < 6 and 17 +3 -4 Myr; the zero-age main sequence mass of the SN is constrained by assuming the star is associated with one of those events, taking into account errors due, for example, to uncertain late-stage stellar evolution and mass loss. This result is consistent with the YSG progenitor scenario. Throughout 2012 other authors presented further panchromatic observations, some of which favoured a compact progenitor while others suggested an intermediate or extended progenitor, emphasising the need for a definitive progenitor identification (e.g., Krauss et al. 2012; Bietenholz et al. 2012; Campana & Immler 2012; Horesh et al. 2012; Sasaki & Ducci 2012). The desired identification was provided by Van Dyk et al. (2013), who reported that the YSG progenitor candidate had disappeared from new HST images. Specifically, at an age of ∼ 641 days SN 2011dh had faded down to 1.30 and 1.39 mag fainter than the YSG progenitor in the HST Wide Field Camera 3 (WFC3) F 555 W and F 814 W passbands, respectively. This result is corroborated by Ergon et al. (2013), who report a significant decline in the B , V , and r -band fluxes between pre-explosion imaging of the YSG progenitor and imaging of the SN at 600+days past explosion. These results clearly point toward the extended YSG progenitor found in archival HST images as the progenitor star of SN 2011dh. In this paper, we present six new spectra of SN 2011dh taken between 201 and 678 days after core collapse, in the nebular phase of its evolution. During the nebular phase, the SN ejecta are optically thin and we can directly observe the products of explosive nucleosynthesis without reprocessing through a photosphere. Our very late-time spectra show that the flux observed by Van Dyk et al. (2013) and Ergon et al. (2013) was produced primarily by the fading SN and not a stellar source. We present models of the nebular emission spectra and detailed analyses of the line profiles and the late-time flux energetics, providing constraints on the progenitor's mass and composition and the geometry of the explosion. We describe our observations and data-reduction procedure in § 2, present our spectra and analysis in § 3, discuss our model spectra in § 4, and conclude in § 5.", "pages": [ 1, 2 ] }, { "title": "2.1 Spectroscopy", "content": "Following its discovery in early June 2011, we began an extensive spectroscopic monitoring campaign of SN 2011dh. Some of our early-time spectra from the Lick and Keck Observatories (including a spectrum obtained only 2.4 days after explosion) have already been presented by Arcavi et al. (2011), and other groups have published their own spectra (Marion et al. 2013; Ergon et al. 2013; Sahu, Anupama & Chakradhari 2013). This study focuses on the nebular phase of SN 2011dh. We collected spectra using both the Lick and Keck Observatories, moving to a larger aperture as the SN faded away. We used the Kast double spectrograph on the Shane 3 m telescope at Lick Observatory (Miller & Stone 1993), the Low Resolution Imaging Spectrometer (LRIS) mounted on the 10 m Keck I telescope (Oke et al. 1995), and the DEep Imaging Multi-Object Spectrograph (DEIMOS) on the 10 m Keck II telescope (Faber et al. 2003) to collect 3, 1, and 2 nebular spectra of SN 2011dh, respectively. Table 1 summarises observing details for these 6 spectra.", "pages": [ 2 ] }, { "title": "2.2 Data Reduction", "content": "All observations were collected and reduced following standard techniques as described by Silverman et al. (2012). All spectra were taken with the slit oriented at the parallactic angle to minimise flux losses caused by atmospheric dispersion (Filippenko 1982). We use a low-order polynomial fit to arc-lamp observations to calibrate the wavelength scale, and we flux calibrate each spectrum with a spline fit to standardstar spectra observed the same night at a similar airmass. In addition, we have removed telluric absorption lines from all spectra. Upon publication, all raw spectra presented in this paper will be made available in electronic format on WISeREP (the Weizmann Interactive Supernova data REPository; Yaron & Gal-Yam 2012). 1", "pages": [ 2, 3 ] }, { "title": "2.3 Distance", "content": "The distance to M51 has been measured through several independent methods with significant scatter among their results. We follow Marion et al. (2013) and adopt D = 8 . 05 ± 0 . 35 Mpc, an average of four of these measures (Tonry et al. 2001; Tully & Fisher 1988; Vink'o et al. 2012; Feldmeier, Ciardullo & Jacoby 1997). All spectra have been deredshifted by M51's recession velocity, 600 km s -1 ( z = 0 . 002, NED; Rush, Malkan & Edelson 1996). M51 is at very low redshift and so we neglect time-dilation effects due to cosmological expansion. Both Arcavi et al. (2011) and Vink'o et al. (2012) use high-resolution spectra to measure the reddening toward M51 using Na I D absorption-line widths. Both find the host-galaxy extinction to be negligible and the Milky Way extinction to be consistent with values measured by Schlegel, Finkbeiner & Davis (1998): E ( B -V ) = 0 . 035 mag. We deredden all spectra by this value prior to analysis, using the reddening law of Cardelli, Clayton & Mathis (1989) and assuming R V = 3 . 1. Note that Ergon et al. (2013) adopted a slightly higher value of E ( B -V ) = 0 . 07 +0 . 07 -0 . 04 mag, corresponding to a ∼ 5-10% difference in absolute flux level across the optical spectrum, not enough to significantly affect the discussion below.", "pages": [ 3 ] }, { "title": "2.4 Absolute Flux Calibration", "content": "Our observation techniques and data-reduction methods record the relative flux with high fidelity, but absolute flux calibrations are a persistent difficulty in long-slit spectroscopy. Variations in atmospheric seeing between fluxstandard observations and science observations can result in varying amounts of flux falling out of the slit and spectral observations are often taken in less-than-photometric conditions with nonnegligible (and possibly varying) levels of cloud cover. Parts of our analysis (see §§ 3.1, 4) require an absolute flux measure, however, so we address this problem by flux calibrating our spectra to late-time photometry of SN 2011dh wherever possible. Tsvetkov et al. (2012) present UBVRI light curves of SN 2011dh extending to just over 300 days; we assume a linear decay in R -band magnitudes beyond ∼ 70 days and perform a maximum-likelihood analysis to estimate the R magnitude of SN 2011dh at the time each spectrum was taken. We chose the R band because of its relatively dense coverage and because several of the strongest nebular lines ([O I ], [Ca II ], Na I , H α ) fall within the passband, making it a good tracer of the SN's decline. We match synthetic photometry of our 201-334 day spectra to these values. All synthetic photometry has been calculated with pysynphot (Laidler et al. 2008). As shown in Figure 1, we find an R -band decline rate of of 0 . 0195 ± 0 . 0006 mag day -1 and a V - decline rate of 0 . 0207 ± 0 . 0009 mag day -1 (reported errors are 68% confidence levels; ∼ 1 σ ). A linear decay in magnitudes is a reasonable assumption so long as emission is primarily driven by the radioactive decay of 56 Co (e.g., Colgate & McKee 1969; Arnett 1996). It is common for SNe Ib/IIb to display decline rates significantly faster than the 56 Co → 56 Fe rate (0.0098 mag day -1 ) - a steep decline rate is reasonably interpreted as evidence for a declining γ -ray trapping fraction in the ejecta (as the ejecta expand and the density drops, more of the γ -rays produced by 56 Co decay escape before depositing their energy). The decline rate of SN 2011dh is slightly faster than those measured for both SN 1993J and SN 2008ax, two well-understood SNe IIb which had decline rates of 0.0157 and 0.0164 mag day -1 , respectively (fit to days ∼ 60-300; Taubenberger et al. 2011). See § 3.1 for a comparison between these early-time nebular decline rates and the flux observed at very late times ( > 600 days). We do not assume that the same decay law holds true out to our last two spectra, at 628 and 678 days after core collapse. Instead, we repeat the analysis described above using photometry from Ergon et al. (2013), who report Nordic Optical Telescope (NOT) observations in V at 601 and 685 days.", "pages": [ 3, 4 ] }, { "title": "3 ANALYSIS", "content": "By 201 days past explosion SN 2011dh was well into the nebular phase, with a spectrum dominated by strong emission lines and little or no continuum. Figure 2 shows our complete spectral sequence of SN 2011dh in the nebular phase with spectra from 201 to 678 days after explosion and a few prominent lines identified, and compares the spectra of SN 2011dh to those of a few other prominent SNe IIb at comparable epochs. Throughout the first year after explosion the nebular spectra of SN 2011dh are dominated by strong [O I ] λλ 6300, 6364 and [Ca II ] λλ 7291, 7324 emission lines, alongside a strong Mg I ] λ 4571 emission line and persistent Na I D and H α lines. Table 2 lists relative line strengths of several prominent lines in the early nebular phase. We measured these fluxes by subtracting a local linear continuum and integrating over each line. Note that the continuum here is not from the photosphere of the SN, but rather is likely a mixture of blended lines, producing a sort of pseudocontinuum. Also, note that this type of integrated flux measurement is by no means exact due to line blending and the approximated local continuum, but care was taken to treat each line similarly and these measures should accurately portray the relativeflux trends. The relative flux of [Ca II ] and [O I ] has been shown to be a useful indicator of progenitor core mass, with smaller [O I ]/[Ca II ] ratios generally indicative of a less massive helium core at the time of explosion (e.g., Fransson & Chevalier 1989; Jerkstrand et al. 2012). SN 2011dh displays an [O I ]/[Ca II ] ratio significantly smaller than that in both SN 1993J and SN 2001ig. The ratio is similar to that in SN 2008ax, which also displayed a similar upward trend throughout the nebular phase (Silverman et al. 2009; Filippenko, Matheson & Barth 1994; Chornock et al. 2011). It therefore appears that SN 2011dh's progenitor He core mass was relatively close to that of SN 2008ax and significantly less than that of both SN 2001ig and SN 1993J. See § 4 for a more thorough analysis. There appears to be a weak blue continuum in the 600+day spectra of SN 2011dh. Maund et al. (2004), in a very high S/N spectrum of SN 1993J taken ∼ 10 yr after explosion, were able to associate a blue continuum (and a detection of the Balmer absorption-line series) with a companion B supergiant, thereby strongly supporting the binary nature of the SN and identifying the components - a K-giant progenitor and a B-giant companion. In the spectrum of SN 2011dh above, however, we cannot attribute the blue continuum to any stellar companion: fitting a RayleighJeans curve to the apparent continuum yields best-fit temperatures much too hot for a stellar source. The blue continuum in SN 2011dh is instead most likely a pseudocontinuum caused by many blended lines. In addition, our spectra are more noisy at the blue end, and the blue rise may be partially caused by increased noise. HST photometry taken near this time provides a slightly redder colour than synthetic photometry from our spectrum: F 555 W -F 814 W = 0 . 69 ± 0 . 03 mag (641 days; Van Dyk et al. 2013), compared to ∼ 0.34 mag from our spectrum (628+678 days). Thus, we tilt our spectrum to match the HST F 555 W -F 814 W colour and re-examine the result for evidence of a stellar companion. Our conclusion is essentially unchanged: even after tilting our spectrum, the blue pseudocontinuum yields unreasonably hot best-fit blackbody temperatures. Interestingly, there is a broad H α emission line in spectra of SN 2011dh through at least 334 days, similar to the emission seen in SN 1993J, (Filippenko, Matheson & Barth 1994), SN 2007Y (Stritzinger et al. 2009), and SN 2008ax (Milisavljevic et al. 2010) around the same time. There is also some indication of very broad H α in the spectra of SN 2011dh at 600+days, though the S/N is low. At late times the H α emission of SN 1993J was unambiguously identified with interactions between the expanding SN shock wave and circumstellar material produced by mass loss from the progenitor (e.g., Patat, Chugai & Mazzali 1995; Houck & Fransson 1996; Matheson et al. 2000). As we discuss in § 3.1, SN 2011dh seems to present us with a more complex situation. SN 2011dh, like SN 2001ig, displayed a relatively strong Mg I ] λ 4571 line - significantly more prominent than Mg I ] in spectra of SN 2008ax (Silverman et al. 2009; Chornock et al. 2011). This is especially apparent around day 334, where Mg I ] emission almost matches the emission in [Ca II ] and [O I ]. At very late times, in the 628+678 day spectrum, the Mg I ] is still quite apparent, though [O I ] and [Ca II ] have faded into the noise. Unfortunately, our 628 and 678 day spectra do not go much blueward of the Mg I ] emission peak; this, together with high noise levels at the blue end, prevents us from measuring the integrated flux reliably at these times. The Na I D flux is also remarkably strong in the 600+ day spectra, as discussed below.", "pages": [ 4 ] }, { "title": "3.1 The Spectrum of SN 2011dh at 600 + Days", "content": "Recent photometry of the site of SN 2011dh taken by HST (Van Dyk et al. 2013) and the Nordic Optical Telescope (NOT; Ergon et al. 2013) provide late-time flux measurements of SN 2011dh. Our latest two spectra, taken around Errors are difficult to estimate for these values, as line edges and continuum levels have been estimated by eye. However, care was taken to treat each line similarly. Measurement errors alone (determined through repeated measurements) are ∼ 5%. the same time, confirm that the optical flux measured by these groups was predominantly from the SN remnant and not, for example, from a binary companion to the progenitor star. 678 days after explosion, SN 2011dh continues to show clear Na I D emission with approximately the same width as at earlier epochs (see Fig. 3). [Ca II ] emission is present but much reduced relative to Na I D, Mg I ] is still relatively strong but is buried in the noise at the blue end of the spec- trum, and there is a very broad feature near 6500 ˚ A - most likely a blend of broad H α and the [O I ] doublet. Ergon et al. (2013) report a slight fading of 2011dh between days 601 and 685: 0 . 009 ± 0 . 0026 magday -1 in V . The V -band decline rate between the last observation reported by Tsvetkov et al. (2012, 19 . 44 ± 0 . 12 mag, 2012 Apr. 4, 310 days) and the first observation by Ergon et al. (2013, 22 . 56 ± 0 . 10 mag, 2013 Jan. 20, 601 days) is ∼ 0.011 mag day -1 . Both of these rates are notably less rapid than the 0 . 021 magday -1 decline rate measured from the 65-310 day photometry by Tsvetkov et al. (2012, see § 2.4); it seems that the flux decline rate is slowing down at t /greaterorsimilar 300 days. Na I D provides the clearest feature in our 600+ day spectra and is unambiguously associated with the SN (see Fig. 3). We measured the integrated flux in the Na I D line for each of our spectra in the nebular phase; the results are shown in Table 3 and Figure 4. Note that the absolute flux calibrations of long-slit spectra are often unreliable. As described in § 2.4, we address this by flux calibrating our spectra to published photometry. Quoted uncertainties include estimated error due to spectral noise, reported photometric errors, and estimated measurement errors, all added in quadrature. As Figure 4 illustrates, the Na I D line flux mirrors the trends in the photometry, falling at the early R -band decline rate through ∼ 300 days but then deviating significantly around 300 or 350 days and fading much more slowly through ∼ 600 days. This slowdown in the flux decay rate is possibly indicating a transition to γ -ray transparency in the ejecta of SN 2011dh. As described in more detail by Arnett (1996), 56 Co radioactivity ( 56 Co → 56 Fe with a half-life of ∼ 77 days) is the dominant source of energy for SNe at these epochs. 56 Co decay produces both γ -rays and high-energy positrons. The kinetic energy of the positrons is very likely to be deposited into the ejecta (and therefore contribute to the nebular line flux), while the fraction of γ -ray energy that gets deposited depends upon the optical depth of the ejecta to γ -rays. As the ejecta expand and the optical depth drops, a larger fraction of the γ -rays escape, carrying their energy with them. As the γ -ray energy deposition fraction drops, the SN fades away faster than the 56 Co rate (0.0098 mag day -1 ), until such time as the ejecta are transparent to γ -rays and approximately all of them escape. At that point, positron energy deposition dominates the energy input of the ejecta and the bolometric flux decline rate is expected to follow the 56 Co rate closely. Broadband photometry should exhibit roughly the same behaviour. Assuming that the ejecta's abundance of neutral sodium is constant and that the exposure to heating does not change significantly, so should the Na I D line flux. The flux decline rates for both the late-time V -band photometry and the Na I D line flux in SN 2011dh are consistent with a transition to a positron-powered ejecta sometime between 300 and 350 days. Because positrons deposit their energy locally (near the decaying 56 Co) but γ -rays deposit energy throughout the ejecta, the transition to a positron-dominated energy input is likely to correspond to a change in the dominant emission lines. In SN 2011dh, we see that Na I D and Mg I ] emission stays strong in the positron-dominated epoch while [O I ] and [Ca II ] fade away. More detailed modeling is necessary to test this scenario, however. Note that the above discussion assumes the bolomet- Photometric normalisations applied to our nebular spectra and resultant absolute flux in the Na I D line. See § 2.4 for a description of how we calculated the photometric estimates. We measured the integrated line flux in the manner described in § 3, and the quoted errors include photometric normalisation errors, spectral noise, and estimated measurement errors added in quadrature. . c To estimate the spectrum near the HST observations, we produce an average of the 628 and 678 day spectra. We flux calibrate both spectra to R = 0 . 0 mag, coadd them, and renormalise the result to the HST photometry. This produces an equally weighted average between the two spectra - we assume the SN is changing slowly at this epoch and that this averaged spectrum is a good measure of the relative flux near the average time (653 days). ric light curve is completely powered by 56 Co. Any additional energy input could be a confounding factor; most importantly, there may be a flux contribution from shockwave interactions with circumstellar gas. The progenitors of SNe IIb are stars that have lost much of their hydrogen envelope, either through radiative winds or through strip- ping by a binary companion. If a significant amount of that material remains nearby in a cloud of circumstellar matter, the expanding SN ejecta will impact it and form a shock boundary. This shocked region produces high-energy photons which are then reprocessed down to optical wavelengths by material in the outer shells of the ejecta, thereby producing broad emission lines and (possibly) a blue pseudocontinuum (for a more complete description of this process, see, e.g., Chevalier & Fransson 1994; Fransson 1984). Late-time emission from circumstellar interaction (CSI) is common in SNe IIn (e.g., Fox et al. 2013), which have lost a majority of their envelope in the years prior to explosion, and it has been observed in other SNe IIb (e.g., Matheson et al. 2000). CSI could be augmenting the Na I D flux discussed above through either Na I D or He I λ 5876 emission. As Figure 3 shows, SN 1993J clearly displayed a shockwavepowered Na I D + He I blend with ∼ 1/3 the flux of the H α line (Matheson et al. 2000). In the Chevalier & Fransson (1994) model, approximately all of the shockwave-powered Na I D flux is emitted from a thin shell at the boundary between the unshocked circumstellar material and the shocked ejecta, while the H α flux comes from both the thin shell and the shocked ejecta. This would imply a more boxy line profile for Na I D than H α (see below for further discussion of the nebular H α line in SN 2011dh). However, the late-time Na I D emission of SN 2011dh has a relatively narrow profile with no evidence of the box-like shape that would be expected if it were mostly CSI-powered, and the profile does not appear to change significantly between the spectra taken at < 1 yr and those taken at > 1 yr. It seems clear that, at 600+ days, the dominant source of Na I D flux remains radioactive decay and not CSI, though there may well be some small amount of shockwave-powered He I and Na I D emission buried in the noise. If SN 2011dh's CSI-powered Na I D + He I flux were a factor of 3 less than the H α flux (as was the case in SN 1993J), we would not expect to be able to distinguish it from the noise in our spectra. Note that the V -band data may, in addition, include flux contributions from a CSI-powered blue pseudocontinuum at these late times - the 600+ d spectra do indicate a faint continuum blueward of 6000 ˚ A. Because of the large time gap between our spectra at 334 d and 628 d, we do not know exactly when this blue pseudocontinuum emerged. It does not seem to evolve significantly between our spectra taken at 628 and 678 days, and so we believe that the flux decline measured around this time by Ergon et al. (2013) (0 . 009 ± 0 . 0026 mag day -1 ) is dominated by the fading 56 Co contribution and not by any evolving CSI-powered flux. Note that V -band photometry likely also includes contributions from the Na I D, [O I ], [Fe II ], and H α lines, whether they are powered by radioactivity or CSI - a rather complex puzzle to decode. However, this complexity does not affect the measurement of individual line fluxes. As described above, the Na I D line indicates that the ejecta of SN 2011dh became fully γ -ray transparent (and therefore powered through positron energy deposition) between 300 and 350 days after core collapse. In contrast with the Na I D line, the nebular H α line appears to be fueled entirely through CSI. Assuming the broad feature near 6563 ˚ A in the 678 day spectrum of SN 2011dh (see Fig. 3) is a broad and boxy H α feature, it exhibits a full width at half-maximum intensity (FWHM) of roughly 21,000-26,000 km s -1 (there are large errors when measuring the line width, as this spectrum has a low S/N and the line is weak). Spectra taken the first month after core collapse show a blueshifted H α absorption component around 15,400 kms -1 to 12,500 km s -1 (velocities at 4 to 14 days; Marion et al. 2013). These measurements mesh with the CSI scenario described above, wherein the unslowed outer ejecta impact the circumstellar material and produce a shell of emitting gas that continues to move outward at its initial expansion velocity. The scenario that SN 2011dh presents to us in its latestage evolution is notably different than that of SN 1993J or SN 1987A. As shown by Suntzeff et al. (1992), the peculiar Type II-P SN 1987A exhibited a continuously declining (yet nonzero) γ -ray opacity until the slowly decaying isotope 57 Co became the dominant source of energy around 800-900 days after explosion ( 57 Co → 57 Fe with a half-life of ∼ 272 days). This indicates that SN 1987A had a significantly higher γ -ray opacity than SN 2011dh. In contrast, CSI became the dominant flux source in SN 1993J around 350 days, when the spectrum became dominated by broad and boxy emission lines (Matheson et al. 2000), and it is impossible to tell when (or if) the radioactive energy deposition entered the 56 Co positron-powered phase. Of course, several questions remain about SN 2011dh and its late-time evolution. Unless it rebrightens, however, the SN is too faint to hope for a significantly higher S/N spectrum than those presented here (our spectrum at 678 days represents an hour of integration time on a clear night with the 10 m Keck II telescope). Continued photometric monitoring should provide more information as the SN evolves.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "3.2 The Oxygen Line Profile and the Ejecta Geometry", "content": "Fransson & Chevalier (1989) showed that, given the reasonable assumptions of homologous expansion and optically thin emission, the profiles of forbidden lines in nebular SN spectra can be used as tracers of the geometry and density profile of the emitting material. The [O I ] λλ 6300, 6364 doublet, specifically, has been used as a diagnostic of ejecta asphericity in Type Ibc/IIb SNe by many studies (e.g., Mazzali et al. 2005; Taubenberger et al. 2009; Milisavljevic et al. 2010; Modjaz et al. 2008; Maurer et al. 2010; Maeda et al. 2008). The [O I ] doublet is generally used because it is consistently one of the strongest lines in nebular SN spectra and is largely isolated and unblended, and oxygen is one of the most abundant elements in strippedenvelope core-collapse SNe. The structure apparent in the [O I ] doublet has often been attributed to either a jet or torus geometry in the ejecta with the diversity of line profiles explained through viewing-angle dependencies (e.g., Mazzali et al. 2005; Maeda et al. 2008; Modjaz et al. 2008), though other explanations have been presented for some SNe (e.g., Maurer et al. 2010; Milisavljevic et al. 2010). The [O I ] profile of SN 2011dh prominently displays multiple peaks and troughs. We explore the geometrical implications of this line profile in two ways. Several studies have previously explored [O I ] line profiles by decomposing the profile into a set of overlapping Gaussian curves, effectively assuming a multi-component Gaussian spatial distribution (e.g., Taubenberger et al. 2009). We performed a similar fit for comparison, but the spatial distribution of emissivity is not necessarily Gaussian; the choice to decompose the profile this way is mainly for convenience. We also ran threedimensional (3D) nebular radiative transfer models for a variety of geometries, attempting to fit the observed line profile with a simple and physically plausible ejecta geometry. [O I ] λλ 6300, 6364 is a doublet with two peaks separated by 64 ˚ A, with their relative intensities determined by the local density of neutral oxygen. The intensity ratio reliably approaches 3:1 in nebular SN spectra (e.g., Chugai 1992; Li & McCray 1992), and we assume this holds true for SN 2011dh at these epochs. Before analysing the [O I ] profile we remove the H α emission by assuming it is symmetric about the rest wavelength (6563 ˚ A) and subtracting a smoothed profile. Note that this nebular spectral analysis is not a well-posed inverse problem; many different geometries could produce the same spectral profile. The results of our Gaussian decomposition of the line profile are shown in the left panel of Figure 5. For each component in our fit we specify the amplitude, position, and width of the 6300 ˚ A line; the properties of the 6364 ˚ A line are then forced. In the spectrum at 268 days past core collapse, our best fit to the [O I ] line requires three such components. There is a broad component blueshifted by ∼ 250 kms -1 , a narrow component blueshifted by ∼ 400 km s -1 , and a second narrow component redshifted by ∼ 1600 kms -1 . Note, however, that the broad component is only needed because the overall line profile is distinctly non-Gaussian. A more nuanced approach (below) provides a good fit to the profile with only two components. The results of our 3D modeling analysis are shown in the right panel of Figure 5. In our models, we specify the emissivity of the [O I ] doublet in each spatial zone and integrate the transfer equation using the Sobolev approximation under the assumption that the ejecta are optically thin (see, e.g., Jeffery & Branch 1990). We decompose the 3D emission into multiple overlapping spherical clumps, each with an exponentially-declining emissivity profile. The primary peak is well fit by a sphere with an emissivity profile characterised by an exponential falloff with e -folding velocity v e = 950kms -1 . To match the position of the peak, we need to offset the entire sphere from the origin toward the observer by ∼ 250 km s -1 . The secondary peak is well fit by placing a second smaller spherical clump along the observer's line of sight but moving away at ∼ 1500 kms -1 . The emissivity profile of this second clump has an e -folding velocity of v e = 300 km s -1 which terminates at 600 km s -1 . The integrated emission from the primary sphere is ∼ 24 times greater than the integrated emission of the smaller clump, though the peak local emissivity of the smaller clump is a factor of ∼ 4 higher than that of the primary sphere. The right panel of Figure 5 shows that this model does a decent job of fitting all features in the [O I ] profile. Though this simple model invokes only two components, the true ejecta geometry could in fact consist of multiple clumps of similar or smaller spatial dimensions. This is because it is only the larger inhomogeneities located along the line of sight that produce noticeable and well-separated features in the line profile. It is unclear whether the clump-like structures we infer from the [O I ] doublet correspond to inhomogeneities in the distribution of the oxygen itself or the 56 Ni that ex- cites it. In 3D core collapse simulations, convective motions during neutrino heating act as seeds for Rayleigh-Taylor instabilities that develop when the shock passes through compositional interfaces (e.g., Hammer, Janka & Muller 2010). This results in fingers of heavier elements, such as 56 Ni, punching out into the overlying layers of lighter elements. Such a picture could explain the irregular line profiles seen in several core-collapse SNe at late times (e.g., Filippenko & Sargent 1989; Matheson et al. 2000), and has been explicitly considered previously for the asymmetry seen in SN 1987A. In particular, the substructure noted in the H α line profile of SN 1987A (the 'Bochum event'; Hanuschik, Thimm & Dachs 1988) has been interpreted as resulting from a relatively high velocity ( ∼ 4700 km s -1 ) 'bullet' of 56 Ni (Utrobin, Chugai & Andronova 1995). A similar geometry could potentially be applicable to SN 2011dh, assuming a sizable (but slower) clump of 56 Ni was mixed into the oxygen layer. More sophisticated 3D nebular spectral modeling will be needed to constrain the geometry in more detail. For example, extending the secondary clump of our model by adding more material to the extreme redshifted edge (making the clump aspherical) would fill in the discrepancies apparent in the line profile near 6340 ˚ A and 6420 ˚ A, and would also more closely resemble the extended structures apparent in the models of Hammer, Janka & Muller (2010, see their Fig. 2). Though the primary emitting sphere in our model is slightly offset from the origin, this may be due to uncertainty in the true SN Doppler velocity rather than an actual asymmetry of the ejecta. Though M51 is almost face-on, Tully (1974) show that the southeast quadrant of M51 (where SN 2011dh occurred) is rotating toward us. The line-ofsight motion is significantly less than 250 km s -1 but the true Doppler velocity of M51 may be lower than the value we used to deredshift our spectra. Adopting z = 0 . 00155 ± 0 . 0002 (Falco et al. 1999) instead of the z = 0 . 002 used in the rest of this paper places the primary emitting component at a blueshift of ∼ 120 km s -1 , within a factor of 2 of the expected line-of-sight rotational velocity of M51 at the SN position. In addition, there are narrow H α lines from the host galaxy superimposed on our spectra (slight oversubtractions are apparent in both the 268 and 334 d spectra; see Figure 2). Assuming the strongest of these lines in our 268 d spectrum was emitted in the rest frame of the SN, we measure a Doppler velocity of 550 ± 160 kms -1 . Adopting this value places the primary component at a blueshift of ∼ 200 km s -1 relative to rest. It appears that the primary emitting component of SN 2011dh is either symmetric with respect to the rest frame or nearly so. Note, as well, that these relatively low Doppler velocities lie near the resolution limit of our spectra. As described by Silverman et al. (2012), our spectra have characteristic wavelength errors of 1 - 2 ˚ A ( /lessorsimilar 90 kms -1 ). The positions of most other nebular lines are consistent with this scenario (with widths of several thousand kms -1 and irregular profiles, it is difficult to determine the centres of nebular SN lines to high precision). The Mg I ] λ 4571 line is an exception, displaying a strong asymmetry and a blueshifted peak (see below). As Figure 6 shows, the components described above persist from 201 to 334 days, and similar components at similar relative positions are apparent in the O I λ 7774 profile and the Mg I ] λ 4571 profile, though the primary component appears to be more blueshifted in Mg I ]. The [Ca II ] λλ 7291, 7324 profile, however, exhibits a simple and singly-peaked profile. Other studies have shown that it is common for Mg I ] and [O I ] to display similarly asymmetric line profiles while [Ca II ] remains relatively symmetric (e.g., Modjaz et al. 2008; Milisavljevic et al. 2010). SN 2011dh's nebular [O I ] profile is not consistent with the often-proposed simple torus model of emitting material. An emission trough due to an overall torus-like geometry of emitting material would fall at the rest wavelength of the line (in the SN rest frame). As Figure 5 shows, SN 2011dh instead displays an emission peak at roughly the rest wavelength; if the main trough in the line profile were associated with the centre of a torus, it would have to be offset from the rest frame of M51 by ∼ 1000 kms -1 , an offset inconsistent with the centres of other nebular emission lines. To explore this further we ran several models similar to the two-clump model described above but with a toroidal component at various viewing angles, and we found no physically plausible toroidal geometries that matched the profile well. Maurer et al. (2010) showed that foreground H α absorption was a reasonable explanation for the double [O I ] peak in SN 2008ax. It is possible that foreground hydrogen absorption is also affecting the oxygen profile in SN 2011dh: early-time spectra indicate a hydrogen expansion velocity of 12,500-15,400 km s -1 (velocities at 14 and 4 days; Marion et al. 2013), and the peak of [O I ] emission is ∼ 12,000 kms -1 (265 ˚ A) blueward of H α . However, explain- ll three bumps in the profile through foreground absorption would require three well-placed hydrogen overdensities, at ∼ 8100, 9600, and 11,100 km s -1 , and would not account for the line profiles of Mg I ] λ 4571 and O I λ 7774. It seems apparent that the [O I ] line-profile asymmetries in SN 2011dh come from distinct emitting components moving relative to each other, each displaying the doublet nature of the line. Additionally, the lack of obvious H α absorption features may indicate a very low hydrogen shell mass.", "pages": [ 8, 9, 10 ] }, { "title": "4 NEBULAR MODELS", "content": "We use a spherically symmetric single-zone non-LTE (local thermodynamic equilibrium) nebular modeling code to further explore SN 2011dh. The code tracks the heating of the nebular ejecta through deposition of γ -rays and positrons produced by radioactive decay. This heating is balanced by line emission to determine both the temperature and ionization state of the nebula. Following methods and ideas first outlined by Axelrod (1980) and Ruiz-Lapuente & Lucy (1992), the code was developed by Mazzali et al. (2001), and has been described in greater detail by, for example, Mazzali et al. (2010) and Mazzali & Hachinger (2012). The code is available in a one-zone version, a stratified version, and a three-dimensional version. The one-zone model provides a rough estimate of the properties of a SN nebular spectrum (e.g., mass and elemental abundances). The stratified model is preferred when comparing a detailed model of the explosion with the data (Mazzali et al. 2007), and the three-dimensional model is useful for strongly asymmetric events (Mazzali et al. 2005). Here, since the profiles of the emission lines do not deviate significantly from the theoretically expected parabolic profiles and developing a complete explosion model is beyond the scope of this paper, we restrict ourselves to the one-zone approach. Massestimate differences between the one-zone and the stratified model are relatively small in cases where the ejecta do not display strong asphericity ( ∼ 20%; Mazzali et al. 2001). The code does not include recombination emission, and therefore neither H α nor the O I λ 7774 recombination line are reproduced. While hydrogen is located outside the carbon-oxygen core and ignoring H α does not affect our result, ignoring the oxygen recombination line can introduce an error, though it should be small (Maurer et al. 2010). Another element of uncertainty is introduced by the fact that silicon does not have strong lines in the optical range. The strongest line, [Si I ] λ 6527, is about one third as strong as Na I D and is swamped by the H α emission. All these effects could lead to an overestimate of the mass. Finally, a major source of uncertainty is the subtraction of the background pseudocontinuum. Our best-fit models to the day 207 and 268 spectra are shown in Figure 7. The spectrum of SN 2011dh changes only slightly between these two epochs, and the two (independent) models are very similar. These models are powered by ∼ 0.07 M /circledot of 56 Ni and exhibit an outer envelope velocity of ∼ 3500 km s -1 with a total enclosed mass of ∼ 0.75 M /circledot . See Table 4 for a detailed listing of the mass composition of the models. Note that these values are sensitive to errors in the determination of the distance to SN 2011dh and errors in the absolute flux calibration of our spectra. Most of the major features of these spectra are matched by the models, including the prominent [Ca II ] λλ 7291, 7324, [O I ] λλ 6300, 6364, Na I D, and Mg I ] λ 4571 lines. Bersten et al. (2012) derived a similar but slightly lower 56 Ni mass of ∼ 0.065 M /circledot from bolometric light curve modeling through the first 80 days, while Sahu, Anupama & Chakradhari (2013) derived a slightly higher mass of ∼ 0.09 M /circledot through an analytic treatment of the bolometric light curve peak. Several other SNe IIb have been modeled in their nebular phase with similar codes, providing a useful set for comparison. Our models of SN 2011dh include ∼ 0.26 M /circledot of oxygen, much less than was needed for SNe 2008ax, 2001ig, and 2003bg ( ∼ 0.51, 0.81, and 1.3 M /circledot , respectively; Maurer et al. 2010; Silverman et al. 2009; Mazzali et al. 2009). The 56 Ni mass required is also relatively low. SN 2011dh had ∼ 0.067 M /circledot of nickel, but as the above authors have shown, SNe 2008ax, 2001ig, and 2003bg required ∼ 0.10, 0.13, and 0.17 M /circledot , respectively. This indicates a relatively low-mass progenitor for SN 2011dh. On the other hand, it is unclear whether the nebular spectra capture all of the carbon-oxygen core. The lowest velocity of the He lines is ∼ 5000 km s -1 , while the width of the emission lines is only ∼ 3500 kms -1 . Material between these two velocities may not be captured by the nebular modelling. Although this would go in the direction of compensating for the overestimate described above, we conservatively assign an error of ∼ ± 50% in our mass estimates. It would be interesting to test the data against realistic explosion models in order to narrow down this uncertainty. This will be the subject of future work. Multiple groups have modeled the nucleosynthetic yields of core-collapse SNe of various zero-age main sequence masses (e.g., Woosley, Langer & Weaver 1995; Mass composition of non-LTE nebular models fit to spectra of SN 2011dh at 207 and 268 days after core collapse. Thielemann, Nomoto & Hashimoto 1996; Nomoto et al. 2006). Though there are some discrepancies between our nebular model and the nucleosynthetic models (and some disagreements between different nucleosynthetic modeling efforts), our models are most consistent with a progenitor mass of 13-15 M /circledot . For example, Thielemann, Nomoto & Hashimoto (1996) predict carbon yields of 0.06, 0.08, and 0.115 M /circledot and oxygen yields of 0.218, 0.433, and 1.48 M /circledot for 13, 15, and 20 M /circledot progenitors, respectively. The values required by our best-fit model, 0.07 M /circledot of carbon and 0.26 M /circledot of oxygen, indicate a 13-15 M /circledot progenitor. Note that not all elements are in such good agreement. This result corroborates the findings of several other groups: the progenitor of SN 2011dh was a relatively lowmass ( ∼ 13-17 M /circledot ) yellow supergiant that likely had its outer envelope stripped away by a binary companion (e.g., Bersten et al. 2012; Benvenuto, Bersten & Nomoto 2012; Maund et al. 2011; Van Dyk et al. 2013; Murphy et al. 2011). SN 2011dh has provided a powerful test of the accuracy of SN progenitor studies through nebular spectra. The clear agreement between the nebular modeling and the results of such varied studies indicates that models of nebular SN spectra provide real and powerful constraints of the progenitor's properties. There is, however, work to be done to understand the discrepancies between modeled nucleosynthetic yields and nebular-spectra models.", "pages": [ 10, 11, 12 ] }, { "title": "5 CONCLUSIONS", "content": "SN 2011dh was a very nearby SN IIb discovered in M51 in early June 2011, providing observers with a valuable opportunity to track the evolution of one of these relatively rare SNe in detail. The nature of SN 2011dh's progenitor star has been much debated. In this paper, we present nebular spectra from 201 to 678 days after explosion as well as new modeling results. We confirm that the progenitor of SN 2011dh was a star with a zero-age main sequence mass of 13-15 M /circledot , in agreement with the photometric identification of a candidate YSG progenitor. In addition, our spectra at ∼ 2 yr show that photometric observations taken near that time are dominated by the fading SN and not, for example, by a background source or a binary companion. We present evidence pointing toward interaction between the expanding SN blastwave and a circumstellar medium, and show that the SN enters the positron-dominated phase by ∼ 1 yr after explosion. Finally, we explore the geometry of the ejecta through the nebular line profiles at day 268, concluding that the ejecta are well fit by a globally spherical model with dense aspherical components or clumps. In addition to the data presented here we have obtained several epochs of spectropolarimetry of SN 2011dh as it evolved. The analysis of those data is beyond the scope of this paper, but they will provide additional constraints on any asymmetry in the explosion of SN 2011dh.", "pages": [ 12 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "Sincere thanks to all of the supernova experts who contributed through discussions, including (but not limited to) Brad Tucker, WeiKang Zheng, Ori Fox, Patrick Kelly, and J. Craig Wheeler (whose keen eye identified a significant typo in the manuscript). We thank the referee for suggestions that helped to improve this paper. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration (NASA); the observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognise and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community; we are most fortunate to have the opportunity to conduct observations from this mountain. This material is partially based upon work supported by a National Science Foundation (NSF) Graduate Research Fellowship to J.B. under Grant No. DGE 1106400. J.M.S. is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1302771. A.V.F. and his SN group at UC Berkeley acknowledge generous support from Gary and Cynthia Bengier, the Richard and Rhoda Goldman Fund, the Christopher R. Redlich Fund, the TABASGO Foundation, and NSF grant AST-1211916. This research has made use of NASA's Astrophysics Data System Bibliographic Services, as well as the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.", "pages": [ 12 ] }, { "title": "REFERENCES", "content": "Arcavi I. et al., 2011, ApJ, 742, L18 Arnett D., 1996, Supernovae and nucleosynthesis: an investigation of the history of matter, from the big bang to the present, Princeton series in astrophysics. Princeton University Press Axelrod T. S., 1980, PhD thesis, Univ. California, Santa Cruz Benvenuto O. G., Bersten M. C., Nomoto K., 2012, ApJ, 762, 74 Bersten M. C. et al., 2012, ApJ, 757, 31 Bietenholz M. F., Brunthaler A., Soderberg A. M., Krauss", "pages": [ 12 ] }, { "title": "14 Shivvers et al.", "content": "Woosley S. E., Langer N., Weaver T. A., 1995, ApJ, 448, 315 Woosley S. E., Pinto P. A., Martin P. G., Weaver T. A., 1987, ApJ, 318, 664 Yaron O., Gal-Yam A., 2012, PASP, 124, 668", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "We report on new VLA radio observations performed at 1.4 GHz and 4.8 GHz with unprecedented sensitivity and angular resolution ( ∼ 1 arcsec) of the cool core cluster A 2626, which is known to possess a radio mini-halo at its center. The most unusual features of A 2626 are two elongated radio features detected in previous observations to the north and south, having morphologies not common to the typical jet-lobe structures in cool cores. In our new sensitive images the two elongated features appears clearly as bright radio arcs, and we discover the presence of a new arc to the west. These radio arcs are not detected at 4.8 GHz, implying a steep ( α > 1) spectrum, and their origin is puzzling. After subtracting the flux density contributed by these discrete features from the total flux measured at low resolution, we estimate a residual 18 . 0 ± 1 . 8 mJy flux density of diffuse radio emission at 1.4 GHz. We therefore confirm the detection of diffuse radio emission, which appears distinct from the discrete radio arcs embedded in it. Although its radio power is lower (1 . 4 × 10 23 WHz -1 ) than previously known, the diffuse emission may still be classified as a radio mini-halo. Key words: Galaxies: clusters: individual: Abell 2626 - Radio continuum: galaxies - galaxies: jets - galaxies: cooling flows", "pages": [ 1 ] }, { "title": "M. Gitti 1 , 2 , 3 /star", "content": "1 Physics and Astronomy Department, University of Bologna, via Ranzani 1, 40127 Bologna, Italy 2 INAF, Astronomical Observatory of Bologna, via Ranzani 1, 40127 Bologna, Italy 3 INAF, Istituto di Radioastronomia di Bologna, via Gobetti 101, I-40129 Bologna, Italy Accepted 2013 August 14", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The central dominant (cD) galaxies of cool core clusters have a high incidence of radio activity, showing the presence of central FR-I radiogalaxies in 70% of the cases (Burns 1990; Best et al. 2007; Mittal et al. 2009). Their behaviour differs from that of quasar: in many low-accretion-rate AGNs almost all the released energy is channelled into jets because the density of the gas surrounding the black hole is not high enough for an efficient radiation (e.g., Churazov et al. 2005). In fact, the importance of these objects has been underestimated for a long time due to their poor optical luminosity, and began to emerge after the discovery, with the X-ray satellite ROSAT , of deficits in the X-ray emission of the Perseus and Cygnus A clusters which are spatially coincident with regions of enhanced synchrotron emission (Boehringer et al. 1993; Carilli et al. 1994). With the advent of the new high-resolution Xray observations performed with Chandra and XMM-Newton , it became clear that the central radio sources have a profound, persistent effect on the ICM - the central hot gas in many cool core systems is not smoothly distributed, but shows instead 'holes' on scales often approximately coincident with lobes of extended radio emission. The most typical configuration is for jets from the central dominant elliptical of a cluster to extend outwards in a bipolar flow, inflating lobes of radio-emitting plasma (radio 'bubbles' ). These lobes push aside the X-ray emitting gas of the cluster atmosphere, thus excavating depressions in the ICM which are detectable as apparent 'cavities' in the X-ray images. Radio galaxies have thus been identified as a primary source of feedback in the hot atmospheres of galaxy clusters and groups (for recent reviews see Gitti et al. 2012; McNamara & Nulsen 2012, and references therein). In some cases, the powerful radio galaxies at the center of cool core clusters are surrounded by diffuse radio emission on scales ∼ 200 -500 kpc having steep radio spectra ( α > 1; S ν ∝ ν -α ). These radio sources, generally referred to as 'radio mini-halos' , are synchrotron emission from GeV electrons diffusing through µ G magnetic fields. Although the central radio galaxy is the obvious candidate for the injection of the population of relativistic electrons, mini-halos do appear quite different from the extended lobes maintained by AGN, therefore their radio emission proves that magnetic fields permeate the ICM and at the same time may be indicative of the presence of diffuse relativistic electrons. In particular, due to the fact that the radiative lifetime of radio-emitting electrons ( ∼ 10 8 yr) is much shorter than any reasonable transport time over the cluster scale, the relativistic electrons responsible for the extended radio emission from mini-halos need be continuously re-energized by various mechanisms associated with turbulence in the ICM (reaccelerated primary electrons), or freshly injected on a cluster-wide scale (e.g. as a result of the decay of charged pions produced in hadronic collisions, secondary electrons). Gitti et al. (2002) developed a theoretical model which accounts for the origin of radio mini-halos as related to electron reacceleration by magnetohydrodynamic (MHD) turbulence, which is amplified by compression in the cool cores. In this model, the necessary energetics to power radio mini-halos is supplied by the cooling flow process itself, through the compressional work done on the ICM and the frozenin magnetic field. Although secondary electron models have been proposed to explain the presence of their persistent, diffuse radio emission on large-scale in the ICM (e.g., Pfrommer & Enßlin 2004; Keshet & Loeb 2010), the observed trend between the radio power of mini-halos and the maximum power of cooling flows (Gitti et al. 2004, 2012) has given support to a primary origin of the relativistic electrons radiating in radio mini-halos, favored also by the successful, detailed application of the Gitti et al. (2002) model to two cool core clusters (Perseus and A 2626, Gitti et al. 2004) and by recent statistical studies (Cassano et al. 2008). However, the origin of the turbulence necessary to trigger the electron reacceleration is still debated. The signatures of minor dynamical activity have recently been detected in some mini-halo clusters, thus suggesting that additional or alternative turbulent energy for the reacceleration may be provided by minor mergers (Gitti et al. 2007) and related gas sloshing mechanism in cool core clusters (Mazzotta & Giacintucci 2008; ZuHone et al. 2013). Radio mini-halos are rare, with only about a dozen objects known so far (Feretti et al. 2012). The criteria initially adopted by Gitti et al. (2004) to select the first sample of mini-halo clusters, i.e. the presence of both a cool core and a diffuse, amorphous radio emission with no direct association with the central radio source, having size comparable to that of the cooling region, are now typically used to identify mini-halos. However, these criteria are somehow arbitrary (in particular, total size, morphology, presence of cool core) and some identifications are still controversial. Furthermore, we stress that the classification of a radio source as a minihalo is not trivial: their detection is complicated by the fact that the diffuse, low surface brightness emission needs to be disentangled from the strong radio emission of the central radio galaxy and of other discrete sources. A 2626 is a low-redshift (z=0.0553, Struble & Rood 1999), regular, relatively poor Abell cluster (Mohr et al. 1996), with a double-nuclei cD elliptical galaxy (IC 5338) showing extended strong emission lines (Johnstone et al. 1987). Early observations with Einstein and ROSAT , then confirmed by Chandra and XMMNewton , indicate that it hosts a moderate cooling flow (White et al. 1991; Rizza et al. 2000; Wong et al. 2008). Early, low resolution radio surveys showed that this cluster contains a central radio source exhibiting a compact unresolved core and a diffuse structure with very steep radio spectrum (Slee & Siegman 1983; Roland et al. 1985; Burns 1990). The compact radio component is associated with the southwest nucleus of the cD galaxy IC 5338 (Owen et al. 1995). More recently, high resolution VLA images of Gitti et al. (2004) highlighted the unusual properties of the A 2626 radio source. These authors found that at 1.4 GHz the central component consists of an unresolved core plus a small jet-like feature pointing to the south-western direction. The extended emission, that at lower resolution appears as a diffuse diamond-shaped emission detected up to ∼ 1 ' from the cluster center, is resolved out and two elongated parallel features of similar brightness and extent are visible. The total flux density of these 'bar' structures is ∼ 6 . 6 mJy, contributing to ∼ 20% of the flux of the diffuse radio emission detected at low resolution. Such symmetric, elongated features are imaged also by low resolution observations at 330 MHz, with a total flux density (including the diffuse emission) of ∼ 1 Jy, whereas no radio emission is detected at the location of the core. Gitti et al. (2004) argue that the two unusual elongated features are distinct from and embedded in the diffuse extended radio emission, which they clas- a radio mini-halo and successfully modeled as radio emission from relativistic electrons reaccelerated by MHD turbulence in the cooling core region. On the other hand, the origin and nature of the two radio bars is not clear as they are not associated to any X-ray cavities (Wong et al. 2008). In this paper we present new high-resolution VLA data of the central radio source in A 2626, whose morphology much complex than that of the standard X-ray radio bubbles seen in other cool core clusters represents a challenge to models for the ICM - radio source interaction. With H 0 = 70 km s -1 Mpc -1 , and Ω M = 1 -Ω Λ = 0 . 3, the luminosity distance of A 2626 is 246.8 Mpc and 1 arcsec corresponds to 1.1 kpc.", "pages": [ 1, 2 ] }, { "title": "2 OBSERVATIONS AND DATA REDUCTION", "content": "We performed new Very Large Array 1 observations of the radio source A 2626 at 1.4 GHz and 4.8 GHz in A- and B-configuration (see Table 1 for details regarding these observations). In all observations the source 3C 48 (0137 + 331) was used as the primary flux density calibrator, while the sources 0016 -002 and 3C 138 (0521 + 166) were used as secondary phase and polarization calibrators, respectively. Data reduction was done using the NRAO AIPS (Astronomical Image Processing System) package, version 31DEC13. Accurate editing of the (u,v) data was applied with the task TVFLG to identify and remove bad visibility points. Images were produced by following the standard procedures: Calibration, Fourier-Transform, Clean and Restore. Self-calibration with the task CALIB was applied to remove residual phase variations. The four datasets, taken at different frequencies and with different configurations, were reduced separately in order to optimize the accuracy of data editing and (self-)calibration of each single observation. In order to fully exploit the relative advantages in terms of angular resolution and sensitivity of the two VLA configurations used for the observations, we then combined together (with the task DBCON ) the resulting (u,v) data in A- and B- configuration taken at the same frequency. On these combined A + B datasets, one at 1.4 GHz and one at 4.8 GHz, we also performed a few more iterations of self-calibration to improve the dynamic range of the images. We produced images in the Stokes parameters I, Q and U at different resolutions by specifying appropriate values of the parameters UVTAPER and ROBUST in the AIPS task IMAGR . The images of the polarized intensity, the fractional polarization and the position angle of polarization were derived from the I, Q and U images. The final maps show the contours of the total intensity. 1 The Very Large Array (VLA) is a facility of the National Radio Astronomy Observatory (NRAO). The NRAO is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. For the only purpose of deriving the total flux density of the radio source, including the diffuse emission, we also analyzed old 1.4 GHz data obtained with the VLA in C-configuration (Proj. code AM735, PI: T. Markovic). In this 3.5 h archival observation, performed in spectral-line mode, the source 3C 48 was used as the primary flux density calibrator, while the source 0204 + 152 was used as secondary phase calibrator. Typical amplitude calibration errors are at 3%, therefore we assume this uncertainty on the flux density measurements.", "pages": [ 2, 3 ] }, { "title": "3 RESULTS", "content": "Figure 1 shows the 1.4 GHz radio image of A2626 at full resolution (restoring beam of 1 '' . 2 × 1 '' . 2), obtained with pure uniform weighting by setting ROBUST=-5 , whereas Figure 2 shows the contour map obtained by tempering the uniform weights with ROBUST=0 (restoring beam of 1 '' . 7 × 1 '' . 6), which slightly improved the dy- image. With these high resolutions it is possible to image the central source and to spot the presence of other discrete features that can contribute to the total flux density and extended morphology detected at lower resolution. The nuclear source, located at RA (J2000): 23 h 36 m 30 s . 5, Dec (J2000): 21 · 08 ' 47 '' . 6, is here fully unveiled: it clearly shows two jets pointing to the northeast-southwest direction, extending out from the unresolved core to a distance of approximately 5 kpc. The total flux density of the core-jets structure is 17 . 7 ± 0 . 5 mJy. In these sensitive images the already known elongated radio features stand out very clearly as bright radio arcs (arc N and arc S in Fig. 1). They are symmetrically located at ∼ 25 kpc to the north and south of the core, showing total longitudinal extensions of approximately 70 and 62 kpc, respectively. Their combined total flux density is 16 . 2 ± 0 . 5 mJy, which is a factor ∼ 2 . 5 higher than previously estimated (Gitti et al. 2004). Furthermore, we identify a new feature which was not detected in the previous observations: a faint (3 . 1 ± 0 . 1 mJy), elongated arc to the west of the core (arc W in Fig. 1), which extending for about 60 kpc appears to ideally connect the western edges of the two radio arcs N-S. No diffuse emission is seen at this high resolution and no significant polarized flux is detected at 1.4 GHz at any resolution. We produced an image at the lower resolution of 4 '' . 4 × 3 '' . 9, obtained with tapered natural weighting by setting ROBUST=5 and UVTAPER=90 (Fig. 3), to better map the extended structures. Diffuse 1.4 GHz emission appears clearly surrounding the nucleus in the region encompassed by the features (arc N, arc S and arc W) discussed above. In order to estimate correctly the flux density of the diffuse emission, we analyzed the archival C-array data and produced a 1.4 GHz map at a resolution of 14 '' . 4 × 12 '' . 6 (not shown here). The total source A 2626 imaged with this low resolution has a well-known diamond-like shape extending for approximately 2 ' . 9 × 2 ' . 0 (156 × 135 kpc), with a total flux density of 55 . 0 ± 1 . 7 mJy (source W excluded), in agreement with previous observations (Ledlow & Owen 1995; Rizza et al. 2000). By subtracting the emission contributed by the discrete features seen at high resolution (core + jets + arcs = 37 . 0 ± 0 . 6 mJy, see Table 2) from the total flux density of the C-array 1.4 GHz emission, we thus measure a residual flux density of diffuse radio emission of 18 . 0 ± 1 . 8 mJy. The more crude approach of measuring directly the flux density inside the contours of the diffuse emission visible in the map in Fig. 3 would provide an estimate of ∼ 5 mJy. However, we note that this is certainly a lower limit as the A + B array data misses the flux contribution of the short baselines which are instead present in the C- array data, thus losing sensitivity to diffuse radio structures. We also attempted to derive the flux of the discrete features from the Carray map directly, getting a rough estimate of ∼ 40 mJy (unresolved central component + arcs). Considering the difficulty in separating the emission from each component at low resolution, leading to big uncertainties, this estimate should be considered not far from the accurate one derived from our high resolution A + B array data. The variety of methods discussed here demonstrates the complexity of disentangling the relative contribution of the diffuse emission and of the discrete sources to the total observed radio emission. Figure 4 shows the 4.8 GHz radio image of A 2626 at a resolution of 2 '' . 3 × 2 '' . 0, obtained with tapered natural weighting by setting ROBUST=5 and UVTAPER=90 . At this higher frequency only the central component is visible, showing an unresolved core plus jet-like features pointing to the northeast-southwest directions 2 . The total flux density is 9 . 8 ± 0 . 3 mJy (jet-like features included). The core appears slightly polarized at a level of ∼ 4%. The three discrete arcs seen at 1.4 GHz are not detected, implying a steep spectral index (see Table 2). We finally note that a discrete source is visible to the west of A 2626 in all radio maps (source W in Fig. 1), located at RA(J2000):23 h 36 m 25 s . 1, Dec(J2000):21 · 09 ' 03 '' , which is associated to the cluster S0 galaxy IC 5337. The morphology of this source suggests that it is a head-tail radio galaxy. The radio results are summarized in Table 2, where we also report the 1.4 GHz monochromatic radio power.", "pages": [ 3, 4 ] }, { "title": "4.1 Radio Arcs", "content": "Since their discovery (Gitti et al. 2004), the nature of the two elongated radio features to the north and south directions (arc N and arc S in Fig. 1) has been a puzzle. Their symmetric positions on each side of the core suggests that they are radio bubbles, but their thin, elongated shapes are unlike those typically observed in cool core clusters. They may represent radio emitting plasma injected by the central source during an earlier active phase, which has then propagated through the cool core region in the form of buoyant subsonic plumes (e.g., Gull & Northover 1973; Churazov et al. 2000; Bruggen & Kaiser 2001). However, such plumes are expected to have a torus-like concavity (Churazov et al. 2000), contrarily to the observed shape in A 2626. Furthermore, by comparison to radio lobes and bubbles associated with other cool core dominant radio galaxies, one might expect these to be regions of reduced X-ray emission surrounded by bright rims. In fact, previous ROSAT observations failed to find strong X-ray deficit in the cool core of A 2626 (Rizza et al. 2000). Recently, this cluster has been studied in more detail with Chandra and XMM-Newton , but yet no X-ray cavities associated with the elongated radio features have been found (Wong et al. 2008). Wong et al. (2008) argue that the lack of obvious correlation between the two symmetric parallel radio features and any structures in the X-ray images may indicate that they are thin tubes parallel to the plane of the sky, or that the radio plasma is mixed with the X-ray gas, rather than displacing it. These authors suggest that jet precession might also provide an alternative explanation of the peculiar radio morphology. If two jets ejected towards the north and south by the southwest cD nucleus are precessing about an axis which is nearly perpendicular to the line-of-sight and are stopped at approximately equal radii from the AGN (at a 'working surface'), radio emission may be produced by particle acceleration, thus originating the elongated structures. The impressive arc-like, symmetric morphology of these features highlighted by the new high resolution radio images may support this interpretation. The discovery of a third elongated feature to the west (arc W in Fig. 1) further complicates the picture. It could represent another radio bubble ejected in a different direction, similarly to what observed in RBS 797 (Gitti et al. 2006), but again the absence of any correlation with the X-ray image and its 'wrong' concavity, as well as the absence of its counterpart to the east, disfavor this interpretation. In the model proposed by Wong et al. (2008), it could also represent the result of particle acceleration produced at a working surface by a third jet ejected to the west. This interpretation would imply the existence of radio jets emanating also from the northeast nucleus (separated by only ∼ 4 kpc from the active southwest nucleus) of the cD galaxy IC 5338, which however does not show a radio core. We stress that the radio arcs have an elongated morphology and steep spectral index (see Table 2). These characteristics are similar to those of cluster radio relics associated to particle reacceleration due to shocks (Feretti et al. 2012). We also note that each arc resembles the morphology of the large-scale structure of the radio source 3C 338, which is disconnected from the core emission having two-sided jets and has been interpreted as a relic structure not related to the present nuclear activity (Giovannini et al. 1998). The presence of three such arcs in A 2626 makes the interpretation even more puzzling. Seen all together, the combined shape of the three arcs suggests that they could trace the symmetric fronts of a single elliptical shock originating from the core. However, we notice again that their concavity is not what one would expect from a shock propagating from the center. If they are cluster radio reliclike sources due to reacceleration at a bow shock, the observed concavity suggests that three distinct shocks should be propagating toward the center from different directions. However, the presence of such symmetric shocks in the atmosphere of a relatively relaxed cluster seems very unlikely, given also that Chandra observations failed to detect any obvious X-ray edge ascribable to shock fronts. Furthermore, relic sources are typically strongly polarized (Feretti et al. 2012), whereas no significant polarized flux is detected in our data (see Sect. 3). It is also possible that the nature of the two bright radio arcs NS is different from that of the fainter arc W, which lacks an obvious counterpart to the east and could be related to the merging of the nearby S0 galaxy IC 5337 (see Fig. 16 of Wong et al. (2008) for a possible correlation with the Chandra hardness ratio map). Therefore the complex morphology of the A 2626 radio source may result from a combination of the scenarios presented above. As it appears evident from this discussion, the new highresolution VLA data are not definitive to solve the puzzle of the origin and nature of the radio arcs in A 2626.", "pages": [ 4, 5 ] }, { "title": "4.2 Diffuse radio emission", "content": "With the new sensitive observations presented here, which improve by a factor ∼ 3 the r.m.s. noise of the published maps (Gitti et al. 2004), we confirm the detection of diffuse 1.4 GHz emission, having a radio power of P 1 . 4 = 1 . 4 × 10 23 WHz -1 . As imaged at this high resolution ( ∼ 4 '' ), the diffuse radio emission appears to be confined in the region encompassed by the three radio arcs, surrounding the nucleus, and shows a fragmented morphology with total size ∼ 60 kpc. Although lower than previously estimated, the radio power of the diffuse emission in A 2626 still follows the trend with the maximum power of cooling flows, which is expected in the framework of the model proposed by Gitti et al. (2002, 2004). This model links the origin of radio mini-halos to radio emission from relativistic electrons reaccelerated by Fermi mechanism associated with MHD turbulence amplified by the compression of the magnetic field in the cooling core region, thus supporting a direct connection between cool cores and radio mini-halos (see Fig. 3b of Gitti et al. (2012) for a recent version of the observed trend). The relativistic electrons responsible for the diffuse emission may also have been reaccelerated by turbulence generated by the sloshing of the cool core gas (Mazzotta & Giacintucci 2008; ZuHone et al. 2013). The presence of three symmetric sloshing sub-clumps in the cluster atmosphere, although unlikely, might also induce local electron (re)acceleration thus explaining the origin of the three radio arcs discussed in Sect. 4.1.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "MGthanks G. Giovannini and D. Dallacasa for helpful advices during the data reduction, and S. Giacintucci and K. Wong for providing comments on the original manuscript. MG acknowledges the financial contribution from contract ASI-I/009/10/0.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Best P. N., von der Linden A., Kauffmann G., Heckman T. M., Kaiser C. R., 2007, MNRAS, 379, 894 Boehringer H., Voges W., Fabian A. C., Edge A. C., Neumann D. M., 1993, MNRAS, 264, L25 Bruggen M., Kaiser C. 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[ { "title": "Temperature of Electron Fluctuations in an Accelerated Vacuum", "content": "Lance Labun ∗ and Johann Rafelski Department of Physics, The University of Arizona, Tucson, AZ 85721, USA ∗ E-mail: [email protected] The electron vacuum fluctuations measured by 〈 ¯ ψψ 〉 do not vanish in an externally applied electric field E . For an exactly constant field, that is for vacuum fluctuations in presence of a constant accelerating force, we show that 〈 ¯ ψψ 〉 has a Boson-like structure with spectral state density tanh -1 ( E/m ) and temperature T M = e E /mπ = a v /π . Considering the vacuum fluctuations of 'classical' gyromagnetic ratio g = 1 particles we find Fermi-like structure with the same spectral state density at a smaller temperature T 1 = a v / 2 π which corresponds to the Unruh temperature of an accelerated observer. Keywords : acceleration; quantum vacuum; nonperturbative QED", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Muller et al. 1 showed that in presence of constant external electromagnetic fields the Heisenberg-Euler effective potential of QED can be cast in the form of a thermal background characterized by a spectral density of states ρ ( E ) and temperature T M Since an electric field accelerates all charged particles, real or virtual, this can be understood as a property of the vacuum under constant global acceleration 2 a v = e E /m . This circumstance has also been discussed by Pauchy Hwang and Kim. 3 Associated with this result is the quantum statistics: spin-1/2 QED exhibits a thermal distribution as though the fluctuating degrees of freedom are bosons, while spin-0 scalar QED exhibits vacuum fluctuations as though the degrees of freedom are fermionic. The former result could be understood as being due to pairing of electron-positron pairs. However the latter case has no obvious explanation. The case of 'accelerated vacuum' parallels that of an 'accelerated observer' traveling in a matter- and field-free spacetime. This observer sees a thermal background characterized by the Unruh temperature The statistics of the thermal distribution are bosonic considering the vacuum of a scalar particle 4,5 and fermionic in the vacuum of a fermi particle. 6 Many readers will expect that there should not be a difference between 'accelerated vacuum' and 'accelerated observer', yet there is a disagreement in relation to particle spin and statistics and the value of the temperatures Eq. (1) & Eq. (2). We will show that these two results Eq. (1) & Eq. (2) are experimentally distinguishable. This implies the ability to determine which is accelerated, the observer or the vacuum state. The question is whether or not one should be able in-principle to determine the accelerated state within the current formulation of quantum field theory. If the conclusion is that one should not be able to determine which is accelerated, then there is additional undiscovered physics content in either the vacuum structure or the Unruh accelerated detector. It is of considerable interest to find a model achieving agreement in the quasithermal properties of the quantum vacuum, in the sense that the accelerated observer registers the same outcome as the 'accelerated' field-filled vacuum. The disagreement in relation to particle spin suggests a closer look at the spin properties of the fluctuations. Altering the magnetic moment of the electron from its Dirac value described by the gyromagnetic ratio g = 2 to the 'classical' spinning-particle value g = 1 achieves agreement with the Unruh temperature and statistics. 7 We summarize the results obtained and consider the properties of QED vacuum condensate 〈 ¯ ψψ 〉 in presence of strong quasi-constant external fields in the thermal framework modeled by the 'classical' g = 1 QED.", "pages": [ 1, 2 ] }, { "title": "2. Effective Action", "content": "Charge convective current and spin magnetic dipole current are conserved independently and thus in QED the associated integral 'charges' - the electric charge and magnetic dipole moment can be prescribed arbitrarily. The gyromagnetic ratio g combines with particle charge and mass in defining the magnitude of the magnetic dipole moment. The dynamics of a particle ψ with general g is generated by where D = i∂ + eA is the covariant derivative, F µν the electromagnetic field strength tensor and σ µν = ( i/ 2)[ γ µ , γ ν ]. Only for the specific case g = 2 can one choose to write this in the Dirac equation form. Any value of the gyromagnetic ratio g can arise, and as long as the quantization of charge is not understood, it is difficult to claim that the value of g arising in a specific simplified dynamical equation is of greater interest than other values. Like charge, g is the subject of experimental and theoretical effort to determine quantum corrections to an input value, the best known being the QED correction to the Dirac particle g = 2 + α/π + . . . . The effect of g on vacuum fluctuations is determined from the effective potential For the Heisenberg-Euler case of a constant electric-only field of strength E , we evaluate the trace using the proper time method 8 in which the generalized degeneracy γ s counts the number and type of degrees of freedom. γ s = 4 when g = 2 for spin-1/2 Dirac electron, or γ s = -2 when g = 0 for a spin-0 electron. The -1 inside the parentheses removes the field-independent constant. Transforming V eff to a statistical format proceeds via meromorphic expansion of the integrand of Eq. (5). 1,7 The finite (regularized and renormalized) effective potential is with integration contour defined by the usual assignment This also makes explicit that the effective potential contains an imaginary part, as will be discussed below. Setting g = 2 for a spin-1/2 (Dirac) electron, cos 2 nπ = 1 for all n , and setting g = 0 for a spin-0 electron, cos nπ = ( -1) n producing an alternating sum. For each case, integrating by parts twice and summing the series yields the results of Muller et al. 1 The exponential weights of the terms in the series in Eq. (6) generate a thermal distribution, and the statistics of the distribution are determined by the phase of the terms in the series. Summing with arbitrary g , the effective potential is The sum over ± ensures the distribution is real so that the imaginary part arises only from the branch cut in the first log factor. Restoring g = 2 and γ s = 4 identifies the spectral density of states Eq. (1) according to V ≡ T ∫ ∞ 0 ln(1 -e -E/T ) ρ ( E ) dE . For g = 1 summing over ± simplifies it to exhibiting in the second log factor a thermal fermionic distribution controlled by the Unruh temperature, T U . The effective potential of a 'classical spinning electron' with g = 1 in a constant field thus has the format of a thermodynamic potential with temperature parameter and statistics in agreement with the expectation of an accelerated observer in the (unaccelerated) vacuum of a fermion field.", "pages": [ 2, 3 ] }, { "title": "3. Condensate", "content": "The quantum fluctuations induced by the external field are measured by the condensate 〈 ¯ ψψ 〉 , which is the difference of the Green's functions in the vacuum with the external field and the vacuum with no field present (0 superscript) 〈T ... 〉 is the vacuum expectation of time-ordered operators, the limit x ' → x is evaluated in the point-splitting procedure to preserve gauge invariance, and the F subscript indicates Feynman boundary conditions. This definition of the condensate displays the implicit definition of the reference, no-field vacuum state. The condensate is related to the effective potential by Note that the differentiation with respect to m improves the convergence properties of Eq. (5). Evaluating the derivative of Eq. (5), we find the formerly logarithmically divergent contribution is finite. Because -m 〈 ¯ ψψ 〉 includes this term quadratic in E (see discussion in 9 ), we use now the meromorphic series The resulting expression for 〈 ¯ ψψ 〉 for arbitrary g is Setting g = 0 ( g = 2) yields a fermionic (bosonic) distribution controlled by T M Identifying the numerator of the integrand tanh -1 ( z ) by analogy with statistical physics 〈 N 〉 /V = ∫ ∞ 0 Γ( E ) dE/ ( e E/T ± 1), the degeneracy of states is the same in each value of g considered. For g = 1, the terms containing cosine in the numerator and denominator of Eq.(13) vanish, leaving the fermi occupancy factor in the denominator with twice the Euler-Heisenberg temperature 2 /T M = 1 /T U . which displays the fermionic occupancy factor in the denominator with a distribution controlled by the Unruh temperature. The fluctuations of a g = 1 'electron' in an external electric field are thus reconciled with the fluctuations expected by an observer accelerated at a v = e E /πm through the (unaccelerated) vacuum of a fermion field.", "pages": [ 4 ] }, { "title": "4. Observable Effects from g = 1", "content": "One observable effect is spontaneous pair production in strong fields discussed by us earlier, 7 and originating in the imaginary part of the effective action Eq. (6). This allows any non-vanishing electric field to spontaneously decay into charged particle pairs. g = 0 , 2 yield the largest total decay probability and are in this sense the best cases for experiment. For g = 1 the reduction in the effective temperature parameter is largest. Due to the exponential dependence, the reduction in the temperature parameter by factor 2 reduces spontaneous pair production below the critical field E c = m 2 /e by many orders of magnitude. The real part of the effective potential generates nonlinear field-field interactions. These interactions are exhibited order-by-order in the field by expanding V eff in a (semi-convergent) power series in ( e E ) 2 n . The power series representation of V eff is obtained by expanding the proper time integrand of Eq. (5) for e E u /lessmuch 1 At each order in ( e E ) 2 , we have separated the numerical coefficients for g = 2 outside the parentheses for ease of comparison to the g = 1 result, For example, the ratio of the coefficients of the ( e E ) 4 terms is V eff ( g = 1) /V eff ( g = 2) /similarequal 7 / 128. We see that nonlinear field-field interactions generated by this potential are suppressed in the g = 1 case relative to the g = 2 (or g = 0) electron. This outcome is consistent with the suppression of the imaginary part of V eff . Experiments seeking nonlinear field effects 10,11 derived from the Euler-Heisenberg effective potential are thus also sensitive to the effective value of g .", "pages": [ 5 ] }, { "title": "5. Discussion and conclusions", "content": "In summary, we have recalled that in a constant electric field E , the electron fluctuations 〈 ¯ ψψ 〉 display a thermal Bose spectrum with temperature T M = e E /mπ = a v /π Eq.(14). This result contrasts with the Fermi spectrum and Unruh temperature T U = a U / 2 π expected from viewing the vacuum fluctuations of the electrons as accelerated. We have calculated 〈 ¯ ψψ 〉 in an electric field for the gyromagnetic ratio g = 0 , 1 , 2. Setting g = 1, as though considering the quantum fluctuations of a 'classical spinning particle', displays the Unruh T U = a/ 2 π and a Fermi spectrum, see Eq. (16). We highlight the functional dependence of light-light scattering on g because it has implications for future experiments. Any (effective) value of g deviating from the Dirac value g = 2 results in a suppression of the rate of light-light scattering. We note that QED is not yet tested near the critical field strength E c = m 2 /e , and in this strong-field regime, we have still to validate the approach to calculating Eq. (5), which is perturbative in α . 12 Even more to the point there are serious questions about validity of QED in this limit. 13 Therefore, the connection which we established to reconcile the two ways of viewing acceleration could forebear forthcoming theoretical developments. An observable, physical difference such as now predicted in Eqs. (2) and (1) provides an in-principle test to determine whether the strong field theory is valid, and/or it is the observer or the vacuum that is accelerated. Being able to determine who is accelerated means a fixed reference frame has been selected and defined as inertial. In quantum theory, the quantum vacuum state is a natural candidate for the fixed reference frame, 13 and here we have recalled that the electron condensate contains in its definition Eq. (10) a specific vacuum state as reference. Experimental observation of quantum vacuum phenomena such as light-light scattering offers an important test of our understanding of the vacuum state canonically selected in quantum field theory and may reveal whether or not it is consistent with the vacuum selected in the Unruh accelerated detector situation. Acknowledgments: L.L. thanks Director Pisin Chen for the opportunity to visit LeCosPA. This work was supported by a grant from the US Department of Energy, DE-FG02-04ER41318.", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "Modern Physics Letters A", "pages": [ 1 ] }, { "title": "RENAT R. ABBYAZOV", "content": "Department of Physics, Ulyanovsk State Pedagogical University named after I.N. Ulyanov, 100 years V.I. Lenin's Birthday Square, 4, 432700 Ulyanovsk, Russia [email protected]", "pages": [ 1 ] }, { "title": "SERGEY V. CHERVON", "content": "Astrophysics and Cosmology Research Unit School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal Private Bag X54 001, Durban 4000, South Africa ∗ [email protected] Received (Day Month Year) Revised (Day Month Year) We show the way of dark matter and dark energy presentation via ansatzs on the kinetic energies of the fields in the two-component chiral cosmological model. To connect a kinetic interaction of dark matter and dark energy with observational data the reconstruction procedure for the chiral metric component h 22 and the potential of (self)interaction V has been developed. The reconstruction of h 22 and V for the early and later inflation have been performed. The proposed model is confronted to Λ CDM model as well. Keywords : Chiral cosmological model; cosmic acceleration; dark energy; dark matter. PACS Nos.: 98.80.-k, 95.36.+x", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The later-time cosmic acceleration of our Universe is strongly supported by observational data. Namely observations of supernovae type Ia 1 , the data from Baryon Acoustic Oscillations (BAO) 2 and Cosmic Microwave Background (CMB) 3 measurements confirm that the Universe is expending with an acceleration at the present time and about 70% of the energy density consists of dark energy in a wide sense 4 , i.e. as the substance which is responsible for an anti-gravity force. In the range with well-known ΛCDM model, which potentially provides correct description of the Universe evolution but suffers from fine-tuning and coincidence ∗ The permanent address: Department of Physics, Ulyanovsk State Pedagogical University named after I.N. Ulyanov, 100 years V.I. Lenin's Birthday Square, 4, 432700 Ulyanovsk, Russia", "pages": [ 1 ] }, { "title": "2 R.R.Abbyazov, S. V. Chervon", "content": "problems, some alternative models were proposed. We will pay attention to the models with presence of scalar fields included in quintessence, phantom and quintom 5 , 6 , 7 , 8 models. A chiral cosmological model (CCM) as a nonlinear sigma model with a potential of (self)interactions 9 has been already used extensively in various areas of gravitation and cosmology 10 , 11 , 12 and in particular for description of the very early Universe 13 , 14 and inflation 15 , 16 . A CCM can be applicable as well to the late-time Universe with dark matter and dark energy domination as it was shown in 17 . The purpose of this article is to put into use the two-component CCM as the model where the dark energy content of the Universe and also the dark matter component are represented by two chiral fields with kinetic and potential interactions 9 . By considering a target space metric in the form we prescribe a kinetic interaction between chiral fields ϕ and χ as a functional dependence h 22 on the fields. The potential interaction will be included into standard potential energy term of the action. There are no enough indications from observations about kinetic interactions between dark sector fields. Therefore we always deal with the problem: what is the functional dependence for the chiral metric component on the fields? First idea is to attract some results from HEP, for example, to consider SO(3) symmetry (by taking h 22 = sin 2 ϕ ) and/or others symmetries for a chiral space. From the other hand one can use some testing kinetic interactions 12 , 17 . Thus we can state that there is no evidence for some preferable functional form of the kinetic interaction contained in the functional form of the h 22 chiral metric component. To avoid this problem we develop here the reconstruction procedure for the chiral metric component h 22 . We ascribe a certain desirable behavior on the kinetic energy of the second chiral field χ and it becomes possible to determine both the target space metric component h 22 and a (self)interacting potential V depending on the first chiral field ϕ . So we can restore a functional dependence the h 22 and V on the scalar field ϕ using observational data. Unfortunately it turns out that the procedure could not be applied for the entirely Universe evolution and we have necessity to consider separately the early and late epochs of the Universe evolution. It will be shown also that a CCM describes dark energy and dark matter in the unified form under special restrictions on the chiral fields (ansatzs). Therefore to include into consideration the present Universe with accelerated expansion it needs to take into account baryonic matter and radiation in the range with a twocomponent CCM. Making confrontation of proposed model predictions with observational data we found the way of a reconstruction of a kinetic interaction term h 22 and the potential V in an exact form. This reconstruction is based on the procedure of finding the best-fit values matching to the astrophysical observations. The structure of the article is like follow. In section 2, we give the basic model equations and discuss their properties including the exact solutions for a pure CCM (without matter and radiation). We derive the Friedmann equation for the proposed model with the aim to make comparison with ΛCDM in section 3. In section 4, we give the details of a fitting procedure outline. We present the way of the reconstruction of the kinetic coupling and potential in section 4. The early and recent Universe approximations are discussed there as well. Section 6 is devoted to the background dynamics of a CCM. Finally in section 7, we discuss the obtained results and consider perspectives for the future investigations.", "pages": [ 2, 3 ] }, { "title": "2. The model equations and their properties", "content": "Recently we proposed a CCM coupling to a perfect fluid 18 with the aim to investigate chiral fields interaction with CDM. For the sake of shortness we termed this model as σCDM to stress its difference from Λ CDM,QCDM and others models. σCDM model presents a generalization of a single scalar field model coupled to CDM in the form of a perfect fluid 19 . The model is described by the action functional Here S ( pf ) stands for the perfect fluid part of the action, h AB = h AB ( ϕ C ) are the target space metric components depending on the scalar fields ϕ C . The line element of a target (chiral) space is We use shortened notations for the partial derivatives with respect to the spacetime coordinates: ∂ϕ A ∂x µ = ∂ µ ϕ A . As usual g µν ( x α ) denotes a space-time metric as a function on the space-time coordinates, so Greek indices α, µ, ... vary in a range from 0 to 3, Latin capital letters A , B,... - take values from 1 to N where N is evidently corresponding to the chiral fields number. The space-time of homogeneous and isotropic Universe is described by a spatially-flat Friedmann - Robertson - Walker (FRW) metric The two-component CCM has a target space metric simplified to 18 The σCDM (2.1) with internal space metric (2.4) includes the models proposed earlier: cold dark matter and cosmological constant (ΛCDM, when h 11 = h 22 = 0 , V = const = Λ) model 5 , 6 , quintessence model (QCDM, when h 11 = 1 , h 22 = 0), phantom model (PhCDM, when h 11 = -1 , h 22 = 0), quintom model (qCDM, when", "pages": [ 3 ] }, { "title": "4 R.R.Abbyazov, S. V. Chervon", "content": "h 11 = 1 , h 22 = -1) 8 , 20 , 21 , 22 . Thus the model under consideration is a generalization of the models investigated earlier and mentioned above. As a first step of our study we consider the system of equations of the twocomponent CCM without a perfect fluid. Using assumptions h 11 = const and h 22 = h 22 ( ϕ ) expressed in (2.4) one can obtain the system of Einstein and chiral field equations Here H = ˙ a a , (˙) = d dt . When first inflationary models were analyzed it was much attention to a very simple case when an inflationary potential V ( φ ) equals to the constant 23 . Moreover this regime is very important because it leads to an exponential expansion of the Universe. Note that a scalar field is equal to a constant value as well in this regime. Let us consider for a minute the case of V = const for the model under consideration (2.5)-(2.8). From (2.8) one can obtain 16 , 24 Combining (2.5) and (2.6) one can obtain the well-known solution of a de Sitter Universe with Hubble parameter and scale factor 9 This solution with some approximation corresponds to the inflationary stage of the Universe evolution. But our intention is to proceed further in time therefore we need to include into consideration radiation and matter to describe the present epoch of the Universe. The method of the exact solutions construction for a CCM (2.5)-(2.8) is based on exploiting an additional degree of freedom (see, for ex. discussion in 14 ). Namely even we fix the potential V ( φ, χ ) there are still four equations with four unknown functions H, ϕ, χ, h 22 ( h 11 can be set equal to ± 1 without the loss of generality 9 ). Nevertheless the equation (2.5) can be obtained from the linear combination of the chiral field equations (2.7)-(2.8), so the equation (2.5) doesn't independent one. Therefore one may insert the symmetry on the target space or can suggest a testing interaction between chiral (dark sector) fields 12 , 25 . Essentially new approach to this issue we propose here as a reconstruction both h 22 and V from observational data. Let us remind that for the scalar field cosmology by introducing the selfinteracting potential V ( φ ) we have two equations with two unknown functions. (The same situation will be if we set the dependence a scalar field on time or if we know the scale factor of the Universe as a function on time 14 ). To solve the system of a CCM interacting with a perfect fluid (or matter) in explicit form is a very difficult task. Therefore we will use an additional freedom connecting with the chiral metric components h 11 and h 22 as a part of a kinetic energy. An interesting approach for a two-fields model with a cross interaction was proposed in the work 19 . To describe a dark matter component it was constructed the special ansatzs for the time derivatives of the scalar fields. In our approach we will use instead some constraints on the kinetic parts of the chiral fields (ansatzs) to obtain a correct description of the present Universe. Now let us turn our attention to a study of the model equations (2.5)-(2.8). It is easy to check that the solution (2.9) for the constant potential will be valid for the case when V = V ( ϕ ) only. By extracting from (2.9) the kinetic energy term for the field χ one can obtain We can ascribe by suggestion h 22 ∼ a -3 dust matter like behavior to the kinetic energy of the field χ . Using the behavior h 22 ∼ a -3 it is easy to see that the second field can be related to the dark matter term provided the restriction to the kinetic energy of the second field χ (ansatz) Let us mention here, that more simple ansatz 1 2 h 22 ˙ χ 2 = Λ ψ = const has been analyzed in 26 and gave possibility to obtain the exact solutions for the two-component CCM. For the kinetic energy of the first field ϕ we can form the ansatz by a simple way 1 2 h 11 ˙ ϕ = B = const. (2.13) Further we will show that this relation is associated with the dark energy component in the present Universe. For convenience let us represent the ansatzs (2.11), (2.13) in the general forms: 2", "pages": [ 4, 5 ] }, { "title": "6 R.R.Abbyazov, S. V. Chervon", "content": "Thus we have f ( a ) = B = const, g ( a ) = Ca -3 in (2.14)-(2.15) and the chiral metric component Let us note that the suggested restrictions above give rise to the exact solution for the CCM describing by equations (2.5)-(2.8). Indeed from ansatzs we can find the solutions for the chiral fields Then from Einstein equations (2.5)-(2.6) we can define the potential The solution for the scale factor can be obtained from the equation It is difficult to find the scale factor in exact view from this general equation, but for the special case assuming C ∗ = 0 and C = 0 (under this assumption the second field χ becomes a constant), we found that the Universe is in the stage with an exponential expansion with a ∝ exp( Bt 2 ).", "pages": [ 6 ] }, { "title": "3. A CCM coupling to barion matter and radiation. Friedmann equation of the model", "content": "Our following task is to connect the energy densities of various species of the Universe to the Hubble parameter. To this end we need to include into Friedmann equation (2.5) the energy density of barion matter ρ b and radiation ρ r . Thus (2.5) for the recent Universe takes the form where ρ σ = 1 2 h 11 ˙ ϕ 2 + 1 2 h 22 ˙ χ 2 + V . Introducing the 'pressure' of chiral fields p σ = 1 2 h 11 ˙ ϕ 2 + 1 2 h 22 ˙ χ 2 -V and using ansatzs (2.14) and (2.15) we can obtain Using (2.18) and extracting the cosmological parameter Λ from V ∗ the energy density, potential and pressure of the two-component CCM can be expressed as Unified dark matter and dark energy description in a chiral cosmological model 7 By standard way (see, for ex. 27 ) one can define a critical density ρ c = 3 H 2 0 8 πG , where H 0 is the Hubble parameter of today expansion H 0 = ˙ a a ( t 0 ) . Herefrom the subscript '0' is related to the present time t 0 when the scale factor a ( t 0 ) = a 0 = 1 . Also we will use the density parameter Ω 0 = ρ ρ c ( t 0 ) and the individual rations Ω i = ρ i ρ c ( t 0 ) for chiral fields, barion matter and radiation. Let us remember that equations of state for radiation and baryons are The energy densities and the contribution to the critical density can be represented as Taking into account (3.2) Friedmann equation (3.1) can be transformed to the normalised Hubble parameter form Making renormalization of the constants we finally obtain the normalised Hubble rate in the form which is suitable for further confronting with observational data where We need to find ˜ Λ at a = a 0 = 1 with the help of Friedmann equation. Cold dark matter (CDM) is included in the model as the kinetic ansatz (2.11) Summing up the notations above we display the final form of the normalised Hubble parameter We propose here the σ CDM model containing the dark energy with variable equation of state. The model is an alternative to ΛCDM model with the cosmological constant and CDM. Let us note that generally speaking the presence of ˜ B in (3.8) may change the values of Ω m 0 and Ω Λ0 , thus they can be distinctive from the corresponding quantities in ΛCDM model. Nevertheless to find the exact values of this distinction we need to perform comparison with the experimental data.", "pages": [ 6, 7 ] }, { "title": "4. Comparison with experimental data", "content": "From the very beginning 28 , 29 supernovae Ia type observations directly indicated an accelerated expansion of the Universe. Observing supernovae luminosity distance d L as a function of a redshift one can infer about an expansion history of the Universe. Here we use one of the most recent compilation of the supernovae sets Union 2.1 1 . The procedure of confronting cosmological model predictions with observations consists of minimizing quantity procedure and calculation as a result best-fit values of the model parameters for Here as usual in the supernovae experimental analysis 30 module distance µ ( z i ) is used. The dependence on the luminosity distance is In order to find more accurate parameter values and to reduce errors significantly it is necessary to supplement the supernovae observations with information about baryonic acoustic oscillations (BAO) 2 and cosmic microwave background (CMB) 3 . BAO χ 2 function is defined as where is an effective distance measure, while is the angular diameter distance 2 . Function χ 2 for CMB is where x i = ( l A , R, z ∗ ) - the vector of quantities which characterizes the cosmological model and ( C -1 ) ij -WMAP7 covariance matrix 3 . Here we use acoustic scale, Unified dark matter and dark energy description in a chiral cosmological model 9 from which first acoustic peak of CMB power spectrum is depending on 4 which has been taken at the moment z ∗ of decoupling of radiation from matter, and on the sound horizon We will use the fitting formula for decoupling moment 31 . Shift parameter R is defined as Minimizing the sum χ 2 joint = χ 2 SN + χ 2 BAO + χ 2 CMB one can find the bestfit ˜ B and ˜ C values. We also keep fixed the radiation and baryonic contributions to the critical density today Ω γ 0 = 2 . 469 · 10 -5 h -2 , Ω b 0 = 0 . 022765 · 10 -2 , h = 0 . 742. Also we take into account a relativistic neutrino in addition to the photon radiation component Ω r 0 = (1 + N eff )Ω γ 0 , where N eff = 3 . 04 - the effective neutrino number 30 . Our results for the best-fit from χ 2 joint minimization are ˜ B = 0 . 00078 , Ω σm 0 = Ω b 0 +2 ˜ C = Ω b 0 +Ω σcdm 0 = 0 . 23398. For ΛCDM model we take best-fit values Ω m 0 = 0 . 27 and Ω Λ0 = 1 -Ω m 0 -Ω r 0 . To avoid confusion between Ω cdm and Ω m in σ CDM and ΛCDM models we put additional index σ in Ω above.", "pages": [ 8, 9 ] }, { "title": "5. The reconstruction of the metric component h 22 and the potential V", "content": "In order to learn more about kinetic and potential interactions between DM and DE we must extend the standard reconstruction of the expansion history of the Universe 32 to a restoration of a functional dependence on the scalar field ϕ for the target space metric component h 22 = h 22 ( ϕ ) and the potential V = V ( ϕ ) of σ CDM. Let us transform the ansatz (2.14) (setting h 11 = 1) Changing variables from t to a and introducing already known for us (3.8) Hubble parameter one can obtain Our next goal is to find the dependence ϕ = ϕ ( a ), so we fix limits of integration from some early epoch a i up to desired time moment, corresponded to a The integral written here cannot be taken in an explicit form. Nevertheless there is a possibility to use some approximations based on a behavior of the different energy densities components with a scale factor. Therefore following the idea of 19 we consider the early a glyph[lessmuch] 1 and recent Universe a ≈ 1 approximations. Let us start from the case a glyph[lessmuch] 1. The Universe is known to be radiation dominated at the very early times. This means that all components contribution in Hubble parameter is negligible in comparison with the radiation term. So such observation makes possible to do integration in We can normalize the scalar field on today's critical density ϕ → ϕ √ ρ c 0 , which gives us ˜ B instead of B in (5.3). Here ϕ ( a i ) = ϕ ( a = a i ), where a i is the fixed value of a scale factor which will be taken equal to 10 -5 and normalized to current value a 0 . It will be helpful for our analysis to introduce ˜ ϕ early = ϕ ( a i ) -√ ˜ B H 0 √ 2Ω r 0 a 2 i = ϕ ( a i ) + const and ϕ early = ϕ ( a i ). Now we have to invert ϕ = ϕ ( a ) dependence to get the a = a ( ϕ ) dependence If we know a we can write down the chiral metric component h 22 as a function on ϕ", "pages": [ 9, 10, 11 ] }, { "title": "12 R.R.Abbyazov, S. V. Chervon", "content": "Taking the constant V 0 = Λ -B in (3.3) one can obtain Thus we have finished the procedure of reconstruction of h 22 and V for the very early epoch of the Universe evolution when a glyph[lessmuch] 1. Next step in consideration is the recent Universe approximation with a ≈ 1. Transforming the Hubble parameter to the form (3.8) we obtain Let us apply the Taylor expansion about (1 -a ) ≈ 0 up to first order terms in (5.5). The result is From this moment we are able to fulfill the reconstruction procedure. Dividing scalar field on √ ρ c 0 once again we come to where It is essential for the subsequent analysis that the best-fit value of ˜ B is known, so we have an opportunity to make calculation of the integral (5.7). Such a type of an integral is calculated by", "pages": [ 12, 13 ] }, { "title": "14 R.R.Abbyazov, S. V. Chervon", "content": "One more issue is about a transition through a = 1. This scale factor value should be explicitly presented in the expression for ϕ With the help of and one can carry out computaions further in more compact form. Plausibility of the early and recent approximations can be deduced from the comparison of H 0 ( ϕ -ϕ early ) and H 0 ( ϕ -ϕ recent ) for the exact (3.8) and approximate (5.6) Hubble parameter expressions and the time limits (for which corresponding approximations are hold on) will be extracted graphically.", "pages": [ 14 ] }, { "title": "Unified dark matter and dark energy description in a chiral cosmological model 15", "content": "The reconstruction is performed as usual when a = a ( ϕ ) is obtained. Using notation of the ˜ ϕ recent in (5.8) we have and We can substitiute this result to h 22 (2.16) and V (3.3) to get", "pages": [ 15 ] }, { "title": "6. Background dynamics of the model", "content": "One of the most important cosmological parameter used for the description of a background evolution is a contribution to a critical density of the Universe. The", "pages": [ 15 ] }, { "title": "16 R.R.Abbyazov, S. V. Chervon", "content": "last is defined as Ω = ρ ρ c . For the chiral fields sector we have Using (3.2) and (3.8) one can obtain In order to understand a general picture of the Universe evolution and to analyze periods of domination by various species of the Universe we represent the residual components of σ CDM model where ˜ H comes from (3.8). The latter quantity is responsible for the late accelerated expansion of the Universe, supported by σ CDM model. In the ΛCDM we have Unified dark matter and dark energy description in a chiral cosmological model 17 Here ˜ H 2 is given by Let us turn our attention to the effective equation of the state parameter It is necessary to take into account expressions for densities and pressures of the chiral fields (3.2), (3.4) and the other components, together with (3.8) Then in the σ CDM model we will have At the same time for ΛCDM model the effective equation of state parameter is Using the definition of the deceleration parameter q broadly used for background dynamics studies we can obtain Presence in the q the second derivative of the scale factor gives us evidence of the transition from decelaration to acceleration epoch at the time when q = 0. The deceleration parameter for chiral sector takes the view where ˜ H 2 is defined in (3.8). For ΛCDM model the expression for deceleration parameter looks like with Hubble parameter taken from (6.1). The evident differences both in numerator and denominator in q σ and q ΛCDM inevitably lead to distinctive evolution of the Universe if it is supported by σ CDM or ΛCDM models. It is known feature of ω eff that a moment of time of deceleration/accelaration transition corresponds to value -1 / 3 crossing. It is also well-known result this time to be exactly equivalent to those obtained from q analysis. We would like to point out here that in ΛCDM model equation of state of the dark energy parameter is equal ω ΛCDM de = -1.", "pages": [ 16, 17 ] }, { "title": "7. Discussion", "content": "Fig.1 shows good agreement of supernovae data with σ CDM model taken with the best-fit parameters values. This fact confirm the validity of proposed model, i.e., σ CDM model does not contradict to observational data and may serve as a good dynamical alternative to ΛCDM. In fig. 2 the confidence contours are depicted. We keep only positive values of parameter ˜ B in order to prevent a crossing of the phantom divide. One can see from the evolution of the individual densities Ω i , deceleration parameters q and effective equation of state parameters ω eff (figs. 3, 4 and 5) that accelerated expansion takes place earlier in the Universe supported by σ CDM. Also one may notice that radiation/matter domination transition occurs earlier in ΛCDM model. These observations are in the full agreement with a smaller total matter amount including cold dark and baryonic components in σ CDM model in comparison to ΛCDM model. The graphical comparison (see fig. 4 and fig. 5) of the a ΛCDM acc and a σacc (taken from the scale factor values corresponding to -1 / 3 and 0 crossing) gives us clear evidence for equality of the transitions to accelerate expansion in corresponding models. This observation is concluded from q and ω eff values and has been already mentioned above. From fig. 6 one can conclude that the early Universe approximation holds for a = 10 -5 up a = 5 · 10 -5 scale factor values. The recent Universe approximation depicted on fig. 7 is true from a = 0 . 8 to a = 1 . 2 values. Let us remind that validity of the early and recent approximations comes from confrontation of H 0 ( ϕ -ϕ early ) and H 0 ( ϕ -ϕ recent ). The deviation for approximated and exact ˜ H is associated with the lost of domination of DE for the early times. In conclusion it needs to stress that we first time reconstructed from observations the kinetic interaction between DM and DE in the form of chiral metric component h 22 for σ CDM. Also we have hope that the reconstruction techniques presented here may be useful for exact solution construction because of obtaining h 22 from observational data.", "pages": [ 18 ] }, { "title": "Acknowledgments", "content": "SVC is thankful to the University of KwaZulu-Natal, the University of Zululand and the NRF for financial support and warm hospitality during his visit in 2012 to South Africa where the part of the work was done. RRA is grateful to participants of the scientific seminars headed by Melnikov V.N.(Institute of Gravitation and Cosmology, Moscow), Rybakov Yu.P. (PFUR, Moscow) and Sushkov S.V. (KFU, Kazan) for valuable comments and criticize.", "pages": [ 18 ] }, { "title": "References", "content": "Unified dark matter and dark energy description in a chiral cosmological model 19", "pages": [ 19 ] } ]

[ { "title": "Sunandan Gangopadhyay a,b †", "content": "Department of Physics, West Bengal State University, Barasat, India b Visiting Associate in Inter University Centre for Astronomy & Astrophysics, Pune, India", "pages": [ 1 ] }, { "title": "Abstract", "content": "We emphasize the importance of the Voros product in defining noncommutative inspired black holes. The computation of entropy for both the noncommutative inspired Schwarzschild and Reissner-Nordstrom black holes show that the area law holds upto order 1 √ θ e -M 2 /θ . The leading correction to the entropy (computed in the tunneling formalism) is shown to be logarithmic. The Komar energy E for these black holes is then obtained and a deviation from the standard identity E = 2 ST H is found at the order √ θe -M 2 /θ . This deviation leads to a nonvanishing Komar energy at the extremal point T H = 0 of these black holes. The Smarr formula is finally worked out for the noncommutative Schwarzschild black hole. Similar features also exist for a deSitter-Schwarzschild geometry. Noncommutative inspired black holes [4, 5] has gained considerable interest recently. In this paper, we discuss some issues concerning them. The main point of interest is that there is no clear cut connection of this type of noncommutativity with standard notions of a noncommutative (NC) spacetime where point-wise multiplications are replaced by appropriate star multiplications. Interestingly, we observe that the Voros star product [6] plays an important role in obtaining the mass and charge densities of a static, spherically symmetric, smeared, charged particle-like gravitational source. In doing so, we have also extended our earlier discussion of the formulation of NC quantum mechanics in two spatial dimensions to three dimensional NC space. We then proceed to derive quantum corrections to the semiclassical Hawking temperature and entropy for these black holes by the tunneling mechanism by going beyond the standard semiclassical approximation [7]. Finally, we examine the status of the relation between the Komar energy [8], entropy and Hawking temperature ( E = 2 ST H ) in the context of these black holes. To address the first issue, one needs to take recourse to the formulation and interpretational aspects of NC quantum mechanics [9, 10]. In two spatial dimensions, we observe that the inner product of the coherent states | z, ¯ z ) (used in the construction of the wave-function of a free point particle) can be calculated by using a deformed completeness relation (involving the Voros product) among the coherent states where the Voros star product between two functions f ( z, ¯ z ) and g ( z, ¯ z ) is defined as The wave-function of the free point particle on the NC plane [11, 9] is given by where the momentum eigenstates are normalised such that ( p ' | p ) = δ ( p ' -p ) and satisfy the completeness relation It turns out that the Voros product plays a vital role in providing a consistent probabilistic interpretation of this wave-function. These observations and interpretations now allow us to write down the overlap of two coherent states | ξ, ¯ ξ ) and | w, ¯ w ) using the completeness relation for the position eigenstates in eq.(1) It is easy to check that satisfies the above equation. A straightforward dimensional lift of this solution from two to three space dimensions immediately motivates one to write down the mass and charge densities of a static, spherically symmetric, smeared, charged particle-like gravitational source in three space dimensions as [1] Interestingly, the formulation of NC quantum mechanics in two spatial dimensions can be generalized to three dimensional NC space also. The Voros star product can be defined in this odd dimensional space by identifying the appropriate basis in which the state has to be represented. In complete analogy with the two dimensional case, one can introduce the position basis in three spatial dimensions through an expansion in momentum basis as [12] which once again satisfy a deformed completeness relation where the Voros star product in three spatial dimensions is given by The overlap of two position states | /vectorx ) and | /vectorx ' ) (using the completeness relation (9)) read which yields by a simple inspection The formalism of NC quantum mechanics in three spatial dimensions, therefore, gives a specific representation of the Dirac delta function in three dimensions since which immediately leads to eq.(7). The above discussion of obtaining the overlap between two position states in three spatial dimensions based on the formalism of NC quantum mechanics is a direct derivation of eq.(7) in contrast to the arguement presented in two dimensions and also clearly brings out the important part played by the Voros product in defining the mass and charge densities of the NC inspired black holes. Solution of Einstein's equations with the above mass density incorporated in the energy-momentum tensor leads to the following NC inspired Schwarzschild black hole metric [4],[5] The event horizon of the black hole can be found by setting g tt ( r h ) = 0 in eq.(14), which yields The large radius regime ( r 2 h 4 θ >> 1) allow us to expand the incomplete gamma function to solve r h by iteration. Keeping upto next to leading order √ θe -M 2 /θ leads to Now for a general stationary, static and spherically symmetric space time, the Hawking temperature ( T H ) is related to the surface gravity ( κ ) by the following relation [13] where the surface gravity of the black hole is given by Hence the Hawking temperature for the NC inspired Schwarzschild black hole upto order √ θe -M 2 /θ is given by The Bekenstein-Hawking entropy can now be calculated from the first law of black hole thermodynamics which reads Hence the Bekenstein-Hawking entropy in the next to leading order in θ is found to be To express the entropy in terms of the NC horizon area ( A θ ), eq.(16) is used to get Comparing equations (21) and (22), we find that at the leading order in θ (i.e. upto order 1 √ θ e -M 2 /θ ), the NC black hole entropy satisfies the area law (in the regime r 2 h 4 θ >> 1) We now look for corrections to the semiclassical area law upto leading order in θ . To do so, we first compute the corrected Hawking temperature ˜ T H . For that we use the tunneling method by going beyond the semiclassical approximation [7]. Considering the massless scalar particle tunneling under the background metric (14), the corrected Hawking temperature is given by Application of the first law of black hole thermodynamics once again with this corrected Hawking temperature, gives the following expression for the corrected entropy/area law : We now move on to solve Einstein's equations with both the mass and charge densities incorporated in the energy-momentum tensor. This leads to the following NC inspired ReissnerNordstrom (RN) black hole metric [5] where The event horizon of the black hole can be found by setting g tt ( r h ) = 0 in (27). Once again in the large radius regime ( r 2 h 4 θ >> 1), we can expand the incomplete gamma function to solve r h by iteration. Keeping upto order √ θe -r 2 0 / (4 θ ) , we obtain where is the horizon radius of the commutative RN black hole. Now using eq(s)(17, 18), we obtain the Hawking temperature for the NC inspired RN black hole (upto order √ θe -r 2 0 / (4 θ ) ) We shall now write down the first law of black hole thermodynamics in the case of a charged black hole to calculate the Bekenstein-Hawking entropy. It reads [3] where Using eq(s)(28, 30, 32), the Bekenstein-Hawking entropy upto order √ θe -r 2 0 / (4 θ ) is found to be In order to express the entropy in terms of the NC horizon area ( A θ ), we use eq.(28) to obtain Comparing eq(s)(33, 34), we find that at the next to leading order in θ , the NC black hole entropy satisfies the area law in the regime r 2 h / (4 θ ) >> 1 To investigate the corrections to the semiclassical area law upto next to leading order in θ , we once again need to compute the corrected Hawking temperature ˜ T H . In this case, it reads [7] Application of the first law of black hole thermodynamics once again with this corrected Hawking temperature leads to the following expression for the corrected entropy/area law : where A θ and S BH are defined in (34) and (35) respectively. Finally, we proceed to investigate the status of the relation between the Komar energy E , entropy S and Hawking temperature T H in the case of these NC inspired black holes. The expression for the Komar energy E for the NC inspired Schwarzschild metric (14) is given by [1] This expression allows one to identify M as the mass of the black hole since E = M in the limit r →∞ . This identification plays an important role as we shall see below. The above expression computed near the event horizon of the black hole 1 upto order √ θe -M 2 /θ gives Finally, using eqs.(19), (21) and (40), we obtain where in the second line we have used eq.(21) to replace M 2 by S/ (4 π ) in the exponent. Interestingly, we observe that the relation E = 2 ST H gets deformed upto order √ θe -M 2 /θ which is consistent with the fact that the area law also gets modified at this order. The deformation gives a nonvanishing Komar energy at the extremal point T H = 0 of these black holes [2]. Also, we have once again managed to write down the deformed relation in terms of the Komar energy E , entropy S and the Hawking temperature T H . Similar features are also present for a de-Sitter Schwarzschild geometry [14]. Eq.(41) can also be written with M being expressed in terms of the black hole parameters S and T H using eq.(40) We name eq.(42) as the Smarr formula [15] for NC inspired Schwarzschild black hole since M has been identified earlier to be the mass of the black hole. The expression for the Komar energy E for the NC inspired RN metric (26) is given by [3] Now using eq(s)(30, 33, 43), we obtain upto order √ θe -r 2 0 / (4 θ ) where in the second line we have used eq.(33) to replace r 2 0 by S/π in the exponent. The above relation is analogous to the deformed relation between Komar energy E , entropy S and semiclassical Hawking temperature T H derived in case of the NC inspired Schwarzschild black hole [2]. However, in this case the relation is not amongst E , S and T H only but also involves the mass M and charge Q of the black hole. In the limit Q → 0, the above result reduces to eq.(41) [2].", "pages": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] }, { "title": "Acknowledgments", "content": "I would like to thank Inter University Centre for Astronomy & Astrophysics, Pune, India for providing facilities.", "pages": [ 10 ] } ]

[ { "title": "Multi-Dimensional Cosmology and DSR-GUP", "content": "K. Zeynali 1 , 2 ∗ , F. Darabi 3 † , and H. Motavalli 2 ‡ 1 Faculty of Medicine, Ardabil University of Medical Sciences (ArUMS), Daneshgah St., Ardabil, Iran. 2 Department of Theoretical Physics and Astrophysics, University of Tabriz, 51666-16471, Tabriz, Iran. 3 Department of Physics, Azarbaijan Shahid Madani University , 53714-161, Tabriz, Iran. November 3, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "A multidimensional cosmology with FRW type metric having 4-dimensional spacetime and d -dimensional Ricci-flat internal space is considered with a higher dimensional cosmological constant. The classical cosmology in commutative and DSR-GUP contexts is studied and the corresponding exact solutions for negative and positive cosmological constants are obtained. In the positive cosmological constant case, it is shown that unlike the commutative as well as GUP cases, in DSR-GUP case both scale factors of internal and external spaces after accelerating phase will inevitably experience decelerating phase leading simultaneously to a big crunch. This demarcation from GUP originates from the difference between the GUP and DSR-GUP algebras. The important result is that unlike GUP which results in eternal acceleration, DSR-GUP at first generates acceleration but prevents the eternal acceleration at late times and turns it into deceleration. PACS numbers: 98.80.Hw; 04.50.+h", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The Generalized Uncertainty Principal (GUP) is a generalization of Heisenberg Uncertainty Principal in the Planck scale where the gravitational effects on quantum gravity may be considerable. This idea, was firstly considered by Mead [1] and then implemented in the context of string theory as a candidate of quantum gravity as well as black hole physics with the prediction of a minimum measurable length [2, 3, 4, 5, 6, 7, 8]. Doubly Special Relativity (DSR) theory [9] as a possible ingredient of the flat space-time limit of the quantum theory of gravity proposed another modification on Heisenberg Uncertainty Principal [10]. Recently the authors in [11] considered these two modification as a limit of a single algebra (DSRGUP). Nowadays, a large amount of interest has been focused on the effects of these modification on system in high energy physics [12]. In a recent paper [13], we have studied a multi-dimensional Cosmology with GUP and obtained the corresponding exact solutions for negative and positive cosmological constants. Especially, for positive cosmological constant, the solutions revealed late time accelerating behavior and internal space stabilized to the sub-Planck size, in good agreement with current observations. Motivated by the interest in DSR-GUP, in the present paper we are interested in studying the effects of DSR-GUP modifications on our multi dimensional cosmology and comparing its results with the GUP case 1 . In section 2, we introduce the notions of GUP and DSR-GUP as well as their corresponding algebras. In section 3, we briefly introduce our cosmological model. In section 4, we first obtain the commutative solutions and then find the DSR-GUP solutions. Finally, in conclusion, we compare the commutative, GUP and DSR-GUP solutions.", "pages": [ 1, 2 ] }, { "title": "2 Generalized uncertainty principal", "content": "The simplest form of the GUP in a one dimensional system can be written as [8] where L Pl ∼ 10 -35 m is the Planck length and β is a constant of order unity. The algebra corresponding to (1) can be written as [8] which reduces to the ordinary one for β → 0. Doubly Special Relativity theories, on the other hand, suggest that the planck scales similar to the light speed are observer independent scales. This is because different observers should not observe quantum gravity effects at different scales [9]. The algebra corresponding to DSR-GUP can be written as [10] which reduces to the ordinary one for L Pl → 0. The authors in [11] showed that by assumption [ x i , x j ] = 0 = [ p i , p j ], the two above algebra (2), (3) can be considered as a single algebra in phase space where a is assumed to be of order unity and p 2 = ∑ p i p i . By definition [11] the equation (4) can be satisfied, where x i 0 and p i 0 are the ordinary position and momenta with [ x i 0 , p j 0 ] = iδ ij and p j 0 = -i ∂ ∂x i 0 . To distinguish between the linear and second order terms in Planck length, we rewrite equation (5) in a more general form Here, the coefficient α indicates the effect of linear term in Planck length and β the effect of second order term in Planck length. So setting α = 0 and α = a, β = 2 a 2 gives back the ordinary GUP algebra (2) and the DSR-GUP algebra (4), respectively. Using (6), we can show that the p 2 term in the any Hamiltonian can be be derived as", "pages": [ 2, 3 ] }, { "title": "3 The Cosmological Model", "content": "We consider a multi-dimensional cosmology in which the space-time is established by a FRW type metric with 4-dimensional space-time and a d-dimensional Ricci-flat internal space [14] where R ( t ) and a ( t ) are the scale factors of the external and internal spaces respectively, and g ( d ) ij is the Ricci-flat metric of the internal space. The Ricci scalar is derived from the metric (8) [14] where a dot represents differentiation with respect to time t . The Einstein-Hilbert action with a (3 + d )-dimensional cosmological constant Λ is written as where k 3+ d is the (3+ d )-dimensional gravitational constant and S Y GH is the York-GibbonsHawking boundary term. By substituting (9) in (10) and dimensional reduction we have where and a 0 is the present time compactification scale of the internal space. We introduce the following change of variables provided v 3+ d = 1 [13] with where R = R ( x 1 , x 2 ) and Φ = Φ( x 1 , x 2 ) are functions of new variables x 1 , x 2 . The above transformations with k = 0 result in the Lagrangian and Hamiltonian as follows where", "pages": [ 3, 4 ] }, { "title": "4.1 Commutative case", "content": "The dynamical variables defined in (14) and their conjugate momenta satisfy the following Poisson bracket algebra [14, 15] where η µν is the two dimensional Minkowski metric. The equations of motion are obtained For a negative cosmological constant ω 2 is positive and Eq.(20) describes the equations of motion for two ordinary uncoupled harmonic oscillators with solutions where A µ and B µ are constants of integration satisfying A µ B µ = 0 due to the Hamiltonian constraint ( H = 0). Using (12) and (14), the solutions for scale factors take the following forms where k 1 and k 2 are arbitrary constants and φ 1 and φ 2 are arbitrary phases. Imposing the Hamiltonian constraint leads to the following relation /negationslash where because of k 1 , k 2 = 0, it results in φ 1 -φ 2 = π 2 . In what follows, we will investigate the behavior of a Universe with one internal dimension ( D = 3+1). By setting φ 1 = π 2 and φ 2 = 0, we obtain The Hubble and deceleration parameters for both R ( t ) and a ( t ) are calculated as The time evolution of R 2 ( t ) and a 2 ( t ) are depicted in Fig.1 (solid lines). According to this behavior, the Universe begins from a big bang at t = 0, expands till t = π 2 ω toward a maximum value, and starts contracting toward a big crunch at t = π ω . As is clear from the figure, as well as the solution R ( t ) in (24), the big bang is initiated by a anti-de Sitter phase in the case of negative cosmological constant. Using the present value of Hubble constant, the age of Universe becomes t present = 1 ω cot -1 ( 2 H 0 ω ) ≈ ω -1 ≈ 10 17 s which is in agreement with current observations 2 . The present Universe is also in tideway to get to maximum and minimum of R 2 ( t ) and a 2 ( t ), respectively, within ∆ t ≈ 0 . 57 ω -1 . We set the initial condition at planck time R ( t Pl ) = a ( t Pl ) and according to Fig.1, we see that during the whole time evolution of Universe ( t Pl ≤ t ≤ π ω -t Pl ), the scale factor of internal space is contracted towards the sizes very smaller than a ( t Pl ), and so can never exceed a ( t Pl ). Moreover, considering the above initial condition results in by which we obtain the following ratio If the present radius of external space be equal to the radius of observed Universe 10 28 cm , then the present radius of internal space becomes about the Planck length (10 -33 cm ) and this justifies the non observability of the extra dimension. For a positive cosmological constant ω 2 is negative, so by replacing ω 2 with -ω 2 in Eq.(20) and using the Hamiltonian constraint the new solutions are obtained where for d = 1 we have and t As in the case of negative cosmological constant, the magnitude of the radius of external to internal spaces is asymptotically ( t → ∞ ) about 10 61 . As is seen in Fig. 2, R ( t ) is an exponentially increasing function of time whereas a ( t ) at first decrease with time till t /similarequal 0 . 88 ω -1 and then increase exponentially. Note that the big bang is initiated by a de Sitter phase in the case of positive cosmological constant. If the age of Universe is taken as ω -1 /similarequal 10 17 s , then we find that at present time we are around the minimum point of a ( t ) and that in the time interval t Pl ≤ t ≤ 141 ω -1 , a ( t ) can never exceeds a ( t Pl ). This indicates that the internal scale factor remains very small, at least for 140 times of the present age of the Universe. The results obtained here with a positive cosmological constant are consistent with the current observations on the acceleration of the Universe. Tho confirm this, we have depicted H R , H a and q R , q a in the figures 3 and 4 (see solid lines). Figure 3 shows that q R becomes negative a little bit earlier than the present age of the Universe namely ωt ∼ 1. This means, the Universe has started its acceleration recently. Fig.4 shows that q a is always negative and has a minimum at the position where q R becomes negative. The figures 3 and 4 indicate that at the beginning of time in both commutative and GUP cases, q R is positive ( R is decelerating) and q a is negative ( a is accelerating). With time evolution, q R approaches the threshold of negative values ( R is less decelerating) while q a approaches to more negative values ( a is highly accelerating). Once q R enters the region of negative values ( R is accelerating), q a reaches its minimum ( a stops its increasing acceleration). Finally, q R becomes more negative ( R is highly accelerating) whereas q a goes to rather less negative values ( a is slowly accelerating). It is interesting to note that the late time behavior of the Universe is more considerable in the GUP case, where both R and a exhibit highly accelerating features.", "pages": [ 5, 6, 7, 8 ] }, { "title": "4.2 DSR-GUP solutions", "content": "In this section, we aim to study this cosmological model in the DSR-GUP context to find effects of new terms in commutation relations on the time evolution of Universe. The new terms in commutation relations can be considered in two view point: first order in Planck length due to DSR theory and second order term in Planck length due to GUP in string t t theory. Following equation (7) and (17) we write perturbed Hamiltonian as where p 2 0 = p 2 10 2 -p 2 20 2 and [ x i 0 , p j 0 ] = iδ ij . Here, we want to investigate the classical version of DSR-GUP algebra. To do this, we must replace the quantum mechanical commutators with the classical poisson bracket as [ P, Q ] → i { P, Q } . Using equation ( 19), the equations of motion can be written as We see that deformed classical equations form a system of nonlinear coupled differential equation, so we need numerical solutions. Setting α = 0 reduces the equations to GUP framework so we can see effect of the second order term in Planck length on the time evolution of Universe. We limit ourselves to the investigation of the effect of first order term in Planck length due to DSR theory. In so doing, we should set β = 0 and α = /epsilon1 , /epsilon1 being a nonvanishing but so small parameter that α 2 /similarequal 0. This removes the third term, second order in Planck length, in the right hand side of the first equation in (34). In the negative cosmological constant framework, ω 2 is positive. Numerical solution of equations (34) shows that the deformed scale factors like commutative and GUP cases have periodic behavior. To compare with the results obtained in GUP [13], we have included the behaviour of the scale factors in GUP as well as commutative cases within the figures. As is seen in Fig.1, the time interval between big bang and big crunch in GUP case is shortened with respect to commutative one while in DSR-GUP case the time interval between big bang and big crunch is longer. In GUP case, the deformed scale factor of the internal space reaches it's minimum value sooner than commutative one while the deformed scale factor of the internal space in DSR-GUP case reaches later than commutative one. The deformed scale factor of the external space in GUP case reaches it's maximum sooner than commutative one and has larger value while in the DSR-GUP case, the deformed scale factor of the external space has smaller value and reaches to it later than commutative one. Replacing ω 2 with -ω 2 in Eq.(34), leads to the corresponding equations in the case of positive cosmological constant. Numerical analysis shows that (Fig.2) at early times the deformed scale factors of internal and the external spaces in both GUP and DSR-GUP cases behave like commutative one, and specifically the Universe is initiated by a de Sitter phase. At later times in the GUP framework, the expanding rate of deformed scale factors are bigger than commutative case while in the DSR-GUP case the deformed scale factors increase slower than commutative case. Moreover, at very late times after experiencing a maximum value, both scale factors decrease towards a big crunch simultaneously. Looking at the behaviour of deceleration parameter in figures 3 and 4 for internal and external scale factors, respectively, shows that unlike the commutative and GUP cases, in DSR-GUP case both scale factors after accelerating phases experience a decelerating phase at late times. This makes a remarkable difference between the DSR-GUP in one hand, and commutative together with GUP cases on the other hand. The difference between GUP and DSR-GUP is due to a relative sign difference in the algebras (2) and (3) corresponding to GUP and DSRGUP. This is interesting because the final fate of our multidimensional cosmology, being accelerated forever or decelerated towards a big crunch, is simply related to a relative sign difference in the quantum algebra corresponding to two different generalized uncertainty principles, GUP and DSR-GUP. Note also that the maximums of scale factors and the temporal location of big crunch in DSR-GUP case (Fig.2) depends on the Planck length: the more smaller Planck length, the more distant big crunch.", "pages": [ 8, 9, 10 ] }, { "title": "5 Discussion and Conclusion", "content": "Wehave studied a multidimensional cosmology having FRW type metric with a 4-dimensional space-time sector and a d -dimensional Ricci-flat internal space subjected to a higher dimensional cosmological constant in the frameworks of commutative and DSR-GUP contexts. The corresponding exact solutions for negative and positive cosmological constants are obtained and compared with each other as well as GUP case. It is shown in DSR-GUP case that for positive cosmological constant, both scale factors of internal and external spaces after accelerating phase, unlike the commutative and GUP cases, will inevitably experience decelerating phase leading simultaneously to a big crunch. This unexpected behaviour originates simply from a negative sign in the DSR-GUP algebra. The important result in this model is that DSR-GUP prevents the eternal acceleration. The exact solutions which we have obtained are the background cosmologies. Although they are interesting in themselves, but it is useful to provide insight on the limits of our approach, and the extent of viability of these solutions. The background cosmological solutions obtained here describe an ideal picture of the Universe and its evolution subject to Ricci flat extra dimensions, generalized uncertainty principle and positive/negative cosmological constant. No real matter is assumed, however, it is possible to interpret the contribution of extra dimensions as a kind of effective matter. Actually, the presence of real matter complicates the calculations, hence our approach is limited to the vacuum with a cosmological constant. We have also limited ourselves to the investigation of the effect of first order term in Planck length which affects the equations of motion due to DSR-GUP algebra. Note, however, that using the generalized uncertainty principle, including the Planck length in the corresponding algebra, does not mean that we have involved directly with quantum gravity. In this paper, we did not pay attention to the issue of perturbations, however it is useful to briefly mention about this subject. In fact, a realistic description of Universe needs the study of homogeneous and isotropic perturbations and the stability of these solutions against the perturbations. In this regard, we may use a set of convenient phase-space variables and write the Friedmann equation in terms of dimensionless density parameters. Then, we obtain a system of differential equations. The behavior of this system of equations in the neighborhood of its stationary point is determined by the corresponding matrix of its linearization. The real parts of its eigenvalues tell us whether the corresponding cosmological solution is stable or unstable with respect to the homogeneous and isotropic perturbations. If the real part of the eigenvalues of a critical point is not zero, the point is said to be hyperbolic. In this case, the dynamical character of the critical point is determined by the sign of the real part of the eigenvalues. If all of them are positive, the point is said to be a repeller , because arbitrarily small deviations from this point will move the system away from this state. If all of them are negative, the point is called an attractor because if we move the system slightly from this point in an arbitrary way, it will return to it. Otherwise, we say the critical point is a saddle point .", "pages": [ 10, 11 ] }, { "title": "Acknowledgment", "content": "The authors would like to thank the anonymous referee for the enlightening comments.", "pages": [ 11 ] } ]

[ { "title": "Quantization of Black Holes entropy and its cosmological consequences", "content": "G. Cristofano 1 , 2 , G. Maiella 1 , 2 and C. Stornaiolo 1 1 Istituto Nazionale di Fisica Nucleare,Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio 6 via Cinthia, 45 - 80126 Napoli 2 Dipartimento di Scienze Fisiche, Universit'a 'Federico II' di Napoli, Complesso Universitario di Monte S. Angelo Edificio 6 via Cinthia, 45 - 80126 Napoli", "pages": [ 1 ] }, { "title": "Abstract", "content": "Starting from a quantization relation for primordial extremal black holes with electric and magnetic charges, it is shown that their entropy is quantized. Furthermore the energy levels spacing for such black holes is derived as a function of the level number n , appearing in the quantization relation. Some interesting cosmological consequences are presented for small values of n . By producing a mismatch between the mass and the charge the black hole temperature is derived and its behavior investigated. Finally extending the quantum relation to Schwarzschild black holes their temperature is found to be in agreement with the Hawking temperature and a simple interpretation of the microscopic degrees of freedom of the black holes is given.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recently [1] it has been shown that primordial black holes masses obey a fundamental quantization condition which basically stems from the Dirac consistency relation for the wave function describing the quantum state of electrically and magnetically charged black holes. What is really far reaching is that the masses of astrophysical self-gravitating structures, as galaxies, clusters of galaxies and superclusters and the same universe mass are correctly described by such a quantization condition, so reveling a strict connection between quantum fluctuations, at work at the very beginning of the universe, and the large scale astrophysical structures observed at present in the universe. Furthermore such astrophysical structures lie on a straight line in a plot of angular momentum J = n /planckover2pi1 of the structure versus its squared mass m 2 with a universal slope G / c . More recently it has been shown that as an outcome of the fundamental quantum relation, also their physical scales are fully derived in terms of the Compton wavelength of its basic constituent, that is the proton [2]. In this letter starting from a numerical coincidence, we derive an analytical expression for the quantum level number n , which allows for an interesting interpretation, More precisely the quantum level n associated to the mass squared m 2 turns out to describe the number of degrees of freedom of the astrophysical structures, that is its entropy, which then appears to be quantized. The concept of energy level spacing is then introduced for black holes as a natural consequence of the quantization relation (14). Also its far reaching cosmological implications are presented, by considering the splitting of the lowest 'energy' states. Furthermore introducing a mismatch ∆ m between the mass and the charge of the black hole its temperature is derived and analyzed in the two interesting limits ∆ m /greatermuch m and ∆ m /lessmuch m . Also the time lapse for the excited state to decay to the stationary one is estimated and it is found to go to infinity for ∆ m going to zero . Finally extending the quantum relation to Schwarzschild black holes and using the connection between the quantum level number n and the entropy of the black hole the first law of thermodynamics for a black hole is enforced and its temperature derived, finding full agreement with the Hawking temperature. More interestingly such an approach allows for a microscopic description of the internal degrees of freedom of the black hole. The letter is organized as follows, in sect.2 we recall the basic quantization relation emphasizing the numerical coincidence of the quantum level number n and the maximal entropy for the universe as a black hole. In sect. 3 we derive the analytical expression for n in terms of the entropy of the astrophysical structure. In sect. 4 the energy level spacing for a quantized black hole is obtained and its cosmological consequences presented. In sect. 5 the temperature of the perturbed extremal black hole is derived and its behavior discussed. In sect. 6 the thermodynamic properties of Schwarzschild black holes are discussed and the Hawking temperature derived together with the interpretation of its internal degrees of freedom. In sect. 7 some comments and conclusions are presented.", "pages": [ 1, 2 ] }, { "title": "2 Black holes quantization", "content": "Let us start from the Reissner-Nordstrom type metric for a black hole of mass m with electric charge Qe and magnetic charge Qm , with the Schwarzschild radius and the other significant lengths where Qe and Qm indicate the electric and magnetic charges of the black hole. Furthermore, by imposing the condition g 00 = 0 we get for the event horizons the following solutions, For extremal black holes r + = r -implying The preceding relations can be derived from a Lagrangian formulation of extremal black holes. Basically there are two approaches to it. A non-supersymmetric [3] and a supersymmetric one [4] in which there are N scalar fields φ i coupled to gauge fields with a dilaton-like coupling. In both cases an e ff ective potential Vef f . ( φ i ) is derived and a so-called attractor mechanism takes place once the following conditions are satisfied. where φ i 0 are critical field values; the matrix has positive eigenvalues. In this context in the N = 2 , 4 supersymmetric theories the above conditions are automatically satisfied. The e ff ective potential VBH for a black hole with electric and magnetic charges in supergravity D = 4, N = 4 is where φ is the dilaton field, a the axion field and only one component of the electric and magnetic charges is considered. We can find a correspondence of this potential in conformal field theory (CFT) where the e ff ective potential is obtained in the CFT description of a quantum Hall fluid, where R 2 c is the compactification radius of the scalar Fubini field [5] and θ is the theta parameter introduced by 't Hooft [6] [7]. There is a striking resemblance between the two potentials (6) and (7), which shows that the black hole physics can be described in terms of a CFT (see also [8]) and the following identifications can be made. We must notice that having considered just one electric and magnetic charge component in eq.(6), the axion field a and then θ in eq. (7) has to be taken equal to zero for stability reasons (see ref. [4]). Then the e ff ective potential we will consider from now on becomes the following, By taking the field φ constant we can determine its value by the stability condition ∂ VBH /∂φ = 0 keeping fixed the charges, obtaining Such a procedure is equivalent to the requirement of the double extremality condition used in [4]. As a consequence the following relation gets satisfied (see in particular equation (5.13) in [4]). Notice that the black hole mass depends only on the strength of the charges, in fact substituting the value at the horizon given by eq (10) into eq. (11) we obtain the following relation By employing Dirac consistency relation [9] for a quantum description for such extremal black holes, we get finally For the lowest allowed mass for an extremal quantum black hole we get That is, according to the condition g 00 = 0, we can suggest that at its very beginning t ≈ 10 -43 s and at the Planck temperature TPlanck = 10 32 K charged black holes were forming, thanks to the balance between the attractive gravitational force and the repulsive electric and magnetic forces. Astrophysical and cosmological observations suggest that the previous quantum relation (14) found for dilatonic charged black holes in their extremal regime correctly describes also the astrophysical and cosmological structures at any scale [1] [2], and suggests that it can be applied to black holes of any class. In section 6 another argument in favor of this last statement will be presented, by studying their thermodynamical properties. In the following sections, starting from the above quantization relation, the black hole entropy will be expressed in terms of the quantum number n , the energy levels for the allowed black holes and its temperature derived.", "pages": [ 2, 3, 4 ] }, { "title": "3 Quantization of black holes entropy", "content": "A further interesting comment can be made regarding the generality of the quantization relation (14), that is there is no remnant of the charges of the allowed quantum black holes, instead there appears its angular momentum n /planckover2pi1 as a quantized quantity. That suggests the quantum relation (14) to be very general and indeed its validity has been proven to extend to astrophysical structures as galaxies, clusters of galaxies, superclusters and the whole universe (scaling hypothesis) [1][2]. That is the quantum relation (14) represents a basic, conceptual link, relating cosmological structures to quantum fluctuations at primordial epochs. It extends up to the whole (observable) universe In fact by using mU = Npmp where mp = 10 -31 g is the proton mass and Np ≈ 10 80 is the number of protons in the observed universe we obtain nU ≈ (10 60 ) 2 = 10 120 , just what it is expected for the universe total action [1]. This huge number seems to be related with the maximal possible entropy of the universe, as if it were a black hole as a whole, with radius equal to the event horizon, as suggested by the holographic principle [13] for a black hole where kB is the Boltzmann constant. We now show that such a coincidence is not merely numerical. Indeed, assuming the generality of equation (14), the quantization of the black hole entropy immediately follows. In fact the explicit expression for the quantum level number n appearing in reference [1] is, apart from a numerical factor, nothing else but the Bekenstein-Hawking entropy for an extremal black hole. To this end the following relation can be easily verified, where /lambdabar Compt . = /planckover2pi1 mc is the reduced Compton wavelength corresponding to the mass m . The above relation implies that the entropy is quantized, due to the appearance of the level quantum number n in the last step. The other interesting result is that the black hole entropy is expressed also by the ratio between the Schwarzschild radius and the Compton wavelength associated to the black hole mass. A consequence of these two statements is that the Schwarzschild radius is an integer number of times the Compton wavelength which then appears to be a fundamental length in black hole physics.", "pages": [ 4, 5 ] }, { "title": "4 Energy level spacing for a quantized black hole", "content": "It is interesting to show that the quantum relation (14) allows us to introduce the concept of energy level spacing between two energy levels for a quantized black hole where EP is the Planck energy. We can evaluate ( mn + 1 -mn ) from equation (14) by deriving the following quantity For n >> 1, mn + 1 ≈ mn and we can write safely and Similar results can be found in a di ff erent context in [10]. We can see that the larger is the mass, the smaller is the energy needed for a transition from one mass to a higher one. That is a black hole of larger mass can 'swallow' almost anything more easily than black holes of smaller masses [11] . If we evaluate relation (20) for the lowest value n = 1 a far reaching cosmological result can be obtained. In fact for n = 1 we obtain the following relation, That is and finally That is the energy level splitting between the 'first excited level' and the 'ground state level' is of the order of the Planck energy. That is below the temperature due to the expansion of the universe, gravitons [14] do not have enough energy to be absorbed from the primordial quantum lowest mass black holes due to quantization and consequently they decouple from matter. In our opinion these high energetic gravitons would travel isotropically in all directions and fill now all the space within the cosmological structures. It would be interesting to detect their 'relics' now and trace back a picture of the universe at Planck time. Further details on that will be reported elsewhere. A further question is now in order, together with its decoupling from matter are we allowed to say that at Tplanck / 2 gravity decouples from the other still unified forces, i.e. the strong, the weak and the electromagnetic forces?", "pages": [ 5, 6 ] }, { "title": "5 Heuristic derivation of black hole temperature and its quantization", "content": "Let us start noticing that the quantization relation given in eq. (14) was derived just under the assumption that black holes, with electric and magnetic charges, were extremal that is for r + = r -which is equivalent to temperature T = 0 for them. Furthermore by changing n , through higher and higher positive integer values the energy levels described in eq.(14) correspond to higher and higher mass black holes, which do not radiate due to its zero temperature; that is such states appear to be stable or stationary. By introducing a mismatch between the mass and the charge, the temperature will be di ff erent from zero according to That is explicitly If we perturb the extremal black hole by changing only its mass according to we obtain for the temperature the following expression, where we defined After imposing the condition m 2 = Q 2 we get, where x = ∆ m / m . It is instructive to consider the two extreme cases x /greatermuch 1 and x /lessmuch 1. For x 1, keeping only the leading terms, one easily gets /greatermuch which is the temperature of a Schwarzschild black hole of mass ∆ m . For the x /lessmuch 1 case, after neglecting smaller terms, one gets or where use has been made of the quantization condition given in eq. (14). Notice that in eq. (32) the deviation of the black hole temperature from the temperature of a Schwarzschild black hole of equal mass is explicitly shown, while in eq. (33) the relation of the temperature with the quantum number n has been evidenced. Let us now determine the time required for a perturbed charged black hole to reach the extremal state. One must solve the di ff erential equation where is the radiation power of a black hole, being A the black hole surface area, σ = π 2 kB / 60 /planckover2pi1 3 c 2 the Stefan-Boltzmann constant. Assuming that the spectrum of the black hole is a black body spectrum ( /epsilon1 = 1), get and eq. (34) becomes Solving this equation it is easy to see that , Notice that the integral on the lefthand side diverges in agreement with the third law of thermodynamics which states that the zero temperature cannot be reached in a finite time. That is equivalent to say that the derivative dx / dt goes to zero very fast and consequently the emission of the Hawking radiation slows down, see eq. (37).", "pages": [ 6, 7, 8 ] }, { "title": "6 Extending the quantum relation to more general black holes", "content": "We are now ready to enforce the statement that the quantum relation obtained for extremal black holes is indeed valid for black holes of any class. Here we will consider just the Schwarzschild ones. To such an extent we will derive the temperature of a black hole, just using relation (14) and the established connection between its entropy and the level quantum number n which is an extension of equation (18) to Schwarzschild black holes. It is interesting to notice that by using these two relations it is possible to enforce the first principle of thermodynamics and derive from it the black hole temperature TBH , which is found to agree with the Hawking temperature, but in such a new context it is seen naturally to be a quantized quantity. First let us start with the assumption that the quantum relation extends to Schwarzschild black holes. Indeed we recall that several authors have tried to introduce a quantization for them, through analogous methods, like quantization of black hole horizon or Ehrenfest quantization of their action, see refs. [10] [11] [12] among others. By di ff erentiating left and right hand in the previous relation we obtain, Furthermore using we get Dividing by the factor 2 Gm and multiplying by c 2 get which states the first law of thermodynamics for a Schwarzschild black hole, with which agrees with the Hawking temperature. Moreover, as it appears in the last step of eq. (46), the black hole temperature TBH is quantized as a consequence of the quantum relation (40). We are now ready to give a microscopic description of the internal degrees of freedom of a black hole, and we will discover that it naturally arises a proposal of other di ff erent scenarios, possibly at work. Let us start from the simplest case of lowest black hole mass corresponding to m = mP . Then eq. (46) specialized for the case n = 1 allows us to give a direct interpretation of the microscopic internal degrees of freedom of the elementary black hole. In fact it is straightforward to get the following relation. the factor 2 kBTP is the average energy associated to a two-dimensional oscillator in thermal equilibrium at Planck temperature TP . Notice that 4 π is the same incommensurability factor appearing in eq. (39) in defining the entropy as the ratio between the black hole area and the elementary area /lscript 2 p . We can now go back to eq. (46) to get a more general result regarding the total energy of a black hole with a generic mass obtaining analogously and the factor 2 kBTBH is the average energy of each two-dimensional harmonic oscillator at thermal equilibrium at temperature TBH and consequently n can be interpreted as the number of oscillators filling the horizon in a two-dimensional array. In addition to the above description of the internal degrees of freedom, we present here two more possible scenarios, among others, 1) mc 2 = 4 π nF 1 / 2 × 1 2 kBTBH × 2 (spin degrees of freedom) × 2 (translational degrees of freedom), where nF 1 / 2 and nF 3 / 2 are the number of fermions of spin s = 1 / 2 and s = 3 / 2 respectively. We must comment that all such scenarios stem from the quantum relation (40) together with the interesting interpretation of the level number n , appearing in it, and in the black hole entropy given by eq. (39).", "pages": [ 8, 9 ] }, { "title": "7 Discussion and conclusions", "content": "In this paper it has been presented a quantization of a black hole entropy from first principles. More precisely it has been shown that it arises from the quantization relation given in (14), which stems from the Dirac relation for electrically and magnetically charged black holes. To obtain this result we used a strict similarity between the e ff ective potentials for extremal black holes and conformal field theory. The link between black holes and conformal field theory was already put in evidence previously in [8], where the properties of the horizon of the BTZ black holes in ADS3 were described in terms of an e ff ective unitary CFT2, with central charge c = 1, realized in terms of the Fubini-Veneziano vertex operators [5]. We must observe that before this presentation the constant /planckover2pi1 appearing in the expression of the black hole entropy was just associated to an 'ad hoc' introduction of a minimal Planck area, in the definition of black hole 'degrees of freedom' from its horizon. Furthermore it has been shown the existence of energy levels for the black holes and their spacing as a decreasing function of the energy level. By considering the lowest lying energy levels it has been evidenced that gravitons decouple as soon as the temperature decreases, due to expansion, below TPlanck / 2 and their 'relics' could be experimentally detected. Also keeping the black hole's charge fixed and increasing its mass the emission temperature has been shown to be equal to the Hawking temperature of a neutral black hole, with the same mass, multiplied by a factor x = ∆ m / m . It is easy to see from eq. ( 30) that for increasing mass, the temperature first increases, reaches a maximum and finally, as in the case of a Schwarzschild black hole, decreases. The variation of mass due to the Hawking radiation is continuous, but di ff erently from the Schwarzschild black hole it takes an infinite time to evaporate all the excess mass. The total evaporation of the black hole could be obtained by bombarding the black hole with charges of the opposite sign until it becomes totally neutral. Moreover, extending the quantum relation to Schwarzschild black holes, its temperature TBH has been derived in a natural way, within our framework, showing agreement with the Hawking temperature and a simple interpretation of the microscopic degrees of freedom of the black hole provided. Even though interesting ideas have been presented here regarding black hole entropy quantization, its energy levels, its temperature quantization and a picture of its microstates, much more has to be done towards the understanding of problems like their interaction and / or the role of primordial black hole in the missing mass problem in the universe. Finally, being the universe at Planck time a quantum mechanical system and knowing that a phase transition is at work at T = TPlanck with the consequent formation of the lowest mass primordial black holes ( m = mPlanck ) one is very tempted to describe the state wave function of the universe at Planck time in terms of a quantum fluid of highly correlated lowest mass black holes, but that will be given elsewhere.", "pages": [ 10 ] } ]

[ { "title": "Fermion-antifermion mixing in gravitational fields", "content": "Giorgio Papini ∗ Department of Physics and Prairie Particle Physics Institute, University of Regina, Regina, Sask, S4S 0A2, Canada and International Institute for Advanced Scientific Studies, 89019 Vietri sul Mare (SA), Italy. (Dated: May 24, 2021) Mixing of fermion and antifermion states occurs in gravitational interactions, leading to nonconservation of fermion number above temperatures determined by the particle masses. We study the evolution of a f , ¯ f system and calculate the cross sections for the reactions f ⇀ ↽ ¯ f . Their values are identical in both directions. However, if ¯ f changes quickly into a lighter antiparticle, then the reaction symmetry is broken, resulting in an increased production of matter over antimatter . Of the fermions that exist as free particles, baryons appear to satisfy, with good accuracy, a number conservation law in the strong, electromagnetic and weak interactions. The same holds true, separately, for the numbers of the lepton families. It is however thought that baryon and lepton numbers are not exactly conserved quantities. This is sometimes invoked as a possible way to explain the value n B /n γ ∼ 10 -9 for the baryon to photon ratio observed in the universe and which is at the heart of the baryosynthesis problem [1]. It is shown below that fermion numbers need not be conserved in gravitational interactions. We consider external gravitational fields for which solutions of the covariant Dirac equation [2-4] and of other covariant wave equations [5-7] can be constructed. These solutions are exact to first order in the metric deviation γ µν = g µν -η µν , where η µν is the Minkowski metric and can be calculated explicitly by means of path integrals if the solution of the free wave equation (corresponding to the same particle of mass m ) and γ µν ( x ) are known. The notations are those of [4, 8], in particular D µ = ∇ µ + i Γ µ ( x ), ∇ µ is the covariant derivative and Γ µ ( x ) the spin connection. A semicolon and a comma are frequently used to indicate covariant and partial derivatives respectively. The first order solutions of (1) are of the form where ψ ( x ) satisfies the usual flat spacetime Dirac equation and is represented here by the positive (particle) and negative (antiparticle) energy solutions ψ ( x ) = u ( /vector k ) e -ik µ x µ and ψ (1) ( x ) = v ( /vector k ) e ik µ x µ respectively, where the spin-up ( ↑ ) and spin-down ( ↓ ) components of the spinors u and v obey the well-known relations [9] The matrices γ ˆ µ in (3) are the usual constant Dirac matrices. The operator ˆ T is given by [4] where and In (6) and (7), the path integrals are taken along the classical world line of the particle starting from a point P and Φ G ∼ O ( γ µν ) is the covariant Berry phase [10]. It follows from (4), γ 5 ≡ iγ ˆ 0 γ ˆ 1 γ ˆ 2 γ ˆ 3 , { γ 5 , γ ˆ µ } = 0, σ ˆ α ˆ β = i 2 [ γ ˆ α , γ ˆ β ] and that if Ψ( x ) = e -ik µ x µ ˆ Tu is a solution of (1), then Ψ (1) ( x ) = e ik µ x µ ˆ T 1 v also is a solution of (1), and ˆ T 1 = γ 5 ˆ Tγ 5 . It is useful to further isolate the gravitational contribution in the vierbein fields by writing e µ ˆ α /similarequal δ µ ˆ α + h µ ˆ α . We therefore obtain where ˆ T 0 = 1 2 m ( m + γ ˆ α k α ) and ˆ T G contains the (first order) gravitational corrections. Similarly, we find where ˆ T 10 = 1 2 m ( m -γ ˆ α k α ) . Notice that ψ ( x ) and ψ (1) ( x ) as well as Ψ( x ) and Ψ (1) ( x ) describe particles of the same mass m as required by CPT -invariance. In general the eigenstates U ± = 1 / √ 2( u ± v ) of γ 5 and the eigenstates u and v of ˆ T 0 are not the same and ˆ T, ˆ T 1 mix u and v , thus producing f ¯ f transitions. A similar mixing phenomenon occurs in the K 0 , ¯ K 0 system where the CP eigenstates are mixed by the weak interaction Hamiltonian resulting in CP violation. It is the fermion number conservation that is violated in our work and the mixing is carried out by the gravitational contributions ˆ T G and ˆ T 1 G . This can be seen as follows. The time evolution of a f, ¯ f system in a gravitational field can be written in the form where | α 0 | 2 + | β 0 | 2 = 1, from which we obtain If at t = 0 the gravitational field is not present, then ˆ T G = 0 , ˆ T 1 G = 0 and α 0 = α (0) , β 0 = β (0). It follows from (12) that as the f , ¯ f system propagates in a gravitational field, transitions f ⇀ ↽ ¯ f can take place. In order to obtain the transition probabilities | α ( t ) | 2 , | β ( t ) | 2 from (12) in a concrete case, we choose for simplicity where f R is the positive helicity eigenvector. The normalizations are 〈 ψ | ψ 〉 = 1, where 〈 ψ | = 〈 ψ † | γ ˆ 0 , 〈 ψ (1) | ψ (1) 〉 = -1 and 〈 ψ | ψ (1) 〉 = 〈 ψ (1) | ψ 〉 = 0. In addition, we need explicit expressions of the metric components for the purpose of calculating ˆ T and ˆ T 1 . We choose the Lense-Thirring metric [11], represented, in its post-Newtonian form, by where and M , R , ω = (0 , 0 , ω ) and J are mass, radius, angular velocity and angular momentum of the source. This metric is non trivial, has no Newtonian counterpart and describes the physically significant case of a rotating source. By using the freedom allowed by local Lorentz transformations, the vierbein field to O ( γ µν ) is Without loss of generality, we consider particles starting from z = -∞ , and propagating, with impact parameter b ≥ R , along x = b , y = 0 in the field of the rotating source. We also set k 3 ≡ k and k 0 ≡ E . We can now return to (12). If originally the system is an antifermion, then α 0 = 0 , β 0 = 1, | Φ( t ) 〉 = ˆ T 1 | ψ (1) 〉 and from (11) we obtain σ = where q i = k ' i + k i , k ' i refers to 〈 ψ | and q 0 = 2 E because the external field is time-independent. The probability of the transition ψ (1) → ψ , which, because of the coupling of spin to gravity, violates ¯ f number conservation, follows from (17) and is In the approximation k /greatermuch m we obtain from (18) P ψ (1) → ψ = (3 / 4) 2 φ 2 , while P ψ (1) → ψ = ( m/ 2 k ) 2 φ 2 for k /lessmuch m . The latter result leads in the limit k → 0 to a well known infrared divergence that will be discussed below. If we choose α 0 = 1 , β 0 = 0, then | β ( t ) | 2 represents the probability for the inverse process ψ → ψ (1) . We find According to (18) and (19), the transitions proceed in both directions with the same probability, with no contribution from the source's rotation when /vector k is in the z -direction, as specified. With the help of (18) we can also calculate the cross section for the ¯ f → f process which corresponds to a change | ∆ L | = 2 for leptons and | ∆ B | = 2 for baryons. We get [12] where use has been made of the relations k f dk f = E f dE f , k 2 f = E 2 f -m 2 = E 2 -m 2 = k 2 , integration over E f has been performed and the index f indicates momentum and energy in the final state. There is, of course, no energy transfer to the external field, since the field is static. We also find from (18) where φ ( ˜ k ) = -( GM/π ) K 0 ( b ˜ k ) is the Fourier transform of φ ( z ) = -GM/ √ z 2 + b 2 , the momentum transfer is ˜ k = | /vector k f -/vector k | = 2 k sin( θ/ 2) and θ is the angle that /vector k f makes with the z -axis. Integrating over the angles, we obtain from (20) the total cross-section 1 π ( - 3 m 2 E k ) ( GM ) [ 4 b 1 2 k 2 + K 2 0 (2 bk ) - K 2 1 (2 bk ) ] . (22) On account of the exponential decay of the Bessel functions, (22) can be approximated by for all physical values of bk . In (23), β and γ are the usual Lorentz factors and k = mβγ . It also follows from (21) and ˆ T = γ 5 ˆ T 1 γ 5 that the transition amplitude 〈 ψ (1) | ˆ T | ψ 〉 leads to the same cross-section (22). From 0 ≤ β ≤ 1, one obtains T ≥ m/κ ≡ T c and also T c /similarequal 1 . 2 · 10 4 K for neutrinos of mass m ∼ 1 eV and T c /similarequal 1 . 1 · 10 13 K for nucleons. Expressing β and γ as temperatures, we can re-write (23) in the form where ˜ T ≡ T/T c . The transitions discussed are forbidden for T < T c , have vanishing value for ˜ T → ∞ and diverge for ˜ T → 1 as expected. This divergence, already met in connection with (18), arises as a consequence of the implied requirement that particles interact without the emission of gravitons (or photons in the electromagnetic case) [9]). This requirement can not in fact be realized physically. Processes in which gravitons with energy less than the energy resolution ∆ /epsilon1 of the apparatus are in fact indistinguishable from those in which gravitons of energy less or equal to ∆ /epsilon1 are also emitted and reabsorbed by the particles. The natural cutoff for the graviton energy is therefore ∆ /epsilon1 . It is also shown in the literature that when the gravitational [13] and electromagnetic [9] fields are quantized, all infrared divergencies disappear. k 3 2 2 Summary and discussion . It is possible for baryons and leptons to achieve the required energies in astrophysical situations. If, for the sake of numerical comparisons, we take GM/b ∼ 0 . 1, which applies in the vicinity of a black hole, and β ∼ 0 . 95, then the cross section value is σ ∼ 8 · 10 -14 cm 2 for 1 eV neutrinos, while we find σ ∼ 8 · 10 -32 cm 2 for baryons of mass m ∼ 1 GeV . It therefore follows from (23) and (24) that, for the same value of β , the baryon number is more likely to be conserved ( σ smaller) than the neutrino number, i.e. the B -violating processes are suppressed relative to those violating L . The actual value of σ depends on ( GM/b ) 2 . It may therefore be useful to re-examine attentively some of the processes that take place close to compact, massive objects. Gravity behaves similarly on the cosmological scale and does not conserve the fermion number at lower momenta and higher temperatures. The values of T c given above indicate that B conservation occurred in the universe at the time of primordial nucleosynthesis, while L conservation is relatively recent, thus resulting in the coexistence of two fermion populations at temperatures T c,L ≤ T ≤ T c,B , with the leptons agitated by the f ⇀ ↽ ¯ f transitions, and the baryons dormant (neglecting, of course, the effect of any other concomitant interactions). The cross section for f ⇀ ↽ ¯ f reactions is the same in both directions, hence the B - and L -violating processes do not produce more fermions than antifermions. The baryosynthesis problem can not therefore be solved using the gravitational non-conservation of fermion number without additional assumptions regarding the initial distributions of matter and antimatter, or the breakdown of thermal equilibrium. There is, however, an important, additional possibility suggested by (23). If, in fact, ¯ f of mass m produced in the reaction f → ¯ f changes rapidly into an antifermion of mass m ' < m , then the cross section for the inverse ¯ f ' → f ' process can no longer re-establish symmetry between matter and antimatter. This reaction cross section does in fact increase, relative to the original one, by a factor σ ' /σ ∼ ( m/m ' ) 2 [ γ ' ( β ' -1 /β ' 2 )] 2 / [ γ ( β -1 /β 2 )] 2 which can be large, depending on ( m/m ' ) 2 and the reaction kinematics. The result is therefore an increase in the production of matter over antimatter . Any further analysis of the mechanism's relevance to baryon asymmetry depends on m and m ' and their experimental signatures and is not pursued here. Scalar-tensor theories of gravitation based on early efforts by a number of authors [14], [15], [16] have been recently considered for the purpose of obtaining a varying gravitational constant that could in principle enhance the strength of gravity [19],[17],[18]. Would the mixing mechanism discussed above be affected by the introduction of a running gravitational constant and concomitant scalar field? Unfortunately, no useful, first order tensor-scalar solutions of (1) exist at present. Nonetheless, a qualitative indication of the role played by a scalar field and running gravitational constant can be obtained from (1) by means of the conformal transformation g µν = G Λ( x ) η µν which separates the scalar from the tensor component of the theory. In this simple case, (1) can be solved [20], the operators ˆ T and ˆ T 1 given by (9) and (10) become and still confirm the presence of the symmetry breaking mechanism. However, in view of the limitations on the structure of Λ( x ) discussed in [20], more precise conclusions must await the detailed study of a meaningful physical model. The introduction of back reaction terms discussed in [8], while increasing the cross-sections over appropriate time intervals, does not alter the symmetry of the original distribution. The back reaction would however increase the cross-section of the process ¯ f ' → f ' discussed above, and of a reaction for which CP violation has been observed, as in the time evolution of a K 0 ¯ K 0 system, if the latter process occurred in the neighborhood of a gravitational source of appropriate strength. The model of back reaction discussed in [8] is based in fact on the introduction of a dissipation term in the wave equation of the particle propagating in an external gravitational field. It is then shown that a gravitational perturbation can grow rapidly, as with a fluid heated from below in which a small disturbance in the wave equation grows rapidly as soon as convection starts. The covariant wave equation of a kaon in the external gravitational field approximation has the solution ϕ ( x ) = e -i Φ G ( x ) ϕ 0 ( x ) where ϕ 0 ( x ) satisfies the equation [6] It follows [8] that the addition to (27) of a dissipation term -2 mξ∂ 0 ϕ 0 where ξ = ε ( m k 0 GM 2 b ) 2 and ε is a dimensionless, arbitrary parameter 0 ≤ ε ≤ 1, transforms the solution of the covariant wave equation into ϕ ( x ) = e -i Φ G ( x ) e mξx 0 ϕ 0 ( x ) and the overlap of the kaon eigenstates K L and K s , which is a measure of CP violation, into 〈 K L | K s 〉 G = 〈 K L | K s 〉 e 2 mξx 0 . Then the increase in CP violation could become large even if the growth of the exponential term were restricted by competing effects [8].", "pages": [ 1, 2, 3, 4 ] } ]

[ { "title": "Thermodynamics in Kaluza-Klein Universe", "content": "M. Sharif ∗ and Rabia Saleem † Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan.", "pages": [ 1 ] }, { "title": "Abstract", "content": "This paper is devoted to check the validity of laws of thermodynamics for Kaluza-Klein universe in the state of thermal equilibrium, composed of dark matter and dark energy. The generalized holographic dark energy and generalized Ricci dark energy models are considered here. It is proved that the first and generalized second law of thermodynamics are valid on the apparent horizon for both of these models. Further, we take a horizon of radius L with modified holographic or Ricci dark energy. We conclude that these models do not obey the first and generalized second law of thermodynamics on the horizon of fixed radius L for a specific range of model parameters. Keywords: Dark energy models; Thermodynamics. PACS: 95.36.+x, 98.80.-k", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The well-established notion is that the universe has entered in the phase of accelerating expansion. Type Ia supernovae [1]-[3], cosmic microwave background radiation (CMBR) [4], Wilkinson microwave anisotropy probe (WMAP) [5] and Sloan digital sky survey (SDSS) [6, 7] has indicated that our universe is flat, homogeneous and isotropic over large scale. This speedy expansion of our universe is due to an antigravity force which is drawing galaxies apart from each other, dubbed as dark energy (DE). Some scientists believe that extra dimensions of space are also responsible for this expansion. The mechanism behind this expansion and the nature of DE is not very much clear. Dark energy having large negative pressure dominates 76% energy density of the universe [8]. The cosmological constant is the most suitable candidate of DE which may be characterized by an equation of state (EoS) parameter, ω = -1. The current value of this constant is 10 -55 cm -2 whereas in particle physics it is 10 120 times greater than this factor, this problem is known as fine-tuning problem [9]. The other serious problem is the cosmic coincidence problem which raised due to the comparison of dark matter (DM) and DE in the present expanding universe. There have been many DE models proposed such as scalar field models and interacting models etc. Quitessence [10, 11], k-essence [12], phantom [13, 14], tachyon [15, 16], and quintom [17, 18] are the scalar field models while the interacting DE models are Chaplygin gas [19, 20], braneworld [21, 22] and holographic DE (HDE) [23, 24]. Unfortunately, this whole class of DE models do not explain the nature and its origin in a comprehensive way. According to recent observations, multidimensional theories may help to resolve such problems of cosmology and astrophysics. The most impressing theory in this scenario is offered firstly by Kaluza [25] and Klein [26] by adding an extra dimension in general relativity (GR), known as Kaluza-Klein (KK) theory. It is basically a five dimensional (5D) theory in which gravity is unified with electromagnetism through this extra dimension. The validity of laws of thermodynamics has been discussed with modified HDE (MHDE) [27]-[31]. Some authors [32]-[35] extended this work to modified gravity theories like f ( R ) , f ( T ), Brans-Dicke (BD) and Horava-Lifshitz theory. Sharif and Khanum [36] checked the validity of generalized second law of thermodynamics (GSLT) in KK universe with interacting MHDE and DM. Recently, Sharif and Jawad [37] explored this work with varying G to investigate the validity of GSLT in the same scenario. Holographic DE model based on the holographic principle, is a good effort in quantum gravity to understand the nature of DE to some extent. According to this principle, a physical system placed inside a spatial region is observed with its area but not within its volume [38]. Cohen et al. [39] argued the cosmological version of this principle, the quantum zero-point energy ( ρ Λ ) of the system having size L (infrared cutoff) cannot exceed the mass of a black hole (BH) with the same size. Mathematically, we get an inequality i.e., L 3 ρ Λ ≤ LM 2 p , where M p is the reduced Planck mass expressed as M p = (8 πG ) -1 2 . This inequality is most suitable for large L with event horizon. The HDE density can be expressed as ρ Λ = 3 c 2 M 2 p L -2 , where 3 c 2 is a dimensionless constant. The HDE in modified version for KK theory is known as MHDE [40] and can be calculated from the ( N + 1)-dimensional mass of the BH [41]. Ricci DE (RDE) [42] is a type of DE obtained by taking square root of the inverse Ricci scalar as its infrared cutoff. Gao et al. [43] explored that the DE is proportional to the Ricci scalar. Some recent work [44]-[46] shows that the RDE model fits well with observational data. Xu et al. [47] gave the generalization of two dynamical DE models, i.e., generalized HDE (GHDE) and generalized RDE (GRDE) models. These two models, combination of ˙ H and H 2 , gave the late time accelerating universe. In [48], similar type of investigation has been done in FRW universe model. In a recent paper [49], we have checked the validity of the first and GSLT for Bianchi I universe model. We have also explored the statefinder, deceleration and Hubble parameters for the same line element [50]. Here we extend the work of [48] to KK universe model with the same scenario. In this paper, we use KK universe in thermal equilibrium composed of DM and DE with GHDE and GRDE models. The paper is designed as follows: In section 2 , the density and pressure for GHDE/GRDE models are calculated. Section 3 is devoted to check the validity of the first and GSLT on the apparent horizon and also by taking GHDE/GRDE as the MHDE/MRDE. In the last section, we summarize the results.", "pages": [ 1, 2, 3 ] }, { "title": "2 Density and Pressure for GHDE and GRDE models", "content": "In this section, we evaluate energy density and pressure for GHDE as well as GRDE models in KK universe. This universe model contains 4-dimensional Einstein field equations and the fifth dimension satisfies the Maxwell field equations. This metric is the simple generalization of the FRW metric to extend the range of observable universe by increasing the dimensions of the universe. The line element of KK model is given by where k denotes the curvature parameter having values +1 , 0 and -1 corresponding to open, flat and closed universe, respectively. The energymomentum tensor for perfect fluid is where P, ρ and V α are the pressure of the fluid, energy density and five velocity vector, respectively. We consider that the fluid is a mixture of DM and DE, thus P and ρ can be written as P = P m + P E and ρ = ρ m + ρ E with P m = 0. The field equations for KK universe become We are interested in flat KK universe so that k = 0 yields the field equations as where Hubble parameter is defined as H = ˙ a a . The conservation equation can be written as Differentiating Eq.(5) and using (7), it follows that Here we assume that there does not exist any sort of interaction between DE and DM, therefore these are separately conserved. Thus the conservation equation (8) yields Solving Eq.(9), the matter energy density is obtained as where ρ m 0 is the constant of integration, known as the present value of DE density and cosmological red shift is z = 1 a -1. The matter density in KK universe decreases more rapidly as compared to FRW universe with the evolution of the universe which is consistent with the current observations. Now, we evaluate energy density and pressure for GHDE and GRDE models as follows.", "pages": [ 3, 4, 5 ] }, { "title": "2.1 Generalized Holographic Dark Energy Model", "content": "The energy density of this model is given as [47] where c 2 is a non-zero arbitrary constant and f ( x ) > 0 such that g ( x ) = γx +(1 -γ ) , 0 ≤ γ ≤ 1. The Ricci scalar is For γ = 0, the energy density of the original HDE is recovered while γ = 1 leads to the original RDE. Using Eq.(13) in (12), it follows that Inserting Eqs.(11) and (14) in (5), we obtain a first order linear differential equation whose solution is with H 0 an integration constant. Differentiating Eq.(15) with respect to t , we get Substituting H 2 and ˙ H in Eqs.(10), (13) and (14), it follows that Equations (17) and (19) represent pressure and energy density in terms of red shift z .", "pages": [ 5, 6 ] }, { "title": "2.2 Generalized Ricci Dark Energy Model", "content": "The energy density of GRDE model is [47] where h ( y ) = δy + (1 -δ ) > 0 , 0 ≤ δ ≤ 1. For δ = 0, the original energy density of the RDE is recovered whereas δ = 1 leads to energy density of the original HDE. Comparing Eqs.(12) and (20), we see that the GRDE reduces to the GHDE and vice versa for δ = 1 -γ . By replacing γ with (1 -δ ) in Eqs.(14)-(19), we obtain similar solutions for GRDE model. This implies that these equations are also solutions of the GRDE model with γ = 1 -δ .", "pages": [ 6 ] }, { "title": "3 First and Generalized Second Law of Thermodynamics", "content": "Firstly, we discuss the validity of the first and GSLT on the apparent horizon. For this purpose, we use the entropy given by Gibb's law [51, 52] where S I , V, P, E A and T A are internal entropy, volume, pressure, internal energy and temperature of the apparent horizon, respectively. In FRW metric, the apparent horizon has the radius Here FRW metric contained in the KK universe is a subspace with compact fifth dimension having similar properties of flat FRW universe on the apparent horizon. The internal energy and volume in extra dimensional system are In flat geometry, the radius of the apparent horizon coincides with Hubble horizon given as The entropy and temperature of the apparent horizon are [53] and entropy in four dimensions takes the form The first law of thermodynamics on the apparent horizon is defined as The energy crossing formula on the apparent horizon for KK universe can be found as follows [54] Inserting L from Eq.(23) in the above equation, we get Using Eqs.(24) and (25), it follows These two equations lead to the following form of the first law of thermodynamics which gives its validity on the apparent horizon for all kinds of energies as it is independent of DE. Now for the GSLT to be satisfied for the apparent horizon, we evaluate the derivative of internal entropy through Eq.(21) as Substituting the values of ˙ V , T A , ˙ ρ and using conservation equation, we get According to SLT, entropy of the thermodynamical system can never be decreased. This is generalized in such a way that the derivative of any entropy is always increasing, i.e., ˙ S I + ˙ S A ≥ 0. Thus we have We conclude that GSLT always holds on the apparent horizon. Notice that these laws always hold on the apparent horizon as it is independent of choice of DE . Further, we take GHDE or GRDE models as the density of MHDE or MRDE to check the validity of the first and GSLT on the horizon having radius L . The MHDE density can be calculated by taking the mass of ( N +1)dimensional BH [41] where A N -1 is the unit N -sphere area, R H is the scale of the BH horizon and G is the gravitational constant in ( N +1)-dimensions related to Planck mass M N +1 . As 8 πG = M -( N -1) N +1 = V N -3 M 2 p , V N -3 is the volume of this space, so M can be written as We can write which implies that For N = 4, it gives Inserting the value of 4-sphere area, we have Comparing this value with the energy density of GHDE, it follows that Substituting the values of H 2 and R from Eqs.(15) and (18) in (35), the expression for L 2 in the form of red shift is Here the temperature, entropy and the total energy crossing on this horizon with radius L is similar to Eqs.(24) and (27), respectively, with the difference that dS L , T L and dE L are used instead of dS A , T A and dE A . We can write /OverDot /OverDot For the first law, we must have -dE L = T L dS L . Equations (27) and (37) imply Since the second term on the right hand side in the above equation is time dependent, so it can never be zero in the evolving universe. Thus /negationslash This indicates that the first law of thermodynamics does not hold for the horizon of radius L . For the validity of GSLT on the horizon of radius L , the derivative of total entropy is as follows According to the GSLT, the total entropy of the thermodynamical system always increases, i.e., 4 ˙ HL 2 ( HL -˙ L ) + π ˙ L ≥ 0 indicating its dependence only on L in the DE model. In GHDE model, the variation of total entropy on the horizon is where L 2 is given in Eq.(36). This expression does not provide any indication, whether it increases or decreases. To get insights, we plot a graph of total entropy ( ˙ S I + ˙ S L ) versus red shift z as shown in Figure 1 . This indicates that ( ˙ S I + ˙ S L ) < 0 and hence the GSLT does not hold on this horizon with radius L for the specific values of the parameters.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4 Concluding Remarks", "content": "We have considered KK universe in compact form in the state of thermal equilibrium, similar to FRW universe by assuming that our universe is filled with DM and DE. Two types of DE models, GHDE and GRDE have been used. It is worth noticing that the GRDE model can be converted to GHDE model by interchanging δ with 1 -γ . Also, the original density of HDE and RDE models is obtained for γ = 0 , δ = 1 and γ = 1 , δ = 0, respectively. The density and pressure for GHDE and GRDE models in terms of red shift z are evaluated. We have investigated the validity of the first and GSLT on the apparent horizon in this scenario. These laws turn out to be independent of the choice of DE models, geometry of BH, and also the fifth dimension. Hence these laws are always satisfied on the apparent horizon for all kinds of DE models. We have also checked that the first law remains invalid on particle as well as on the event horizon while GSLT holds on the particle horizon only. It is worth mentioning here that KK universe in non-compact form also gives the same results as the variation along the fifth dimension is negligible. Further, we have considered the GHDE and GRDE as the MHDE and MRDE and found L in terms of z , to check the validity of these laws on the horizon whose radius is denoted by L . It is concluded that the first law of thermodynamics does not hold on the horizon of radius L for both DE models. The GSLT always holds on this horizon in the range γ, δ ∈ (0 , 0 . 1) for GHDE and GRDE, respectively, but it remains invalid for γ, δ ∈ (0 . 1 , 1). It is worth mentioning here that our results on the apparent as well as on the horizon of radius L are consistent with FRW universe [48].", "pages": [ 11 ] } ]

[ { "title": "A Study on Anisotropy in the Arrival Directions of Ultra-High-Energy Cosmic Rays Observed by Pierre Auger Observatory", "content": "Hang Bae Kim ∗ Department of Physics and The Research Institute of Natural Science, Hanyang University, Seoul 133-791, Korea We study the anisotropy in the arrival directions of PAO UHECRs, using the point source correlational angular distance distribution. The result shows that the anisotropy is characterized by one prominent excess region and one void region. The excess region is located near the Centaurus A direction, supporting that the Centaurus A is a promising UHECR source. The void region near the south pole direction may be used to limit the diffuse isotropic background contribution. PACS numbers: 98.70.Sa Keywords: ultra high energy cosmic rays, anisotropy, Centaurus A", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The cosmic ray with energy of order of 10 20 eV was first reported in almost 50 years ago [1]. Then, the observation of cosmic microwave background radiation followed [2]. Soon after, Greisen [3], Zatsepin and Kuzmin [4] pointed out that the interaction with the cosmic microwave background would cause the energy loss and limit the distance that such high energy cosmic rays could travel. This would result in the suppression in the cosmic ray energy spectrum above the so called GZK cutoff E GZK ∼ 4 × 10 19 eV, if the sources are distributed over the whole universe. If this suppression is true, as indicated by recent observations [57], it implies that UHECR with energies above the GZK cutoff mostly come from relatively close extragalactic sources within the GZK radius r GZK ∼ 100 Mpc. One consequence of this would be anisotropy in the arrival directions of ultra-high-energy cosmic rays (UHECR), since the matter within the GZK radius is distributed inhomogeneously and the UHECR sources are more or less correlated with the matter distribution. Therefore, the existence of anisotropy is an important clue for tracing the origin of UHECR. Searches for anisotropy in the UHECR arrival directions have been done by using many different methods. For the PAO data, the anisotropy manifested itself as a correlation between the UHECR arrival directions and the locations of active galactic nuclei (AGN) [8, 9]. When we extend the correlation study to the galaxy distribution, the conclusion is less clear than in the AGN case [10]. Besides, instead of being based on the astrophysical objects, the anisotropy search based on the auto-clustering did not provide a strong evidence of anisotropy [11]. Surely the result of anisotropy study depends on methodology. We need an method appropriate for the purpose we consider and the amount of data we have. One serious huddle in the anisotropy search in the UHECR arrival directions is that cosmic rays can be strongly affected by the intergalactic magnetic fields. UHECR have such high energy that the intergalactic magnetic fields could not completely erase the anisotropy arising from the inhomogeneous distribution of sources. However, the auto-clustering at small angles could be significantly weakened. In the presence of such magnetic fields, the better way to see the clustering due to the strong point source would be to examine the point-wise clustering up to large angles. The existence of a point source would manifest itself through the local clustering of observed UHECRs about the location of the source. This is nothing but a general principle for finding point sources of various bands of radiation in astronomy. In the case of UHECR, what is different from other astronomical particles is that the spreading could be much larger and the number of data is small. This paper reports this pointwise clustering test for the arrival directions of UHECR observed by Pierre Auger Observatory (PAO), aiming for both the anisotropy study and the point source search. To see whether the local clusterings exist in the observed PAO data, we sweep the whole sky covered by the PAO exposure by taking each point as the reference point for the clustering of arrival directions. We take the angular distance distribution of UHECRs about the reference point and compare it with the one expected from the isotropic distribution. We calculate the p-value by applying the Kuiper test on the angular distance distribution. The small p-value implies the departure from isotropy about the reference point. The departure from the isotropy can arise either from the excess (local clustering) or the deficit (local void). In Sec. 2, we describe the detail of statistical method we use, including the criterion for this excess/deficit decision based on the Kuiper test. In Sec. 3, we present the results for the PAO UHECR arrival directions, characterizing anisotropy by one strong excess region and one void region. We discuss the implications of the results and conclude in Sec. 4.", "pages": [ 2, 3 ] }, { "title": "II. ANISOTROPY STUDY USING THE SINGLE SOURCE CADD METHOD", "content": "There are many ways to test the isotropy of the spherical data. First, we may check the multipole moments. We can also use the auto-correlation of arrival directions, which is good for checking if there are small scale clusterings or some regularities. Here we use the test method suitable for hunting for the point sources of UHECR. We take any one point on the sphere as a reference point and examine the distribution of arrival directions about that point. The simple way is to look at the angular distance distribution of arrival directions from the reference point. In Refs. [12, 13], we developed the simple comparison method for the UHECR arrival direction distributions, where the two-dimensional UHECR arrival direction distributions on the sphere is reduced to onedimensional probability distributions of some sort, so that they can be compared by using the standard Kolmogorov-Smirnov (KS) test or its variants. For the correlation test, we adopted the reduction methods called the correlational angular distance distribution (CADD). The method used in this paper is simply the special case of CADD, where the reference point is taken as if it is a single source. When averaged over the all point sources, this is similar to Ripley's K function, a well-known second-order summary characteristic for the spatial pattern. We do not take the average as we focus on the local clustering, not on the overall clustering characteristic. Here we present briefly the basic ideas of the reduction method and how to calculate the p-value for the isotropy. The correlational angular distance distribution is the probability distribution of the angular distances of all pairs of UHECR arrival directions and the point source directions. As we consider the reference point as a single source, CADD is just the probability distribution of the angular distances of UHECR arrival directions from the reference point: θ i ≡ cos -1 (ˆ r i · ˆ R ) , where ˆ r i ( i = 1 , . . . , N ) are the UHECR arrival directions and ˆ R is the reference direction. For the comparison of CADD obtained from the data and that from the isotropic distribution, we can apply KS test or its variants such as Kuiper test. In this analysis, we use Kuiper test because it seems most suitable for our purpose and the probability function of its statistic is available in analytic form. The Kuiper test is based on the cumulative probability distribution (CPD), S N ( x ) = ∫ x p ( x ' ) dx ' and the Kuiper statistic D K is the sum of maximum difference above and below two CPDs, where From the KP statistic D K , the probability that CADD of the observed data is obtained from the model under consideration can be estimated using the Monte-Carlo simulations in general. When the data in the distribution are all independently sampled, as in our case, the following approximate probability formula is available: where Q KP ( λ ) = 2 ∑ ∞ j =1 (4 j 2 λ 2 -1) e -2 j 2 λ 2 and N e = N 1 N 2 / ( N 1 + N 2 ) is the effective number of data. Now, N 1 = N O , the number of observed UHECR data and N 2 = N S , the number of mock UHECR data obtained from the isotropic distribution. We can make the expected distribution more accurate by increasing the number of mock data N S . In the limit N S →∞ , the effective number of data is simply N e = N O . For a given reference point, we obtain two single point CADDs to be compared, one from the observed PAO data and the other expected from the isotropic distribution. Then we calculate the p-value using the formula (3). The small p-value indicates that the distribution of arrival directions in view of the given reference point significantly differs from the isotropic distribution. The departure from isotropy can be either the local excess or the local deficit of observed UHECR around the reference point compared to the isotropic distribution. For small p-value, to decide whether it is the excess or the deficit, we consider the following: The Kuiper test uses the maximum differences of the observed distribution above and below the expected distribution, D K+ and D K -. Let the angular distances at which D K+ and D K -are attained be θ + and θ -, respectively. Then, the order of the angular values θ + and θ -can be used for this purpose. If θ + < θ -, it is probably the excess. If θ + > θ -, it is probably the deficit. Of course, there can be a subtlety that small excess or deficit very near the reference point can be missed. But, this simple rule could catch the overall behavior correctly in most cases.", "pages": [ 3, 4 ] }, { "title": "III. RESULTS FOR UHECRS OBSERVED BY PAO", "content": "We use the UHECR data set released in 2010 by PAO [8]. It contains 69 UHECR with energy higher than 5 . 5 × 10 19 eV. Their arrival directions are shown as black dots in Fig. 2. The PAO site has the latitude λ = -35 . 20 · and the zenith angle cut of the data is θ m = 60 · . We use the geometric exposure function, which is known to work well for the cosmic rays with energy higher than the GZK cutoff. We sweep the whole sky covered by the PAO exposure, by taking each point as the reference point for the reduction of the arrival direction distribution to single point CADD. For illustration of our method, we show in Fig. 1 the cases of two reference directions whose p-values are smallest among excess regions and deficit regions. The maximal excess point is ( α = 192 . 75 · , δ = -38 . 77 · ) and the maximal deficit point is ( α = 81 . 69 · , δ = -67 . 90 · ), which are marked by the + symbols in Fig. 2. The left panels show the CADD with angular bin size of 10 · . Compared to the CADD of isotropic background (green dashed lines), the excess and the deficit of the PAO data (black solid lines) at small angles are clearly seen. The right panels show the corresponding CPD, to which we apply the Kuiper test and obtain the p-values P excess , min = 2 . 0 × 10 -4 and P deficit , min = 1 . 2 × 10 -3 , respectively. The vertical bars represent the sizes of D K+ , D K -and the angular distances θ + , θ -at which they are attained. For the excess at small angles, we have θ + < θ -. For the deficit at small angles, the order is reversed. It confirms that we can tell the excess region and the deficit region from the order of θ + and θ -. However, one caveat is in order. Because θ + and θ -are determined from the whole distribution over small to large angles, sometimes small clustering at small angles can be overlooked. An example can be found around α = 90 · , δ = -15 · region. It seems that there is a small clustering there, but the region is classified into the deficit region due to the stronger deficit at middle angles. In Fig. 2, we show our main result, the map of p-values obtained by the single source CADD method for the PAO UHECR data. The black dots are the arrival directions of PAO data and graded colors represent the p-value bands as indicated in the p-value scale bar. Our criterion for the excess or deficit region is that the p-value of that region is smaller than 0 . 0455. To those regions we apply the excess/deficit criterion explained in the previous section to further classify them into the excess or the deficit regions. They are depicted in red color or in blue color for distinction. An overall feature is that we have one large excess region and one large void region. The large excess region is centered around the location of Centaurus A ( α = 201 . 37 · , δ = -43 . 02 · , marked by the × symbol in Fig. 2). The maximal excess point ( α = 192 . 75 · , δ = -38 . 77 · , marked by the + symbol in Fig. 2), which is located inside this region, has the p-value P excess , min = 2 . 0 × 10 -4 . The Centaurus A position yields the p-value P excess , Cen A = 1 . 2 × 10 -3 . The large deficit region is located near the south pole and it contains the maximal deficit point ( α = 81 . 69 · , δ = -67 . 90 · , marked by the + symbol in Fig. 2), whose p-value is P deficit , min = 1 . 2 × 10 -3 . Both the excess region near Centaurus A and the deficit region near the south pole confirms that the arrival direction distribution of PAO UHECR is anisotropic.", "pages": [ 4, 5, 6 ] }, { "title": "IV. CONCLUSION", "content": "The anisotropy in the arrival directions of UHECRs observed by PAO was firstly noted in the correlation with AGNs. The observed arrival directions showed more correlation with AGNs than the isotropic distribution. It was measured by the number of correlated events, which lie within a fixed angular distance from AGNs. But since the number of AGNs is larger than the number of observed UHECRs, it is obvious that all AGNs are not the sources of UHECRs. The method adopted in this paper is quite helpful for the search of point sources of UHECR, since it scans the sky pointwisely detecting the deviation from isotropy in the arrival directions. The excess region may be an indication of UHECR source in that region. In this regard, the large excess region located near Centaurus A supports the hypothesis that Centaurus A is a promising source of UHECR [8, 14]. This fact can be used to infer the fraction of Centaurus A contribution to the whole observed UHECRs and to estimate the size of intergalactic magnetic fields in the vicinity of Centaurus A [15]. The distinguishing feature in our results of anisotropy study is the identification of a void region near the south pole. The important implication of the existence of void region in the arrival directions of observed UHECRs is that it limits the number of UHECRs contributed by the isotropic background, which is presumably considered to be the contribution from the sources outside of the GZK radius. A detailed study on this limit is in progress. In conclusion, we adopted the point source correlational angular distance distrusting to study the anisotropy in the arrival directions of ultra-high energy cosmic rays. Our method reveals that anisotropy in the arrival directions of UHECR observed by PAO is characterized by one prominent excess region and one void region, which confirms that the arrival direction distribution is anisotropic in the sense that it is hard to obtain from the isotropic distribution. The excess region is located near the Centaurus A direction supports that the Centaurus A is a promising UHECR source. The void region near the south pole direction may be used to limit the diffuse isotropic background contribution, that is, the contribution from outside of the GZK radius.", "pages": [ 6, 7 ] }, { "title": "ACKNOWLEDGMENT", "content": "This research was supported by Basic Science Research Program through the National Research Foundation (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2008381).", "pages": [ 7 ] } ]

[ { "title": "Schwarzschild-de Sitter Metric and Inertial Beltrami Coordinates", "content": "Li-Feng Sun, Mu-Lin Yan ∗ , Ya Deng, Wei Huang † , Sen Hu ‡ Wu Wen-Tsun Key Lab of Mathematics of Chinese Academy of Sciences, School of Mathematical Sciences, and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China August 27, 2013", "pages": [ 1 ] }, { "title": "Abstract", "content": "Under consideration of coordinate conditions, we get the Schwarzschild-Beltramide Sitter (S-BdS) metric solution of the Einstein field equations with a cosmological constant Λ. A brief review to the de Sitter invariant special relativity (dS-SR), and de Sitter general relativity (dS-GR, or GR with a Λ) is presented. The Beltrami metric B µν provides inertial reference frame for the dS-spacetime. By examining the Schwarzschild-de Sitter (S-dS) metric g ( M ) µν existed in literatures since 1918, we find that the existed S-dS metric g ( M ) µν describes some mixing effects of gravity and inertial-force, instead of a pure gravity effect arisen from 'solar mass' M in dS-GR. In this paper, we solve the vacuum Einstein equation of dS-GR, with the requirement of gravity-free metric g ( M ) µν | M → 0 = B µν . In this way we find S-BdS solution of dS-GR, written in inertial Beltrami coordinates. This is a new form of S-dS metric. Its physical meaning and possible applications are discussed. PACS numbers: 04.20.Jb; 11.30.Cp; 98.80.Jk Key words: Classical general relativity, Exact solutions, Special Relativity, de Sitter spacetime symmetry, Beltrami metric, Mathematical and relativistic aspects of cosmology.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Discussions to de Sitter(dS) spacetimes have attracted much interests recently. The reasons are multiple. Two of them are: (1) The recent observations in cosmology show that our universe is in accelerated expansion (see, e.g., [1] and references within), which implies that the universe is probably asymptotically dS spacetime with positive cosmological constant Λ; (2) Just as weakening the fifth axiom leads to non-Euclidean geometry, giving up Einstein's Euclidean assumption on the rest rigid ruler and clock in special relativity leads to other kind of Special Relativity (SR) on the dS-spacetime with dS-radius R [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. We call it dS-SR. Localizing the spacetime symmetry in inertial frames of dS-SR, we can reach gravitational field theory with a cosmological constant Λ ≡ 3 /R 2 [10, 11]. Such theory is just de Sitter General Relativity (dS-GR) with Λ. In this paper we try to solve the equation of dS-GR in vacuum, and discuss Schwarzschild-de Sitter metric in inertial Beltrami spacetime coordinates. 1 Existence of inertial coordinate system is the foundation of special relativity. And existence of local inertial coordinate system is one of GR-principles. What we say the inertial coordinate system here is that in which the inertial motion law for free particles holds. Namely, in a maximally symmetric spacetime with a specific metric g µν , if the free particle motion is inertial, we could call such sort of g µν inertial metric . There are two inertial metrics: Minkowski spacetime metric η µν = diag { 1 , -1 , -1 , -1 } , and Beltrami metric (see Appendix Eq.(60)). It is easy to check η µν is inertial. The Landau-Lifshitz action [12] for free particle with mass m 0 in Minkowski spacetime is: and then From δA = 0 (the least action principle, or the free-particle motion law along geodesic line), the equation of motion of the particle reads and then ˙ x = constant . We conclude that the the inertial motion law for free particle holds in Minkowski spacetime, and metric η µν is inertial. Similarly, it can also be proved that the Betrami metric B µν ( x ) is also inertial via straightforward calculations (see Appendix: Eqs.(66)-(80)). The fact that B µν ( x ) is inertial led to the discovery of de Sitter invariant special relativity [2, 3, 8]. Physically, it is useful and meaningful to find out GR-solutions for empty spacetime, which approach to the metric of the inertial system when the gravity vanishes. A typical example is usual Schwarzschild solution of GR without Λ. To empty spacetime with T µν = 0, the Einstein equation reads where R µν is Ricci tensor. The Schwarzschild solution of (4) in spherical space coordinates r, θ, φ is where M is 'solar' mass. It is essential that when M → 0, the Schwarzschild metric approaches Minkowski ( Mink ) metric, which provides inertial reference frames. Explicitly, when M → 0, from (4), we have (Note, in Cartesian space coordinates { x 1 , x 2 , x 3 } , ds 2 Mink = η µν dx µ dx ν where { η µν } = diag { 1 , -1 , -1 , -1 } and x 0 = ct . η µν is a solution of (4)). This fact indicates that when the gravity disappears, the spacetime becomes Minkowski's. Thanks to this outstanding property, one can use the Schwarzschild metric to achieve the calculations of effects such as the motion in a centrally symmetric gravitational field to verify GR (see, e.g.,[12], pp.306 ). From the above we learned that the Schwarzschild solution structures in GR rely on two essential properties: (a) the metric satisfies the Einstein equation in empty spacetime with T µν = 0; (b) when the gravity disappear due to M → 0, the metric tends to empty spacetimemetric of inertial coordinate system. To dS-GR (or GR with a cosmologic constant Λ, see Eq.(16) in below), differing from (4), the corresponding Einstein field equations for empty spacetime are which will be derived below (see Eq.(20)). Obviously η µν is no longer the solution of (7). We should find a metric which satisfies (7), and meanwhile the motion of free particle in the spacetime with this metric is inertial. In [11], we have obtained the solution of this problem: (see also Appendix A) where R 2 = 3 / Λ and σ ( x ) ≡ 1 -η µν x µ x ν /R 2 . B µν ( x ) is called Beltrami metric. We also call both η µν and B µν ( x ) inertial metrics . The SR based on η µν is usual Einstein SR (E-SR), and the one based on B µν ( x ) is dS-SR. In this case, one may ask what is the Schwarzschild-de Sitter metric of dS-GR written in inertial Beltrami coordinates? Namely, a metric satisfies both (7) and the requirement that when the gravity disappears (corresponding to 'solar mass' M → 0) it tends to Beltrami metric B µν ( x ). In general relativity, a suitable choice of the coordinate system is often useful to solve actual problems or make actual predictions. Similarly, the metric within asymptotic inertial Beltrami-spacetime frame is a new and useful metric. We call such metricSchwarzschild-de Sitter metric in inertial Beltrami coordinates, or Schwarzschild-Beltrami-de Sitter metric. The aim of this paper is to solve this problem. The paper is organized as follows. In section 2, we briefly review the dS-SR, and construct dS-GR via localizing the global dS spacetime symmetry in dS-SR. We show that Beltrami metric plays an essential role for charactering the inertial systems of dS spacetime; In section 3, we reexamine the old Schwarzschild-de Sitter metric existed in literatures since 1918, and show that it contains both effects of gravity and effects of non-inertial forces. After that we solve the vacuum Einstein equation of dS-GR under the requirement that the metric must purely reflect gravity effect. In other words, our new solution is Schwarzschild-de Sitter metric in inertial Beltrami coordinates. In section 4, we sum up the main point of this paper and briefly discuss the physical meaning of our new solution presented in the paper. In Appendix A, more interpretations on Beltrami metrics and dS-SR are presented.", "pages": [ 2, 3 ] }, { "title": "2 de Sitter Special Relativity, de Sitter General Relativity and Beltrami Metric", "content": "We start with a brief review to de Sitter Special Relativity (dS-SR) and de Sitter General Relativity (dS-GR). The Lagrangian for a free particle in dS-SR has been shown in [8]: (see Appendix A) where ˙ x µ = d dt x µ , B µν ( x ) is Beltrami metric: (see Appendix A) with constant R the radius of the pseudo-sphere in dS -space which is related to the cosmological constant via R = √ 3 / Λ. The Euler-Lagrangian equation reads Substituting (9) into the Euler-Lagrangian equation (11) and after a long but straightforward calculation, we obtain [8] (see Appendix A) This result indicates that the free particle in the Beltrami space-time B ≡ { x µ , g µν ( x ) = B µν ( x ) } moves along straight line and with constant coordinate velocities. Namely the inertial motion law for free particles holds true in the space-time B , and hence the inertial reference frame can be set in B (see Appendix A). The coordinates, which would be used for both dS-SR and dS-GR, are the inertial Beltrami coordinates { x µ } . When we transform from one initial Beltrami frame x µ to another Beltrami frame ˜ x µ with the origin of the new frame a µ in the original frame, the transformations between them with 10 parameters are as follows Under this transformation, the metric B µν is preserved [8]: ˜ ˜ The ten parameters in (13) are 4 space-time transition parameters a µ , 3 boost parameters β i and 3 space rotation parameters α i (Euler angles). They are constants and space-time independent. Therefore the dS-SR transformations (13) are global. According to the gauge principle, the localization of global symmetry will yield gauge field theory. As is well known that the external spacetime gauge theory is gravitational field theory [13, 14]. Like to localize the global Poincar'e (or inhomogeneous Lorentz) group transformation, the global transformation of (13) can also been localized via a µ → a µ ( x ) , β i → β i ( x ) , α i → α i ( x ). Thus, localized transformation of (13) reads where f µ ( x ) are four arbitrary functions of x . Hence, (15) represents a general spacetime coordinates transformation, or curvilinear coordinates transformation. Assuming the spacetime is torsion-free just like Einstein did in GR, the affine connection here is also Christoffel symbol: Γ λ µν = Γ λ νµ . Now let us determine the action of gravity fields S G ≡ ∫ d 4 x √ -g G ( x ) in empty spacetime, where G ( x ) is a scalar. To determine G ( x ) we should also consider the fact that the equation of the gravitational field must contain derivatives of the 'potentials' (i.e., g µν ( x )) no higher than the second order (just as is the case for the electromagnetic field). From the Riemann geometry, it is found that only R and trivial constant Λ ≡ constant satisfies all requirements. Therefore, G ( x ) = a ( R 2Λ) where a is also a constant. In Gaussian system of units , a = -c 3 / (16 πG ) where G = 6 . 67 × 10 -8 cm 3 · gm -1 · sec -2 is the universal gravitational constant. Thus we obtain the action of gauge gravity in empty spacetime: From δS G = 0, we obtain dS-SR tells us that to empty spacetime the metric must be Beltrami metric (10). Namely, one solution of (17) is required to be g µν = B µν . Then the value of constant Λ is determined to be and (17) becomes This is the basic equation of the dS-GR in empty spacetime, which is different from the usual GR's (see (4)).", "pages": [ 4, 5 ] }, { "title": "3.1 Schwarzschild-de Sitter solution in non-inertial system", "content": "The simplest vacuum solution of Einstein's equation with a positive cosmological constant were derived by Kottler (1918), Weyl(1919), Trefftz (1922)[15]. It is actually a spherical Schwarzchild-de Sitter solution of dS-GR. We call that solution S-dS metric, which is [15]: where M is 'solar' mass, and { x µ N } = { ct N , r N , θ N , φ N } (subindex N is short for Noninertial system, which will be proved below) represent the S-dS spacetime coordinates. When M → 0, we get empty de Sitter spacetime metric: Let us explore the question whether the empty de Sitter spacetime metric g (0) µν is a metric of spacetime with inertial frame or not. Namely, we should pursue whether the motion of free particles in de Sitter spacetime with metric g (0) µν is inertial or not. For this, we consider the expression of g (0) µν ( y ) in the Cartesian space coordinates y i with i = { 1 , 2 , 3 } . From (22), and noting y 0 = x 0 N = ct N , y 1 = r N sin θ N cos φ N , y 2 = r N sin θ N sin φ N , y 3 = r N cos θ N , with η ij = diag {-1 , -1 , -1 } , we have Note there is no boost in { x µ N → y µ } , so it is not a transformation between reference systems. g (0) µν ( x N ) | x N → y = g (0) µν ( y ) is nothing, but only a variable change. Comparing (23) with (10), we find: /negationslash This fact indicates that g (0) µν ( y ) is generally not a spacetime metric of inertial reference systems. In other words, the coordinates { y µ } are not an inertial coordinate system. To be more concrete, let's see the motion of free particle in S-dS spacetime. The Landau-Lifshitz Lagrangian L N ( y i , ˙ y i ) for a free particle in S-dS is where y = y 1 i + y 2 j + y 3 k and ˙ y ≡ d y /dt N = ˙ y 1 i + ˙ y 2 j + ˙ y 3 k . From δS = -m 0 c δ ∫ ds = δ ∫ dt N L ( y i , ˙ y i ) = 0, we have Substituting (25) into (26), we can easily obtain the equation of motion f (y i , ˙ y i , y i ) = 0. For our purpose, an explicit consideration of one-dimensional motion of the particle is enough. Namely, setting y 2 = y 3 = 0 and ignoring the motion of j , k directions, the equation of motion f (y i , ˙ y i , y i ) = 0 becomes: /negationslash /negationslash This equation explicitly indicates that there exist inertial forces in de Sitter spacetime with metric g (0) µν , which make the particle's acceleration in direction i to be non-zero, i.e., y 1 = 0. Consequently, we conclude that the empty de Sitter spacetime metric g (0) µν ( x N ) is not a metric of spacetime within inertial reference systems. Or, in short, g (0) µν ( x N ) is non-inertial.", "pages": [ 5, 6, 7 ] }, { "title": "3.2 Coordinate transformation between non-inertial and inertial systems", "content": "We have shown in the above that the Schwartzschild-de Sitter metric g ( M ) µν ( x N ) within asymptotically non-inertial framework described by g (0) µν ( x N ) has been derived by [15] from It has been addressed in last subsection that g (0) µν ( x N ) is non-inertial. It is meaningful to find the S-dS metric within asymptotic inertial spacetime frame. Namely, we should solve the equation as follows Here, we use B g ( M ) µν ( x ) to denote new S-dS metric with non-zero 'solar mass' M , which is called Schwarzschild-Beltrami-de Sitter metric hereafter, or S-BdS metric in short. The equation of (31) means that B g ( M ) µν ( x ) must satisfy the requirement that the empty spacetime metric (i.e., the metric B g (0) µν ( x )) is inertial. This condition ensures B g ( M ) µν ( x ) to be desired new S-dS metric within inertial Beltrami coordinates. Since all of g ( M ) µν ( x N ) , g (0) µν ( x N ) and B g (0) µν ( x ) ≡ B µν ( x ) have already been known, the reference system transformation T between spacetimes { x µ N } ≡ { ct N , r N , θ N , φ N } and { x µ } ≡ { ct, r, θ, φ } can be derived from the tensor-transformation properties of g (0) µν ( x N ) and B µν ( x ) (see (36) below). Then the desired result of B g ( M ) µν ( x ) will be determined by means of B g ( M ) = T ' g ( M ) T . We sketch the logic in following diagram (32). Now let us derive the reference system transformation T between spacetimes { x µ N } and { x µ } . In spherical coordinates, and from equations (22) and (10) we have where σ = R 2 -c 2 t 2 + r 2 R 2 , and the following spherical coordinates expression of Beltrami metric has been used: where subindex Bel means Beltrami. Under reference system transformation between { x µ N } and { x µ } the transformation from g (0) αβ ( x N ) to B µν ( x ) reads which can be rewritten in matrix form, where matrices B ≡ { B µν ( x ) } , g (0) ≡ { g (0) αβ ( x N ) } , T ≡ { ∂x β N ∂x ν } , and T ' is the transpose of the matrix T . To simplify the problem, we assume T has the form of Then from (37) we have Assume ∂t N ∂r = 0, to solve ∂t N ∂t , ∂r N ∂t and ∂r N ∂r , let the first three equations of (39) can be reduced to From (44) (note 1 -c 2 t 2 R 2 = σ -r 2 R 2 ), and r ' N ≡ ∂r N ∂r , we get a special solution of the coordinate transformation between non-inertial and inertial systems From (46), Finally, we get a coordinate transformation to inertial spacetime frame It's consistant with [3].", "pages": [ 7, 8, 9 ] }, { "title": "3.3 Schwarzschild-Beltrami-de Sitter Metric", "content": "Under the transformation T of equations (50) -(53), the S-dS metric (21) transforms to S-BdS metric: The solution is Substituting (50) -(53) into (54), we finally obtain the desired S-BdS metric as follows This is a new metric of dS-GR, and serves as main result of this paper. It is a metric written in inertial Beltrami coordinates. It is straightforward to check that it satisfies the Einstein field equation of dS-GR in empty spacetime, (19). It is also easy to see that when R →∞ , B g ( M ) µν ( x ) of (55) coincides with Schwarzschild metric of (5); when M → 0, it coincides with Beltrami metric of (35); and when R →∞ and M → 0, it goes back to Minkowski metric of (6).", "pages": [ 9, 10 ] }, { "title": "4 Summary and Discussion", "content": "In this paper we start with a brief review to the de Sitter invariant special relativity (dSSR), and construct de Sitter general relativity (dS-GR) via localizing the global de Sitter spacetime symmetry, which is equivalent to the GR with a cosmology constant Λ = 3 /R 2 . We emphasized that the Beltrami metric B µν in corresponding Beltrami coordinates plays an essential role to characterize the dS-spacetime with inertial reference frames in both dS-SR and dS-GR. Namely the motions of free particles in Beltrami coordinates are inertial (i.e., along straight or say geodesic line with uniform velocity). Existence of inertial reference systems is the foundation of special relativity. And existence of local inertial reference systems is one of GR-principles. Physically, it is useful and meaningful to find out GRsolutions for empty spacetime, which approach to the metric of the inertial system when the gravity vanishes. After reexamining the Schwarzschild-de Sitter (S-dS) metric g ( M ) µν existed in literatures sine 1918, we find that when the gravity arisen from 'solar mass' M from g ( M ) µν disappears, the metric g (0) µν is not equal to the Beltrami metric B µν , and then the motion of a free particle in g (0) µν -spacetime is non-inertial, or violates the so called inertial motion law. This means that the existed S-dS metric g ( M ) µν is in non-inertial g (0) µν -spacetime. So g ( M ) µν describes some mixing effects of gravity and inertial-force, instead of a purely gravity effect arisen from 'solar mass' M . As is well known that, in appropriate local inertial Minkowski spacetime coordinates in which the inertial motion law holds, the ordinary Schwarzschild metric in usual GR is a description of pure gravities due to M . Generally, the predictions in inertial reference systems play essential role to clarify the corresponding physics conceptually. Almost all valuable predictions for experiments in particle physics, for instance, are formulated in expressions in inertial systems. In GR, the remarkable calculations of the motion of a planet in the gravitational field of the Sun, and of the bent of the light ray under the influence of the gravity of the Sun, and etc are achieved in terms of Schwarzschild metric which is just a GR-prediction in inertial framework. Therefore, it is necessary and useful to find out the Schwarzschild-de Sitter metric written in inertial Beltrami coordinates in dS-GR, or in short Schwarzschild-Beltrami-de Sitter (S-BdS) metric B g ( M ) µν ( x ). We provide the result as in (55). This is the main result of this paper. We would like to mention that if parameter R is finite (instead of infinite), both the motion of a planet in the gravitational field of the Sun and the bent of the light ray under the influence of the Sun will have to use B g ( M ) µν ( x ) to do such calculations. This project is in progress [16]. In addition, the dS-black hole physics should also be based on B g ( M ) µν ( x ) instead of the former g ( M ) µν ( x N ). However, it goes beyond the scope of this paper, and remains to be open at present stage. Finally, we would like to mention that B g ( M ) µν ( x ) is a time dependent metric. Since such a dependence is via a ratio of c 2 t 2 /R 2 , and R is a cosmologic huge length scale [1], usually c 2 t 2 /R 2 << 1 and ordinary Schwarzschild metric in GR could be thought as a leading approximation of B g ( M ) µν ( x ).", "pages": [ 10, 11 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This work is partially supported by National Natural Science Foundation of China under Grant No. 10975128 and No. 11031005, by Chinese Universities Scientific Fund under Grant No. WK0010000030, and by the Wu Wen-Tsun Key Laboratory of Mathematics at USTC of Chinese Academy of Sciences.", "pages": [ 11 ] }, { "title": "A Beltrami Metric and de Sitter Invariant Special Relativity", "content": "In this Appendix we present some explicit calculations related to de Sitter invariant Special Relativity (dS-SR), and some interpretations to reference [8].", "pages": [ 11 ] }, { "title": "1. Beltrami metric:", "content": "We derive the expression of Beltrami metric (10) in the text. We consider a 4dimensional pseudo-sphere (or hyperboloid) S Λ embedded in a 5-dimensional Minkowski spacetime with metric η AB = diag (1 , -1 , -1 , -1 , -1): where index A, B = { 0 , 1 , 2 , 3 , 5 } , R 2 := 3Λ -1 and Λ is the cosmological constant. S Λ is also called de Sitter pseudo-spherical surface with radii R . Defining /negationslash and treating x µ are Cartesian-type coordinates of a 4-dimensional spacetime with metric g µν ( x ) ≡ B µν ( x ), denoting this 4-dimensional spacetime as B Λ (call it Beltrami spacetime), we derive B µν ( x ) by means of the geodesic projection of {S Λ ↦→ B Λ } (see Figure 1). From the definition (56), we have where Substituting them into Eq.(58), we have Then, we obtain the Beltrami metric as follows The first Newtonian law is the foundation of the relativity. This law claims that the free particle moves with uniform velocity and along straight line. There exist systems of reference in which the first Newtonian motion law holds. Such reference systems are defined to be inertial . And the Newtonian motion law is always called the inertial moving law . If two reference systems move uniformly relative to each other, and if one of them is an inertial system, then clearly the other is also inertial. Experiment, e.g., the observations in the Galileo-boat which moves uniformly, shows that the so-called principle of relativity is valid. According to this principle all the law of nature are identical in all inertial systems of reference. Theorem 1 : The motion of particle with mass m 0 and described by the following Lagrangian Since ξ A,B ∈ S Λ , and from (57) and (56), it is easy to obtain: satisfy the first Newtonian motion law, or the motion is inertial . In (61), the Cartesian expression of the velocity is as follows where i · i = j · j = k · k = 1, and i · j = i · k = j · k = 0. Proof : By means of the Euler-Lagrangian equation (where ∂/∂ x ≡ ∇ := ( ∂/∂x 1 ) i + ( ∂/∂x 2 ) j + ( ∂/∂x 3 ) k and etc) and L = L Newton we obtain Theorem 2 : The motion of particle in Minkowski spacetime described by is inertial. The proof is the same as above, because both L Newton and L Einstein are coordinates x i -independent. Generally, any x -free and time t -free Lagrangian functions L ( ˙ x ) can always reach the result of (64). However, when Lagrangian function is time-dependent that rule will become invalid. A useful example is as follows: /negationslash where a constant Λ = 0. The stick-to-itive readers can verify the following identity via straightforward calculations from (66): Noting that the Euler-Lagrange equation (63) reads and substituting (67) to (68), we have Since we have /negationslash which indicates that the particle motion described by Lagrangian function (66) is inertial, and the first Newton motion law holds. Thus, the corresponding inertial reference systems can be built. Noting it is essential and remarkable that a new kind of Special Relativity based on L Λ (66) serving as an extension of the Einstein's Special Relativity (E-SR) may exist.", "pages": [ 11, 12, 13, 14 ] }, { "title": "3. de Sitter invariant Special Relativity (dS-SR):", "content": "Following the Landau-Lifshitz formulation of Lagrangian [12] (see (65)), we examine the motion of free particle in the spacetime with Beltrami metric (60). From Eq.(9) in text we derive its expression in Cartesian coordinates. Setting up the time t = x 0 /c , B µν ( x ) can be rewritten as follows where Substituting eqs.(74)-(77) into (73), we obtain the Lagrangian for free particle in B Λ : By using Cartesian notations (62) and expressions of (59) (75) (76) (77), the explicit expression of Lagrangian (78) is: where x 2 = ( x · x ). Noting Eq.(18) in text, R 2 = 3 / Λ, and comparing L dS with L Λ ( t, x , ˙ x ) of (66), we find which is the Lagrangian for free particle mechanics of dS-SR. Since (72), when | R | → ∞ , the dS-SR goes back to E-SR. B µν . When space rotations were neglected temporarily for simplify, the transformation both due to a Lorentz-like boost and a space-transition in the x 1 direction with parameters β = ˙ x 1 /c and a 1 respectively and due to a time transition with parameter a 0 can be explicitly written as follows: where γ = 1 / √ 1 -β 2 . It is easy to check when R →∞ the above transformation goes back to Poincar'e transformation (or inhomogeneous Lorentz group ISO (1 , 3) transformation) in E-SR. The external spacetime symmetry of dS-SR is SO (4 , 1). According to Neother theorem, the corresponding 10-Noether charges are energy E , momentums p i , boost charges K i and angular-momentums L i . All have been derived in [8]. The results are as follows Noether charges for Lorentz boost : K i = m 0 Γ c ( x i -t ˙ x i ) Charges for space -transitions (momenta) : p i = m 0 Γ˙ x i , Charge for time -transition (energy) : E = m 0 c 2 Γ (82) Charges for rotations in space (angularmomenta) : L i = /epsilon1 i jk x j p k , where the Lorentz factor of dS-SR is: It can be checked that ˙ E = ˙ p i = ˙ K i = ˙ L i = 0 under the equation of motion x i = 0 (or x = 0). [8]", "pages": [ 14, 15 ] } ]

[ { "title": "Unimodular Constraint on global scale Invariance", "content": "Naveen K. Singh ∗ Theory Division, Physical Research Laboratory, Navrangpura, Ahmedabad - 380 009, India Abstract: We study a model with global scale invariance within the framework of unimodular gravity. The global scale invariant gravitational action which follows the unimodular general coordinate transformations is considered without invoking any scalar field. This is generalization of conformal theory described in the Ref. [1]. The possible solutions for the gravitational potential under static linear field approximation are discussed. The new modified solution has additional corrections to the Schwarzschild solution which describe the galactic rotational curve. A comparative study of unimodular theory with conformal theory is also presented. Furthermore, the cosmological solution is studied and it is shown that the unimodular constraint preserve the de Sitter solution explaining the dark energy of the universe.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Surprisingly, we live in the era of late time acceleration of universe [2,3]. The acceleration in expansion of universe offers the possibility of new physics of the unknown component 'dark energy'. The dark matter, whose physical nature is only partially known, is also one of the interesting subjects of research in cosmology. Dark matter was introduced by Fritz Zwicky in year 1933 [4, 5] to account for evidence of missing mass in Coma cluster. The more interesting issue to notice is that dark matter and dark energy dominate the energy density of the universe. The observed dark energy and dark matter contribute approximately 72 . 8% and 22 . 7% respectively to the total energy content of the universe. There are number of scalar field theories such as Quintessence, K-essence, Chaplygin gas model and theory of modified gravity such as f ( R ), DGP model, etc. to describe the dark side of universe. In the recent literature [6-8], authors discuss the absence of the dark matter in the vicinity of solar system. The dark matter might be internal property of the space and hence might be explained by the modified theory of gravity [1,9-14]. In this paper, such a theory of modified gravity is considered which has global scale invariance along with the unimodular constraint to explain the galactic rotational curve and acceleration of universe expansion. The scale symmetry prevents any dimensionful parameter in the action and hence might give a possible explanation for the cosmological constant problem [15-21]. This symmetry is broken if the theory contains any dimensionful parameters, such as particle masses, cosmological constant, gravitational constant etc.. Although scale invariance is generally believed to be anomalous, it is possible to maintain this symmetry in the full quantum theory if the symmetry is broken by a soft mechanism [18-21]. It has been shown [20-23] that scale invariance has its other advantages in explaining the dark sector of the universe. The local scale invariance explains the current era of the universe [22], favoring the Λ CDM model. In [22,23] the symmetry breaking mechanism of local scale invariance generates the dark energy and vector field which acts as dark matter field. The global scale transformation is given by x µ → Λ x µ where Λ is a constant parameter. The above transformations make the action like ∫ R 2 √ -gd 4 x or ∫ φ 2 R √ -gd 4 x invariant. The theories of higher order invariants in the action were initially started by Weyl in 1919 and by Eddington in 1923. One of the advantage of f ( R ) theory is it describes the early universe [24-26]. Further, it also explains the late time acceleration [14,27] in the expansion of universe and might be an alternative for the dark matter [1, 9-14]. The theory follows the principle of covariance similar as the Einstein-Hilbert action. It modifies the Einstein equation as well as the gravitational potential. The only quadratic terms of the curvature scalar in the action preserves the global scale invariance. It is reasonable to take only quadratic terms of the curvature scalar in the action, since the solution could include the Schwarzschild solution and the corrections in addition [1]. Using the Gauss-Bonnet identity, we may write such global scale invariant action as [28] In this paper we generalize the conformal theory [1] by imposing the unimodular constraint. Conformal theory is one of the special case of (1), where γ = -1 / 3 [1]. In subsection (1.1), the field equation of metric is derived. In section 2, the corresponding equation for the unimodular theory is given. The field equation is solved under static linear field approximation and corresponding galactic rotational curves are discussed in section 2. The cosmological solution with unimodular constraint is discussed in section (3). The last section (4) contains the discussion and conclusions.", "pages": [ 1, 2 ] }, { "title": "1.1 Field Equation", "content": "The variation of action 1 gives the field equation as Here W 1 µν and W 2 µν are the terms corresponding to the variation of R 2 and R ρσ R ρσ and these are given as respectively.", "pages": [ 2 ] }, { "title": "2 Unimodular Gravity", "content": "Unimodular gravity was introduced in [29,30] and has been reviewd in [31]. The theory is subclass of general theory of relativity but with a constraint in addition, i.e., the determinant of the metric is not dynamical; g µν δg µν = 0. The motivation of the unimodular gravity is to solve the cosmological constant problem as we don't have any such term in the action. Further, in the reference [32], authors discuss dynamics of expansion of universe with the unimodular theory of gravity taking the dynamical part of determinant of metric as a separate scalar field. However, in this paper, any scalar field is not considered. The condition g µν δg µν = 0 modifies the Einstein equation as following [30,31], where κ is coupling constant, T µν is energy momentum tensor of source field and T is its trace. The Eq. 4 is traceless part of the Einstein equation. The variation of action 1 gives following field equation The same procedure of constraint of unimodular gravity over the action given in Eq. 1 gives the following field equation, where W = W α α is trace of tensor W αβ . The Eq. 6 is the traceless part of Eq. 5. Here g µν δg µν = 0 is used, i.e., the action does not have any constant term.", "pages": [ 2, 3 ] }, { "title": "2.1 Vacuum Solution for the Conformal Theory", "content": "For γ = -1 / 3, 1 /B of W rr component of Eq. (6) gives the following equation where the metric is given by, The exact vacuum of Eq. (7) may be written as [1] where, C 1 , C 2 and C 3 are constants. Now in next subsection we generalize this for general γ .", "pages": [ 3 ] }, { "title": "2.2 Vacuum Solution for Unimodular Gravity", "content": "In this subsection, we solve for the gravitational potential with unimodular gravity considering the line element (8). -(1 /B ) of t -t component, -B of r -r component and 1 /r 2 of θ -θ component of field Eq. 6 are given by and respectively. Now, considering linear approximation, i.e., B ( r ) ≈ 1+ φ ( r ), we have following three equations and respectively. The solution of Eq. 13 is given by Plugging this solution, either in Eq. 14 or 15, we get the same constraint over the constants which is given as follows Now, we have different solution for allowed values of γ and other constants. The constraint Eq. 17 gives one of the case where γ = -1 / 3. For this value we get /negationslash which is same solution as in Eq. (9) for the conformal theory. However, for γ = -1 / 3, we have C 3 = 0 and which implies For these values of γ the solutions are given by respectively. The solution 22 or 23 with the data of galactic rotational curve [33,34] for the Milky Way galaxy is plotted in Fig 1. For the large scale, data is taken from the simulation II given in the table (3) of the Ref. [33] and for small scale the data is taken from the table (2) of the Ref. [34]. The effective velocity of star may be written as where f = 9 × 10 6 to make velocity unit as (100 Km/sec ). For the best fit, the values of constants are The plot is shown by the dotted line. The further plot of the solution (21) with solid line is shown in Fig. 1. The values of constants for this case are as following The plot for the conformal theory is also shown with dashed line, where the gravitational potential is given by the Eq. (9) and for the best fit the values of the constants are given by The values of χ 2 min per degree of freedom for the best fit for the solution (9), (21) and (22) are given by 3 . 19, 5 . 54 and 6 . 15 respectively. However, the solution (21) gives the best fit for the scale > 15 Kpc as shown in the Fig. (1). For the large scale > 15 Kpc , we find χ 2 min for the conformal theory; Eq. (9) as 2 . 25 whereas for the case of unimodular gravity; Eq. (22) and (21) it is as 1 . 09 and 0 . 77 respectively. Hence for the large scale, the theory of unimodular gravity describes the galactic rotational curve with the best fit.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Cosmological Solution", "content": "It is known to us that Gauss-Bonnet action explains acceleration in the expansion of the universe [35-37] . Further, in the modified theory of gravity f ( R ) = R 2 , we have exact de-Sitter solution [14] for the vacuum. In this section, we test it explicitly as now the action has the unimodular constraint in addition. For the FRW metric [ -1 , a 2 , a 2 , a 2 ], where a is scale factor of the universe, Eq. (6) gives the same equation for 0 -0 and i -j components and it is given by The Eq. (28) may be written as independent of the parameter γ as Looking over Eq. (29), one may conclude for the exact de-Sitter solution, which explain the acceleration in the expansion of universe, where H 0 is Hubble constant. Hence, the de Sitter solution satisfies both the conformal theory and the theory of unimodular gravity.", "pages": [ 6, 7 ] }, { "title": "4 Discussion and Conclusions", "content": "A scale invariant model of higher order invariant in the action is presented. The unimodular constraint on the theory is also considered. Scale invariance allows only quadratic terms of curvature scalar in the action, whereas consideration of unimodular theory in addition constrain on the values of the parameter of the resulting theory. It is shown that for the parameter γ = -1 / 2 and 1, the solution of the gravitational potential includes the Schwarzschild solution as well as the term corresponding to the integration constant. The solution for this case explains the galactic rotational curve, but the corresponding gravitational field increases as distance increases whereas for γ = -2 / 3, the solution has one more term proportional to r 4 so that the velocity or corresponding gravitational field decreases after ∼ 42 Kpc . Furthermore, the solution of conformal theory is recovered for γ = -1 / 3. The conformal solution has a lighter bump at ∼ 30 Kpc . Hence, the unimodular theory of gravity has good behavior for the large scale rather than that of conformal theory. The proper scale invariant matter source term in the action might describe the rotational curve for the low range. We will proceed it further in the future publication. The theory is interesting as it does not require the dark matter which has not been observed in the solar neighborhood so far. Furthermore, the de Sitter solution is also obtained for the considered theory explaining the dynamics of current era.", "pages": [ 7 ] }, { "title": "Acknowledgements", "content": "I thank Subhendra Mohanty, Pankaj Jain and Girish Chakrabarty for useful discussions.", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "Mem. S.A.It. Vol. , 1 c GLYPH<13> SAIt 2013 Memorie della", "pages": [ 1 ] }, { "title": "The CO5BOLD Analysis Tool", "content": "S. Wedemeyer Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway e-mail: [email protected] Abstract. The interactive IDL-based CO5BOLD Analysis Tool (CAT) was developed to facilitate an easy and quick analysis of numerical simulation data produced with the 2D / 3D radiation magnetohydrodynamics code CO5BOLD. The basic mode of operation is the display and analysis of cross-sections through a model either as 2D slices or 1D graphs. A wide range of physical quantities can be selected. Further features include the export of models into VAPOR format or the output of images and animations. A short overview including scientific analysis examples is given. Key words. Sun: photosphere; Radiative transfer", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The growing computational resources allow for increasingly larger and more detailed numerical simulations of stellar atmospheres, resulting in a considerably large amount of data. The production of advanced comprehensive models must therefore be accompanied by the development of e GLYPH<14> cient analysis and visualization software that is capable of handling the produced large data sets. Here, the CO 5 BOLD Analysis Tool (abbreviated CAT, Fig. 1) is described. It is designed for an interactive analysis of 2D and 3D model atmospheres, which are produced with CO 5 BOLD - a widely used state-of-the-art code for the simulation of stellar atmospheres (Freytag et al. 2012).", "pages": [ 1 ] }, { "title": "2. Program overview", "content": "CAThas an interactive graphical user interface, which is is programmed in IDL (see Fig. 2). The primary mode of operation is the display Send o GLYPH<11> print requests to : S. Wedemeyer and analysis of slices through 3D (or 2D) models. CAT can handle files with multiple snapshots ( .full ). Settings and sessions. After the installation, adjustments such as the default window size and paths can be made and saved. These standard settings will be restored every time CAT is started. Settings that are connected to a particular analysis session can be saved and re- stored as session files. It is possible to define a default session, which is automatically loaded when CAT starts. Slice display The projection plane can be chosen perpendicular to the axes, resulting in the three planes x-y, x-z, and y-z in a 3D model. The position within a plane can be changed either by clicking in the displayed image or through the widgets in the control panel. CAT now also provides a zoom function in a separate tab, in which the magnification and the currently displayed region can be selected (see Fig. 3 for a close-up region). Di GLYPH<11> erent physical quantities can be chosen in the data tab (see Fig. 2), ranging from the basic quantities contained in the model file to more advanced quantities like, e.g., the spatial components of the electric current density (e.g., Scha GLYPH<11> enberger et al. 2005). The chosen quantity can then be combined with a mathematical operation, e.g. the logarithm, which is useful for quantities that cover many orders of magnitude in the displayed slice (e.g., the mass density in the x-z plane). By default, a colorcoded 2D slice is displayed. Alternatively, the data in the current projection plane can be shown as surface plot, contour plot or line plot. Color-coded slices can be overlaid with contours (incl. a reference contour at optical depth unity) and / or a vector field. For the latter the spatial components of the velocity or the magnetic field in the selected projection plane can be drawn as vectors or streamlines, while the perpendicular component is not considered. The length of the vectors can be chosen directly in the data tab with more options being available in the options menu . Measurements. The new measure tab provides di GLYPH<11> erent tools for (i) measuring distances between two points or along a polygon path, (ii) defining markers, and (iii) producing and exporting contour paths. The markers, which are sets of coordinates, and the contours can be interactively selected, modified and exported for further processing. These tools can be used for, e.g., defining 3D structures or for manually tracking features in space and time. Images and animation. The displayed images can be output in various image formats. The animation tool provides an interactive way to produce MPEG videos or image sequences. It can either be a time animation for a series of model snapshots contained in one or more multi-snapshot file(s) or a spatial scan through a selected model snapshot along one of the spatial axes. Export. CAT o GLYPH<11> ers several options to export model data. The currently loaded simulation snapshot can be written out in UIO format, even when it is part of a multi-snapshot ( .full ) file. This feature is useful if a simulation terminated without producing an .end file or if this file is corrupted for some reason. In this case CAT can be used to extract the last time step from the corresponding .full file for using it as start model for the next simulation run. Model snapshots can also be saved as plane-parallel models. In a coming version, it will be possible to save only parts of the model and to extract data into IDL savefiles for further analysis. The 3D visualization tool, which is integrated in CAT, can give a first 3D impression but has limited functionality (see Fig. 4). The VAPOR tool (Clyne et al. 2007; Clyne & Rast 2005) allows for a more comprehensive analysis of the 3D structure, in particular for vector fields like velocity and magnetic field. For that reason, CAT o GLYPH<11> ers the possibility to export CO 5 BOLD data into VAPOR format.", "pages": [ 1, 2, 3 ] }, { "title": "3. Data analysis examples", "content": "CAT has been repeatedly proved useful in the past. It provides an easy way of monitoring the evolution of an ongoing simulation. CAT can also help to quickly reveal what is wrong in cases where a simulation terminates due to problems at a specific grid cell. Next to these 'care-taking' tasks, CAT as been essential for the detailed scientific analysis in many cases. Swirls and tornadoes. One example is the discovery of ring-like structures with increased horizontal velocity at chromospheric heights in 3D magnetohydrodynamic (MHD) models of the Sun. Similar rings were previously observed as so-called 'chromospheric swirls' with the Swedish 1-m Solar Telescope (Wedemeyer-Bohm & Rouppe van der Voort 2009). The rings of enhanced velocity, which appeared prominently in CAT (see Fig. 3), suggested that chromospheric swirls are the observational signature of rotating magnetic field structures. This finding finally led to a more comprehensive study of this phenomenon, then known as 'magnetic tornadoes' (WedemeyerBohm et al. 2012). M-type dwarf stars. CAT has been extensively used for the development of 3D MHD atmosphere models of M-type dwarf stars that extend from the upper convection zone into the chromosphere (Wedemeyer et al. 2012). The resulting set of models with di GLYPH<11> erent initial magnetic fields with di GLYPH<11> erent field strengths and di GLYPH<11> erent topologies (e.g., homogeneous vertical or mixed polarities) will be analysed in more detail in forthcoming publications.", "pages": [ 3, 4 ] }, { "title": "4. Outlook", "content": "Since the first version in 2002, the development of CAT has been driven by the need for additional functionality for the analysis of increasingly complex models. Also in the future, CAT will be continuously extended. The feedback of users is therefore very welcome. Possible extensions in future releases could include the ability to read the data formats of other MHD codes and to display and analyse slices at arbitrary angles. Acknowledgements. The author likes to thank the organisers of the 2 nd CO 5 BOLD Workshop (CW 2 ), which was held in Heidelberg, Germany, in 2012.", "pages": [ 4 ] }, { "title": "References", "content": "Clyne, J., Mininni, P., Norton, A., & Rast, M. 2007, New J. Phys, 9, 1 Clyne, J. & Rast, M. 2005, in Proceedings of Visualization and Data Analysis 2005, ed. J. C. G. M. T. B. K. Edited by Erbacher, Robert F.; Roberts, 284-294 Freytag, B., Ste GLYPH<11> en, M., Ludwig, H.-G., et al. 2012, Journal of Computational Physics, 231, 919 Scha GLYPH<11> enberger, W., Wedemeyer-Bohm, S., Steiner, O., & Freytag, B. 2005, in ESA Special Publication, Vol. 596, Chromospheric and Coronal Magnetic Fields, ed. D. E. Innes, A. Lagg, & S. A. Solanki Wedemeyer, S., Ludwig, H.-G., & Steiner, O. 2012, ArXiv e-prints Wedemeyer-Bohm, S. & Rouppe van der Voort, L. 2009, A&A, 507, L9 Wedemeyer-Bohm, S., Scullion, E., Steiner, O., et al. 2012, Nature, 486, 505", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "Mem. S.A.It. Vol. 75, 282 c © SAIt 2008", "pages": [ 1 ] }, { "title": "Chemical element abundances in the outer halo globular cluster M 75", "content": "N. Kacharov and A. Koch Landessternwarte, Zentrum fur Astronomie der Universitat Heidelberg, Konigstuhl 12, D69117 Heidelberg, Germany e-mail: [email protected] Abstract. We present the first comprehensive abundance study of the massive, outer halo globular cluster (GC) M 75 (NGC 6864). This unique system shows a very extended trimodal horizontal branch (HB), but no other clues for multiple populations have been detected in its colour-magnitude diagram (CMD). Based on high-resolution spectroscopic observations of 16 red giant stars, we derived the abundances of a large variety of α , p-capture, iron-peak, and n-capture elements. We found that the cluster is metal-rich ([Fe / H] = -1 . 16 ± 0 . 02 dex, [ α / Fe] = + 0 . 30 ± 0 . 02 dex), and shows a marginal spread in [Fe / H] of 0 . 07 dex, typical of most GCs of similar luminosity. We detected significant variations of O, Na, and Al among our sample, suggesting three di ff erent populations. Additionally, the two most Na-rich stars are also significantly Ba-enhanced, indicating a fourth population of stars. Curiously, most stars in M 75 (excluding the two Ba-rich stars) show a predominant r-process enrichment pattern, which is unusual at the cluster's high metallicity. We compare the abundance properties of M 75 and NGC 1851 (a GC very similar to M 75 in terms of age, metallicity, and HB morphology) and draw conclusions on M 75's possible formation scenarios. Key words. Stars: abundances - Globular clusters: individual: M 75 - Galaxy: halo -", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "With ages of 10 - 14 Gyr, Globular GCs are amongst the oldest stellar systems in the Milky Way. Long time considered as simple stellar populations, nowadays we recognise their complex star formation histories through precise abundance analyses of a variety of chemical elements in individual cluster members. Although we do not necessarily see clues for multiple populations in all GC CMDs, all of them studied to date present significant spreads and certain anticorrelations between their light and α -element abundances (Gratton Send o ff print requests to : N. Kacharov et al. 2012a). These are tightly linked to the 'second parameter' e ff ect, which needs to explain discordant Horizontal Branch (HB) morphologies at any given metallicity (D'Antona et al. 2002). CNO or He-content variations amongst the stars in a GC are responsible for the appearance of very extended HBs and are accompanied by variations of the p-capture elements (Gratton et al. 2011, 2012b). We obtained high-resolution (R ∼ 30000) spectra of 16 bright giant stars in M 75 with the MIKE spectrograph mounted at the 6.5-m Magellan telescope aiming at a comprehensive abundance analysis of this GC. It is located at a Galactocentric distance of 15 kpc, which tenants the transition region between the inner and outer Milky Way halo. Its younger age ( ∼ 10 Gyr; Catelan et al. 2002) and high metallicity ([Fe / H] = -1 . 16 dex) are compatible with the properties of the outer halo GC system and suggest a possible extragalactic origin. On the other hand, M 75 is amongst the most concentrated GCs ( c = log( rt / rc ) = 1 . 80), which could be contrasted to the extended and loose clusters in the outer halo (Koch & Cot'e 2010; Koch et al. 2009). This unique GC also has a trimodal HB, which is not explicable under canonical stellar evolutionary models (Catelan et al. 2002). Apart from the well separated red HB (RHB) and blue HB (BHB), its CMD shows a distinct third extension of a very blue, faint tail. Thus, it is very important to assess possible multiple populations, which could be related to the peculiar HB morphology and to look for peculiarities in its chemical composition, which might reveal clues about its origin and early evolution.", "pages": [ 1, 2 ] }, { "title": "2. Light elements in M 75 - clues for multiple populations", "content": "We detected significant variations in the starto-star abundances of Na, O, and Al in M 75 but not in Mg, which would be typical for bright GCs (Fig. 1). The stars with [Na / Fe] < 0 . 1 dex are considered to be the remainder of the primordial first generation (FG) of stars. They are similar to the stars in the Galactic halo. In contrast, stars from the second generation (SG) show considerably higher Naand lower O-abundances but they are indistinct in terms of other chemical elements. We also found that the two most Na-rich stars are Ba-enhanced by 0 . 4 and 0 . 6 dex, respectively, above the cluster's mean solar [Ba / Fe] ratio, indicative of s-process enrichment from AGB stars. The extended blue tail of M 75's HB is at odds with our findings of only a moderate Na-O anticorrelation, which so far lacks an extreme population on the RGB, characterised by extremely O-poor stars. The Na-O anticorrelation of M 75 better resembles the less massive GC M 4 than that of NGC 1851 (Fig. 2). The latter GC is often thought as M 75's twin in terms of age, luminosity, and HB morphology. However, its CMD shows double RGB and SGB, which are not found in M 75. The less extended Na-O anticorrelation of M 75 suggests lower mass AGB polluters (Carretta et al. 2009), which is further supported by the presence of Ba-enhanced stars in this cluster. Atomic Number The Na-O anticorrelation (although not very extended) is consistent with three di ff erent stellar populations in M 75. Complemented with the two Ba-rich stars, we suggest the presence of four chemically distinct populations in this GC (Fig. 1).", "pages": [ 2, 3 ] }, { "title": "3. n-capture elements in M 75 s-process deficient primordial stars", "content": "M 75 has an extremely low [Ba / Eu] ratio of -0 . 63 dex and seems to be one of the rarer cases of a GC compatible with predominant rprocess production (Fig. 3). An exception are the lighter n-capture elements Rb, Y, and Zr, which are more consistent with scaled solar r + s-process production. These elements, however, are associated with the weak s-process, which appears in massive (M ∼ 20M /circledot ) stars on similar timescales as the r-process production from SNe II. The best fit to the production of the elements from Ba to Th in M 75 is found for (scaled solar) pure r-process enrichment, plus only 10% of the (scaled solar) s-process yields. This means that only a small number of AGB stars contributed to the enrichment of the primordial cloud from which M 75 formed. Pure r-process enhancement is typical for metal poor halo field and GC stars with [Fe / H] < 2 . 0 dex (e.g. Sneden et al. 2000), but there are also examples of GCs and halo field stars with predominant r-process enrichment at higher metallicities, e.g. NGC 3201 (Gonzalez & Wallerstein 1998) and Pal 3 (Koch et al. 2009) with [Fe / H] -1 . 6 dex. Most similar to M 75 in this respect is the GC M 5 with [Fe / H] = -1 . 3 dex and [Ba / Eu] = -0 . 60 dex (Yong et al. 2008), see Fig. 4.", "pages": [ 3 ] }, { "title": "4. M 75 and its place in the Galaxy", "content": "In order to place M 75 amongst other GCs and halo field stars, we plotted four key abundance ratios, important to trace the chemical evolution of any stellar population. The compilation of Galactic stars is taken from Venn et al. (2004), and the mean abundances of various GCs are from Pritzl et al. (2005), complemented with more recent results for NGC 1851 (Carretta et al. 2011), M 5 (Yong et al. 2008), and the outermost GCs Pal 3 (Koch et al. 2009) and Pal 4 (Koch & Cot'e 2010). The mean (Mg, Ca, Si)-abundance is representative of M 75's α -content, [Ba / Y] represents the main s- to weak s-process ratio, [Y / Eu] - the weak s- to main r-process, and [Ba / Eu] - the main s- to main r-process. The chemical abundances of M 75 are fully compatible with the bulk of Galactic GCs and halo field stars, which rules out possible extragalactic origin and accretion on a later stage to the Milky Way halo.", "pages": [ 3 ] }, { "title": "5. Conclusions", "content": "M75 likely hosts four chemically distinct stellar populations formed on a short timescale. The two most Na-rich stars are also Baenhanced, which prompts that the main polluters, which enriched the SG, also included intermediate mass AGB stars. The moderate NaOanticorrelation and the lack of significant Mg variations are at odds with the very extended HB of M 75. The n-capture elements pattern is consistent with predominant r-process enrichment with a marginal contribution (about 10% of the scaled solar yields) of s-process enriched material, typical for the most metal poor GCs and halo field stars, but not excluded at higher metallicities. The overall abundances of M 75 are consistent with the abundances of other in- ner and outer halo GCs and field stars, which suggests a similar origin with the bulk of Milky Way Globular clusters. Acknowledgements. The authors acknowledge the Deutsche Forschungsgemeinschaft for funding from Emmy-Noether grant Ko 4161 / 1. NK is grateful to F. D'Antona and R. G. Gratton for helpful discussions and to the committee headed by Father Funes, which decided to award this work with the Rome Globular Cluster Prize 2012.", "pages": [ 4 ] }, { "title": "References", "content": "Burris, D. L., Pilachowski, C. A., Armandro ff ,", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "Mem. S.A.It. Vol. 84, 1 c © SAIt 2008", "pages": [ 1 ] }, { "title": "Tackling the soft X-ray excess in AGN with variability studies", "content": "A.M.Lohfink 1 , C. S. Reynolds 1 , R. F. Mushotzky 1 , and M. A. Nowak 2 e-mail: [email protected] Abstract. The origin of the soft X-ray excess in AGN has been a mystery ever since its discovery. We present how the time variability of this spectral component can point towards its origin. Using the powerful technique of multi-epoch fitting, we study how the soft excess in a given object depends on other parameters of the continuum and the accretion disk possibly hinting at its nature. As an example, we present results from this technique applied to the Seyfert galaxy Mrk 841. We study all (3) XMM and some of the Suzaku pointings available and find that the source displays an impressive variability in the soft X-ray band on the timescale of years. We study several common soft excess models and their ability to physically consistently explain this spectral variability. Mrk 841 is found to show a distinct variability pattern that can be best explained by the soft excess originating mostly from a thermal Comptonization component. The variability timescale can be constrained to be on the order of a few days. Key words. galaxies: individual(Mrk 841) - X-rays: galaxies - galaxies: nuclei - galaxies: Seyfert -black hole physics", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The X-ray spectrum from most Seyfert galaxies can be characterized by a power law continuum with reflection from the accretion disk and the torus. However, there is one portion of the X-ray spectrum that is mysterious even after a decade of XMM and Suzaku observations: the soft X-ray excess ('soft excess' hereafter), i.e. the excess flux at soft X-ray energies with regard to a power law continuum. This feature is common in active galactic nuclei (AGN) X- Send o ff print requests to : A. Lohfink ray spectra but its physical nature is completely uncertain. The spectral models able to describe the soft excess are highly degenerate (Page et al. 2004; Lohfink et al. 2012) and even high resolution spectroscopy obtained with Chandra and XMM has not yielded any additional insights or been able to break the degeneracies (Turner et al. 2001). Due to these modeling difficulties, the physical origin of this soft excess is highly uncertain. Understanding its nature is crucial because of its potentially large luminosity (depending on its exact shape) and the influence it has on the detailed spectral shape of the continuum. For example, uncertainty in the soft excess shape can be an important source of systematic error for the spin parameter of the black hole (see recent analysis by Nardini et al. 2011 and Lohfink et al. 2012). Ever since its discovery over 25 years ago (Singh et al. 1985), there has been a debate about what is producing this excess and there are many distinct ideas as to its physical nature. While the currently two most popular ideas are blurred ionized reflection from the inner parts of the accretion disk (Gierli'nski & Done 2004; Crummy et al. 2006), and Comptonization components (Ross et al. 1992) many more have been proposed. For radio-loud objects and narrow line Seyferts an additional power law or broken power law can describe the soft excess rather well (Kataoka et al. 2007). Physically this power law component can be associated with an optically thick Comptonization component (Papadakis et al. 2010) or a jet component (Chatterjee et al. 2009). A variety of methods have been used to determine the nature of the soft excess is and have narrowed down the options to only two possibilities. Moreover the study of AGN samples and multi-wavelength studies have led to further insights into the properties of the soft excess. From the CAIXA sample we learned that no correlation of strength of the soft excess with black hole mass or luminosity of the AGN exists (Bianchi et al. 2009). Multiwavelength studies have revealed a possible dependency of the UV slope with the soft excess strength and shape (Walter & Fink 1993; Atlee & Mathur 2009). While this is valuable information, to make further progress regarding the soft excess a new approach is needed. An analysis of the luminous Seyfert 1 galaxy Fairall 9 by Lohfink et al. (2012) already emphasized that the soft excess is variable. Assembling the most recent Suzaku key programme spectra of the Seyfert 1 galaxy Mrk841 together with all the archival XMM data available, we confirm this variability (Fig. 1) also in Mrk 841. While individual objects have been studied in the past, it is important to note that these mostly have been studies of single epoch pointings, ignoring the variability.", "pages": [ 1, 2 ] }, { "title": "2. Spectral Analysis", "content": "The major unresolved question regarding the soft excess is whether the soft excess is characterized by a separate spectral component (e.g. Noda et al. 2012) or is part of some broad band component, like ionized reflection. In order to address this question and eventually determine the physical nature of the soft excess, a new approach is needed. Here we utilize the time variability of the soft excess to test which model leads to the best description. From previous analyses it is clear that the X-ray spectrum of Mrk 841 can be well described by a 2-zone warm absorber, a continuum (modeled as a power law), cold and ionized reflection, and a soft excess. For the soft excess we consider three di ff erent possibilities: a) the soft excess is caused by the blurred ionized reflection in the spectrum, b) the soft excess is a superposition of multiple ionized layers of the accretion disk or c) the soft excess is an additional thermal Comptonization component. The di ff erent ideas were tested performing a multi-epoch fit to all the datasets shown in Fig. 1, where the key parameters such as spin, accretion disk inclination and iron abundance are tied during the fitting. A single ionized reflector leads to an unacceptable fit and will not be further discussed. The remaining two models lead to somewhat comparable fit qualities with the Comptonization being statistically preferred. For a fit including an additional Comptonization component at soft energies to model the soft excess only the Comptony parameter can be constrained not kT and τ separately. We observe a correlation between the soft X-ray flux and the Comptony parameter (Fig. 2). The more soft X-ray flux the smaller the Comptony . Another correlation exists between the hard X-ray photon index and the Comptony of the soft Compton component (Fig. 3). The overall spectrum becomes steeper as the Comptony parameter decreases.", "pages": [ 2, 3 ] }, { "title": "2.2. ReflectionModel", "content": "Based on the correlations seen for a model including a soft Compton component one would expect to also see correlations for a model with just blurred ionized reflection. Contrary to the expectations however there is no correlation between the soft X-ray flux and the photon index of the overall spectrum (Fig. 4). Moreover, one would expect a correlation between the ionization states of the reflectors which is not the case either (Fig. 5). In fact at times the 'outer' reflector is more ionized than the 'inner' reflector pointing towards clear problems in the modeling. The spectrum where such an inconsistency happens is the brightest one displayed in Fig. 1. We note that this spectrum also does not show the usual reflection signatures, such as a reflection hump towards higher energies.", "pages": [ 3 ] }, { "title": "3. Summary & Conclusions", "content": "From the analysis of all archival XMM data and the newest Suzaku data for Mrk 841 it is evident that the soft X-ray excess is highly variable by a factor of 2 or more. Assuming that an additional Comptonization component is the correct model and Noda et al. (2012) is correct, it is possible to estimate the variability timescale of the excess to larger than 2 days but less than 7 days. We discover that the soft excess seems to follow a distinct variability pattern. The lack of any clear correlation for the reflection modeling of the soft excess could either be caused by data of insu ffi cient quality to yield sensible constraints and break possible degeneracies or by the assumptions made to construct the model. The approximation of an ionization gradient in an accretion disk by only two ionization zones is only a rough approximation. To be certain that this idea can be excluded a better model will be necessary. However, to learn we need to study more objects to learn whether this is a general property of the soft excess. The origin of soft excess can be further narrowed down by studying simultaneous SEDs and doing broad band fits (e.g. using Nustar ) as the spectral models di ff er significantly at hard X-rays.", "pages": [ 3 ] }, { "title": "References", "content": "Atlee, D. W. & Mathur, S. 2009, ApJ, 703, 1597 Bianchi, S., Guainazzi, M., Matt, G., Fonseca Bonilla, N., & Ponti, G. 2009, A&A,, 495, 421 Brinkmann, W., Papadakis, I. E., & Raeth, C. 2007, A&A,, 465, 107 Chatterjee, R., Marscher, A. P., Jorstad, S. G., et al. 2009, ApJ,, 704, 1689 Crummy, J., Fabian, A. C., Gallo, L., & Ross, R. R. 2006, MNRAS,, 365, 1067 Gierli'nski, M. & Done, C. 2004, MNRAS,, 349, L7 Kataoka, J., Reeves, J. N., Iwasawa, K., et al. 2007, PASJ,, 59, 279 Lohfink, A. M., Reynolds, C. S., Miller, J. M., et al. 2012, ApJ,, 758, 67 Nardini, E., Fabian, A. C., Reis, R. C., & Walton, D. J. 2011, MNRAS,, 410, 1251 Noda, H., Makishima, K., Nakazawa, K., et al. 2012, ArXiv e-prints Page, K. L., Schartel, N., Turner, M. J. L., & O'Brien, P. T. 2004, MNRAS,, 352, 523 Papadakis, I. E., Brinkmann, W., Gliozzi, M., et al. 2010, A&A,, 510, A65 + Ross, R. R., Fabian, A. C., & Mineshige, S. 1992, MNRAS, 258, 189 Singh, K. P., Garmire, G. P., & Nousek, J., 1985, ApJ, 297, 633 Turner, T. J., George, I. M., Yaqoob, T., et al. 2001, ApJL,, 548, L13 Walter, R. & Fink, H. H. 1993, A&A, 274, 105", "pages": [ 3, 4 ] } ]

[ { "title": "ABSTRACT", "content": "Mem. S.A.It. Vol. 75, 282 c © SAIt 2008", "pages": [ 1 ] }, { "title": "A Near-Infrared Spectroscopic Study of Young Field Ultracool Dwarfs: Additional Analysis", "content": "K. N. Allers 1 and Michael C. Liu 2 Abstract. We present additional analysis of the classification system presented in Allers & Liu (2013). We refer the reader to Allers & Liu (2013) for a detailed discussion of our nearIR spectral type and gravity classification system. Here, we address questions and comments from participants of the Brown Dwarfs Come of Age meeting. In particular, we examine the e ff ects of binarity and metallicity on our classification system. We also present our classification of Pleiades brown dwarfs using published spectra. Lastly, we determine SpTs and calculate gravity-sensitive indices for the BT-Settl atmospheric models and compare them to observations. Key words. brown dwarfs - infrared: stars - planets and satellites: atmospheres - stars: low-mass", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In Allers & Liu (2013), hereinafter A13, we present a method for classifying the spectral types (SpTs) and surface gravities of ultracool dwarfs. The SpT classification utilizes both visual comparison to field standards and SpT-sensitive indices. A13 also present new gravity-sensitive indices which measure FeH, VO, alkali line and H -band continuum features. Using gravity-sensitive indices and line EWs, A13 propose three near-IR gravity classes: fld -g for objects with normal field dwarf gravities, vl -g for objects with strong spectral signatures of youth (ages ∼ 1030 Myr), & int -g for objects with intermediate spectral signatures of youth (ages ∼ 50200 Myr).", "pages": [ 1 ] }, { "title": "2. Binarity", "content": "Unresolved binarity can cause peculiarities in the near-IR spectra of brown dwarfs, which are apparent even at low spectral resolution (R ≈ 100). Spectral peculiarity has been used to identify candidate brown dwarf binaries (e.g., Burgasser et al. 2010). Young, low-gravity objects also show signs of spectral peculiarity, which raises two interesting questions: 1) could the spectral peculiarities we attribute to low-gravity be mimicked by unresolved binarity of normal field dwarfs? and 2) to what extent could binarity a ff ect our classification of young, low-gravity, ultracool dwarfs? To test if the spectra of unresolved field dwarf binaries could show evidence of youth in our indices, we combined the spectra of the field dwarf near-IR standards from Kirkpatrick et al. (2010) to create artificial binaries. We first scaled the spectra of the field standards using SpT - M J relations from Dupuy & Liu (2012) so that the spectra were in units of absolute flux. We then co-added two scaled spectra for all possible pairings of standards to create 434 field composite binary spectra. Using the method of A13, we determined the nearIR SpTs for each binary and found that we properly classified all of our artificial M4-L6 (the applicable range of the A13 method) field dwarf binaries as having normal fld -g gravities. We conclude that normal field dwarf binaries are unlikely to contaminate spectroscopic samples of young-low gravity objects. To test the e ff ects of binarity on our classification of low-gravity objects, we created artificial composite binary spectra by combining the low-resolution spectra of young objects in the A13 sample having published parallax values. Table 1 lists the particular spectra we used. We created the artificial low-gravity binary spectra in a manner similar to that used to create artificial field dwarf binary spectra, except that we scaled each low-gravity spectrum to absolute flux units using published parallaxes and JHK mags. We then determined the SpTs and gravities of the artificial low-gravity binary spectra using the methods outlined in A13. The SpTs of the artificial binaries were found to agree with the near-IR SpTs of the primary star to within 1 subtype. The gravity classifications for 54 of the 55 low-gravity artificial binaries agreed with the gravity classifications of the low-resolution spectra of the primaries. The only simulated binary whose classification did not agree with its primary was 2M 0032-44 + 2M 0355 + 11, which we classify as L1 int -g . Overall, it appears that binarity does not significantly a ff ect our SpT or gravity classifications.", "pages": [ 1, 2 ] }, { "title": "3. Metallicity", "content": "In A13, we did not consider the e ff ects of metallicity when determining the SpTs and gravity classifications for our sample. Our gravity-sensitive indices measure the depths of FeH, alkali line (Na and K) and VO features, which in addition to being gravity dependent, are sensitive to metallicity (e.g., Mann et al. Near-IR spectral types and gravities from A13. References: D02 = Dahn et al. (2002); M03 = Ducourant et al. (2008); F12 = Faherty et al. (2012); L13 = Liu et al. (2013); W13 = Weinberger et al. (2013) 2013; Kirkpatrick et al. 2010). Figure 1 compares the spectrum of a mildly metal-poor object (2M 0041 + 35; Burgasser et al. 2004) to the spectra of young, dusty, and normal field ultracool dwarfs of similar optical SpT. The A13 classification system types this object as an L0 fld -g , in good agreement with its optical spectral classification. Not all subdwarfs are well classified by the A13 system, however. Figure 2 compares the spectra of low-gravity, normal and subdwarf L3-L3.5 objects. Although the subdwarf, SDSS 1256-02 (Burgasser et al. 2009), is classified as fld -g , its near-IR SpT is determined to be M6, in stark contrast to its optical type of sdL3.5. We often determined near-IR SpTs of subdwarfs that are significantly earlier than their published optical SpTs. Thus, if one suspects a spectrum could be low metallicity, extreme caution should be used when determining near-IR SpTs. Figure 3 displays the indices calculated for subdwarf spectra, all of which are classified as fld -g . We note that the A13 study included several 'dusty' brown dwarfs, whose spectral peculiarities could be due to a metal-rich photosphere (Looper et al. 2008), all of which were classified as fld -g . Thus, it does not appear that high or low metallicity ultracool dwarfs would be misclassified by A13 as having low gravity.", "pages": [ 2, 3 ] }, { "title": "4. Pleiades Brown Dwarfs", "content": "In A13, we claim that our classification system can identify low-gravity brown dwarfs with ages /lessorsimilar 200 Myr. To test this, we classified spec- a All spectra from Bihain et al. (2010). b SpT determined using the method described in A13. c Gravity Scores are listed in the following order: FeH, VO, alkali lines, H -band continuum shape. See A13 for details. tra for ultracool Pleiades dwarfs from Bihain et al. (2010). We note that many of the spectra in Bihain et al. (2010) have low S / N ( /lessorsimilar 20) compared to the spectra in the A13 sample. Table 2 shows the results of our classification. We calculate SpTs for the objects that are in agreement with the Bihain et al. (2010) SpTs to within ± 1 subtype. We classify all of the Pleiades objects as having low-gravity (and most as having vl -g ). It is interesting to note that among Pleiades spectra of similar SpT, the features indicating youth vary among the objects (as indicated by which features receive scores of '2' in Table 2), with the caveat that some calculated indices have low S / N (Figure 3). This supports the conclusion of A13 that objects of the same age and SpT may have different spectral signatures of youth.", "pages": [ 3 ] }, { "title": "5. Atmospheric Models", "content": "Atmospheric models are calculated for various values of log(g), which could allow us to tie our gravity classifications to particular log(g) values. Figure 4 shows the index values calculated for the BT-Settl (AGSS2009) atmospheric models (Allard et al. 2012). To place the models on the diagram, we first smoothed and resampled them to have resolution similar to the prism spectra in the A13 sample. We then treat the model spectra as if they were the spectra of brown dwarfs, determining SpTs and calculating their gravity sensitive indices using the method described in A13. A detailed comparison between our spectra and the BT-Settl models is beyond the scope of this work, but a couple of trends became apparent from our index calculations. Evolutionary models (Chabrier et al. 2000) predict that log(g) = 3.5, 4.5, & 5.5 corresponds to ages of ∼ 5, 50, & 5000 Myr for 1800-2600 K objects. The model FeH z index values agree fairly well with observations, as do the KI J indices. The H -Cont index values of the models are significantly higher than observations of objects of similar predicted surface gravity. The VO z index for all of the models lie well below the field dwarfs sequence (gray shaded area in Figure 3).", "pages": [ 3, 4 ] }, { "title": "6. Conclusions", "content": "In conclusion, we have found that binarity and metallicity are unlikely to a ff ect our gravity classifications of young brown dwarfs. We note, however, that our near-IR spectral types for low-metallicity objects do not show good agreement with their published optical spectral types. We have applied the A13 classification method to spectra of Pleiades objects from Bihain et al. (2010), and find that we classify all of the spectra as having low-gravity, with most being classified as vl -g . A comparison of indices calculated from the BT-Settl model atmospheres shows that the models reproduce the observed FeH z and KI J index values reasonably well. Model VO z index values, however, are much lower than observations, and model H -Cont indices are higher than observations. Acknowledgements. We are grateful to the organizers of the Brown Dwarfs Come of Age meeting for giving us the opportunity to present our work. We also thank the participants of the meeting for their helpful and thought-provoking comments, which motivated the discussion presented in this manuscript. This research has benefited from the M, L, and T dwarf compendium housed at DwarfArchives.org and maintained by Chris Gelino, Davy Kirkpatrick, and Adam Burgasser as well as from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http: // www.browndwarfs.org / spexprism. This work was supported by NSF grants AST-0407441 and AST-0507833 as well as NASA Grant NNX07AI83G.", "pages": [ 4, 5 ] }, { "title": "References", "content": "Allard, F., Homeier, D., & Freytag, B. 2012, Royal Society of London Philosophical Transactions Series A, 370, 2765 Looper, D. L., Burgasser, A. J., Kirkpatrick, Weinberger, A. J., Anglada-Escud'e, G., & Boss, A. P. 2013, ApJ, 762, 118", "pages": [ 5 ] } ]

[ { "title": "Flow Through Randomly Curved Manifolds", "content": "M. Mendoza, 1, ∗ S. Succi, 2, † and H. J. Herrmann 1, 3, ‡ 1 ETH Zurich, Computational Physics for Engineering Materials, Institute for Building Materials, Schafmattstrasse 6, HIF, CH-8093 Zurich (Switzerland) 2 Istituto per le Applicazioni del Calcolo C.N.R., Via dei Taurini, 19 00185, Rome (Italy), and Freiburg Institute for Advanced Studies, Albertstrasse, 19, D-79104, Freiburg, (Germany) 3 Departamento de F'ısica, Universidade Federal do Cear'a, Campus do Pici, 60455-760 Fortaleza, Cear'a, (Brazil) (Dated: June 28, 2018) We have found that the relation between the flow through campylotic (generically curved) media, consisting of randomly located curvature perturbations, and the average Ricci scalar of the system exhibits two distinct functional expressions (hysteresis), depending on whether the typical spatial extent of the curvature perturbation lies above or below the critical value maximizing the overall Ricci curvature. Furthermore, the flow through such systems as a function of the number of curvature perturbations presents a sublinear behavior for large concentrations due to the interference between curvature perturbations that, consequently, produces a less curved space. For the purpose of this study, we have developed and validated a lattice kinetic model capable of describing fluid flow in arbitrarily curved manifolds, which allows to deal with highly complex spaces in a very compact and efficient way. PACS numbers: 47.11.-j, 02.40.-k, 95.30.Sf Many systems in Nature present either intrinsic spatial curvature, e.g. curved space, due to presence of stars and other interestellar media [1], or geometric confinement constraining the degrees of freedom of particles moving on such media, e.g. flow on soap films [2], solar photosphere [3], flow between two rotating cylinders and spheres [4-6], to name but a few. In general, these systems force a fluid to move along non-straight trajectories (curved geodesics), leading to the upsurge of non-inertial forces. We will denote such systems as Campylotic , from the greek word καµπ ' υλoς for curved, media. Due to the arbitrary trajectories that particles through a campylotic medium can take, depending on the complexity of the curved space, the flow through these media can present very unusual new transport properties. Campylotic media play a prominent role in all applications where metric curvature has a major impact on the flow structure and topology; biology, astrophysics and cosmology offering perhaps the most natural examples. Indeed, for several special cases, the flow through simple campylotic media has already been studied, e.g. Taylor-Couette flow, which was originally formulated between two concentric, rotating cylinders [4, 5], and later extended to the case of spheres [7]. However, beyond very simple geometries, the flow through more complicated structures, like randomly located stars or many biological systems, to the best of our knowledge, has never systematically been addressed before on quantitative grounds. Since, in general, this class of flows lacks analytical solutions, their study is inherently dependent on the availability of appropriate numerical methods. Flows in complex geometries, such as cars or airplanes, make a time-honored mainstream of computational fluid dynamics (CFD), a discipline which has made tremendous progress for the last decades [8, 9]. However, campylotic media set a major challenge even to the most sophisticated CFD methods, because the geometrical complexity is often such to command very high spatial accuracy to resolve the most acute metric and topological features of the flow. Therefore, in this work, we also present a new lattice kinetic scheme that can handle flows in virtually arbitrary complex manifolds in a very natural and elegant way, by resorting to a covariant formulation of the lattice Boltzmann (LB) kinetic equation in general coordinates. The method is validated quantitatively for very simple campylotic media by calculating the critical Reynolds number for the onset of the Taylor-Couette instability in concentric cylinders and spheres [5-7, 10], and applied to the case of two concentric tori. In this Letter, by using the new numerical scheme, we simulate the flow through campylotic media consisting of randomly distributed spatial curvature perturbations (see Fig. 1). The flow is characterized by the number of curvature perturbations and the average Ricci scalar of the space. The campylotic media explored in this work are static, in the sense that the metric tensor and curvature are prescribed at the outset once and for all, and do not evolve self-consistently with the flow. The latter case, which is a major mainstream of current numerical relativity [11, 12], makes a very interesting subject for future extensions of this work. In order to study the campylotic media, we develop a lattice kinetic approach in general geometries, taking into account the metric tensor g ij and the Christoffel symbols Γ i kj . The former characterizes the way to measure distances in space, while the latter is responsible for the non-inertial forces. The corresponding hydrodynamic equations can be obtained by replacing the partial derivatives by covariant ones, in both, the mass continuity and the momentum conservation equations. After some algebraic manipulations, the hydrodynamic equations read as follows: ∂ t ρ +( ρu i ) ; i = 0, and ∂ t ( ρu i ) + T ij ; j = 0, where the notation ; i denotes the covariant derivative with respect to spatial component i (further details are given in the Supplementary Material [13]). The energy tensor T ij is given by, T ij = Pg ij + ρu i u j -µ ( g lj u i ; l + g il u j ; l + g ij u l ; l ), where P is the hydrostatic pressure, u i the i -th contravariant component of the velocity, g ij the inverse of the metric tensor, ρ is the density of the fluid, and µ is the dynamic shear viscosity. Since lattice Boltzmann methods are based on kinetic theory, we construct our model by writing the MaxwellBoltzmann distribution and the Boltzmann equation in general geometries. The former takes the form [14]: where g is the determinant of the metric g ij , and θ is the normalized temperature. The macroscopic and microscopic velocities, u i and ξ i are both normalized with the speed of sound c s = √ k B T 0 /m , k B being the Boltzmann constant, T 0 the typical temperature, and m the mass of the particles. Note that the metric tensor appears explicitly in the distribution function, due to the fact that the kinetic energy is a quadratic function of the velocity, u i u i = g ij u i u j . To recover the macroscopic fluid dynamic equations, we have to extract the moments from the equilibrium distribution function. The four first moments of the Maxwellian distribution function on a manifold are given by, These moments are sufficient to reproduce the mass and the momentum conservation equations. Here, for simplicity we have used dξ to denote dξ 1 dξ 2 dξ 3 and the Jacobian of the integration is already included in the Maxwell Boltzmann distribution, through the determinant term √ g . In the absence of external forces, in the standard theory of the Boltzmann equation, the single particle distribution function f ( x i , ξ i , t ) evolves, according to the equation, ∂ t f + ξ i ∂ i f = C ( f ), where C is the collision term, which, using the BGK approximation, can be written as, C = -(1 /τ )( f -f eq ), with the single relaxation time τ . This equation can be obtained from a more general expression, df/dt = C ( f ), where the total time derivative now includes a streaming term in velocity space due to external forces, df dt = ∂ t f + dx i dt ∂ i f + dp i dt ∂ p i f , with p i the i -th contravariant component of the momentum of the particles. Using the definition of velocity, ξ i = dx i /dt , and due to the fact that the particles in our fluid move along geodesics, which implies the equation of motion we can write the Boltzmann equation as [15], where we have used the definition of the momentum, p i = mξ i . Note that the third term of the left hand side carries all the information on non-inertial forces. Thus, all the ingredients required to model a fluid in general geometries within the Boltzmann equation are now in place. Note that the Christoffel symbols and metric tensor are arbitrary and therefore we can model the fluid flow in curved spaces, whose metric tensor is very complicated and/or only known numerically. Since the contravariant components of the velocity are free of space-dependent metric factors, they lend themselves to standard lattice Boltzmann discretization of velocity space. All the metric and non-inertial information is conveyed into the generalized local equilibria and forcing term, respectively. These features are key to the LB formulation in general manifolds. As an additional feature, complex boundary conditions related to a specific geometry, e.g. surface of sphere, in many cases, can be treated exactly by cubic cells in the contravariant coordinate frame, thereby avoiding stair-case approximations typical of cartesian grids. The details of the discretization of this model on a lattice can be found in the Supplementary Material [13]. To provide numerical validation of our model, we study the flow through one of the simplest campylotic medium, the Taylor-Coutte instability in three different geometries, i.e. two concentric rotating cylinders, spheres and 1 tori, respectively. Full details of the validation are given in the Supplementary Material[13]. In Figure 2, we report the critical Reynolds number as a function of the aspect ratio η = a/b , where a and b are the minor and major radii, respectively. As one can appreciate, for the cylindrical geometry we obtain excellent agreement with analytical theory [5], and a similar match with experimental data [7] is found for the spherical case. We have also computed the torque coefficient, and found reasonable agreement, within a few percent, with experimental data [16, 17]. For the case of two concentric rotating tori, the critical Reynolds numbers for different configurations can also be observed in Fig. 2, showing values around 10% larger than for the case of cylinders. Further details can be found in the Supplementary Material [13]. Next, we move to a genuinely campylotic medium, consisting of randomly located curvature perturbations. To this purpose, we define a coordinate system ( x, y, z ), such that its metric tensor takes the form: g ij = δ ij (1 -a 0 ∑ N n =1 exp( -r n /r 0 )), where n labels each local curvature perturbation located at /vectorr n = ( x n , y n , z n ), N is the total number of perturbations, r n = | /vectorr n | , and r 0 characterizes the size of the deformation. Note that the coefficient a 0 can be either signed, depending on whether a positive or negative curvature is imposed, respectively. In our study, we have chosen to work with positive values of a 0 , due to the analogy with a system of randomly located stars, which produce deformations in the metric tensor of spacetime[1]. The Christoffel symbols are R calculated numerically. The flux is calculated by the geometrical relation, Φ = ∫ S ρu x √ g xx gdS , where S is the cross section at the location where the measurements are taken. Since the fluid dynamic equations only contain the metric tensor and its first derivatives (via the Christoffel symbol), and due to the fact that particles move along geodesics according to Eq. (3), it is natural to expect that the flow could be characterized by a quantity that contains the metric tensor and its first derivatives. Although the Christoffel symbols Γ i jk meet this requirement, they are not components of a tensor, and therefore they are not invariant under a coordinate system transformation (physics should not depend on the choice of the coordinate system). An invariant, or tensor, that can be used to characterize the system is the Ricci tensor R ik . In this work, we use the Ricci scalar (curvature scalar) which can be calculated from the Ricci tensor, R ij , by contraction of the indices, R ' = g ij R ij . The relation between the metric tensor and Christoffel symbols and the Ricci tensor can be found in the Supplementary Material [13]. To study this particular system, we use a lattice size L x × L y × L z of 128 × 64 × 64, and τ = 1. All quantities will be expressed in numerical units. To drive the fluid through the medium, we add an external force along the x -component, which in all simulations takes the value, f ext = 5 × 10 -5 . The flux in flat space, i.e. in the absence of curvature perturbations is denoted by Φ 0 . Shown in Fig. 1, are the velocity streamlines, the Ricci scalar R ' and the high-curvature locations, represented by gray isosurfaces. Note that the streamlines are very complex, as the flow can orbit around the spheres before continuing its trajectory [18, 19]. Also we can see how the curvature perturbations interact, creating non-spherical shaped isosurfaces. Fig. 3 shows the flux reduction Φ 0 -Φ, as function of the number of curvature perturbations, N . We observe that the flux Φ decreases with N . This effect is due to the interplay between the longer trajectories that particles must take and the acceleration due to the non-inertial forces, see Eq. (3). Note that, in general, for systems with different configurations (e.g. negative a 0 ), we could expect that the combination of the two effects might lead to higher flux by increasing N . We also see that the flux depends linearly on N for low concentration of curvature perturbations, and only sublinearly at higher concentrations. This is due to the fact that at low concentration, the average distance between curvature perturbations is large, and consequently each perturbation adds up as a single modification to the total spatial curvature. However, as the concentration is increased, the curvature perturbations start to interfere with each other and consequently the space becomes less curved (decrease of the overall Ricci curvature). The flux is found to obey the following law, where A 1 = 163 ± 2 and N 0 = (1 . 54 ± 0 . 03) × 10 4 are fitting parameters. The parameter N 0 denotes a characteristic number of curvature perturbations, above which the sublinear behavior sets in ( N /greaterorsimilar N 0 ). In the inset of Fig. 3, we observe that by fixing the number of curvature perturbations N = 1024 and the strength a 0 = 0 . 01, and changing the range of the perturbation, r 0 , the difference Φ 0 -Φ presents a maximum for a given r 0 ∼ r c . Furthermore, another interesting result is that the average curvature, here defined as R = -10 8 < R ' > (where < ... > means average over space), shows the same qualitative behavior. Since by increasing r 0 the metric tensor components decrease monotonically, this maximum is due to the Christoffel symbols (or non-inertial forces), which can be characterized via R . However, the maxima are slightly shifted, due to the fact that the Ricci scalar does not uniquely determine the metric tensor and Christoffel symbols, the quantities that play a key role in the fluid dynamic equations. Taking into account this effect, we can plot the flux reduction Φ 0 -Φ as a function of R , and find that, indeed, for r 0 < r c , the flux decreases by increasing the average curvature R with a different law than for the case of large values of r 0 > r c (see inset of Fig. 4). This gives rise to a hysteresis-shaped curve, the reason for this hysteresis being that the metric tensor is different for r 0 < r c and r 0 > r c , even if R takes the same value. However, in both cases, the system shows that higher values of the average curvature R always result in a lower flux. The behavior of the flux for r 0 < r c is well represented by the following law: and for the case of r 0 > r c , where R 0 = 5 . 2 ± 0 . 1, A 2 = 50 ± 2, A 3 = 154 ± 4, and Φ ' = 5 ± 1. The quantity R 0 is related to the maximum curvature achieved by the system and the intersection of the two laws (see Fig. 4). The other interesting quantity is Φ ' , which represents the difference of flux between r 0 /greatermuch r c and r 0 /lessmuch r c , when the curvature scalar becomes zero, and it is due to the fact that in both cases, although the space has no curvature, it has nonetheless different metric tensors. Summarizing, we have explored the laws that rule the flow through campylotic media consisting of randomly distributed curvature perturbations, and shown that, for the configurations studied in this Letter, curved spaces invariably support less flux than flat spaces. Furthermore, the flux can be characterized by the Ricci scalar, a geometrical invariant that contains the metric tensor and Christoffel symbols, the quantities that appear in the fluid dynamics equations. The trajectories of the flow can become very complicated due to the total curvature of the medium, presenting, in some cases, orbits winding several times around regions with high curvature. The present method opens the possibility to apply the actual model to astrophysical systems, where the curvature of space is due to the presence of stars and other interstellar material. We have not considered time curvature, since its contribution remains sub-dominant unless mass is made extremely large. To calculate the flux in campylotic media, we have developed a new lattice Boltzmann model to simulate fluid dynamics in general non-cartesian manifolds. The model has been successfully validated on the Taylor-Couette instability for the case of two concentric cylinders and spheres, the inner rotating with a given speed and the outer being fixed. We also studied the Taylor-Couette instability in two concentric rotating tori, finding that the critical Reynolds number for the onset of the instability is about ten percent larger than the one for the cylinder. By solving the Navier-Stokes equations in contravariant coordinates, which can be represented on a cubic lattice precisely in the format requested by the lattice Boltzmann formulation, the present model opens up the possibility to study fluid dynamics in complex manifolds by retaining the outstanding simplicity and computational efficiency of the standard lattice Boltzmann method in cartesian coordinates. The case of dynamically adaptive campylotic media, in which the metric tensor and curvature would evolve self-consistently together with the flow, makes a very interesting subject for future extensions of the present lattice kinetic method in the direction of numerical relativity [20, 21]. The authors are grateful for the financial support of the Eidgenossische Technische Hochschule Zurich (ETHZ) under Grant No. 06 11-1.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "Supplementary Material", "content": "We show the details of the new lattice kinetic model to study campylotic media, and include a respective validation by studying the Taylor-Couette instability for the case of two concentric rotating cylinders, spheres and tori. We also implement a convergence study showing that the model presents nearly second order convergence, and introduce basic relations in differential geometry like the calculation of covariant derivatives and the Ricci tensor.", "pages": [ 5 ] }, { "title": "Covariant derivative and Ricci Tensor", "content": "The formulation of fluid equations in general coordinates implies the replacement of partial derivatives with the corresponding covariant ones. Given a vector A i , the covariant derivative is defined by where Γ i jk is the Christoffel symbol associated with the curvature of the metric manifold, namely Γ i jk = 1 2 g im ( ∂g jm ∂x k + ∂g km ∂x j -∂g jk ∂x m ). For an arbitrary tensor of second order, the covariant derivative is given by Here and throughout, according to Einstein's convention, repeated indices are summed upon. The Ricci tensor R ik is related with the metric tensor and Christoffel symbols by the relation,", "pages": [ 5 ] }, { "title": "Tensor Hermite Polynomials", "content": "The Lattice Boltzmann formulation in general geometries makes strong reliance on Hermite expansion of the kinetic distribution function. The first three Hermite polynomials are, where we have used the Kronecker delta δ ij .", "pages": [ 5 ] }, { "title": "Hermite polynomials expansion", "content": "Let us expand the distribution function f ( x i , ξ i , t ) in the form, where the coefficients a ( n ) are n -th order space-time dependent tensors, and H ( n ) are the tensorial Hermite polynomials of n -th order. The weights w ( ξ ) are defined as: The coefficients a ( n ) can be calculated with the relation, To recover the correct hydrodynamic equations, the model must be built in such a way as to recover the first four moments, given by, The fourth one ensures that the dissipation term achieves the correct form. To this purpose, we need to expand the distribution function at least up to the third order Hermite polynomial (The explicit expression of the Hermite polynomials have been given above). Thus, using Eq. (14), and replacing the Maxwell-Boltzmann distribution for a manifold, we obtain: Next, by taking θ = 1 (isothermal limit), we obtain the coefficients of the expansion, as follows: Therefore, the truncated equilibrium distribution function up to third order, using Eq. (12), reads as follows: With this, we have expanded the equilibrium distribution function up to third order in Hermite polynomials. Next, we need to expand also the forcing term, Γ i jk ξ j ξ k ∂ ξ i f , in the Boltzmann equation, Due to the fact that the distribution function can be written using Eq. (12), and invoking the properties of the Hermite polynomials, we can write the forcing term as, where we have introduced the notation, F i = -Γ i jk ξ j ξ k . Then, replacing the coefficients from Eq. (17), and the corresponding Hermite polynomials, Eq. (11), we obtain the forcing term, With every term expressed as a series of Hermite polynomials, all is in place to proceed with the LB discretiza- tion according to standard Hermite-Gauss projection of the continuum Boltzmann equation.", "pages": [ 5, 6 ] }, { "title": "Lattice Discretization", "content": "In order to formulate a corresponding lattice Boltzmann model, we implement an expansion of the MaxwellBoltzmann distribution in Hermite polynomials, so as to recover the moments of the distribution function up to third order in velocities, as it is needed to correctly reproduce the dissipation term in the hydrodynamic equations. The expansion of the Maxwell-Boltzmann distribution was introduced by Grad in his 13 moment system [22]. Since this expansion is performed in velocity space, and the metric only depends on the spatial coordinates, we expect such an expansion to preserve its validity also in the case of a general manifold. We have followed a similar procedure as the one described in Refs. [23, 24]. For the discretization of the Maxwell Boltzmann distribution (16) and the Boltzmann equation (19), we need a discrete velocity configuration supporting the expansion up to third order in Hermite polynomials. Our scheme is based on the D 3 Q 41 lattice proposed in Ref. [25], which corresponds to the minimum configuration supporting third-order isotropy in three spatial dimensions, along with a H-theorem for future entropic extensions [26] of the present work. In the following, we shall use the notation c i λ to denote the i -th contravariant component of the vector numbered λ . Thus, the discrete Boltzmann equation for our model takes the form, f λ ( x i + c i λ δt, t + δt ) -f λ ( x i , t ) = -δt τ ( f λ -f eq λ )+ δt F λ , where F λ is the forcing term, which contains the Christoffel symbols, and f eq λ is the discrete form of the Maxwell-Boltzmann distribution, Eq. (16). The relevant physical information about the fluid and the geometry of the system is contained in these two terms. The macroscopic variables are obtained according to the relations, ρ = ∑ 41 λ =0 f λ , ρu i = ∑ 41 λ =0 f λ c i λ . The shear viscosity of the fluid can also be calculated as µ = ρ ( τ -1 / 2) c 2 s δt . In the following, we shall use the notation c i λ to denote the vector number λ and the contravariant component i . The cell configuration D 3 Q 41 has the discrete velocity vectors: (0 , 0 , 0), ( ± 1 , 0 , 0), ( ± 1 , ± 1 , 0), ( ± 1 , ± 1 , ± 1), ( ± 3 , 0 , 0), (0 , ± 3 , 0), (0 , 0 , ± 3), and ( ± 3 , ± 3 , ± 3). The speed of sound for this configuration is c 2 s = 1 -√ 2 / 5. With this setup, and taking into account that the vectors ξ i and u i are normalized by the speed of sound, we obtain the following equilibrium distribution, where the weights w λ are defined as, w (0 , 0 , 0) = 2 2025 (5045 -1507 √ 10), w (1 , 0 , 0) = 37 5 √ 10 -91 40 , w (1 , 1 , 0) = 1 50 (55 -17 √ 10), w (1 , 1 , 1) = 1 1600 (233 √ 10 -730), w (3 , 0 , 0) = 1 16200 (295 -92 √ 10), and w (3 , 3 , 3) = 1 129600 (130 -41 √ 10). The discrete Boltzmann equation for our model takes the form, where we have introduced the forcing term, with F i λ = -Γ i jk ξ j λ ξ k λ and δ kl is the Kronecker delta. In the presence of an external force F ext , this simply extends to F i λ → F i λ + F i ext λ . In order to recover the correct macroscopic fluid equations, via a Chapman-Enskog expansion, the other moments, Eq. (15), also need to be reproduced. A straightforward calculation shows that the equilibrium distribution function f eq λ meets the requirement. The shear viscosity of the fluid can also be calculated as µ = ρ ( τ -1 / 2) c 2 s δt . In this way one can calculate the fluid motion in spaces with arbitrary local curvatures.", "pages": [ 7 ] }, { "title": "Convergence Study", "content": "To check the convergence of the model, we simulate the Poiseuille profile for the velocity on a two-dimensional ring. For this purpose, we use the metric tensor in polar coordinates, g rr = 1, g θθ = r 2 , and g zz = 1, where r is the radial coordinate, θ is the azimuthal angle, and z the axial coordinate. Thus, the non-vanishing Christoffel symbols for this metric are given by, Γ r θθ = -r , and Γ θ rθ = Γ θ θr = 1 /r . Our system consists of a two-dimensional ring with inner radius a and outer one b . On this ring, we impose a constant force f a in the θ -direction. For the simulation we choose τ = 0 . 6. The forcing term f a is set to 0 . 05. All numbers are expressed in numerical units. The inner radius of the ring is taken as a = 1 . 0 and the outer radius as b = 1 . 064. We have taken periodic boundary conditions in the direction θ and z , and free boundary conditions at r = 1 . 0 and r = 1 . 064. To obtain a quantitative measure of the convergence we use the Richardson extrapolation method [27, 28]. In this method, given any quantity A ( δx ) that depends on 10 a size step δx , we can make an estimation of order n of the exact solution A by using with errors O ( δx n +1 ) of order n + 1. Thus the relative error between the value A ( δx ) and the 'exact' solution A can be calculated by ∣ ∣ In our case, the quantity A is the fluid density ρ , when the fluid reaches the steady state, and we set up n = 2. Indeed, the relative error with respect to the 'exact solution' decreases rapidly with increasing grid resolution (see Fig. 5) and we can see that the present scheme exhibits a near second-order convergence. This is basically in line with the convergence properties of classical LB schemes.", "pages": [ 7, 8 ] }, { "title": "Validation", "content": "To provide numerical validation of our model we study the Taylor-Couette instability, which develops between two concentric rotating cylinders. We calculate the critical Reynolds number, Re c , which characterizes the transition between stable Couette flow and Taylor vortex flow. To this purpose, we use the metric tensor for cylindrical coordinates ( r, θ, z ), g rr = 1, g θθ = r 2 , and g zz = 1, where r is the radial coordinate, θ is the azimuthal angle, and z the axial coordinate. Thus, the non-vanishing Christoffel symbols for this metric are given by Γ r θθ = -r , and Γ θ rθ = Γ θ θr = 1 /r . In our system, the inner cylinder has radius a and the outer one radius b . We performed several simulations, by varying the Reynolds number for different aspect ratios η = a/b . The Reynolds number, assuming that the outer cylinder is fixed, can be defined as Re = ( aδ/ν ) dθ/dt where dθ/dt is the angular speed of the inner cylinder and δ = b -a . The inner radius a is always set to a = 1, and for a given value of η , the outer radius b and δ are calculated. In order to vary Re , at fixed η , we change the angular velocity of the inner cylinder. For this simulation, we use a rectangular lattice of 128 × 1 × 256 cells and choose τ = 1 (all values are given in numerical units). We use periodic boundary conditions in the θ and z coordinates. At r = a and r = b boundaries, we have used free boundary conditions, together with a condition to impose the respective angular velocity at each boundary by evaluating the equilibrium function with those values. Note that the boundary conditions can be implemented as if they referred to a cartesian geometry, due to the use of contravariant coordinates, leading to an approximation-free representation of curved geometries. For smooth manifolds, the new scheme is about three times slower than a standard cartesian version, which is mainly due to the calculation of the metric and curvature terms, as well as to the use of third order equilibria to enhance stability. Clearly, the advantage of the present scheme lies in the treatment of complex manifolds which would require very high cartesian grid resolution. In Fig. 2, we can observe the critical Reynolds number as a function of η , as predicted by the simulation and compared with the theoretical values from Ref. [5], finding excellent agreement. We have implemented the same simulation using a lattice size 64 × 1 × 256 cells in order to study the influence of the boundary conditions, and we found an error of around 2 . 5% respect to the theoretical values, which is a clear evidence of the sensitivity to the boundary condition implementation. We have been able to simulate Reynolds number of around 7000 by using τ = 0 . 55. Note that our model works in contravariant coordinates and due to the presence of a metric tensor, the time step is not necessarily unity. For this reason, even when the relaxation time is not small, the computed kinematic viscosity can achieve very small values, leading to large Reynolds numbers. For the case of two rotating spheres, we consider the inner sphere with radius a and the outer one with radius b . We use standard spherical coordinates ( r, φ, θ ), being r the radial, φ the azimuthal, and θ the polar coordinates. The non-vanishing components of the metric tensor are g rr = 1, g φφ = r 2 sin 2 ( θ ), and g θθ = r 2 . The Christoffel symbols can be calculated from the metric tensor by using standard differential geometry relations. Note that our simulation region does not include the poles because there, the determinant of the metric tensor becomes zero and therefore it is not possible to calculate its inverse. To circumvent this problem, we simulate the region θ ∈ ( π/ 6 , 5 π/ 6). We set τ = 0 . 8 and use a lattice of size 32 × 1 × 384. In order to vary the Reynolds number, we change the azimuthal velocity dφ/dt . The boundary conditions have been chosen periodic for the case of φ , and fixed for the case of r and θ . In the inset (left) of Fig. 2, we show the critical Reynolds number for different configurations which is in good agreement with the experimental values given in Ref. [7]. In this figure, we can also observe the radial and polar components of the velocity, and see that there are two small vortices located at the equator and two large ones at high and low latitudes, in agreement with experiments and other numerical simulations [6, 7, 10]. We have also measured the torque coefficient defined by, where σ rφ is the shear stress tensor, which in the context of lattice kinetic theory can be calculated by, The torque coefficient is then computed via the following relation [17] In Fig. 6, we show the comparison between our results, the theory for Re → 0, and the experiments. We find good agreement with the experiments. The small discrepancy can be due to the approximation taken in Eq. (29) and the implementation of the boundary condition. In order to study the Taylor-Couette instability for the case of two concentric rotating tori, which to our knowledge has never been done before, we use a lattice of size 64 × 128 × 64 cells in the orthogonal coordinate system of the torus, ( r, u, v ), being r the radial, u the axial, and v the tangential coordinates. The Christoffel symbols and the components of the metric tensor can be readily calculated from differential geometry relations. The major radius of the tori has been taken as 4 . 0, in numerical units. The other parameters are the same as in the previous simulations, and to vary the Reynolds number we change the tangential velocity dv/dt . In this case, a and b are the minor radii of the inner and outer tori, respectively. We use periodic boundary conditions for the coordinates u and v , and fixed boundaries for r . In addition, the critical Reynolds numbers for different configurations can be observed in Fig. 2, showing values around 10% larger than for the case of cylinders. Philosophical Transactions of the Royal Society of London. Series A", "pages": [ 8, 9, 10 ] } ]

[ { "title": "Exploring the origin, the nature and the dynamical behaviour of distant stars in galaxy models", "content": "Euaggelos E. Zotos Received: 3 June 2013 / Accepted: 7 July 2013 / Published online: 13 August 2013 Abstract We explore the regular or chaotic nature of orbits moving in the meridional plane of an axially symmetric galactic gravitational model with a disk, a dense spherical nucleus and some additional perturbing terms corresponding to influence from nearby galaxies. In order to obtain this we use the Smaller ALingment Index (SALI) technique integrating extensive samples of orbits. Of particular interest is the study of distant, remote stars moving in large galactocentric orbits. Our extensive numerical experiments indicate that the majority of the distant stars perform chaotic orbits. However, there are also distant stars displaying regular motion as well. Most distant stars are ejected into the galactic halo on approaching the dense and massive nucleus. We study the influence of some important parameters of the dynamical system, such as the mass of the nucleus and the angular momentum, by computing in each case the percentage of regular and chaotic orbits. A second order polynomial relationship connects the mass of the nucleus and the critical angular momentum of the distant star. Some heuristic semitheoretical arguments to explain and justify the numerically derived outcomes are also given. Our numerical calculations suggest that the majority of distant stars spend their orbital time in the halo where it is easy to be observed. We present evidence that the main cause for driving stars to distant orbits is the presence of the dense nucleus combined with the perturbation caused by nearby galaxies. The origin of young O and B stars observed in the halo is also discussed. Keywords Galaxies: kinematics and dynamics; Numerical methods", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The Milky Way is believed to be a spiral galaxy, and the best 'educated guess' is that it is a barred Sb to Sc type of galaxy (e.g., [4, 19]). However, since we are inside the Milky Way, it has been proved very di GLYPH<14> cult to properly characterize its structure [18]. While the greater part of the mass of the Milky Way lies in the relatively thin, circularly symmetric plane or disk, there are three other recognized components of the galaxy, each marked by distinct patterns of spatial distribution, motions and stellar types. These are the disk, the halo and the nucleus (see Fig. 1).", "pages": [ 1 ] }, { "title": "(1). The galactic disk:", "content": "The disk consists of stars distributed in the thin, rotating, circularly symmetric plane that has an approximate diameter of 30 kpc and a thickness of about 400 to 500 pc. Most disk stars are relatively old, although the disk is also the site of present star formation as evidenced by the young open clusters and associations. The estimated present conversion rate of interstellar material to new stars is only about 1 solar mass (M GLYPH<12> ) per year. The Sun is a disk star about 8.5 kpc from the center of the galaxy. All these stars, old to young, are fairly homogeneous in their chemical composition, which is similar to that of the Sun. The disk also contains essentially all the galaxy's content of interstellar material, but the gas and dust are concentrated to a much thinner thickness than the stars; half the interstellar material is within about 25 pc of the central plane. Within the interstellar material, denser regions contract to form new stars. In the local region of the disk, the position of young O and B stars, young open clusters, young Cepheid variables, and HII regions associated with recent star formation reveal that star formation does not occur randomly in the plane but in a spiral pattern analogous to the spiral arms found in other disk galaxies. The disk of the galaxy is in dynamical equilibrium, with the inward pull of gravity balanced by motion in circular orbits. The disk is fairly rapidly rotating with a uniform velocity about 220 km / s. Over most of the radial extent of the disk, this circular velocity is reasonably independent of the distance outward from the center of the galaxy.", "pages": [ 1, 2 ] }, { "title": "(2). The galactic halo:", "content": "Some stars and star clusters (globular clusters) form the halo component of the galaxy. They surround and interpenetrate the disk, and are thinly distributed in a more or less spherical (or spheroidal) shape symmetrically around the center of the Milky Way. The halo is traced out to about 50 kpc , but there is no sharp edge to the galaxy; the density of stars simply fades away until they are no longer detectable. The halo's greatest concentration is at its center, where the cumulative light of its stars becomes comparable to that of the disk stars. This region is called the (nuclear) bulge of the galaxy; its spatial distribution is somewhat more flattened than the whole halo. There is also evidence that the stars in the bulge have slightly greater abundances of heavy elements than stars at greater distances from the center of the galaxy. The halo stars consist of old, faint, red main sequence stars or old, red giant stars, considered to be among the first stars to have formed in the galaxy. Their distribution in space and their extremely elongated orbits around the center of the galaxy suggest that they were formed during one of the galaxy's initial collapse phases. Forming before there had been significant thermonuclear processing of materials in the cores of stars, these stars came from interstellar matter with few heavy elements. As a result, they are metal poor. At the time of their formation, conditions also supported the formation of star clusters that had about 10 6 M GLYPH<12> of material, the globular clusters. Today there exists no interstellar medium of any consequence in the halo and hence no current star formation there. The lack of dust in the halo means that this part of the galaxy is transparent, making observation of the rest of the universe possible. Halo stars can easily be discovered by proper motion studies. In extreme cases, these stars have motions nearly radial to the center of the galaxy, hence at right angles to the circular motion of the Sun. Their net relative motion to the Sun therefore is large, and they are discovered as highvelocity stars, although their true space velocities are not necessarily great. Detailed study of the motions of distant halo stars and the globular clusters shows that the net rotation of the halo is small. Random motions of the halo stars prevent the halo from collapsing under the e GLYPH<11> ect of the gravity of the whole galaxy.", "pages": [ 2 ] }, { "title": "(3). The galactic nucleus:", "content": "The nucleus is considered to be a distinct component of the galaxy. It is not only the central region of the galaxy where the densest distribution of stars (about 5 GLYPH<2> 10 4 stars per cubic parsec compared to about 1 star per cubic parsec in the vicinity of the Sun) of both the halo and disk occurs, but it is also the site of violent and energetic activity. The very center of the galaxy harbors objects or phenomena that are not found elsewhere in the galaxy. This is evidenced by a high flux of infrared, radio, and extremely short wavelength gamma radiation coming from the center, a specific infrared source known as Sagittarius A. Infrared emissions in this region show that a high density of cooler stars exists there, in excess of what would be expected from extrapolating the normal distribution of halo and disk stars to the center. The nucleus is also exceptionally bright in radio radiation produced by the interaction of high-velocity charged particles with a weak magnetic field (synchrotron radiation). Of greater significance is the variable emission of gamma rays, particularly at an energy of 0.5 MeV. This gamma-ray emission line has only one source which is the mutual annihilation of electrons with anti-electrons, or positrons, the source of which in the center has yet to be identified. Theoretical attempts to explain these phenomena suggest a total mass involved of 10 6 -10 7 M GLYPH<12> in a region perhaps a few parsecs in diameter. This could be in the form of a single object, a massive black hole; similar massive objects appear to exist in the centers of other galaxies that exhibit active nuclei (AGNs). By the standards of such active galaxies, however, the nucleus of the Milky Way is a quiet place, although interpretations of the observed radiation suggest the existence of huge clouds of warm dust, rings of molecular gas, and other complex features. Recent data derived from observations indicate that the vast majority of distant stars are located in the halo and they are old. Since these stars are among the oldest stars in the galaxy they o GLYPH<11> er great insight into the formation and also the early evolution of the Milky Way galaxy. Nevertheless, many scientists claim that young O and B stars have been also identified in the galactic halo [14]. Naturally the following question arises: Do we have a low level of star formation in the halo, or the observed young stars are in fact disk stars that have been ejected from the galactic plane? If so, what exactly mechanism has driven these young stars in large galactocentric orbits? One of the main targets of this research is to provide some possible answers to these interesting questions, continuing in much more detail the initial work presented in [27]. The present paper is organized as follows: In Section 2 we present the structure and the properties of our galactic model. In Section 3 we provide a brief description of the computational methods we used in order to explore the regular or chaotic nature of orbits. In the following Section, we investigate how the mass of the nucleus and the angular momentum influences the character of the orbits. Moreover in the same section, we search for the origin of young stars, observed in the galactic halo. In Section 5 we try to find a numerical relationship connecting the critical value of the angular momentum with the mass of the nucleus. Then, we present some heuristic semi-theoretical arguments, in order to support and explain the numerically obtained outcomes. We conclude with Section 6, where the discussion and the conclusions of this research are presented.", "pages": [ 2, 3 ] }, { "title": "2 Presentation and properties of the galactic model", "content": "The importance of galactocentric orbits in studies of the Milky Way is well known. The usual approach has been to assume a stationary and axially symmetric potential of the Milky Way with three main contributors: disk, nucleus and halo. In this way galactocentric orbits of many objects have been calculated [1,5,15,26]. Projected on the so-called meridional plane these orbits show a variety of shapes. In most cases one finds box-like orbits, however there are also quite di GLYPH<11> erent orbits. The main objective of the present research work is to investigate the dynamical properties of the motion of stars in the meridional plane of an axially symmetric disk galaxy with an additional spherical nucleus. For this purpose, we use the cylindrical coordinates ( R ; GLYPH<30>; z ), where z is the axis of the symmetry. The total potential V ( R ; z ) in our model is the sum of a disk-halo potential V d, a central spherical component V n and some additional perturbation terms V p. The first one is a generalization of the Miyamoto-Nagai potential [25] (see also [9] and [12]) Here G is the gravitational constant, M d is the mass of the disk, GLYPH<11> is the scale length of the disk, h corresponds to the disk's scale height, while b is the core radius of the disk-halo component. Moreover, in order to describe the spherically symmetric nucleus we use a Plummer potential where M n and c n are the mass and the scale length of the nucleus, respectively. This potential has been used several times in the past to model the central mass component of a galaxy (e.g., [20,21]). At this point, we must point out that the nucleus is not intended to represent a black hole nor any other compact object; but a bulge. Therefore, all the relativistic phenomena that may occur in the central region of a galaxy are completely out of the scope of this research. The potential containing the perturbation terms corresponding to influence and interactions from nearby galaxies is given by where k , GLYPH<21> and GLYPH<12> are parameters. Similar perturbing terms had been used in [36] in order to describe and model how nearby galaxies influence the dynamical behavior of stars. The following lines of arguments justify our choice: it is well known, that in galaxies possessing dense and massive nuclei [7,9,10,38] low angular momentum stars, moving close to the galactic plane are scattered to much higher scale heights upon approaching the central nucleus thus displaying chaotic motion. In a previous work [38], we presented the properties of low angular momentum stars and also we provided useful correlations between the chaotic motion of stars and some important quantities of the dynamical system such as the angular momentum and the mass of the nucleus. The novelty in the current work is the study of the dynamical behavior of stars moving in 'distant orbits'. Here we should clarify, that by the term distant orbits we refer to orbits of stars that reach large galactocentric distances, of order of about 30 kpc or more. In order to have a better estimation of the situation, we agree to call distant orbits all the orbits that obtain galactocentric distances larger than 10 kpc. This is the exact reason for introducing the additional perturbation terms in our model. We will see later on, that these terms are the basic factor which leads stars to distant orbits. It is well known, that in axially symmetric systems like ours, the circular velocity in the galactic plane z = 0, is a very important physical quantity. The total potential V ( R ; z ) describing the properties of motion in our model consists of three components: a disk, a spherical nucleus and some additional perturbing terms. Therefore, the total circular velocity emerges by summing the contributions from the three distinct component; GLYPH<18> ( R ) = GLYPH<18> d( R ) + GLYPH<18> n( R ) + GLYPH<18> p( R ). In Fig. 2 we see the evolution of the three components of the circular velocity as a function of the distance R from the galactic center. In particular, the red curve is the contribution from the spherical nucleus, the blue curve is the contribution from the disk, while the green curve corresponds to the perturbing terms. It is seen, that each contribution prevails in di GLYPH<11> erent distances form the galactic center. In particular, at small distances when R . 3 kpc, the contribution from the spherical nucleus dominates, while at mediocre distances, 3 < R < 10 kpc, the disk contribution is the dominant factor. On the other hand, at large galactocentric distances R > 10 kpc, or in other words outside the main body of the galaxy, we observe that the circular velocity corresponding to the perturbing terms exhibits a very sharp increase being henceforth the prevailing term. Here, we should point out, that the particular values of the parameters entering the perturbation potential, which are given below, were chosen so that our model is as realistic as possible. This is true indeed, because the evolution of GLYPH<18> p( R ) shown in Fig. 2 has a real physical meaning; it exists only in large galactocentric distances ( R > 10 kpc), where the influence from nearby galaxies is strong, while inside the main galaxy is zeroed. Taking into account that the total potential V ( R ; z ) is indeed axially symmetric, we know that in this case the z -component of the angular momentum Lz is conserved. With this restriction, orbits can be described by means of the effective potential (e.g., [3]) The L 2 z = (2 R 2 ) term represents a centrifugal barrier and only orbits with small value of Lz are allowed to pass near the axis of symmetry. Thus, the three-dimensional motion is effectively reduced to a two-dimensional motion in the meridional plane ( R ; z ), which rotates non-uniformly around the axis of symmetry according to where of course the dot indicates derivative with respect to time. The equations of motion on the meridional plane are while the equations governing the evolution of a deviation vector GLYPH<14> w = ( GLYPH<14> R ; GLYPH<14> z ; GLYPH<14> ˙ R ; GLYPH<14> ˙ z ) which joins the corresponding phase space points of two initially nearby orbits, needed for the calculation of the standard indicators of chaos, are given by the variational equations ( ˙ GLYPH<14> ( R ) = ˙ GLYPH<14> z ) = GLYPH<14> R z GLYPH<14> ˙ ˙ ; ; The corresponding Hamiltonian to the e GLYPH<11> ective potential given in Eq. (5) can be written as where ˙ R and ˙ z are the momenta per unit mass conjugate to R and z respectively, while E is the numerical value of the Hamiltonian (star's total energy), which is conserved. Therefore, all orbits are restricted to the area in the meridional plane satisfying E GLYPH<21> V e GLYPH<11> . Weuse a system of galactic units where the unit of length is 1 kpc, the unit of velocity is 10 km s GLYPH<0> 1 , and G = 1. Thus, the unit of mass results 2 : 325 GLYPH<2> 10 7 M GLYPH<12> , that of time is 0 : 9778 GLYPH<2> 10 8 yr, the unit of angular momentum (per unit mass) is 10 km GLYPH<0> 1 kpc s GLYPH<0> 1 , and the unit of energy (per unit mass) is 100 km 2 s GLYPH<0> 2 . We use throughout the paper the following values: M d = 12000, b = 8, GLYPH<11> = 3 and h = 0 : 1, cn = 0 : 25, k = GLYPH<0> 0 : 35, GLYPH<21> = 0 : 01 and GLYPH<12> = 0 : 01. The mass of the nucleus and the angular momentum on the other hand, are treated as parameters. A plot of the iso-potential curves for our galactic model when M n = 400 and L z = 20 is presented in Fig. 3. The vertical, red, dashed line at R = 10 kpc marks the horizontal theoretical limit of the galaxy's main body. We observe, that there are three distinct types of contours: (i) blue isopotential curves which are confined inside the main body of the galaxy, (ii) purple iso-potential curves which correspond to values of energy possessed only by distant stars and therefore, reach to large galactocentric distances ( R max GLYPH<29> 10 kpc) and (iii) black iso-potential curves that combine the two above-mentioned cases. In our research, we shall study the dynamical properties of stars at an energy level corresponding to the last case. In particular, we will use the value E = GLYPH<0> 1100 which remains constant throughout the paper. The iso-potential curve for this value of the energy is shown in green color in Fig. 3. Moreover, in the same plot we see a red dot which indicates the main solution of the system This particular point in the meridional ( R ; z ) plane is indeed a critical point, since it separates the first two families of iso-potential curves (contours around the main body of the galaxy and contours corresponding to distant orbits). Here we should notice, that this critical point is very close to the theoretical boundary ( R = 10 kpc) of the main body of the galaxy.", "pages": [ 3, 4, 5 ] }, { "title": "3 Computational methods", "content": "When studying the orbital structure of a dynamical system, knowing whether an orbit is regular or chaotic is an issue of significant importance. Over the years, several dynamical indicators have been developed in order to determine the nature of orbits. In our case, we chose to use the Smaller ALingment Index (SALI) method. The SALI [33,34] is undoubtedly a very fast, reliable and e GLYPH<11> ective tool, which is defined as where d GLYPH<0> GLYPH<17> k w1 ( t ) GLYPH<0> w2 ( t ) k and d + GLYPH<17> k w1 ( t ) + w2 ( t ) k are the alignments indices, while w1 ( t ) and w2 ( t ), are two deviations vectors which initially point in two random directions. For distinguishing between ordered and chaotic motion, all we have to do is to compute the SALI for a relatively short time interval of numerical integration tmax . More precisely, we track simultaneously the time-evolution of the main orbit itself as well as the two deviation vectors w1 ( t ) and w2 ( t ) in order to compute the SALI. The variational equations (8), as usual, are used for the evolution and computation of the deviation vectors. The time-evolution of SALI strongly depends on the nature of the computed orbit since when the orbit is regular the SALI exhibits small fluctuations around non zero values, while on the other hand, in the case of chaotic orbits the SALI after a small transient period it tends exponentially to zero approaching the limit of the accuracy of the computer (10 GLYPH<0> 16 ). Therefore, the particular time-evolution of the SALI allow us to distinguish fast and safely between regular and chaotic motion. Nevertheless, we have to define a specific numerical threshold value for determining the transition from regularity to chaos. After conducting extensive numerical experiments, integrating many sets of orbits, we conclude that a safe threshold value for the SALI taking into account the total integration time of 10 4 time units is the value 10 GLYPH<0> 8 . In order to decide whether an orbit is regular or chaotic, one may use the usual method according to which we check after a certain and predefined time interval of numerical integration, if the value of SALI has become less than the established threshold value. Therefore, if SALI GLYPH<20> 10 GLYPH<0> 8 the orbit is chaotic, while if SALI > 10 GLYPH<0> 8 the orbit is regular. In Therefore, the distinction between regular and chaotic motion is clear and beyond any doubt when using the SALI method. For the study of our models, we need to define the sample of orbits whose properties (chaos or regularity) we will identify. The best method for this purpose, would have been to choose the sets of initial conditions of the orbits from a distribution function of the models. This, however, is not available so, we define, for each set of values of the parameters of the potential, a grid of initial conditions ( R 0 ; ˙ R 0) regularly distributed in the area allowed by the value of the energy. In each grid the step separation of the initial conditions along the R and ˙ R axis was controlled in such a way that always there are at least 2 GLYPH<2> 10 4 orbits. For each initial condition, we integrated the equations of motion (7) as well as the variational equations (8) using a double precision Bulirsch-Stoer FORTRAN algorithm [28] with a small time step of order of 10 GLYPH<0> 2 , which is su GLYPH<14> cient enough for the desired accuracy of our computations (i.e. our results practically do not change by halving the time step). In all cases, the energy integral (Eq. (9)) was conserved better than one part in 10 GLYPH<0> 10 , although for most orbits it was better than one part in 10 GLYPH<0> 11 . In our study, each orbit was integrated numerically for a time interval of 10 4 time units (10 12 yr), which corresponds to a time span of the order of hundreds of orbital periods and about 100 Hubble times. The particular choice of the total integration time is an element of great importance, especially in the case of the so called 'sticky orbits' (i.e., chaotic orbits that behave as regular ones during long periods of time). A sticky orbit could be easily misclassified as regular by any chaos indicator 1 , if the total integration interval is too small, so that the orbit do not have enough time in order to reveal its true chaotic character. Thus, all the sets of orbits of a given grid were integrated, as we already said, for 10 4 time units, thus avoiding sticky orbits with a stickiness at least of the order of 100 Hubble times. All the sticky orbits which do not show any signs of chaoticity for 10 4 time units are counted as regular ones, since that vast sticky periods are completely out of scope of our research.", "pages": [ 5, 6 ] }, { "title": "4 Numerical results", "content": "In this section, we will complement the classical method of the ( R ; ˙ R ), z = 0, ˙ z > 0 Poincar'e Surface of Section (PSS) [22], in an attempt to visually distinguish the regular or chaotic nature of motion. We use the initial conditions mentioned in the previous section in order to build the respective PSSs, taking values inside the limiting curve defined by Fig. 4a depicts the phase plane when L z = 20 and M n = 50. One can observe a large unified chaotic sea, while there are also several islands of invariant curves corresponding to regular motion. In fact, there are two distinct areas of ordered orbits: (a) orbits that circulate inside only the main body of the galaxy and (b) orbits that live entirely outside the main body of the galaxy. In particular, there are three (b) (b) di GLYPH<11> erent types regarding the first kind of regular orbits: (i) 2:1 banana-type orbits corresponding to the invariant curves surrounding the central periodic point, (ii) box orbits that are located mainly outside of the 2:1 resonant orbits and (iii) 1:1 open linear orbits form the double set of elongated islands outside the 2:1 resonance. On the other hand, we see that all the chaotic orbits can reach large galactocentric distances ( R max ' 33 kpc) and therefore, they are distant orbits. Throughout the paper the vertical, red, dashed line at R = 10 kpc marks the horizontal theoretical limit of the galaxy's main body, while the outermost black thick curve is the ZVC. In Fig. 4b we present the structure of the PSS when L z = 20 and M n = 400, that is the case of a model with a more massive central nucleus. It is evident, that there are many di GLYPH<11> erences with respect to Fig. 4a which focus exclusively on the main body of the galaxy. The most visible di GLYPH<11> erences is the growth of the region occupied by chaotic orbits, the decrease of the percentage of 1:1 resonant orbits and the increase in the allowed radial velocity ˙ R of the stars near the center of the galaxy. On the contrary, we could argue that the structure of the PSS when R > 10 kpc exhibits insignificant di GLYPH<11> erences with respect to that shown in Fig. 4a. Here we must note, that both PSSs shown in Fig. 4(a-b) describe the nature of motion of test particles (stars) of low angular momentum with a variable mass of the nucleus. to small islands of invariant curves embedded in the chaotic sea which the PSS method has di GLYPH<14> culties detecting them. One of the most important parameters that influences significantly the orbital structure at the meridional plane is the angular momentum Lz . In this case, we let Lz vary, while fixing M n = 100. Fig. 6a presents the ( R ; ˙ R ) phase plane when Lz = 5 which correspond to motion of low angular momentum stars. We observe, that the overall structure of the PSS is very similar to the phase planes discussed previously. Again, there is a large unified chaotic sea which surrounds all the islands of stability. Inside the main body of galaxy there are four di GLYPH<11> erent types of regular orbits: (i) 2:1 banana-type orbits which correspond to the invariant curves surrounding the central periodic point in the corresponding PSS, (ii) box orbits that are situated mainly outside of the 2:1 resonant orbits, (iii) 1:1 open linear orbits form the double set of elongated islands in the PSS and (iv) 4:3 resonant orbits which correspond to the triple set of islands of invariant curves. Looking at Fig. 6b, corresponding to Lz = 60, that is the case of high angular momentum stars, it is evident that the amount of chaos in the main body of the galaxy is smaller, while once more, the area where R > 10 kpc remains almost una GLYPH<11> ected by the change of the value of the angular momentum. However, it is worth noticing that, at the highest angular momentum, the 1:1 resonance has been disappeared from the phase plane. From Fig. 6(a-b) we can draw two important conclusions: (i) increasing Lz causes a decreasing of the chaotic region inside the main body of the galaxy and (ii) the permissible area on the ( R ; ˙ R ) phase plane is reduced as we increase the value of Lz . Here we must point out, that both PSSs shown in Fig. 6(a-b) describe the character of orbits of stars in galactic models with a mediocre mass of nucleus and a variable angular momentum. Figs. 7(a-b) show the grids of orbits corresponding to the PSSs of Figs. 6(a-b) respectively. Once more, the excellent agreement between the two methods (PSS and SALI) is more than obvious. In Fig. 8(a-d) we present four characteristic examples of the basic types of regular orbits that are encountered inside the main body of the galaxy. In all cases, the values of all the parameters are as in Fig. 4b except for the orbit shown in Fig. 8d, where the values of the parameters are as in Fig. 6a. As expected, all orbits circulate close to the center of the galaxy and therefore, not only they do not reach large galactocentric distances but they do not even approach the boundary of the main galaxy's body. In fact, for all four orbits shown in Fig. 8(a-b) we have that Rmax ' 5 kpc. In order to have a better view of these orbits we provide at the upper left part of each sub-panel a magnification of the area on the ( R ; z ) plane occupied by every orbit. The box orbit shown in Fig. 8a was computed until t = 100 time units, while the parent periodic orbits were computed until one period has completed. The curve circumscribing each orbit is the limiting curve in the ( R ; z ) meridional plane defined as V e GLYPH<11> ( R ; z ) = E . In Table 1 we provide the exact type and the initial conditions for each of the depicted orbits; for the resonant cases, the initial conditions and the period T per correspond to the parent periodic orbit. Note that every resonance n : m is expressed in such a way that m is equal to the total number of islands of invariant curves produced in the ( R ; ˙ R ) phase plane by the corresponding orbit. Things are quite di GLYPH<11> erent in Fig. 9(a-d) where we observe four typical examples of distant orbits which reach to high galactocentric distances. The orbits presented in Fig. 9(a-c) are resonant periodic orbits which exist outside the main galactic body, spent all their orbital time into the halo and therefore, exhibit large galactocentric distances. On the other hand, in Fig. 9d we see a chaotic which stays near the galactic plane inside the main galaxy, while at large galactocentric distances it gains considerable height, since it consumes the vast majority of its orbital time into the halo. This particular but nevertheless interesting behavior of the orbit is dictated by the structure of the limiting curve (ZVC). It is of particular interest to note, that all the resonant periodic orbits shown in Fig. 9(a-c) develop high departures from the galactic plane, in other words high values of z coordinates and therefore, move on much deeper into the halo than the chaotic orbit. The initial conditions, the period and the exact type of the orbits are given in Table 1. All orbits were com- puted for a time interval equivalent to one period, except of the chaotic orbit shown in Fig. 9d which was integrated for 200 time units. In Fig. 10 we present the evolution of the z component of the chaotic orbit shown in Fig. 9d for a time interval of 10 4 time units. Red dots indicate the time points when the test particle (star) is inside the main body of the galaxy ( R < 10 kpc), while green dots correspond to time points where the star is moving into the halo reaching large galactocentric distances. We observe, that the star moves randomly inside and outside the main galaxy. However, there are long time intervals of order of about 3000 time units, or even more, in which the star moves entirely outside the main body. Our numerical experiments indicate, that all stars moving in chaotic orbits in our dynamical system spent most of their orbital period (about 90%) into the galactic halo, thus reaching large galactocentric distances and also high values of the z coordinate. We explained earlier, that in order to study the character of orbits in our galactic model, we integrated a set of initial conditions in each phase plane. Thus, calculating from all the sets of initial conditions the percentage of the chaotic orbits, we are able to follow how this fraction varies as a function of the mass of the nucleus and the angular momentum. The discrimination between the chaotic and regular orbits is that SALI < 10 GLYPH<0> 8 for chaotic orbits, while SALI GLYPH<21> 10 GLYPH<0> 8 for regular ones. In the latter range, one does of course include the 'sticky' chaotic orbits. To study how the mass of the nucleus M n influences the level of chaos, we let it vary while fixing all the other parameters of our model. We chose L z = 5 ; 20 ; 40 and 60 for the angular momentum and integrated orbits in the meridional plane for the set M n = f 0 ; 50 ; 100 ; :::; 500 g . Fig. 11a shows the evolution of the percentage of the area covered by chaotic orbits in the ( R ; ˙ R ) phase planes as a function of the mass of the nucleus, for the four values of the angular momentum. We observe, that in general terms, the percentage of chaotic orbits increases as the mass of the nucleus increases in all four cases. However, the value of the angular momentum strongly a GLYPH<11> ects the exact way of the evolution. It is clear form Fig. 11a that for low angular momentum stars the percentage of chaotic orbits increases rapidly when the mass of the nucleus is small enough. This is true until M n ' 250. We observe, that a further increase of the mass of the nucleus does not practically a GLYPH<11> ect the amount of chaos. On the other hand, the evolution of the chaotic percentage follow an entirely di GLYPH<11> erent path in the case of high angular momentum stars. For instance, when L z = 60 there is an almost linear increase of the chaotic percentage. Now we proceed in order to investigate how the angular momentum L z influences the amount of chaos in our galaxy model. Again, we let it vary while fixing all the other parameters of our galactic model, choosing M n = 50 ; 200 ; 350 and 500 as fiducial values for the mass of the nucleus, and integrating orbits in the meridional plane for the set L z = f 1 ; 5 ; 10 ; 15 ; :::; 60 g . In Fig. 11b we present the evolution of the percentage of chaotic orbits in the ( R ; ˙ R ) phase planes as a function of the angular momentum, for the four values of the nucleus. It is evident, that the amount of chaos decreases with increasing angular momentum. With a more closer look at the diagram, we see that the more massive is the nucleus the more chaos is observed in the galaxy. For low values of angular momentum ( L z GLYPH<20> 15) the chaotic percentage decreases sharply, while for larger values of L z it remains almost the same. Nevertheless, we have to point out that the four trend lines corresponding to the evolution of the chaotic percentage of the four di GLYPH<11> erent values of M n do not intersect, but they lined up one below the other following the reduction of the mass of the nucleus. Before closing this section, we would like to make some comments regarding the existence of distant stars in the galactic halo. The mechanism responsible for the presence of all distant stars in the halo, is mainly the perturbation due to nearby galaxies. In fact, in a galaxy model without the extra perturbing terms, that is when k = GLYPH<21> = 0, there are no stars moving in remote, distant orbits. On the other hand, in a galaxy model where both the perturbing terms and the massive nucleus are present, the possibility of a star to have been ejected from the galactic plane is more than evident. Extensive numerical experiments, not presented here, indicate that the required time for a star to be ejected from the galactic plane thus moving in a distant orbit is about 50 time units (5 GLYPH<2> 10 5 yr), while the time needed for a distant star to return again inside the main bode of the galaxy is about 300 time units (3 GLYPH<2> 10 10 yr). On this basis, is becomes clear that the total number of distant stars in our galaxy model must be increasing. Our galactic model suggests that a large portion of distant stars are indeed disk stars which have been scattered o GLYPH<11> the galactic plane into the halo moving in chaotic orbits. At this point, we would like to make clear that all these stars are not young O, B stars. In fact, the majority of these stars are very old. It is well known, that the central bulges in spiral galaxies contain old stars, while the disks of spirals contain a mixture of young and old stars. On the other hand, taking into account that the average age of B type stars is of order of several billion years, it seems more possible for those stars to have been formed into the halo. Another possible explanation justifying the presence of the small number of O, B stars at large z coordinates, could be the evolution in binary stellar systems with the subsequent mass exchange through a Roche-lobe flow [23,24]. Finally, one should not overlook the possibility where stars are trapped in distant orbits upon their formation (see Figs. 9(a-d)). These stars are old halo stars, probably RR-Lyrae type stars. This argument is strongly supported by data derived from observations, where faint RR-Lyrae type stars were discovered at large galactocentric distances up to 30 to 35 kpc from the galactic center [30-32].", "pages": [ 6, 7, 8, 9, 10, 11 ] }, { "title": "5 A semi-theoretical approach", "content": "In this section, we shall try to find if there is a relationship between the critical value of the angular momentum L zc and the mass of the nucleus and if so, then try to explain and justify it using elementary semi-theoretical arguments. By the term 'critical value of the angular momentum' we refer to the maximum value of the angular momentum for which stars are scattered o GLYPH<11> the galactic plane into the halo thus displaying distant chaotic orbits, for a given value of the mass of the nucleus. A plot showing the relationship between L zc and M n is presented in Fig. 12. In order to obtain this correlation, we integrated numerically a large number of orbits. These orbits were started at R 0 = 8 : 5 kpc, z 0 = 0, with zero radial velocity ˙ R 0, while the initial value of the vertical velocity ˙ z 0 was always obtained from the energy integral (9). The numerically found results are indicated by dots, while the solid line which joins them is the best polynomial fit. In our case, there is a second degree polynomial dependence between the mass of the nucleus and the critical value of the angular momentum. Specifically, the best fitting curve is represented by the equation which is indeed a second degree polynomial in L zc. Orbits with values of the parameters on the lower right part of the ( L z ; M n) plane including the line correspond to regular orbits that are confined inside the main body of the galaxy ( R max GLYPH<20> 10 kpc), while orbits with values of the parameters on the upper left part of the same plane lead to chaotic orbits obtaining large galactocentric distances. Now we are going to reproduce the form of Eq. (13) by combining some semi-theoretical arguments together with numerical evidence. We know from previous works, that the primary cause driving a star in large galactocentric orbit is the radial force F R near the nucleus. On approaching the central nucleus there is a change in the star's angular momentum in the R direction given by where m is the mass of the star, h F R i is the average force acting along the R direction near the nucleus and GLYPH<1> t is the duration of the encounter. It was observed, that the test particle's (star) deflection far from the galactic center proceeds in each time cumulatively, a little more with each successive pass near the central nucleus and not with a single dramatic encounter. Let us assume that the star is ejected to a distant orbit after n ( n > 1) passes, when the total change in the momentum in the R direction is of order of m GLYPH<29>GLYPH<30> , where GLYPH<29>GLYPH<30> is the tangential velocity of the star near the nucleus. Therefore we have in Eq. (15), we find Elementary numerical calculations reveal that, near the nucleus, where R = R 0 < 1 and z GLYPH<25> 0 the h F R i force is repulsive and can be written in the following form because all the higher order terms are practically negligible near the nucleus. Inserting this value of h F R i in Eq. (17) and after rearranging the order of the terms we find where a 1 and a 2 are constants. This is because the values of R 0 and T c are about the same for the range of values of the mass of the nucleus and the angular momentum used in order to produce the diagram shown in Fig. 12. Eq. (19) shows indeed a second degree polynomial relationship between M n and L zc but it is not complete yet. Numerical experiments indicate, that when L zc ! 0 there must be a minimum value M n0 of the mass of the nucleus in order to drive the star to a distant orbit (see near the origin at the plot in Fig. 12). Inserting this additional term in Eq. (19), we finally obtain Thus it is evident, that the form of Eq. (20) is exactly the same as the numerically obtained relationship given by Eq. (13).", "pages": [ 11, 12 ] }, { "title": "6 Discussion and conclusions", "content": "The main objective of this research work, was the investigation of the character of orbits of stars in the meridional plane of an axially symmetric galactic gravitational model consisting of a disk, a dense spherical nucleus and some additional perturbing terms corresponding to interaction from nearby galaxies. Here, we have to point out that a large variety of dynamical models describing global motion in galaxies are also available in the literature (e.g., [2,13,29,35]). Moreover, one must not forget the local galactic models which are mainly maid up of perturbed harmonic oscillators (e.g., [8,11,16,17,37,39,40]). The reader can find many illuminating information on dynamical galactic models in [6]. Here we have to point out, that there are only but a few similar research works on the nature of distant stars in galaxies. Therefore, we believe that our work makes a significant contribution to our so far knowledge of stars moving in large galactocentric distances under the perturbation of nearby galaxies. In order to estimate the chaoticity of our models, we chose a dense grid of initial conditions in the ( R ; ˙ R ) phase plane, regularly distributed in the area allowed by the value of the energy and defined by the ZVC. All the samples of the orbits were integrated numerically and the regular or chaotic nature of each orbit had been determined by computing the SALI. In our galactic model three main types of orbits appear: (i) regular orbits that are confined inside the main body of the galaxy, (ii) distant regular orbits located at large galactocentric distances and (iii) chaotic orbits that spent the majority of their orbital time in large distances rather than inside the main galactic body and close to the central nucleus. One of the most interesting and surely novel aspects of this research was the study of the dynamical properties of the distant stars. Our results can be summarized as follows: polynomial law had been reproduced and therefore justified using elementary semi-theoretical arguments. According to current data, it is evident that the majority of distant stars are in fact old stars. Therefore, the presence of young O and B stars in the galactic halo is, by all means, a very interesting issue that needs to be explored and clarified. So far, there are two possible scenarios that can, in a way, explain the young O and B distant stars observed in large galactocentric distances. According to the first one, these stars might have been formed initially into the halo. On the other hand, the second scenario suggests the evolution in binary stellar systems with the consequent mass exchange through a Roche-lobe flow. Taking all facts into account we strongly believe, that more observational data are needed, in order to be able to determine, once and for all, not only the origin but also the mysterious nature of the young distant O and B stars.", "pages": [ 12, 13 ] }, { "title": "Acknowledgments", "content": "I would like to express my warmest thanks to the two anonymous referees for the careful reading of the manuscript and for all the aptly suggestions and comments which improved both the quality and the clarity of the paper.", "pages": [ 13 ] } ]

[ { "title": "Gravity/CFT correspondence for three dimensional Einstein gravity with a conformal scalar field", "content": "M. Hasanpour ∗ , F. Loran † and H. Razaghian ‡ Department of Physics, Isfahan University of Technology, Isfahan, 84156-83111, Iran", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the three dimensional Einstein gravity conformally coupled to a scalar field. Solutions of this theory are geometries with vanishing scalar curvature. We consider solutions with a constant scalar field which corresponds to an infinite Newton's constant. There is a class of solutions with possible curvature singularities which asymptotic symmetries are given by two copies of the Virasoro algebra. We argue that the central charge of the corresponding CFT is infinite. Furthermore, we construct a family of Schwarzschild solutions which can be conformally mapped to the Mart'ınezZanelli solution of Einstein's equations with a negative cosmological constant coupled to conformal scalar field.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Asymptotic symmetries of the BTZ black holes [1, 2] which are solutions of Einstein gravity with a negative cosmological constant Λ = -l -2 are given by two copies of the Virasoro algebra. The central charge of the corresponding CFT is given by the Brown-Henneaux formula [3], where l is radius of the AdS space and G is Newton's constant in three dimensions. The CFT is believed to be the dual picture of the corresponding quantum gravity [4]. One reason in support of the duality is the Strominger's observation [5] who showed that the BekensteinHawking entropy is correctly given by the Cardy formula [6]. In principle, the dual CFT is supposed to describe the physics of the gravity side, including the scattering processes, Hawking radiation, back reactions, and specially the problem of closed timelike curves in the BTZ solutions and orbifold (orientifold) singularities [7, 8, 9, 10, 11]. In this paper, we study the 3D Einstein gravity conformally coupled to a scalar field ψ . This action gives the bosonic matter term in supergravity coupled to M2-branes [12, 13, 14]. Classical solutions of this theory are geometries with vanishing scalar curvature. In this theory Newton's constant is scaled by the factor (1 -πGψ 2 ) -1 . Thus for ψ 2 = 1 /πG Newton's constant is effectively infinite. We show that there is a class of black solutions that similar to AdS 3 end on a cylindrical conformal boundary, and the asymptotic symmetries of such geometries are given by two copies of the Virasoro algebra. Motivated by this observation, we try to extend AdS/CFT correspondence to these asymptotically flat geometries. There are two main difficulties in this problem. First of all, the black hole solutions emerge at the critical point where the Planck mass is effectively vanishing. Thus in this case, Einstein gravity cannot be considered as a viable effective theory of gravity, since for vanishing Planck mass, all gravitational excitations, whatever they are, become massless at least at tree level. From a practical point of view, the first step to identify the dual CFT is to find a reliable extension of the existing methods in the literature, to identify the gravitational Noether charges corresponding to asymptotic symmetries. In this paper, we argue that similar to AdS/CFT correspondence [15], one can still identify the expectation value of the boundary stress tensor with the Brown-York tensor [16] as it encapsulates the asymptotic geometry, and use the mass scale defined by ψ 2 = 1 /πG instead of the vanishing Planck mass given by the effectively infinite Newton's constant. The second problem is as follows: the Virasoro generators generically scale the size of the conformal boundary. We argue that the only consistent value for the central charge of the boundary CFT is consequently infinity. We compute this central charge by three methods; by studying the transformation properties of the Brown-York tensor and computing the Schwartzian derivative, by computing the Weyl anomaly, and by using the formula given in [17, 18]. The central charge appears to be where R is the radius of the boundary which, unlike the AdS geometry, is infinite in this case. We show that if the weights of the CFT states corresponding to black hole solutions are assumed to be given by the Noether charges corresponding to asymptotic symmetries, then the Cardy formula gives a finite entropy. Of course as we discuss in section 3.2.6 it is questionable to identify this entropy with the Wald's entropy. These observations provide new problems in 3D gravity/CFT 2 correspondence and the present work should be considered as a first step towards understanding a corner of it i.e. the gravity/CFT correspondence for Einstein gravity with conformally coupled matter field at the critical point. Three primary problems that come to mind are the following. First of all, although in AdS/CFT correspondence it is known that the AdS radius l /greatermuch l Planck and thus c /greatermuch 1, such CFTs are not thoroughly studied in comparison with the ordinary c ∼ O (1) CFTs. Now we have the signature of a CFT with an infinite central charge. As we review in section 3.2.5, tensionless string theory [21] is an example of such a CFT. Thus from the gravity/CFT correspondence point of view, one seeks a general description of CFT's partition function as a function of 1 /c . The second problem is to understand the critical point ψ 2 = 1 /πG in the frame work of [12, 13, 14] which could also lead to a realization of the CFT in terms of M2-brane configurations. Finally, the black hole solutions of section 2 possess curvature singularity located behind the event horizon. Existence of the curvature singularity makes these geometries essentially different from the locally AdS (i.e. BTZ) geometries and their flat limits studied for example in [19, 20]. Thus one needs to address the curvature singularities in terms of the boundary CFT. The second class of solutions that we study in this paper are the Schwarzschild solutions. These solutions can be conformally mapped to the Mart'ınez-Zanelli solutions [22] of Einstein's equations with a negative cosmological constant coupled to a conformal scalar field. AdS/CFT correspondence is studied for the Mart'ınez-Zanelli solution [23, 24], which has also a curvature singularity behind a horizon. The fact that this solution can be conformally mapped to the Schwarzschild solution is a strong evidence for the necessity of extending the AdS/CFT correspondence to Einstein gravity without a negative cosmological constant. The first example of a such a theory is a CFT that gives the string spectrum in flat space [25]. The organization of this paper is as follows. In section 2, we review the Einstein gravity in 2+1 dimensions conformally coupled to a scalar field ψ and study the stationary static solutions for ψ = 1 / √ πG . In section 3 asymptotic symmetries of a class of solutions with a cylindrical conformal boundary are studied. In section 4 we study the Schwarzschild and the Mart'ınez-Zanelli solutions. Our results are summarized in section 5. General stationary static solutions of the theory are given in Appendix A where we show that all stationary static solutions with an r -dependent matter field ψ ( r ) asymptote to ψ = √ 1 /πG . Gravitational Noether charges and the corresponding central charge are computed in appendix B. Stability of the Schwarzschild solution against linear perturbations is studied in appendix C.", "pages": [ 2, 3, 4 ] }, { "title": "2 The Model", "content": "Einstein gravity in 2+1 dimensions conformally coupled to a scalar field ψ is given by the action where κ = 8 πG . Conformal coupling refers to the fact that the matter term in the action is invariant under conformal transformations, Under this transformation, where ∇ stands for covariant derivative. Adding a cosmological constant (Λ) term to the action (2.1), the Einstein field equations become where G µν = R µν -1 2 Rg µν is the Einstein tensor and is the energy momentum tensor. One can show that in general, R = 6Λ on-shell. This follows from the fact that by the conformal symmetry of the matter term, the corresponding energy momentum tensor, given in Eq.(2.5) is traceless on-shell, Consequently any solution of the action (2.1), for which R = 0, can be conformally mapped to a solution of the same theory with a cosmological constant Λ, if An example is given in section 4. Motivated by this observation, we study, in the following, axisymmetric stationary static solutions of action (2.1). General axisymmetric stationary static solution with an r -dependent matter field ψ ( r ) is discussed in Appendix A. In the following we concentrate on solutions with ψ = ± √ 8 /κ . Field equations (2.4) with Λ = 0 corresponding to action (2.1), where become trivial in this case and the only condition on the metric comes from the field equation for ψ which for a nonvanishing constant ψ implies that In general, in three dimensions the Riemann tensor can be determined in terms of the Ricci tensor and the metric tensor by the following identity [26], Given that R = 0 there is only one other invariant quantity is the Kretschmann scalar,", "pages": [ 4, 5, 6 ] }, { "title": "2.1 Axisymmetric Stationary Static Solutions", "content": "The ansatz for stationary static solutions in ( t, r, φ ) coordinate system is given by a diagonal metric For this ansatz, the equation R = 0 can be easily solved to obtain It is known that Einstein gravity has no Black hole solutions [27, 28] while Einstein gravity with a negative cosmological constant enjoys the wide class of (orientifolded) BTZ black hole solutions [29, 30, 31]. Since all such solutions are locally AdS , they do not possess a curvature singularity which is a common phenomenon in higher dimensional black objects. On the contrary, there are many black objects there in Eq.(2.15) with curvature singularities. In general, a black hole solution is identified by an event horizon where n ( r ) = 0 and a region of infinite red-shift where f ( r ) = 0. Eq.(2.15) simplifies the search for possible black hole solutions. An example of such a solution is given by which for a > 0 is a black hole with an event horizon located at r = a . Since the curvature singularities at r = 0 and r = 2 a 3 are covered by the event horizon. For a < 0 there is a naked curvature singularity at r = 0. As the second example consider the geometry For this solution Thus the curvature singularity at r 2 = a 2 is behind the event horizon at r 2 = 2 a 2 . This solution can be extended from r ∈ R + to the r ∈ R region. In the next section we discuss gravity/CFT correspondence for this background. Thus it is worth mentioning that similar to BTZ black holes, the region behind the curvature singularity at r 2 = a 2 can be removed by folding the geometry right there [30, 31]. The corresponding folded geometry is given by the following metric, where Φ = r 2 -a 2 , and θ ( x ) is a step function. 1 This Z 2 folding is accompanied by insertion of a δ -function source at Φ = 0 [32], since there is a jump in the curvature given by [33], Here, [Γ ρ µν ] denotes the jump in the Levi-Civita connections, and g is the determinant of the metric.", "pages": [ 6, 7 ] }, { "title": "3 Gravity/CFT duality", "content": "The second example studied in section 2.1 is a special member of an infinite class of solutions identified by the asymptotic geometry, This geometry is, by itself, a solution corresponding to a = 0 in Eq.(2.19). Similar to AdS 3 , this geometry ends on a conformal boundary, the ( t, φ ) cylinder. For AdS 3 , it known that the asymptotic symmetry are given by two copies of Virasoro algebra with a central charge proportional to the AdS 3 radius [3], which is conjecturally related to a CFT on the cylinder. In the following we examine the asymptotic symmetries of the geometry (3.1).", "pages": [ 7 ] }, { "title": "3.1 Asymptotic symmetry", "content": "The asymptotic symmetry group of (3.1) is given by two copies of Virasoro algebra. 2 To see this in the Brown-Henneaux approach [3], one needs to change the coordinate system. Consider for example, the new radial coordinate x = r 6 , in terms of which, the geometry (3.1) becomes and assume the following boundary condition for the fluctuations around the geometry, The general diffeomorphism preserving the boundary condition (3.3) is given by where Thus the generators of the asymptotic symmetry group are, where σ ± = t ± φ and ∂ ± = 1 2 ( ∂ t ± ∂ φ ). The existence of such a symmetry group is not a surprise since the conformal boundary of the geometry (3.1) is the conformal boundary of an AdS 3 space. 3 To identify the boundary CFT, one needs to determine the corresponding central charge. In order to do this, we note that deformations by h rr ∼ O ( r ) effectively scale the volume of the ( t, φ ) cylinder since g rr = r . Classically such deformations are not observable on the conformal boundary. But quantum mechanically they effectively scale the central charge which is the vacuum energy in units of the volume of the ( t, φ ) torus. To have a meaningful CFT the cental charge should be invariant under such scalings. Thus it is either vanishing or infinite. c = 0 is outside the domain of gravity/CFT correspondence. 4 But the c → ∞ case can be understood from the gravity side. First of all, in AdS/CFT correspondence the central charge is given by the Brown-Henneaux formula where R = -6 l -2 . Thus the R = 0 solution correspond to c →∞ . 5 Furthermore in a CFT [34] As c → ∞ , all finite excitations on the cylinder become effectively degenerate. This is in agreement with the fact that for the Einstein gravity conformally coupled to the scalar ψ = √ 8 /κ , the action is vanishing on-shell, and consequently all solutions have equal contribution to the partition function. A dual CFT with c → ∞ can also be understood in the following way. By AdS/CFT correspondence, we know that a CFT 2 with a finite central charge has a dual gravity picture in an AdS 3 space with radius l c = (2 G/ 3) c . This implies that if Einstein gravity with a conformal matter field is dual to a CFT with the central charge c , then it is also dual to Einstein gravity with the cosmological constant Λ c = -l -2 c . Such a duality is not reasonable while there is no length scale other than G in Einstein gravity with a conformal matter field. Although it sounds meaningful but it is not a 'proof' yet. To determine the central charge one needs to compute the corresponding anomaly explicitly. One also needs to identify the geometry dual to the vacuum state which energy is -c/ 24. In AdS/CFT correspondence, the primary fields of the CFT correspond to the family of the locally AdS solutions including the (orientifolded) AdS space, (orientifolded) BTZ and (orientifolded) self-dual orbifolds [35, 31]. In principle the CFT will tell us about the BTZ singularities, the closed time-like curves and the δ -functions sources of the orientifolded solutions. If the picture obtained so far is correct, the c → ∞ CFT would account for the whole family of solutions with R = 0 that asymptote to the geometry (3.1), including the folded geometry (2.21) and its odd δ -function source (2.23). As the final comment we recall that the c → ∞ CFT is, in principle, included in any CFT with a finite central charge. To see this, recall that for any integer N , the subalgebra L n ≡ N -1 L nN of a Virasoro algebra generated by L n with central charge c , is a Virasoro algebra with the central charge Nc . 6 Obviously the c →∞ CFT corresponds to orbifolding a generic CFT by N →∞ . 7", "pages": [ 7, 8, 9 ] }, { "title": "3.2 Charges", "content": "In this section we compute the central charge of Virasoro algebra (3.6), and the entropy of the black hole solutions (2.19). For this purpose we first consider a more general asymptotic geometry in three dimensions given by the line element where in which /lscript is a parameter characterizing the length scale of the solution. For asymptotically AdS geometry, z = -1 and /lscript is the radius of the AdS space. For z = 1 2 and /lscript = 1, this is the geometry (3.1). In general, in order to construct the asymptotic symmetries, one has to define a new radial coordinate x by In fact one can show that the asymptotic symmetries are trivial for b ( z +1) ≥ 1 2 . 8 Therefore for z = -1 (asymptotic AdS geometry), one assumes that /negationslash For example, in Eq.(3.2) where z = 1 / 2 we have assumed b = 1 / 6. Scalar quantities such as the central charge are independent of the choice of radial coordinate, so their values do not depend on b . In terms of the new radial coordinate x the asymptotic line element is where and", "pages": [ 9, 10 ] }, { "title": "3.2.1 Asymptotic symmetry", "content": "Consider the vector and define the asymptotic conformal Killing vector in which ρ = -b ( z +1) α such that δg xx = 0. 9 One can verify that for where for example, α ' = ∂ φ α, and ˙ α = ∂ T α , Furthermore, and ξ generates the following diffeomorphism in which In the following we compute the corresponding charges and the central charge. Similar results are obtained in Appendix B using the formula given in [17].", "pages": [ 10, 11 ] }, { "title": "3.2.2 Boundary stress tensor", "content": "Consider the Brown-York stress tensor [16] defined by where K µν is the extrinsic curvature of the boundary defined by n µ is the outward pointing unit vector to the boundary, and the boundary metric γ µν is defined by the ADM-like decomposition of the metric For Einstein gravity minimally coupled to matter fields, one can show that after subtracting the 'vacuum' contribution to τ , where D µ is the covariant derivative compatible with γ µν and T µν is the matter field stress tensor. Consequently the charges of the spacetime are encoded in the Brown-York stress tensor [16]. Inspired by this result and noting that the essence of the Brown-York tensor is the geometry, we postulate that for any spacetime (3.9), the charges corresponding to the asymptotic symmetries (3.20) are given by the regularized tensor 10 where K (0) µν is the extrinsic curvature of the 'vacuum' solution γ µν = r 2 γ (0) µν . 11 For the asymptotic geometry one can verify that The first identity is proven in [16]. The second identity is trivial for K µν = K (0) µν . In other cases there is a trace anomaly which is related to the central charge of the CFT on the boundary. We discuss it in the following. To prove the third identity, note that For the metric and Thus, Eq.(3.28) can be used to show that in which R is the radius of the boundary. By definition The above mentioned properties of τ reg follows from Eq.(3.32) and (3.22). We postulate that τ reg µν corresponds to the CFT stress tensor.", "pages": [ 11, 12 ] }, { "title": "3.2.3 Central charge", "content": "In general given a CFT on a plane with metric ds 2 = -dw + dw -the diffeomorphism results in where T µν is the CFT stress tenor. Assuming that (the dimensionless coordinates on the boundary) w ± are given by then Eq.(3.22) gives Assuming that T µν = τ reg µν one obtains /negationslash This result strictly depends on the choice made in (the right column of) (3.18) which in particular works for z = 0. This is not a flaw in the model since uniqueness of the asymptotic symmetry is not claimed so far. In the following, we compute the central charge in terms of the trace anomaly and obtain the same result for general values of z . The stress tensor of a CFT 2 has a trace anomaly To calculate the trace anomaly one can use the identity where G µν is the Einstein tensor. For a geometry given by (3.22), Using Eqs.(3.29), (3.30) and (3.31), one obtains On the other hand, Eq.(3.27) gives Consequently, /negationslash In Eq.(3.38) the same result was obtained for z = 0. Thus it is legitimate to assume the validity of this result for general values of z and classify the asymptotic spacetimes with respect to the corresponding central charges,  Recall that this is the central charge of the Virasoro algebra (3.20) of gravitational charges corresponding to symmetries of the asymptotic geometry (3.9). The black hole solution (2.19) corresponds to z = 1 / 2 and thus the correpsonding central charge is infinite. In appendix B we obtain the same value for the central charge by using the formula given in [17], where we also compute the Noether charges corresponding to symmetries of the asymptotic geometry. It is worth mentioning that the formula (3.45) for z ≥ -1 (including the BTZ black hole and the black hole geometry (2.19) ) is consistent with the c -theorem [41]. Recall that holography implies that an IR cut-off r IR corresponds to a UV cut-off Λ UV = r IR on the CFT side [42]. At such a cut-off Eq.(3.38) gives a finite central charge which, for z ≥ -1, is an increasing function of the UV cut-off Λ UV = r IR with a fixed point c →∞ at Λ UV →∞ . We discuss a related topic at the end of section 3.2.6.", "pages": [ 13, 14 ] }, { "title": "3.2.4 Mass", "content": "The mass of a solution can be defined by This definition is motivated by following facts: τ reg tt is the energy density and the 'volume' equals 2 π R . In order to show that this definition is the correct one, we compute the mass for a geometry given by Eq.(3.29) in which Using Eq.(3.32) one verifies that, It is useful to give the geometry corresponding to (3.48), in the ansatz (2.14). Defining a new radial coordinate ρ 2 = r 2 +4 GM/lscript 2 one verifies that the asymptotic geometry is given by This is a key result. For z = -1 this is the static BTZ and for z = 1 2 this is the asymptotic geometry for the black hole solution (2.19). It is interesting to verify this 'holographic prediction' for black hole solutions to other theories of 3 D gravity with asymptotic geometry (3.9). The geodesic equation for a point particle initially at rest at radial infinity, implies that for z < 1, the mass term produces a repulsive force.", "pages": [ 14, 15 ] }, { "title": "3.2.5 Conformal matter at critical value", "content": "In section 2, for Einstein gravity conformally coupled to a scalar field at the critical value ψ = ( πG ) -1 / 2 we found a black hole solution (2.19) which asymptotic geometry is given by Eq.(3.51) with z = 1 2 and the corresponding central charge (3.45) is infinite. For this solution the Planck mass is effectively zero In string theory this limit corresponds to c →∞ [21]. In fact, in WZW models, where k is the level of current algebra and g ∨ is the dual Coxeter number of G . The critical level is given by k = g ∨ .", "pages": [ 15 ] }, { "title": "3.2.6 Entropy", "content": "The black hole solution (2.19) has a finite mass and a finite Hawking temperature, In Eq.(2.19), /lscript = 1. In principle, one can use the first law of thermodynamics to compute the canonical entropy of the black hole, This entropy is finite while the microcanonical entropy given by the Cardy formula where ∆ = ¯ ∆ = M/lscript 2 , is infinite. The point is that the black hole solution (2.19) has a negative heat capacity as can be seen from equation (3.55). Thus it never comes to equilibrium with an infinite heat bath. Thus canonical entropy is not well defined in this case [43]. Before closing this section, we report an observation for which we do not have a clear justification. As we show in Eq.(B.9), the Noether charge for symmetries of the asymptotic geometry (3.51) is given by Consequently one may assign the following weights to the corresponding CFT state In this case, the Cardy formula gives a finite entropy which agrees with the 'naive' area law where r + is given by g ρρ in the asymptotic geometry (3.51), We call formula (3.62) naive because the area of event horizon is given by A defined in Eq.(3.57) which is not in general equal to 2 πr + . It should be noted that r + is not even the actual radius of event horizon unless z = -1 (BTZ geometry). Seemingly, this result implies that the boundary CFT observes the asymptotic geometry (3.51) and interprets it as a black hole geometry with an event horizon located at r + . Conceptually this is a reasonable statement since r → ∞ corresponds to the IR limit on the gravity side of gravity/CFT correspondence, and in the IR limit, an observer naturally probes the asymptotic geometry and is blind to the details of the spacetime structure.", "pages": [ 15, 16 ] }, { "title": "4 2+1 dimensional Schwarzschild solution", "content": "In this section we study the Schwarzschild solution. As we show in the following, although this geometry does not belong to the class of solutions studied in the previous section, it plays a role in gravity/CFT correspondence since it can be conformally mapped to the Mart'ınez-Zanelli solution [22]. The Schwarzschild solutions is given by The curvature singularity in is hidden by an event horizon if a < 0. This condition also results in a downward gravitational pull of the black hole.", "pages": [ 17 ] }, { "title": "4.1 Rotating Solutions", "content": "The rotating Schwarzschild solution is given by the metric in which The Kerr solution is given by the metric, where, Σ( r ) = r 2 and ∆( r ) = (1 + Q 2 ) r 2 -2 Mr + a 2 , and a = J M in which J is the total angular momentum. For Q = 0 this solution corresponds to the geometry at the equator ( θ = π 2 ) of the Kerr solution in four dimensions. For these solutions,", "pages": [ 17 ] }, { "title": "4.2 The Mart'ınez-Zanelli solution", "content": "The Mart'ınez-Zanelli solution is a black hole solution of Einstein gravity with a negative cosmological constant -l -2 conformally coupled to a massless scalar field. The solution is given by the metric and where ρ 0 > 0 denotes the radius of the event horizon. To show that this solution can be obtained from the Schwarzschild solution (4.1) by a conformal transformation, g µν → ω 2 ( r ) g µν , one may rewrite the Schwarzschild solution as where α and λ are, for the moment, arbitrary constants. By performing the above conformal transformation on the Schwarzschild solution one obtains a spacetime for which Solving this equation for R = -6 /l 2 , a solutions is where β is determined by the Einstein field equation T µν is the matter stress tensor Eq.(2.5), and For ω ( r ) given by Eq.(4.12), the Ricci scalar is R = -8 / (9 λ 2 a 2 ) which determines λ 2 = 4 l 2 / (27 a 2 ). Defining ρ 2 ≡ g φφ = r 2 ω 2 ( r ), (which directly converts Eq.(4.14) to Eq.(4.8)), the Mart'ınez-Zanelli metric is obtained for α = λ -1 and ρ 0 = -3 a . The conformal map between the Schwarzschild solution and the Mart'ınez-Zanelli solution is specially useful in studying scattering in the Mart'ınez-Zanelli background. Scattering in the Schwarzschild background is thoroughly studied in the literature. Using the conformal map above, all those results can be applied to the Mart'ınez-Zanelli background. The relation between the Schwarzschild solution and the Mart'ınez-Zanelli solution can be analyzed from a different point of view. The Schwarzschild spacetime and the Mart'ınezZanelli spacetime are static and stationary, i.e. in both cases [45], For gravity conformally coupled to matter field the Einstein equation, indicates that R = 6Λ. Solving this equation for f ( r ) one obtains f ( r ) = -Λ r 2 + A r + B . Solving for the matter field ψ , one obtains, It is clear that for Λ < 0 this is the Mart'ınez-Zanelli solution while for Λ = 0, Eq.(4.17) retrieves the Schwarzschild solution ψ = √ 8 /κ . The conformal factor that gives the conformal map between these solutions can be easily obtained: the equations, give dρ = ω 2 ( r ) dr and using the definition ρ ≡ rω ( r ), one obtains the conformal factor given by Eq.(4.17). Stability of the Schwarzschild solution against linear perturbations is studied in appendix C.", "pages": [ 17, 18, 19 ] }, { "title": "5 Summary", "content": "We studied 3D Einstein gravity conformally coupled to a massless scalar field ψ . Solutions of this theory are geometries with vanishing Ricci scalar. We studied stationary static solutions with ψ = √ 8 /κ including the Schwarzschild solution. We explicitly showed that the Schwarzschild solution can be conformally mapped to the Mart'ınez-Zanelli solution, and similar to it, the Schwarzschild geometry is unstable against linear perturbations. Furthermore, we observed that R = 0 has an infinite class of stationary static solutions which similar to AdS 3 end on a cylindrical conformal boundary. Following the BrownHenneaux approach, we showed that the asymptotic symmetries of these solutions are given by two copies of the Virasoro algebra. We argued that the central charge of the dual CFT is infinite. In fact, a finite central charge is not consistent with the asymptotic symmetries where deformations that scale the g rr component of the metric are allowed. Furthermore we argued that an infinite central charge is consistent with considerations concerning the semiclassical partition function. Using three different methods we obtained the following value for the central charge in which K (0) denotes the extrinsic curvature of the boundary.", "pages": [ 19 ] }, { "title": "Acknowledgement", "content": "We would like to thank M. M. Sheikh-Jabbari and H. Soltanpanahi for valuable comments and discussions.", "pages": [ 20 ] }, { "title": "A General axisymmetric stationary static solutions", "content": "In this section we give the most general static stationary solution of Einstein gravity in three dimensions coupled to a conformal scalar field, given by action (2.1). The corresponding Einstein field equations are where G µν is the Einstein tensor and T µν is the matter stress tensor given in Eq.(2.5), and the matter field equation is given by Eq.(2.10).", "pages": [ 20 ] }, { "title": "Constant ψ", "content": "/negationslash In this case either ψ = √ 8 /κ and R = 0 or ψ = √ 8 /κ and R µν = 0. We studied the first case in section 2. The second case is the traditional Einstein equation for vacuum, because as can be readily seen in the action (2.1), a nonvanishing constant ψ only changes the Newton's constant.", "pages": [ 20 ] }, { "title": "r -dependent solution", "content": "It is well worth studying this case since it is essentially an example of Einstein gravity with varying Newton's constant. Even for a varying ψ ( r ), the Ricci scalar is vanishing on-shell because T µ µ = 0 by construction. For the metric ansatz, the most general solution of Einstein field equations is where ψ ( r ) is given by the following equation, in which d > 0 and c are arbitrary constants. Since c ↔-c corresponds to ψ ↔-ψ which is a symmetry of the action, one can take c ∈ R + . c = 0 is not allowed because for c = 0, ( rψ 2 ) ' -2 κ = 0 and thus gives a singularity in Eq.(A.3). In fact, independent of Eqs.(A.3), one can show that the only solution of equation ( rψ 2 ) ' -2 κ = 0 that solves the Einstein field equations is ψ = √ 8 /κ . For c > 0, Eq.(A.4) implies that asymptotically, The geometry (3.1) corresponds to c = 2.", "pages": [ 20, 21 ] }, { "title": "B Noether Charges", "content": "In section 2, we observed that Einstein-Hilbert action with conformal matter at the critical value ψ 0 = ± √ 8 κ -1 admits black hole solutions. So far we have treated as the effective coupling constant, and considered the value ψ 0 as the critical point where the Planck mass vanishes. In this section, in order to obtain the conserved charges corresponding to asymptotic Killing vectors [17, 18] we rewrite the action in the Einstein frame. Using the conformal transformation g → exp(2 ω ) g where 12 one obtains, Since the conformal transformation is given by e ω , the black hole solutions of section 2 for which e -ω = 0, are somehow hidden in action (B.3). From this point of view, using action (B.3) to compute the charges of the black hole solutions by the method of [17] is questionable. Nevertheless, as we show in the following, the central charge obtained by this method is consistent with the result of section 3. It is also interesting to note that action (B.3) can be supplemented by a Gibbons-Hawking term in the usual way, which gives the Brown-York stress-tensor used in section 3. In [17] it is shown that for Einstein gravity, the gravitational conserved charge corresponding to an asymptotic Killing vector ξ is given by 13 ¯ g denote the background metric (the vacuum solution), h µν is the first order deviation of the solution from the background geometry, i.e. g µν = ¯ g µν + h µν + O ( h 2 ) and in which where For ξ = ( ξ t , ξ r , ξ φ ) and one finds where R is the radius of the boundary defined in Eq.(3.33), Naively, Eq.(B.9) implies that for the black hole solutions studied in section 3, where z = 1 / 2, Q ξ = 0, but as we discussed in section 3 (see Eq.(3.36)), the correct value of the charges is given by Thus ˜ Q ∂ t = M/ 4. Furthermore one can compute the central charge for the asymptotic Killing vectors. In order to use the formula [17] to compute the central charge, we consider the vector, where instead of the asymptotic conformal Killing vectors considered in section 3. Since one verifies that similar to section 3, the correct fall off condition for δg xt ∼ δg xφ ∼ xg xx requires that b ( z +1) < 1 / 2. Recall that Of course the value of the central charge is independent of the choice of radial coordinate i.e. it is independent of the value of b . The difference between the δg µν generated by (B.13) and the one corresponding to the conformal Killing vectors considered in section 3 is that here, Recall that the conformal transformation considered in Eq.(3.17) is tuned to make δg xx = 0. Equations (B.12)-(B.15) give which is the value obtained in section 3.", "pages": [ 21, 22, 23 ] }, { "title": "C Stability of the Schwarzschild solution", "content": "The Mart'ınez-Zanelli solution is unstable against linear circularly symmetric perturbations [44]. To study the stability of the Schwarzschild solution against linear perturbations, we consider the most general perturbed metric, where f ( t, r, φ ) = (1 -2 M r ) + F ( t, r, φ ). Furthermore we assume that ψ ( t, r, φ ) = √ 8 κ + ξ ( t, r, φ ). Linearizing the Einstein equations with respect to U ( t, r, φ ) , F ( t, r, φ ), H ( t, r, φ ) and ξ ( t, r, φ ) one obtains, Eqs.(C.3)-(C.5) give, The only possible solution of these equations is, In this case, the equation of motion of ψ simplifies as which also gives R = 0. Near the horizon this equation simplifies as follows, /negationslash where x ≡ r -2 M . Assume that H ( t, r, φ ) = 0, F ( t, r, φ ) = /epsilon1 ( t, φ ) x p , and U ( t, r, φ ) = M p -q u ( t, φ ) x q , where p, q ≥ 0. If q = p -1 then the first term in Eq.(C.11) implies that u ( t, φ ) = 0, and consequently, Thus, modes corresponding to 0 < p < 1 which radial slope diverges on the event horizon, grow with time. This shows that the Schwarzschild solution is unstable against such perturbations. It is useful to note that this condition is also sufficient to prevent curvature singularities at the event horizon. For q = p -1 one obtains, which gives In principle, u ( t, φ ) = α/epsilon1 ( t, φ ). Thus for α < -(1 + 1 2 q +1 ), these modes also grow with time.", "pages": [ 23, 24 ] } ]

[ { "title": "Laplace-Beltrami operator and exact solutions for branes", "content": "A. A. Zheltukhin a,b ∗ Royal Institute of Technology and Stockholm University Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden", "pages": [ 1 ] }, { "title": "Abstract", "content": "Proposed is a new approach to finding exact solutions of nonlinear p -brane equations in D -dimensional Minkowski space based on the use of various initial value constraints. It is shown that the constraints ∆ ( p ) /vectorx = 0 and ∆ ( p ) /vectorx = -Λ( t, σ r ) /vectorx give two sets of exact solutions.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Branes are fundamental constituents of string theory [1], but not so much is known about their internal structure encoded in the nonlinear PDEs [2-14]. As a result, the emphasis in the investigations is shifted to exploring various particular solutions and the physics based on them. There is a progress in search for spinning membranes ( p = 2) with spherical/toroidal topology embedded in flat and curved AdS p × S q backgrounds (see e.g. [15-20] ). Extension of these results to the case p = 3 and complexified backgrounds with symmetry groups such as SU ( n ) × SU ( m ) × SU ( k ) was done in [20], where radial stability of three-branes was established. Analysis of spinning branes with higher p , as well as finding other particular solutions of brane equations is an open problem. On this way an important observation was done by Hoppe in [21], where the U (1)-invariant anzats reducing the membrane equations in D = 5 Minkowski space to the system of two-dimensional nonlinear equations was proposed. The particular solutions of the Hoppe equations which describe collapsing or spinning flat tori in D = 5 were found in [22] and their connection with the geometric approach, Abel and pendulum differential equations was established in [23]. The extension of the membrane anzats describing the Abelian U (1) p invariant p -branes revealed exact hyperelliptic solutions for flat p -tori embedded into D = (2 p +1)-dimensional Minkowski space [24]. Exact solutions corresponding to spinning p -branes in D = (2 p +1)-dimensional Minkowski space were found in [25]. Here we make an attempt to understand the above-mentioned exact p -brane solutions on the base of a general approach which allows to find new exact solutions. The approach uses the wave representation of p -brane equations on the ( p +1)-dimensional worldvolume Σ p +1 . In the orthogonal gauge these wave equations are reduced to the ones including Laplace-Beltrami operator ∆ ( p ) on the hypersurface Σ p . We propose to classify the brane solutions exploring various initial value constraints imposed on ∆ ( p ) /vectorx . We show that the harmonicity constraints ∆ ( p ) /vectorx = 0 pick up the solutions describing spinning p -branes which include the spinning anzats [25] in the case D = 2 p + 1. These solutions include the infinite p -branes with the shape of hyperplanes which are reduced to p -dimensional domain walls with the constant brane energy density in the static limit. Found also are periodic solutions describing closed spinning folded p -branes with a singular metric, which generalize the folded string solutions [26], [27] to the case of p -branes. The effect of the formation of singularities for closed strings and membranes was also discussed in [28]. Further, the present paper reveals that the harmonicity constraints ∆ ( p ) /vectorx = -Λ( t, σ r ) /vectorx select the exact solutions with Λ = p /vector R 2 ( t ) describing closed p -branes with their hypersurface Σ p lying on the collapsing sphere S D -2 with the time-dependent radius equal to √ /vector R 2 . The nonlinear equation for R ( t ) turns out to be exactly solvable for any dimension D of the Minkowski space and results in hyperelliptic functions. In the case D = 2 p +1 these solutions are reduced to the degenerate anzats [24] with all equal radii of the corresponding p -tori. The presence of such collapsing solutions generated by the deformed harmonicity constraint is a common property of closed membranes and p -branes independent of the Minkowski space dimension D ≥ p +1.", "pages": [ 1, 2, 3 ] }, { "title": "2 Worldvolume wave equations for branes", "content": "The Dirac action for a p-brane without boundaries is defined by the integral in the dimensionless worldvolume parameters ξ α ( α = 0 , . . . , p ). The components x m = ( t, /vectorx ) of the brane world vector in the D-dimensional Minkowski space with the signature η mn = (+ , -, . . . , -) have the dimension of length, and the dimension of tension T is L -( p +1) . The induced metric G αβ := ∂ α x m ∂ β x m is presented in S by its determinant G . After splitting the parameters ξ α := ( τ, σ r ) the Euler-Lagrange equations and ( p +1) primary constraints generated by S take the form where P m is the energy-momentum density of the brane. It is convenient to use the orthogonal gauge simplifying the metric G αβ with ˙ /vectorx := ∂ t /vectorx = L -1 ∂ τ /vectorx . As a result, the constraint ˜ U (3) represents P 0 as and it becomes the Hamiltonian density H 0 of the p-brane since ˙ P 0 = 0 in view of Eq.(2). Using the definition of P 0 (2) and G ττ = 1 /L 2 (1 -˙ /vectorx 2 ) = 1 /L 2 G tt we express P 0 as a function of the velocity ˙ /vectorx Taking into account this expression and definition (2) one can present /vector P and its evolution equation (2) in the form previously used in [28], [22] and [24], Then Eqs. (7) produce the second-order PDE for /vectorx These equations may be presented in the canonical Hamiltonian form where H 0 is the integrated Hamiltonian density H 0 ≡ P 0 The presence of square root in (9) points to the presence of the known residual symmetry preserving the orthogonal gauge (4) and generated by the constraints ˜ T r (3) reduced to the form The freedom allows to impose p additional time-independent conditions on /vectorx and its space-like derivatives. The presented description does not restrict space-time and brane worldvolume dimensions ( D,p ) and p < D . Alternatively, we present p -brane Eqs. (2) as the reparametrization invariant wave equation for x m on the ( p +1)-dim. brane worldvolume Σ p +1 where /square ( p +1) := 1 √ | G | ∂ α √ | G | G αβ ∂ β is the Laplace-Beltrami operator. Using the relation ∂ α ln √ | G | = Γ β αβ , where Γ γ αβ are the Cristoffel symbols generated by the metric G αβ of Σ p +1 , one can express Eqs.(12) as the vanishing covariant divergence of the worldvolume vector x m,α where x m,α := G αβ ∂ β x m and ∇ α x m,α ≡ ∂ α x m,α + Γ α βα x m,β . Eqs.(13) are presented as the continuity equations for the components of Noether current T mα := T √ | G | G αβ ∂ β x m generated by the global translation symmetry of the Minkowski target space. Below, we shall use the wave representation (12) in a fixed gauge to develop a way for construction of some exact solutions of the brane equations.", "pages": [ 3, 4, 5 ] }, { "title": "3 Laplace-Beltrami operator and Noether identities for p-branes", "content": "Using the gauge (4) one can extract the Laplace-Beltrami operator ∆ ( p ) , associated with the p-brane hypersurface Σ p , from the operator /square ( p +1) where g rs := ∂ r /vectorx · ∂ s /vectorx is the induced metric on Σ p . The use of the LB operator ∆ ( p ) allows to present Eqs. (12) as the system of ( D -1) equations Taking into account the relation 1 2 ∂ r (1 -˙ /vectorx 2 ) = ( ¨ /vectorx∂ r /vectorx ) , following from the orthogonality conditions (11), we rewrite the system (15) in the form where the following condensed notations are used Eqs. (16) show equality between two invariants of the residual diffeomorphisms (10) of Σ p one of which is ∆ ( p ) /vectorx , including only the space-like derivatives of /vectorx , and the other capturing all time-like derivatives of /vectorx . I equals the metric component G tt = 1 / (1 -˙ /vectorx 2 ) multiplied by the l.h.s. of Eqs.(16) equal to projection of the acceleration ¨ /vectorx on the directions orthogonal to Σ p . This follows from the identities that imply that ( /vectorx ,r ∆ ( p ) /vectorx ) ≡ 0 which are a consequence of the formula The covariant derivative ∇ r /vectorx ,s := ∂ r /vectorx ,s +Γ s rq /vectorx ,q contains the Cristoffel symbols Γ r ps constructed from the metric tensor g rs of the brane hypersurface Σ p . Indeed, the representation (20) multiplied by the vectors /vectorx ,r results in in view of the well-known relation g sq dg qs = dln | g | . The derived identities (19) extracted from Eqs.(16) are the Noether identities associated with the residual gauge symmetry (10) of the p -brane equations. From the physical point of view the brane Eqs.(16) mean that the constituent of ¨ /vectorx orthogonal to Σ p is parallel to ∆ ( p ) /vectorx , and therefore the forces orthogonal to the brane hypersurface are represented by the vector ∆ ( p ) /vectorx . The geometric interpretation of the invariant I allows to express the brane equations (16) in the equivalent form where Π ik is the projection operator on the local vectors /vectorn ⊥ orthogonal to the tangent vectors /vectorx ,r of Σ p . Then the property of orthogonality of ∆ ( p ) /vectorx to Σ p is encoded by the conditions showing that ∆ ( p ) /vectorx is an eigenvector of the projection operator Π ik similarly to the Euclidean vectors /vectorn ⊥ and ˙ /vectorx The presence of p Noether identities (19) proves that ( D -1) brane equations (16) contain only ( D -p -1) independent equations generated by the projections of (16) on the vectors /vectorn ⊥ ( t, σ r ) orthogonal to the tangent hyperplane spanned by the vectors ∂ r /vectorx at the point ( t, σ r ) where the subindex = p +1 , p +3 , ..., D 1 takes ( D p Using G αβ ⊥ ---1) values. (4) one can present Eqs.(26) in an equivalent form recognized as the minimality conditions for the worldvolume Σ p +1 embedded in the D -dimensional Minkowski space expressed via the covariant traces of the second fundamental form W ⊥ αβ of the brane worldvolume Σ p +1 . In the considered orthogonal gauge (4) the ( p + 1)-st Noether identity, associated with the freedom in τ -reparametrizations of Σ p +1 , reduces to the energy density conservation ˙ P 0 = 0. It can be seen when analyzing the projection of (16) on the vector ˙ /vectorx . Really, taking into account the relations one can present the projection of Eqs.(16) on ˙ /vectorx as or, after using definition (6) for the energy density P 0 , in the form Eq. (32) is satisfied in view of the above-proved energy conservation law.", "pages": [ 5, 6, 7 ] }, { "title": "4 Solvable p -brane motions with ∆ ( p ) /vectorx = 0", "content": "The interpretation of ∆ ( p ) /vectorx as the vector encoding forces orthogonal to Σ p may be used for exploring admissible motions of branes. On this way it is natural to study the motions in the absence of forces orthogonal to the brane hypersurface Σ p . These motions are fixed by the harmonicity conditions which must be considered as the initial value constraints for brane Eqs.(16). Since the constraints (33) have to be preserved in time the corresponding brane evolution must obey the following equations as it follows from Eqs.(24). It is easy to see that Eqs. (34) have a particular solution that coincides with the general solution of the system which describes the motions in the balance of forces acting on the brane. The general solution of evolution Eqs. (35) is linear in time as it follows from the orthogonality conditions (11). Then harmonicity conditions (35) are transformed to constraints for the initial values /vectorx 0 ( σ r ) and /vectorv 0 ( σ r ). The static p -branes are described by the particular solution and the harmonicity conditions yield the initial data constraints for the brane shape /vectorx 0 ( σ r ). The static brane energy density P ( stat ) 0 = T √ | g | and it can realize the ground state of p -brane, as its kinetic energy vanishes. Let us note that an antipode of the static brane is the one moving with the maximum velocity equals the velocity of light, i.e. ˙ /vectorx 2 = 1. In this case Eqs. (16) are reduced to the above-discussed equation ¨ /vectorx = 0, but with arbitrary ∆ ( p ) /vectorx . The branes moving with the velocity of light have zero tension and degenerate metric (4) of their worldvolumes [8]. The discussed examples of particular solutions confirm correctness of the proposed approach for exploring solutions of Eqs. (22). So, one can apply it for studying the general solution of (34) describing tensionfull branes characterized by (1 -˙ /vectorx 2 ) > 0. Generally Eqs.(34) capture the whole set of motions characterized by zero projections of the acceleration ¨ /vectorx on the directions orthogonal to Σ p The forces acting on the brane are tangent to Σ p and produce acceleration orthogonal to the velocity ˙ /vectorx , respectively. Combining the time-independence of both the squared velocity ˙ /vectorx 2 and the energy density we obtain the formula which shows time-independence of the brane volume, i.e. ˙ g = 0. These conditions are characteristic of spinning p -branes with their elastic force compensated by the centrifugal force. This proves that the solutions of the equations ∆ ( p ) /vectorx = 0 must describe spinning p -branes. To find such solutions in explicit form we restrict ourselves by the case when spinning p -branes evolve in odddimensional Minkowski space with the fixed dimension D = (2 p +1). In this case we have p independent components of /vectorx ( t, σ r ) remaining after the solution of the p orthogonality constraints ( ˙ /vectorx · ∂ r /vectorx ) = 0. In view of the above-derived p Noether identities we have just p (= 2 p -p ) independent equations for p remaining degrees of freedom of /vectorx ( t, σ r ). In addition there are p σ -dependent diffeomorphisms (10) which can be used to fix σ -dependence of these DOF. Finally, the brane equations are reduced to the system of p usual differential equations for p functions independent of σ r . A possible way to accomplish such a type of reduction is, e.g. to separate t and σ variables in each component of the vector /vectorx ( t, σ r ) with subsequent exclusion of gauge and non-propagating DOF using p orthogonality conditions (4) and p additional gauge conditions for the remaining diffeomorphisms (10). This strategy was realized in [25], where the discussed 2 p -dimensional Euclidean vector /vectorx ( t, σ r ) of spinning p -brane was presented as the generalization of the membrane anzatses studied in [21] and [22] which gives a solution of orthogonality constraints (4) with the propagating DOFs represented by the polar angles θ a ( t ). This anzats gives Keeping in mind constraint (38) we obtain the following solution for θ a ( t ) where θ 0 a and ω a are the integration constants with a=1,2,...,p. As a result, the energy density of spinning p -brane P 0 (40) is defined by the following function of the velocity components ω a q a ( σ r ) This time-independent energy density turns into the density P ( stat ) 0 of a static brane in the limiting case of all the vanishing frequencies: ω a = 0. The separation between t and σ r variables realized by anzats (41) turns out to be a sufficient condition for exact solvability of Eqs.(41). Indeed, the substitution of (41) into (16) reduces these 2 p nonlinear PDEs for the components of /vectorx to p PDEs for the p components of q ( σ r ) := ( q 1 , .., q p ). Because ∆ ( p ) q a = 0, as a consequence of ∆ ( p ) x m = 0, Eqs. (44) are satisfied if there is exact cancellation between all its terms. The cancellation occurs when the conditions for the induced metric g rs on Σ p generated by (41) are satisfied. These conditions express the space-like part of metric (4) exactly in the form connecting its with the components of the p -bein e a r attached to the hypersurface Σ p . As a result, the partial derivatives q a ,r coincide with the p -bein e a r and conditions (45) may be presented in the equivalent form as The worldvolume metric G αβ on Σ p +1 generated by anzats (41) is given by which yields the following squared interval ds 2 p +1 on Σ p +1 This shows that in terms of the new coordinates q a ( σ r ) the hypersurface Σ p metric g rs becomes independent of σ r . For infinite p -branes without boundary conditions and -∞ < σ r < + ∞ one can choose the following gauge for the residual symmetry (10) where k ∼ T -1 p +1 is an arbitrary constant with the dimension of length. This choice results in the constant diagonal matrices for p -bein e a r and metric g r,s which solve the considered harmonic equations ∆ ( p ) q a ( σ r ) = ∆ ( p ) /vectorx ( t, σ r ) = 0. It proves that the initial value constraints ∆ ( p ) /vectorx = 0 select exact solutions of Eqs.(22) describing spinning branes with the shape of p -dim. hyperplanes The energy density of the infinite spinning branes is given by and one can see that the condition kω ∼ ωT -1 p +1 → 0 has to be satisfied when | σ a | → ∞ to preserve the real value of P 0 . This demands ω → 0 when the tension T is fixed, and thus P 0 (51) becomes a constant ∼ T 1 p +1 resulting in the divergent total energy in the static limit because of the infinite integration range in the parameters σ a . The static solutions may be treated as domain 'hyperwalls' generalizing the well-known two-dimensional domain walls which appear as solutions in various physical models. The integration range in σ can be made a compact by considering closed or open branes with the corresponding boundary conditons. Below we consider the case of closed spinning p -branes described by the anzats (41).", "pages": [ 8, 9, 10, 11, 12 ] }, { "title": "5 Folded p -branes as solutions of ∆ ( p ) /vectorx = 0", "content": "The change of gauge conditions (49) into the ones considered in [25] where each of the functions q a is a monotonic continuous function of only the variable σ r with r = a , gives more general solutions for conditions (45) with the diagonal matrices q a,r and g rs , and factorized determinant of g rs . The radial components q a ( σ r ) (53) and metric (54) are the solutions of eqs. ∆ ( p ) /vectorx = 0. To verify the statement it is enough to prove that these q -coordinates are the solutions of the reduced harmonic equations This becomes evident after the substitution of (54) into (55) resulting in The latter equations are satisfied in view of cancellation of the derivative ' q a which is only one function depending on σ a in the fraction ∏ ' q b ' q a . It is clear, that the mapping (53) with regular monotonic q -functions describes the same infinite p -dimensional hyperplanes as the solution (49). However, the replacement of the monotonic q -functions by the periodic ones with isolated nonregular points in g rs (54) yields solutions of Eqs. ∆ ( p ) /vectorx = 0 describing compact folded p -branes. The solutions generalize ones describing the folded strings [26], [28], [27] to the case of p -branes. The folds arise as a result of the one-parametric dependence of the functions q a ( σ a ) (53) applied to describe closed p -brane by the generalized anzats (51) with the initial data θ 0 a = 0 at t = 0 and the density energy (43) given by In the case of closed p -branes their σ -parameters are bounded: σ r ∈ [0 , 2 π ], and therefore each of the functions q a ( σ r ) from (57) has to be a periodic one: q a (0) = q a (2 π ). Next we see that at any moment t the world vector /vectorx T ( t, σ r ) (57) is produced from /vector x 0 T ( σ r ) = ( q 1 , 0 , q 2 , 0 , . . . , q p , 0) by the time-parametrized rotations belonging to the diagonal subgroup U (1) p of the group SO (2 p ). This subgroup is composed of the time-dependent rotations in the planes x 1 x 2 , x 3 x 4 ,..., x 2 p -1 x 2 p about the angles θ a = θ 0 a + ω a t , respectively. Thus, the p -brane worldvolume is formed by the rotations of the closed p -brane initially embedded into the p -dim. subspace spanned by all odd coordinate axises of the considered 2 p -dim. Euclidean space. These rotations preserve the initial brane shape. So, the periodicity conditions for q 1 with respect to σ 1 , q 2 with respect to σ 2 , etc. will be satisfied if the p -brane is initially folded up along each of the odd coordinate axises. A simple example of the solution is given by the symmetrically folded closed p -brane with the functions q a ( σ a ) = k | π -σ a | which realize the conditions q a (0) = q a (2 π ) by the bending formation at σ a = π which create additional forces orthogonal to Σ p around these points. The latters fix the lines (planes) on the brane hypersurface Σ p along which it is bent. For the folded membrane ( p = 2) embedded into 4-dim. Euclidean space its image may be visualized as a double-folded sheet of paper forming a stack of four equal small squares originated from the original unfolded square with the side length equal to 2 kπ . The functions q a ( σ a ) in (59) are continuous ones, but their derivatives have the jump discontinuity equal to 2 = 1 -( -1) at σ a = π . These jumps result in the indefiniteness of the induced metric (54) at these points. The change of the parametrization (59) by smooths out the derivative jumps at σ a = π . The flat metric g rs (54) vanish at these points, as well as the energy density P 0 (58) (if p a =1 q 2 a ω 2 a = 1). ∑ A more general parametrization producing ( n 1 , n 2 , . . . , n p ) singular points for g rs defined by the functions ( q 1 , q 2 , . . . , q p ) (54), respectively, may be choosen in the form similar to the one considered in [26] /negationslash with the set ( n 1 , n 2 , . . . , n p ) treated as the topological winding numbers. So, anzats (57) with the periodic q -functions gives exact solutions of ∆ ( p ) /vectorx = 0 with isolated singularities in g rs and describe initially folded branes. The brane worldvolume Σ p +1 associated with the initially folded hypersurface Σ p is produced by its rotations as a whole realized by the above mentioned Abelian group U (1) × U (1) × . . . × U (1) ≡ U (1) p . The corresponding rotation angles θ a are treated as the generalized cyclic coordinates of the Hamiltonian density (5) corresponding to the energy density P 0 (43). The momenta j a conjugate to the generalized coordinates θ a are given by Then the corresponding Hamiltonian p -brane density takes the form The momenta (62) are integrals of the motion proportional to the conserved energy density P 0 . The values j a are the components of the angular momentum density associated with the generators of rotations in the planes x 1 x 2 , x 3 x 4 ,..., x 2 p -1 x 2 p which form the above-discussed Abelian group U (1) p . They may be presented as explicit functions of the non-propagating brane coordinates q a ( σ r ) and their derivatives We conclude that the choice of the initial value constraints in the form of harmonicity conditions (33) selects the regular or singular g rs given by the solutions of Eqs.(22) describing infinite or compact folded spinning p -branes.", "pages": [ 12, 13, 14, 15 ] }, { "title": "6 Solvable p -brane motions with ∆ ( p ) /vectorx = -Λ /vectorx", "content": "In the previous section we have found that the harmonicity equations ∆ ( p ) /vectorx = 0 treated as the initial value constraints provide the exact solutions [25] of brane equations. One can conjecture that specially constructed deformations of the harmonicity conditions may reveal other exact solutions. This proposal is compatible with the specific form of brane Eqs.(22), where the shift of the factor G tt to their l.h.s. leaves only ∆ ( p ) /vectorx in the r.h.s.. Therefore, the time derivatives of /vectorx are concentrated in the l.h.s. of (22). Using various initial value constraints, including ∆ ( p ) /vectorx in combination with /vectorx and ˙ /vectorx , one can generate various evolution equations. It may occur that some of these evolution equations are exactly solvable like in the case ∆ ( p ) /vectorx = 0. The constraint deformations are under control of the Noether identities demanding ∆ ( p ) /vectorx to be an eigenvector of the projection operator Π ik , as it follows from (24). Variation of the constraints will result in deformations of the brane shape selfconsistent with the evolution equations. As an example realizing this proposal and generalizing the solutions [24] we consider the following invariant deformation of the harmonicity conditions where Λ is an arbitrary function invariant under diffeomorphisms of the hypersurface Σ p . The substitution of (65) into (22) yields the evolution equation accompanied with the constraints (67) for the initial value for this evolution equation. Due to the Noether identities we obtain that the projections of Eqs.(66-67) on /vectorx ,r result in the following rotationally invariant constraint which shows that the p -brane hypersurface Σ p resides on the ( D -2)-dimensional sphere of the radius R = √ /vector R 2 ( t ) embedded into the ( D -1)-dimensional Euclidean space. The projections of (66-67) on ˙ /vectorx yield the equations fixing the unknown function Λ. The projections of (66-67) on /vectorx give the relation where l is the integration constant with the dimension [ l ] = L . The latter relation in combination with (69) yields the closed equation for ξ := /vector R 2 ( t ) The first integral of Eq. (72) is given by the relation expressed in terms of the new dimensionless variable ζ ( t ) := √ ξ l ≡ √ /vector R 2 l substituted instead of /vector R 2 ( t ). Then the first integral is presented as after the transition to the new rescaled time variable η := √ 2 t l . For the case p = 2 corresponding to membrane Eq. (73) is the defining equation for the Jacobi elliptic cosine cn ( η, k ) with the elliptic modulus k = 1 √ 2 . For p > 2 the exact solution of (73) is given by the hyperelliptic integral generalizing the elliptic membrane solution to p -branes with arbitrary p . Thus, we obtain exact solution for the length √ /vector R 2 ( t ) of /vectorx without any gauge fixing for the symmetry (10) and the restiction D = 2 p +1 [24]. Then the generalized harmonicity conditions (65) take the form with the known function /vector R 2 ( t ) depending only on time. The σ -independence of /vectorx 2 = /vector R 2 ( t ) results in the σ r -independence of ˙ /vectorx 2 , as it follows from (71) and the fact that the second term in the l.h.s. of (66) vanishes. As a result, Eqs. (66) and (67) are reduced to two connected subsystems with the evolution equations describing 2 p -dim. oscillator with time-dependent frequency given by the (hyper)elliptic function of time. To find all the components of the vector /vectorx we must solve Eqs.(76) and (77). Since the length of /vectorx is σ -independent, this dependence concentrates in the direction cosines of /vectorx . This suggests representation of /vectorx in the form x i ( t, σ r ) = O ik ( t, σ r ) R k ( t ) , where O ik ∈ SO ( D -1) group of rotations of ( D -1)-dimensional subspace of the Minkowski space. In view of the time independence of | ˙ /vectorx | , the time derivative of this representation for /vectorx shows that the matrix O is also timeindependent. This observation results in the separation of variables Similarly to the spinning brane case we restrict ourselves by (2 p + 1)-dim. Minkowski space and choose the matrix O ik from the Abelian subgroup O (2) p of the group SO (2 p ). Then /vectorx takes the form of the anzats [24] Contrary to the spinning anzats (41), considering its polar angles to be propagating DOF, here we have the radial coordinates q ( t ) = ( q 1 , .., q p ) as the propagating DOF. Anzats (79) yields the following expressions for the lengths of /vectorx and ˙ /vectorx and for the worldvolume metric G αβ on Σ p +1 , respectively where θ a,r ≡ ∂ r θ a . The corresponding squared interval ds 2 p +1 is given by Representation (82) shows that in the new coordinates θ a ( σ r ), used instead of σ r , the metric on Σ p becomes independent of σ r with p Killing vector fields represented by the derivatives ∂ ∂θ a . Thus, anzats (79) describes p-dimensional torus S 1 × S 1 × . . . × S 1 with zero curvature and the timedependent radii q a . This anzats reduces the number of degrees of freedom to p carried by the radial coordinates q a which obey reduced Eqs.(76) with their first integral equal to 1 -˙ q 2 = ( q 2 ( t ) l 2 ) p . The substitution of expressions (80) in Eqs.(69) regenerate Eq. (73) and its hyperelliptic solution (74) with q 2 substituted for /vectorx 2 , e.d. η ( t ) = √ q 2 l . The substitution of anzats (79) into Eqs. (77) transforms them to homogeneouos equations for the components θ a which are equivalent to for each a . The equations are easily solved in the gauge θ a = δ ar σ r [24] where σ r -independent metric g rs ( t ) takes the following diagonal form and transforms Eqs. (85) to identities. Eqs. (84) reduce to the conditions which mean coincidence of all q a -functions: q a ( t ) ≡ q ( t ). From the geometrical point of view the coincidence condition picks up the case of degenerate p -torus with equal radii [24]. In view of the above constraints, the system of p tangled equations (83) shrinks to the single exactly solvable nonlinear equation with the above-studied first integral given by The change of variables ˜ ζ = √ pq l , η = √ 2 t l transforms Eq. (91) into Eq. (73) and its solution is given by the considered hyperelliptic integral (74) Thus, we proved that the deformation (65) of the harmonicity conditions selects the exact solution which describes collapsing p -brane with the shape of the degenerate p -torus [24].", "pages": [ 15, 16, 17, 18, 19 ] }, { "title": "7 Summary", "content": "A new approach to the problem of exact solvability of nonlinear p -brane equations and constraints in D -dimensional Minkowski space was considered. The approach is based on the connection between the initial value problem for the brane equations and their exact solutions. The p -brane equations, initially written in the form o f ( p +1)-dimensional worldvolume wave equations, were reduced in the orthogonal gauge to p -dimensional equations with their r.h.s. presented by ∆ ( p ) /vectorx and l.h.s. equal to the brane acceleration projection on the directions orthogonal to its hypersurface Σ p . The Noether identities associated with the diffeomorphisms of the brane worldvolume Σ p +1 were derived and used for the choice of the admissible constraints for the initial data. Two types of such constraints were studied and the corresponding exact solutions were obtained. The first of them considers the harmonicity constraints ∆ ( p ) /vectorx = 0 which select spinning p -branes. In the case D = 2 p +1 the harmonicity constraints are exactly solved by the anzatz previously considered in [25]. These solutions include either regular solutions for g rs describing infinite p -branes with the shape of p -dimensional hyperplanes or nonregular g rs associated with folded compact p -branes. The case of the infinite branes includes static p -branes with the constant density of energy treated as p-dimensional domain walls. The second set is picked up by the deformed harmonicity conditions ∆ ( p ) = -Λ /vectorx and describes closed p -brane lying on a collapsing sphere S D -2 embedded into ( D -1)-dimensional Euclidean subspace of D -dimensional Minkowski space with arbitrary D > 4. The time-dependent radius of the sphere is presented by hyperelliptic functions. In the particular case of odd D = 2 p + 1 the p -brane hypersurface Σ p turns out to be isometric to flat collapsing p -dimensional torus which coincides with the exact solution [24]. The described spinning or collapsing 5-branes ( p = 5) give exact solutions of D = 11 M/string theory and it is interesting to understand the physics associated with them. Extension of the proposed approach to the case of opened p -branes with various boundary conditions as well as its generalization to the case of known cosmological backgrounds seems to be interesting.", "pages": [ 20 ] }, { "title": "Acknowledgments", "content": "The author is grateful to Physics Department of Stockholm University and Nordic Institute for Theoretical NORDITA for kind hospitality and financial support. The results were presented at the Conference STDE-2012 in honor of Vladimir Aleksandrovich Marchenko's 90th birthday [29].", "pages": [ 20 ] } ]

[ { "title": "ABSTRACT", "content": "CQUeST-2012-0549", "pages": [ 1 ] }, { "title": "Extremal Black Holes and Holographic C-Theorem", "content": "Yongjoon Kwon 1 , Soonkeon Nam 2 , Jong-Dae Park 3 , Sang-Heon Yi 4 Department of Physics and Research Institute of Basic Science, Kyung Hee University, Seoul 130-701, Korea 1 , 2 , 3 4 Center for Quantum Spacetime, Sogang University, Seoul 121-741, Korea", "pages": [ 1 ] }, { "title": "Abstract", "content": "We found Bogomol'nyi type of the first order differential equations in three dimensional Einstein gravity and the effective second order ones in new massive gravity when an interacting scalar field is minimally coupled. Using these equations in Einstein gravity, we obtain analytic solutions corresponding to extremally rotating hairy black holes. We also obtain perturbatively extremal black hole solutions in new massive gravity using these lower order differential equations. All these solutions have the anti de-Sitter spaces as their asymptotic geometries and as the near horizon ones. This feature of solutions interpolating two anti de-Sitter spaces leads to the construction of holographic c-theorem in these cases. Since our lower order equations reduce naturally to the well-known equations for domain walls, our results can be regarded as the natural extension of domain walls to more generic cases.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recently, there has been much interest in c -theorem in various dimensions. The content of the theorem is that a certain central charge function, which is regarded as counting the number of degrees of freedom, should be a monotonically decreasing function along a Wilsonian renormalization group(RG) flow. The interesting recent developments include the formal 'proof' [1] of four-dimensional c -(or a -)theorem long after its conjecture [2], the discovery of the relation between central charge and the entanglement entropy(EE) [3], and the identification of the conjectured free energy maximization in three-dimensional field theory named as F-maximization with the internal space volume minimization known as Z-minimization [4]. Furthermore, the content of this theorem was constructed holographically in Einstein gravity through the AdS/CFT correspondence [5, 6] and extended recently to higher derivative gravity [7]. The pioneering work on all these developments in c-theorem was done by Zamolodchikov a few decades ago [8], who argued that c-theorem is natural intuitively in the sense of RG flow from UV to IR and proved rigorously that it holds in two dimensions under the assumption on generic properties of field theory like unitarity, conformal invariance, etc. In general, it seems natural that a kind of c -theorem holds in any kind of sensible unitary field theories. Contrary to this simple intuition, it is challenging to prove this theorem because its nature requires essentially the non-perturbative method and furthermore its proof or verification depends strongly on the spacetime dimensions, in the works that have been done so far. There have been some attempts to overcome this situation. For instance, the relation between central charge and EE was shown in two dimensions, and then two-dimensional c-theorem is rederived by using the property of EE. This construction is extended to higher dimensions and argued to be a generic proof of c-theorem. Other attempts are the holographic construction of c -theorem in various gravity models, which reveal that the conjectured c -theorem is consistent with the holographic construction. In four dimensions there are two central charges called as a and c , which are the coefficients of Euler and Weyl density in the trace anomaly formula. It was conjectured and proved recently that a is the relevant central charge consistent with the monotonic flow property. The holographic construction is very appealing since it reproduces this result nicely and it can be extended to various other dimensions including odd ones. Though the holographic construction is very useful to understand c-theorem uniformly in various dimensions, its construction is restricted usually to the simplest form of domain wall metric in the gravity side. In the context of the AdS/CFT correspondence, domain wall solutions in gravity with two asymptotic AdS spaces correspond to RG flow trajectory between two conformal points in dual field theory. In this holographic construction, appropriate central charge flow functions, which coincide with central charges at the two conformal points, are constructed by using metric functions. And then their monotonicity is verified through the equations of motion and null energy condition on matters which is imposed as a sensible condition in the gravity side. This is the content of the so-called holographic c -theorem. Even though this holographic c -theorem may be checked without the explicit domain wall solutions, it is more satisfactory to obtain the analytic domain wall solutions consistent with holographic c -theorem. Interestingly, it has been shown that the domain wall metric in Einstein gravity satisfies Bogomol'nyi type of the first order differential equations which were derived by minimizing a certain energy functional through complete squares [9] or by introducing a certain 'fake' supersymmetry or supergravity [10, 11]. Using these first order equations, various analytic forms of domain wall solutions have been obtained and shown to be consistent with holographic c -theorem in various dimensions. One may ask whether this domain wall geometry is the unique candidate as the dual to the RG flow in boundary field theory. It is rather clear that any geometry with two asymptotic AdS spaces is viable as the dual to the RG flow and as the background for holographic c -theorem. However, it is not easy to obtain such non-trivial background geometry analytically in gravity since matters play important roles and hinder analytic treatments. In this regard, three-dimensional gravity is exceptional since various analytic solutions are found including black holes with scalar hairs [12]. Indeed, there is the realization of this idea by using more complicated three-dimensional geometry than domain walls, which turns out to be consistent with holographic c -theorem [13]. The relevant geometry is given by the extremally rotating threedimensional AdS black holes which interpolate between AdS space at the asymptotic infinity and near horizon AdS geometry. The existence of the analytic black hole solutions allows the explicit realization of holographic RG flow via Hamilton-Jacobi formalism and the check of the holographic c -theorem. However, the shortcoming in these extremal black hole solutions given in Ref. [13], compared to the domain wall solutions, is the fact that one needs to make a specific 'ad hoc' choice of the scalar potential to obtain analytic results. This point is even amplified when one considers higher curvature gravity like new massive gravity(NMG) [14] which is recently introduced as a nonlinear completion of Pauli-Fierz linear massive graviton theory and shown to be consistent with a simple form of a holographic c -theorem [15]. It becomes very difficult to choose 'ad hoc' scalar potential in the NMG case, which is contrasted to the fact that domain walls satisfy first order differential equations even in NMG and allow analytic results [16]. In the context of the AdS/CFT correspondence, the generic nature of holographic construction seems to imply that there exists a more unified and systematic approach to these extremal black holes with two AdS asymptotics. It is natural to suspect the existence of some reduced differential equations for extremal black holes as for domain walls. One of main results in this paper is the discovery of such differential equations for three-dimensional AdS black holes in Einstein gravity and in NMG. We also show that such equations are enough for the consistency with the holographic c -theorem when a certain central charge flow function is chosen. This paper is organized as follows. In the next section we find the Bogomol'nyi type of first order differential equations, which solves the full equations of motion, in three-dimensional Einstein gravity interacting with a scalar field. It turns out that these restricted first order equations of motion represent extremally rotating black holes with scalar hairs. By solving these first order equations of motion in a more or less systematic way, we obtain some analytic hairy black hole solutions which include the case given in Ref. [13] as a special case. In section three we consider new massive gravity as another gravity theory to obtain reduced differential equations for extremal AdS black holes. As in the Einstein gravity case, it is shown that Bogomol'nyi type of lower order differential equations can be obtained which include domain wall solutions as special cases. By solving these equations asymptotically, we show that there are extremally rotating black hole solutions consistent with a holographic c -theorem. In section four, we consider the holographic c -theorem in our setup and show that it holds generically by using reduced lower order equations of motions. In the final section, we summarize our results with some comments and discuss open issues.", "pages": [ 2, 3, 4 ] }, { "title": "2 Extremal Black Hole Solutions in Einstein Gravity", "content": "In this section we consider three-dimensional Einstein gravity with a minimally coupled interacting scalar field. We find Bogomol'nyi type of first order differential equations which solve full equations of motion. This can be regarded as the extension of first order equations for domain walls [9, 10] to more generic cases. It turns out that the simplest solutions of these equations, which are given by a constant scalar field, correspond to extremal BTZ black holes [17]. After showing that these equations describe the extremally rotating black holes, we obtain analytic solutions of some hairy black holes in a systematic way.", "pages": [ 4 ] }, { "title": "2.1 First order equations of motion", "content": "In the convention of mostly plus signs for the metric with the convention of curvature tensors as [ ∇ µ ∇ ν ] V ρ = R µνρσ V σ and R µν = g αβ R αµβν , our starting action for Einstein gravity with a minimally coupled scalar field is given by of which the equations of motions(EOM) are composed of scalar field equation and the metric field equations as follows ; where To find asymptotically AdS black hole solutions with an interacting scalar field in three dimensions, let us take our metric ansatz in AdS-Schwarzschild-like coordinates as where L denotes the radius of asymptotic AdS space. Asymptotically AdS black holes in these coordinates mean that the asymptotic conditions on the functions A ( r ) , B ( r ) , C ( r ) are given as follows; ∣ ∣ ∣ Note that these boundary conditions are Brown-Henneaux type which allow us to apply the standard central charge extraction by Brown-Henneaux method [18]. The equations of motion even in this case turn out to be complicated non-linear differential equations. For instance, the EOM for the scalar field is given by where ' denotes differentiation with respect to the radial coordinate r . The EOM for metric, 0 = E µν , are relegated to appendix A. To obtain analytic solutions of complicated full EOM, it is very convenient to introduce the so-called 'superpotential' method which is originally applied to the domain wall solutions. Historically, the terminology of superpotential is chosen in analogy with supergravity expression for a scalar potential. When the scalar potential is represented by the so-called superpotential W as the appropriate first order differential equations, which solve full EOM, are given by These are motivated by similar expression in the domain wall case. This form of differential equations, which we call as reduced EOM, is a considerable simplification compared to the original EOM, though restricted solutions among all possible ones can be obtained from these reduced EOM. Specifically, the last equation can be solved as where the integration constants C ∓ can take any values consistently with asymptotic AdS space. As a trivial solution of our first order equations, let us consider the constant potential case, V = -2 /L 2 with a constant scalar field. In this case superpotential is given by W = 2 and then one obtains which, with boundary conditions, leads to the following solution; These metric functions represent the extremal BTZ black holes. Although C ∓ = ± 1 corresponds to the most familiar form of extremal BTZ black holes, any value of the constant, C ∓ , leads to the extremal BTZ black holes. In fact, it is more useful to take C ∓ = 0 to simplify some computations in our case 1 . To see the convenience of this choice, let us consider the near horizon geometry of extremal BTZ black holes given by r → r H . Using a new radial coordinate ρ = 4( r -r H ), one can easily identify the metric of the near horizon geometry as which is the well-known metric form [19] of the self-dual orbifold of AdS 3 with the radius L . Note that this geometry leads to zero Hawking temperature and so dual field theory to extremal BTZ can be thought to be at zero temperature. This near horizon geometry is interpreted as dual to the discrete light cone quantization(DLCQ) of two-dimensional conformal field theory(CFT) and is related to chiral two-dimensional CFT [20]. From now on we will choose the integration constant as C ∓ = 0 so that C is given by The usual choice of C ∓ = ± 1 can be recovered by a simple coordinate transformation: θ → θ + C ∓ t . One may expect that all the solutions of our first order equations correspond to some kind of extremal black holes as can be inferred by the fact that the trivial solutions represent the extremal BTZ black holes. This expectation is also natural in analogy with charged extremal black hole solutions in supergravity, in which those black holes are described by first order equations which can be derived by Killing spinor equations. As in the case of domain walls, we anticipate that some 'fake' Killing spinor equations might lead to our first order equations. In the next section we present perturbative analysis of these first order equations and show that black hole solutions are indeed extremal.", "pages": [ 4, 5, 6 ] }, { "title": "2.2 Extremally rotating black holes", "content": "It is very convenient in solving the first order reduced EOM to take φ as coordinates and r as a function of φ , instead of the original form. Then the first order reduced EOM can be rewritten where ∂ φ denotes differentiation with respect to φ variable. One can see that the first two (or last two) equations can be immediately integrated in terms of r and W and lead to solutions of metric functions A and B as By inserting the expression of A and B functions in the remaining equation, one obtains the differential equation for r ( φ ) or for W ( φ ) as which can be integrated as The physical meaning of this constant ∆ 0 will be given shortly after discussing the near horizon geometry of our black hole solutions, which will also be related to the conserved charges. One can see that the metric functions A and B are now completely determined, in terms of the superpotential W and the constant ∆ 0 , as This form of the metric function B shows us explicitly that e -B outside the horizon is a regular function of ( r -r H ) as long as W can be written as a regular function of ( r -r H ). Before going ahead to present the analytic form of some hairy AdS black hole solutions, let us consider the asymptotic and near horizon behaviors of black hole solutions. According to the boundary conditions and physical consideration, one may take as where φ ∞ denotes the value of scalar field φ at the asymptotic infinity r = ∞ . By solving the above differential equation (14) with (15) perturbatively, one can see that n = 2, k = 1 and obtain where the last equation is obtained by inverting r as a function of φ and using φ 2 1 = 4 ω 1 . Note that the superpotential value outside the horizon becomes always greater than its asymptotic value: W ( φ ) ≥ W ( φ ∞ )(= 2). This fact can be checked explicitly from the analytic solutions presented in the following. Even in these perturbative solutions, the power of the first order reduced EOM shows up as the complete determination of a 2 and b 2 in terms of ω 1 and ∆ 0 . Contrary to this, only the combination of a 1 + b 1 = -ω 1 is determined perturbatively by the original second order EOM. Now, we would like to show that the first order reduced EOM represent extremal black holes by analyzing near horizon geometry of black hole solutions of the reduced EOM. By assuming the existence of the horizon, which is given by e -B ( r H ) = 0, with the expression of the metric function B in Eq. (15), one can see that the constant ∆ 0 in Eq. (14) is determined as By solving Eq. (14) perturbatively near the horizon, one can see that the superpotential W and the radial coordinate can be taken as regular functions of φ as where h 0 is a certain constant 2 . Accordingly, metric functions A and B are given by Eq. (15) as where s 0 is a certain non-vanishing constant and · · · denotes some regular functions of ( r -r H ). Note that the constant s 0 is related to the interval of integration in the expression of A given in Eq. (15). This form of the metric function B on the near horizon shows us the extremality of black hole solutions of our first order reduced EOM: /negationslash By introducing a new radial coordinate ρ ≡ s 0 W 2 ( φ H )( r -r H ), which is appropriate on the near horizon region, one may identify the near horizon geometry as One may introduce the AdS scale ¯ L on the near horizon geometry through V ( φ H ) = -2 / ¯ L 2 , which is related to the superpotential as V ( φ H ) = -W 2 ( φ H ) / 2 L 2 since ∂ φ W ( φ H ) = 0 at the horizon. Interestingly, in terms of this scale ¯ L and ¯ r H = ˆ r H / 2 the near horizon geometry may be written in the form of which is just the metric for the self-dual orbifold of AdS 3 with the radius ¯ L . Now, one can identify ¯ L with the superpotential value at the horizon W ( φ H ) or the constant ∆ 0 as This explains the physical meaning of the integration constant ∆ 0 , which is related to the information about the near horizon geometry. This result will be consistent with the holographic c -theorem, as will be discussed in the section 4. Now, let us consider some physical quantities related to these extremal black holes. One will see that ∆ 0 is directly related to the conserved charges of black holes. Using the above explicit asymptotic expressions of the metric and the scalar field, one can obtain masses and angular momenta of black holes, for instance, through the so-called Abbott-Deser-Tekin(ADT) method [22]. Note that, in this ADT approach, one does not need to compute contributions separately from the metric and the scalar field, which is contrary to the quasi-local charge method given in Refs. [12, 13]. Since conserved charges depend on the coordinates, we need to specify those concretely. Here, we will choose those as C ∓ = ± 1 which gives us the standard metric form of BTZ black holes. That is to say, the background metric for ADT charge computation, which is AdS 3 space, is taken in our coordinates as Then, masses and angular momenta of these black holes for the Killing vectors ξ T = 1 L ∂ ∂t and ξ R = ± ∂ ∂θ are given by the so-called ADT charge Q µν as 3 Therefore, masses and the total angular momenta satisfy the extremal relation in these black holes as ML = ± J . This relation strongly suggests that any black hole solution obtained from our reduced EOM is stable since the bound for angular momentum is saturated. Now, we argue that the inequality W ( φ H ) ≥ W ( φ ∞ ) = 2 holds in general, which is an important ingredient to show the consistency with the holographic c -theorem. On general ground, it is natural to think that masses of hairy black holes deformed from BTZ black holes by a scalar field are always greater than those of hairless BTZ black holes since the scalar hair produces additional positive contribution to masses. By accepting this assumption on mass inequality between extremal hairy black holes and extremal BTZ black holes, one can see that The Bekenstein-Hawking-Wald entropy of the above extremal black holes can be read from the Wald formula or the area law as and the Hawking temperature of these black holes are always zero because of the extremality. This nature of zero Hawking temperature also indicates the stability of the hairy extremal black holes. The angular velocity of these black holes at the horizon r H is given by Since the angular velocity of these black holes with C ∓ = ± 1 is given by Ω H = ± (1 /L ) and the Hawking temperature T H is zero, one can check that the first law of black hole thermodynamics is satisfied trivially.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "2.3 Analytic solutions", "content": "In this section, we present analytic solutions of our first order reduced EOM. According to the given setup, one can try to solve the last differential equation (14) to obtain r = r ( φ ) for the given superpotential W . Then, one can determine metric functions A , B and C just by Eqs. (11) and (15). In principle, this is the correct way to obtain solutions. However, it is not easy to obtain analytic solutions in this way. Therefore, we take a slightly different route: we will try to solve this equation by taking r as an appropriate function of φ . To find exact solutions of the above first order reduced EOM, let us try the simplest choice for r 2 just as the asymptotic form itself: which imiplies that Inserting this ansatz in the reduced differential equation (14), one can see that the superpotential W is given by where the constant α is related to ∆ 0 in Eq. (14) as ∆ 0 = 4 αω 1 and β is an arbitrary constant. The metric functions A , B and C are determined in terms of φ through Eq. (15), which can be converted to the functions of r as where we have rescaled the time coordinate as usual to absorb the integration constant appropriately such that the asymptotic boundary condition on A is satisfied. Asymptotic boundary conditions on metric functions also lead to 2 α + β/ 2 = 1. Now, let us impose the existence of horizon through e -B ( r H ) = 0, which leads to Then, one can obtain α , β in terms of r 2 H = 4 ω 1 / ( φ H -φ ∞ ) 2 as At last, one can see that black hole solutions given by A ( r ) , B ( r ) , C ( r ) and φ ( r ) are nothing but those in Ref. [13], which we have obtained in a different way using the first order reduced EOM. One can identify the near horizon geometry of these black holes with the self-dual orbifold of AdS 3 with the radius ¯ L which is rescaled from the asymptotic radius L as To see this fact, one may use a new radial coordinate ρ defined by and then one can explicitly check that the metric takes the form given in Eq.(23). This shows us that the above hairy extremal AdS black holes interpolate between the asymptotic AdS 3 with the radius L and the self-dual orbifold of AdS 3 with the radius ¯ L . Now, let us take a look at another analytic solution of our first order equations. Under the successful reproduction of known solutions, one may try another choice for r 2 as Following the same procedure in the above, one obtains where we have used the Eq.(14) to obtain this result. The constant α is related to ∆ 0 (= -12 αω 1 ), as in the previous case. Then, metric functions A , B and C are given by Asymptotic boundary conditions on A and B with the existence of the horizon lead to These are new extremal AdS black hole solutions as far as the authors know. By introducing one can show that the near horizon geometry is given by self-dual orbifold of AdS 3 as the same form with Eq.(23) with ¯ L and ¯ r H defined by As the final example, we take the following ansatz Then, the superpotential and metric functions are given by and the constants α and β are determined as Using the radial coordinate ρ in this case as one can check that the near horizon geometry is once again given by self-dual orbifold of AdS 3 in Eq. (23) with the radius ¯ L and ¯ r H Some comments for the above black hole solutions are in order. Firstly, one may note that there is a new free parameter in the above solutions denoted as ω 1 which is related to the scalar field value at the horizon, φ H , and does not exist in extremal BTZ black holes. As is obvious from our method, this parameter ω 1 is also related to the coefficient of the leading term in the superpotential W . Secondly, one may note that the above solutions are the extension of extremal BTZ black holes to hairy cases and reduce to extremal BTZ black hole solutions when the scalar field is turned off. To see this explicitly, one should take ω 1 → 0 with the frozen scalar field φ = φ ∞ . Then, all the above expressions of black hole solutions reduce to those of extremal BTZ black holes. This reveals that the presence of a scalar field may produce diverse hairy black hole solutions via a scalar potential, which reduce to the same BTZ black holes when a scalar field is turned off. This point will also be important to understand the nature of black hole solutions in NMG, which are presented only in the perturbative form in the next section. One may wonder about using the so-called Fefferman-Graham coordinates in this case. That is to say, a new radial coordinate η may be introduced by This gives us the so-called FG coordinates useful in later sections and corresponds to taking the following form of the metric ansatz In these coordinates, most of the first order reduced EOM in the AdS-Schwarzschild coordinates remain as first order differential equations 4 (recall that r ≡ e R ( η ) ) where the dot denotes the differentiation with respect to the radial coordinate η . However, the first order differential equation for B ( r ) is transformed to the second order one for R ( η ) (or for λ ≡ e 2 R ) as Note that this second order differential equations is equivalent to E θθ = 0 in these coordinates and corresponds to Eq. (13) in the ( r, t, θ ) coordinates. This means that the reduced EOM may be taken by the first order equations for φ , A + R and C given in Eq. (43), together with E θθ = 0. Interestingly, the second differential equation for R (or E θθ = 0) can be integrated into the first order form as which corresponds to the Eq. (14) in ( r, t, θ ) coordinates. It is also interesting to note that this integrated first order equation is automatically satisfied for domain wall solutions which are included in our reduced EOM as the special case given by This is consistent with our interpretation of the constant ∆ 0 as related to the near horizon of black holes, which should be absent in domain walls. Though these coordinates cover only outside the horizon and do not seem so useful in obtaining analytic solutions of AdS black holes in Einstein gravity, those simplify some computations and turn out to be particularly useful in NMG and in the holographic c -theorem, which are presented in next sections.", "pages": [ 10, 11, 12, 13, 14 ] }, { "title": "3 Extremal Black Hole Solutions in NMG", "content": "In this section we obtain lower order reduced EOM in NMG and then perturbative black hole solutions in NMG with an interacting scalar field, which reduce to extremal BTZ black holes when the scalar field is turned off. In this section, we will confine ourselves to the two derivative theory for the scalar field, since the higher derivative terms for scalar field lead to ghost instability and so they are more difficult to be analyzed. One may think that the higher derivative terms for gravity is also problematic. However, as was analyzed in several works [7][15][16], higher derivative terms for gravity on an appropriate background can be treated effectively just like two derivative theory. In the following we will follow this approach and consider only two derivative terms for the scalar field.", "pages": [ 14 ] }, { "title": "3.1 New massive gravity with a scalar field", "content": "New massive gravity(NMG) is a three-dimensional higher curvature gravity introduced as the covariant completion of Pauli-Fierz massive graviton theory [14]. Later it was recognized that NMG is more or less the unique extension of Einstein gravity consistent with the holographic c-theorem [7, 15]. In our convention, the Lagrangian of NMG with a scalar field is given by where σ takes 1 or -1. The parameter m 2 can take positive or negative values, and K is a specific combination of scalar curvature square and Ricci tensor square defined by The equations of motion(EOM) of NMG are given by where and K µν , using D µ as a covariant derivative with respect to g µν , is defined by The equation of motion for the scalar field φ takes the same form with Einstein gravity given in Eq.(5). We will focus on σ = 1 case and we set σ = 1 in the following. In order for the positive central charge of dual CFT with the truncation of ghost modes [21], we also focus on the positive m 2 .", "pages": [ 14, 15 ] }, { "title": "3.2 Black hole solutions in new massive gravity", "content": "As was done in the domain walls in NMG [16], let us introduce the superpotential in NMG such that the scalar potential is given by This generalized form of potential in terms of superpotential was first considered by Low and Zee in the context of scalar field coupled to higher derivative gravity [11]. Motivated by results in Einstein gravity as given in Eq.(43), let us take the first order equations for A ( η ) and C ( η ) As in Einstein gravity, the last equation for C can be trivially integrated and may be omitted in the following. One can check that these equations solve scalar EOM, E φ = 0 and metric EOM, E µν = 0 except E θθ = 0, even in NMG. Explicitly in these coordinates, the scalar EOM and metric EOM except E θθ = 0 can be shown to be satisfied as follows: Using λ ≡ e 2 R , one can represent E θθ as where H 1 and H 2 are some functions of W and its derivative with respect to φ given by Now, one can see that the equation E θθ = 0 can be integrated as where ˜ ∆ 0 denotes the integration constant and Ψ is defined as The physical meaning of the constant ˜ ∆ 0 turns out to be similar to that of ∆ 0 in Einstein gravity. That is to say, it is related to the conserved charges of black holes and their near horizon geometry, which will be shown in the below. Note also that we have reduced the fourth order EOM effectively to the second order one. By transforming ( η, t, θ ) coordinates to ( φ, t, θ ) coordinates, (which corresponds to taking φ as the radial coordinate instead of η ) as in Einstein gravity, which corresponds to the following change of variables where we have used the reduced EOM for φ in the second equality. Through this transformation, Ψ is represented as which should satisfy the differential equation (52). When Ψ is obtained, metric functions A and B can be given by These expressions for metric functions in ( r, t, θ ) coordinates come from the first order reduced EOM for φ and A + R in (49), through the substitution of ∂/∂η and R by e -B ∂/∂r and ln r . The differential equation for Ψ is a nonlinear inhomogeneous equation. It is not easy to obtain its solution analytically except a trivial case. Therefore, we try to obtain asymptotic series form of solutions in NMG case, which might be sufficiently illuminating for discussion of holographic c-theorem in this case. As alluded in the previous section, the scalar hairy black holes with asymptotic AdS space and with the near horizon AdS space would correspond to the class of black holes which reduce to extremal BTZ black holes even in NMG. Before doing these perturbative analysis, let us consider the cases which allow analytic results. Firstly, as in the Einstein gravity case, domain wall solutions correspond to which allow analytic results and were studied in Ref. [16]. Secondly, as a trivial example, let us check that extremal BTZ black holes are solutions of the above differential equation of Ψ. By taking W = 2, the first order reduced EOM for the scalar field φ , and metric function ˙ A + ˙ R , lead to where /lscript is defined by The second order differential equation for Ψ gives us a constant Ψ as where we have introduced the constant r H as 2 r 2 H ≡ ˜ ∆ 0 [1 + 1 / 2 m 2 L 2 ] -1 . This gives us e 2 R = e 2 η + r 2 H , which corresponds to the well-known extremal BTZ black hole solutions in NMG with the horizon radius r H . To see nontrivial solutions one may try to solve the above equations for a given superpotential. However, one can see that the resulting solution for Ψ is given by a complicated function. Furthermore, it is not easy to obtain analytic form of metric and scalar field in this way. As in Einstein gravity, we perform the perturbative calculation at the asymptotic infinity and on the near horizon. This analysis already reveals important features of black hole solutions and is sufficient to verify that those black holes are extremal ones. Since the methodology is completely identical with Einstein gravity case, we briefly present the intermediate steps. In summary, let us consider the following asymptotic expansions for metric variables, the superpotential and the scalar field: where q denotes It turns out that ˜ a 1 and ˜ b 1 satisfy ˜ a 1 + ˜ b 1 = -˜ ω 1 and ˜ φ 2 1 = 4 q ˜ ω 1 . By using the expansion of Ψ in terms of r as and by solving the Eq. (52) perturbatively in terms of r , one obtains the following One can also obtain through Eq. (55) Note that this takes the form of hairy deformation from BTZ black holes given in the Eq. (57). Masses and angular momenta of these black holes can be obtained by the ADT method. Using the results given in Ref. [23], one can see that which satisfy the extremal condition ML = ± J , as in Einstein gravity (see Ref. [23] for more details about ADT charges in NMG and how these give the above results.). As in Einstein gravity, it is straightforward to argue in NMG that the inequality, W ( φ H ) ≥ W ( φ ∞ ) holds in general from mass inequality between hairy deformed extremal BTZ black holes and hairless ones, M ( hair ) ≥ M ( BTZ ). Now, let us consider the expansions on the near horizon. By doing the perturbative analysis on the near horizon, one can see that By expanding the radial coordinate r and the superpotential W in terms of φ as which is important to see the relation between ˜ ∆ 0 and W ( φ H ), one can also obtain metric functions as, through the perturbative analysis, where ˜ s 0 is a certain non-vanishing constant related to the specific black holes or the interval of integral. As in Einstein gravity, one can show that and can see that the extremality condition (21) is fulfilled. This result shows us that the black holes under the consideration are extremal ones, indeed. We also obtain the same results through the same expansions with original EOM without using the superpotential W (see appendix B.) Through the above analysis on the near horizon geometry of our extremal black holes, one can see that the Bekenstein-Hawking-Wald entropy of these extremal black holes in NMG are given by One can also verify that these are consistent with the first law of black hole thermodynamics trivially as in Einstein gravity. Let us consider the domain walls in this perturbative approach. By taking the asymptotic form of the superpotential W in the same form with that in Einstein gravity as one can obtain asymptotic expansion of various variables in NMG as where ˜ a 1 and ˜ b 1 satisfy q ˜ a 1 + ˜ b 1 = -ω 1 and ˜ φ 2 1 = 4˜ ω 1 . For a generic expansion of Ψ in terms of φ , one can see that all the coefficients of Ψ vanish by solving the Eq. (52) perturbatively, i.e. which corresponds to the domain wall case. This short computation shows us that the asymptotic form of the superpotential for domain wall solutions in NMG should be taken differently from those of black holes and partially explains to us why the reduced EOM for domain walls are different from those for black holes.", "pages": [ 15, 16, 17, 18, 19 ] }, { "title": "4 Holographic C-Theorem", "content": "In this section we consider the holographic c -theorem in the context of extremal AdS black holes in three dimensional Einstein gravity and in NMG. By constructing central charge flow functions holographically, one finds some non-trivial checks of the consistency between central charge expressions and parameters in extremal AdS black hole solutions. A central charge flow function of the boundary field theory dual to Einstein gravity may be introduced as which gives us the central charge values on the asymptotic AdS space and on the near horizon geometry as where we have introduced a IR scale L IR ≡ 2 L/ W ( φ H ). Note that the superpotential value at the horizon, W ( φ H ), is always greater than its asymptotic value W ( φ ∞ ) = 2, which implies that c UV ≥ c IR . Moreover, c UV takes the standard value of two-dimensional boundary field theory dual to AdS 3 with the radius L , which can be obtained by Brown-Henneaux method [18] or the standard AdS/CFT dictionary [25, 26, 27]. Furthermore, in the domain wall limit, Eq. (44), the central charge flow function reduces to the well-known form C ( η ) = 3 L/ 2 G · 1 /A ( η ). In the standard dictionary of the AdS/CFT correspondence, one identifies the IR scale L IR with the near horizon scale ¯ L from the geometry. To verify the consistency of our choice of central charge flow functions, one needs to check that the central charge flow functions reproduce the central charges of dual conformal field theories even at the IR conformal point. Since ¯ L is already determined by black hole parameters and L IR is done by the superpotential values at the horizon, two results should be matched in order for the self-consistency of our construction. As was shown in Eq. (24), the expressions of ¯ L is indeed identical with L IR as To see this explicitly for analytic solutions, one may note that for each ansatz of the radial coordinate r in terms of φ the superpotential values at the horizon, W ( φ H ), are given respectively by which are consistent with the general analysis as can be seen from Eqs.(33), (37) and (41). Since we have verified that our central charge flow functions lead to the correct central charges at conformal end points, we turn to show their monotonic properties. According to the AdS/CFT correspondence, it is well known that the scale in the RG flow of dual field theory corresponds to the radial coordinate in the so-called Fefferman-Graham coordinates in the gravity side. To show the monotonic property of the above central charge function along the RG flow in the dual field theory, one needs to consider the derivative of the above central charge function with respect to the radial coordinate in the FG coordinates. The radial coordinate η introduced in Eq. (42) forms the FG coordinates together with ( θ, t ) in our case. Note that η →∞ corresponds to the asymptotic infinity (or UV) and η → 0 does to the near horizon (or IR). Now, one can see that where we have used the first order equation for ˙ φ which is given in these coordinate as ˙ φ = -∂ φ W . This result means that the central charge is always increased when η becomes increased (or when the energy scale is increased), and this can be regarded as the holographic construction of two dimensional c -theorem beyond the domain wall geometry. Central charge flow function in NMG may be defined as which gives us central charge values at UV and IR as where we have introduced L IR = 2 L/ W ( φ H ) as in the case of Einstein gravity. One may note that, on the contrary to Einstein gravity, c UV and c IR are not proportional to the cosmological constants at the asymptotic infinity and at the near horizon. It is quite useful to check ¯ L = L IR for the whole consistency of our results. Indeed, one can see that this is the case by the asymptotic analysis given in the previous section as was shown in Eq. (64). The monotonic property of the chosen central charge flow functions is anticipated in NMG. Using the first order equation for scalar field given in Eq. (49), one can verify this anticipation as which represents the holographic construction of c -theorem dual to extremal black holes in NMG. Now, we give some comments about the relation between holographic c -theorem and null energy condition on matters. For domain walls, one usually choose a central charge flow function as a function of metric variables and then relate directly the derivative of central charge flow functions, through EOM, to the null energy condition on matters. On the contrary, the connection between the holographic c -theorem and null energy condition is more or less indirect in our case, because the chosen central charge function depends on superpotential not metric variables. By taking null vectors on our geometry in FG coordinates as N ± = ( N t ± , N η ± , N θ ± ) = (1 /L ) ( ± 1 , e -R , -e -A ) , one may check that null energy condition on a scalar field is satisfied as Only after using reduced EOM and not the original EOM, one may relate ˙ C to T µν N µ ± N ν ± . But this is somewhat indirect connection to the null energy condition compared to the case of domain walls, which use only the original EOM. This indicates that explicit solutions of EOM may be needed to verify the holographic c -theorem for more complicated geometry with two asymptotic AdS spaces as the reduced first order EOM are necessary to show those theorems in our black hole cases. In connection with the holographic c -theorem, it is interesting to consider the entropy of boundary dual CFT by Cardy formula. Since the bulk contains two AdS 3 spaces at the asymptotic infinity and at the near horizon, there are two entropy functions, S UV and S IR . One of the natural questions about these entropies is that there is any relation to the Bekenstein-HawkingWald entropy of extremal black holes. The well-known relations between conserved charges in the bulk gravity and energies in the dual CFT at the asymptotic boundary are given by [28] where E L /E R are left/right energy in the dual CFT and related to the so-called left/right temperatures as E L/R = ( π 2 L/ 6) T 2 L/R . The Cardy formula gives us the entropy S UV in terms of these left/right temperatures as Note that c = c L = c R , since we are dealing with the parity even theories. The extremal condition in the black hole side means that one of the left and right energy should vanish in the dual CFT, which can be deduced from the conserved charge relations. For definiteness, let us consider ML = J case. After some computation, one can see that the entropy of dual CFT at the asymptotic boundary is not less than that the Bekenstein-Hawking-Wald entropy of extremal black holes as 5 Note that the inequality between S UV and S BHW is the direct consequence of the inequality between the superpotential values W ( φ H ) ≥ W ( φ ∞ ) = 2, which can be verified explicitly in analytic solutions and was argued to be the case in general through mass inequalities between mass of hairy deformed extremal BTZ black holes and hairless ones. In Einstein gravity, mass and angular momentum at the horizon, ¯ M and ¯ J as quasi-local quantities were computed and shown to satisfy ¯ M ¯ L = ¯ J = J [13] for specific extremal black holes. This computation for quasi-local quantities on the near horizon requires only the near horizon fall-off behavior of various variables, our extremal black holes would satisfy the same relations. One of the very interesting results in this quasi-local computation is that the angular momentum is invariant from the asymptotic infinity to the near horizon, which was also observed in thermodynamic approach to black holes in a finite region [30]. This invariance of angular momentum in Einstein gravity leads to the entropy of dual CFT on the near horizon through Cardy formula as One may interpret this invariance of angular momentum along the bulk radial direction in the dual CFT side as follows. Along the RG flow in dual field theory, the running of central charge is the consequence of the change of the AdS radius, while the bulk gravitational constant, G , or the coupling m 2 of higher curvature terms are invariant. From the bulk perspective in our three-dimensional case, the AdS radius is the unique candidate for the running variable, which is related to the scalar field. Based on dimensional reasoning, one can see that the dual CFT energy E L/R , which are computed from gravity, should scale inversely to the AdS radius without anomalous scaling. Therefore, the angular momentum is invariant with scaling of the AdS radius. By assuming the validity of the angular momentum invariance, one can compute the entropy of dual CFT on the near horizon as It is very interesting to observe that the entropy of CFT on the near horizon is also identical with the Bekenstein-Hawking-Wald entropy of extremal black holes, as S IR = S BHW , even in NMG 6 . In summary for all cases, one can see that the following inequalities always hold 7", "pages": [ 19, 20, 21, 22, 23 ] }, { "title": "5 Conclusion", "content": "We have discovered Bogomol'nyi type of lower order differential equations for Einstein gravity and for NMG, which we called reduced EOM, in the presence of a minimally coupled scalar field. More explicitly, the first order reduced EOM in Einstein gravity is given by Eq. (7) and the reduced EOM in NMG by Eqs. (49) and (52). Using these equations, we have obtained various analytic hairy black hole solutions which include the previously known example as a special case. We also showed that all these solutions are consistent with the holographic c -theorem. The asymptotic space of all our solutions is AdS space with the radius L and the near horizon geometry is the so-called self-dual orbifold of AdS 3 with the radius ¯ L . After showing that the simplest case of our black hole solutions corresponds to the well-known extremal BTZ black holes, the extremality of our black hole solutions are shown explicitly. We also showed that our reduced EOM implies generically the extremality of any solution of these equations under some mild assumptions, the validity of the power series expansion. In Einstein gravity we have presented several analytic solutions with some generic perturbative treatment and in NMG we have done some perturbative solutions. We have also identified various physical quantities of extremal black hole solutions and shown that conserved charges of these black holes are related directly to the integration constant ∆ 0 and ˜ ∆ 0 of the reduced EOM. Motivated from the domain wall case, we have proposed the holographic central charge flow functions and verified that they coincide with central charges at the conformal end points and that they also satisfy the anticipated monotonic properties. We have also performed the consistency checks on the central charge flow functions by showing that the near horizon scale ¯ L , which is read from the geometry, can be identified with the IR scale L IR in the central charge flow functions. There are some future directions to pursue further. First of all, our reduced EOM are just rewritten down in analogy with the domain wall case. It would be very interesting to derive our reduced EOM by complete squaring of a certain energy functional or by using fake supersymmetry. It is notable that the second order original EOM in Einstein gravity reduce to the first order ones while all of the reduced EOM in NMG are not first order ones. This is clearly contrasted to the domain wall case which is also described by the first order ones even in NMG. It would be very interesting to reproduce conserved charges for our extremal black holes consistently in Einstein gravity and NMG by other methods. Another interesting direction is to verify the stability of our extremal black holes directly. Though we showed that all the black hole solutions from our reduced EOM are extremal and argued that they are all stable by the generic thermodynamic consideration, we did not perform any dynamical stability analysis through some perturbations. It is also interesting to prove the inequality W ( φ H ) ≥ W ( φ ∞ ) rigorously for any sensible black hole solutions, which is argued to be the case on physical ground. It would be also interesting to verify or disprove our conjecture about the invariance of angular momentum along RG flow in general. In Einstein gravity, the invariance is rather established in various cases. However, it is not checked explicitly in other gravity theories like NMG. It is also valuable to find more solutions or to study the complete integrability of our first order reduced EOM in Einstein gravity. Another intriguing direction is the study on couplings with more than one scalar field. For instance, it might be possible to obtain vortex black hole solutions when a complex scalar field is coupled. Finally, it would be very interesting to extend the correspondence between DLCQ of two-dimensional field theory and extremal black holes to our hairy cases.", "pages": [ 23, 24 ] }, { "title": "Acknowledgments", "content": "S.N and S.H.Y were supported by the National Research Foundation(NRF) of Korea grant funded by the Korea government(MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number 2005-0049409. S.H.Y was supported by Basic Science Research Program through the NRF of Korea funded by the MEST(2012R1A1A2004410). S.H.Y would like to thank Prof. Won Tae Kim for some useful discussions. J.D.P was supported by a grant from the Kyung Hee University in 2009(KHU-20110060) and Basic Science Research Program through the NRF of Korea funded by the MEST(2012R1A1A2008020). S.N was supported by Basic Science Research Program through the NRF of Korea funded by the MEST(No.2012-0001955). Y.K was supported by the NRF of Korea grant funded by the Korean government(MEST) (NRF-2011-355-C00027). YK would like to thank Dr. Hyojoong Kim for some helpful discussions.", "pages": [ 25 ] }, { "title": "A Equations of motion in NMG for some coordinate systems", "content": "In this section, we present EOM in NMG only since EOM in Einstein gravity can be obtained by taking m 2 L 2 →∞ in the following expressions of EOM in NMG. For our metric ansatz (3) with the relation (11), we present the metric EOM in the following since the scalar EOM is already given in Eq. (5). where ' denotes differentiation with respect to the radial coordinate r . For the metric ansatz (42) with the relation C = A -R , which is written in the so-called FG coordinates, the scalar field equation and EOM in NMG are given by where ˙ denotes differentiation with respect to the new radial coordinate η .", "pages": [ 26, 27 ] }, { "title": "B Perturbative calulation in the original EOM", "content": "In this section, we perform some perturbative calculations in the original EOM, which reveal that full EOM need to be organized in terms of a certain formal expansion parameter /epsilon1 . This formal expansion parameter /epsilon1 is just the book-keeping one to retain the correct order in EOM and obtain consistent results, which should be set to be unity at the end of calculation. Only the relevant orders of /epsilon1 will be represented in the expansion expressions below. Furthermore, we do not use the trick exchanging the role of the radial coordinate and the scalar field φ . All the variable are expanded in terms of the radial coordinate r . Therefore, this calculation may be regarded as the independent check of our results given in the main text. In the following, we will use ( t, r, θ ) coordinate to show our results. At the asymptotic of extremal black holes, we can consider following expansion of various variables in NMG as With the above expansion and the parameter /epsilon1 , the expressions for the scalar field equation and the metric EOM in NMG are obtained as where q ≡ 1 -1 / 2 m 2 L 2 . Through 0 = E φ and 0 = E µν , we obtain the following relation which is consistent with the results from reduced EOM in NMG. However, as we have alluded to in the main text, the reduced EOM can be integrated partially with the integration constant ˜ ∆ 0 , the constants ˜ a 1 and ˜ b 1 can be determined completely in terms of ˜ ∆ 0 and ˜ ω 1 , while that is not possible in the perturbative calculation in the full EOM. On the near horizon, we have the expansion of various variables as With the above expansion, the expressions for scalar field equation and the metric EOM can be written as From 0 = E φ and 0 = E µν , we obtain which is also consistent with the results obtained from the reduced EOM. Note that there is another possibility such that b 2 -16 m 2 L 2 + W ( φ H ) 2 = 0, which has no Einstein limit and seems to indicate the existence of solutions different from those for the reduced EOM.", "pages": [ 27, 28 ] } ]

[ { "title": "Frank F errari", "content": "Service de Physique Th'eorique et Math'ematique Universit'e Libre de Bruxelles and International Solvay Institutes Campus de la Plaine, CP 231, B-1050 Bruxelles, Belgique [email protected] We provide an elementary systematic discussion of single-trace matrix actions and of the group of matrix reparameterization that acts on them. The action of this group yields a generalized notion of gauge invariance which encompasses ordinary diffeomorphism and gauge invariances. We apply the formalism to non-abelian Dbrane actions in arbitrary supergravity backgrounds, providing in particular explicit checks of the consistency of Myers' formulas with supergravity gauge invariances. We also draw interesting consequences for emergent space models based on the study of matrix effective actions. For example, in the case of the AdS 5 × S 5 background, we explain how the standard tensor transformation laws of the supergravity fields under ordinary diffeomorphisms emerge from the D-instanton effective action in this background.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recently, the author has proposed a general strategy to build calculable models of emergent space, based on a slightly modified version of the usual AdS/CFT correspondence [1]: instead of considering the scattering of closed string modes off a large number N of background branes, which yields ordinary gauge invariant correlators from the point of view of the worldvolume theory, one considers the scattering of a fixed number k of probe D-branes. The pre-geometric, microscopic theory on the probe branes in the presence of the background branes can be solved at large N [2]. The result is an effective action S ( X ), expressed in terms of Hermitian matrix variables X i of size k × k , whose fluctuations are suppressed at large N . These matrices can be interpreted as being the emergent matrix target space coordinates for the probe branes embedded in the ten-dimensional supergravity background sourced by the background branes. The action S ( X ) should then correspond to the full nonabelian action for the probe branes in this emergent background. By studying the expansion of S ( X ) around diagonal configurations, the full supergravity background can actually be read off unambiguously, by comparing with the known form of the non-abelian action for D-branes in a general background [3]. For example, if the probe branes are D-instantons in the type IIB theory, the large N action S ( X ) is a single-trace function of the matrices X and its expansion takes the general form The cyclicity of the trace implies that the coefficients c i 1 ··· i n satisfy the cyclicity condition c i 1 ··· i n = c i n i 1 ··· i n -1 . On the other hand, from Myers' formulas [3], we can relate certain combinations of the coefficients c i 1 ··· i n to the supergravity fields [1], where φ , B , G , C 0 , C 2 and C 4 are the dilaton, the Kalb-Ramond two-form, the string frame metric and the Ramond-Ramond form potentials respectively. By matching the coefficients c i 1 ··· i n , computed from the large N solution of the microscopic model for the D-instanton in the presence of N D3-branes, with (1.3), (1.4), (1.5) and (1.6), the full AdS 5 × S 5 background was derived in [1]. For instance, the coefficient c [ ijklm ] allows to find the non-trivial five-form field strength F 5 = d C 4 which, quite remarkably from the point of view of the microscopic model, turns out to be self-dual and normalized consistently with the Dirac-quantized D3-brane charge in string theory. The emergent geometry point of view that we have just outlined raises many questions on the general properties of the matrix action S ( X ). The aim of the present technical note is to address some of these questions, bringing a better understanding of general properties of non-abelian D-brane actions and providing interesting consistency checks of the approach introduced in [1]. A basic set of questions is related to the general form of the expansion (1.2). The coefficients given by the equations (1.3)-(1.6) have rather non-trivial and surprising properties. For example, the coefficients c [ ijk ] and c [ ijklm ] automatically satisfy the constraints One may assume that these properties are accidents of the leading glyph[lscript] 2 s = 2 πα ' approximation used in Myers'. Indeed, formulas (1.3), (1.4), (1.5) and (1.6) are corrected, in general, by higher derivarive terms generated by the small glyph[lscript] 2 s expansion of appropriate disk string diagrams. When evaluated on highly supersymmetric backgrounds like the AdS 5 × S 5 background studied in [1], these corrections do vanish, but they will not on an arbitrary background. It thus came as a surprise to the author when calculations made in rather complicated examples [4, 5, 6], including cases with no supersymmetry at all, were found to yield results consistent with (1.7). This led us to study the most general form of the Taylor expansion of a single-trace matrix function. Even though this is a rather elementary question with a quite useful solution, yielding approximation-independent constraints on the hard-to-compute (see e.g. [3, 7] and references therein) single-trace effective potential in any matrix theory, we have not been able to find a systematic discussion in the literature and we thus provide one in Section 2. In particular, we show that the conditions (1.7) and their generalizations to higher orders must always be valid, for any single-trace function S ( X ). This has an interesting consequence: the formulas (1.4) and (1.6) can be used to define in a very natural way the supergravity p -form fields to all orders in glyph[lscript] 2 s , and even at finite glyph[lscript] 2 s . We also show that many combinations of the coefficients c i 1 ··· i n that are not listed in (1.3)-(1.6) can actually be expressed in terms of (1.3)-(1.6), consistently with the rather complicated-looking form of Myers' formulas for these coefficients (see for example the equation (5.8) in [1]). Another set of interesting questions is related to the physical content of the matrix actions S ( X ). In other words, what are the natural gauge invariances of a general matrix action? The most traditional point of view on this problem is to start from the known diffeomorphism and p -form gauge invariances and try to check and/or impose them on the non-abelian D-brane actions [8, 9, 10]. This is quite non-trivial. For example, in [8] it is clearly explained that the ordinary group of diffeomorphisms cannot act consistently on non-commuting matrix coordinates X i , a result that will follow straightforwardly from our analysis in Section 3. The invariance under the p -form gauge symmetries is also quite involved [9, 10], since it is the p -form potentials, not the gauge invariant field strengths, that enter into the non-abelian D-brane actions. As was shown in [9], by focusing on the Chern-Simons part of the action, consistency requires that the matrix coordinates must transform non-trivially under the p -form gauge symmetries, a rather surprising result rooted in the non-commutative nature of the space-time coordinates. We provide in Sections 4 and 5 explicit checks of the consistency of Myers' action for D-instantons and D-particles with the p -form gauge invariances, taking into account both the Dirac-Born-Infeld and the Chern-Simons parts of the action. Another point of view on the gauge invariances of the action S ( X ), which is most natural in the emerging space framework, is to start with no prejudice and study in which cases two different sets of coefficients ( c i 1 ··· i n ) n ≥ 0 and ( c ' i 1 ··· i n ) n ≥ 0 describe the same physics and thus should be considered to be equivalent. The most general transformation laws one can consider are associated with the most general redefinitions of the matrix variables X i that are consistent with the single-trace property of the action. The set of all these transformations defines a large group G D-geom of gauge transformations, which we may call the gauge group of D-geometry, and which we study in Section 3. This group contains the ordinary diffeomorphisms and supergravity gauge invariances but also more general, background-dependent gauge transformations that are crucial for a proper interpretation of the non-abelian D-brane actions. These general transformations will be derived and discussed in Sections 4 and 5. An example of application is to show, in Section 4, that G D-geom acts on the AdS 5 × S 5 metric and five-form field strength F 5 derived in [1] in a very simple way: the full action of G D-geom corresponds in this case to the usual tensor transformation laws of the metric and F 5 under the group of ordinary diffeomorphisms induced by the action of G D-geom on the set of commuting matrices. This result nicely shows that the tensor properties of the metric and F 5 can also be considered to be emergent properties following from the microscopic description given in [1]. Similar results can be straightforwardly derived for the emerging geometries found in [4, 5, 6]. The plan of the paper is as follows. In Section 2, we study the Taylor expansion of single-trace matrix functions. In Section 3, we define and study the gauge group G D-geom . In Section 4, we derive the G D-geom gauge transformations and apply the results to the case of D-instantons. In Section 5, we briefly present the generalization of the discussion of Sections 2, 3 and 4 to the case of quantum mechanical actions and D-particles. Finally, we have included an Appendix on tensor symmetries in which we review calculational techniques that are heavily used throughout the main text and that we have implemented in Mathematica.", "pages": [ 2, 3, 4 ] }, { "title": "2 The Taylor expansion", "content": "Let us start with a Definition: A function S ( X ) of k × k Hermitian matrices X 1 , . . . , X d is said to be single-trace if its expansion around an arbitrary diagonal configuration takes the form for a set of cyclic coefficients A similar definition can be given for single-trace actions S ( X ) depending of matrixvalued fields X 1 , . . . X d . Note that the cyclicity conditions on the coefficients can always be imposed, without loss of generality, from the cyclicity property of the trace. All tree-level open string effective actions are single-trace, because they can be computed from disk diagrams for which the contraction of Chan-Paton factors automatically yields a single-trace structure. In this Section, we are going to discuss the most general consistent form of the Taylor expansion (2.2), for an arbitrary single-trace function (or potential) S . The generalization to higher dimensional action is straightforward, see e.g. the discussion in Section 5. The fact that the coefficients c i 1 ··· i n cannot be chosen arbitrarily is already clear in the trivial abelian case k = 1, for which (2.2) is the usual Taylor expansion of a function of d commuting real variables. All the higher order coefficients c i 1 ··· i n , n ≥ 1, are then fixed in terms of the zeroth order coefficient as c i 1 ··· i n = ∂ i 1 ··· i n c .", "pages": [ 5 ] }, { "title": "2.1 The consistency conditions", "content": "The fundamental consistency condition on the expansion (2.2) is the invariance of the action under the simultaneous shifts that leave X i = x i I + glyph[epsilon1] i unchanged. This condition ensures, at least formally, that the expansions (2.2) around arbitrary points x all define the same function S , independently of the points x around which we expand. Let us emphasize that the symmetry under the shifts (2.4) is a completely general consistency requirement and does not assume the existence of additional structures, like a metric, or even a notion of diffeomorphism invariance, on the manifold spanned by the coordinates x . A very similar notion of base-point independence was used in [8], assuming the existence of a metric and diffeomorphism invariance, in order to ensure the consistency of an expansion in Riemann normal coordinates around arbitrary points. The invariance of the action under arbitrary finite shifts (2.4) follows from its invariance under infinitesimal shifts. Taking into account (2.3), the invariance under infinitesimal shifts is equivalent to the conditions It is convenient to rewrite these equations in the language explained in the Appendix. Let us introduce the idempotent, belonging to the group algebra of S n +1 , defined by where Z ' n is the cyclic subgroup of S n +1 generated by the cycle (2 · · · n + 1). This idempotent acts on tensors 1 of rank n +1 and the conditions (2.6) are simply", "pages": [ 5, 6 ] }, { "title": "2.2 Simple consequences", "content": "Let us symmetrize (2.6) with respect to the indices i 1 , . . . , i n . This yields This equation can be further simplified because for a tensor satisfying (2.3). Equation (2.9) thus implies which can be easily solved recursively to yield This simple result is the non-abelian version of the usual Taylor expansion for functions of commuting variables. Let us now assume that n is odd and consider the completely antisymmetric combination of (2.6) in the n +1 indices i, i 1 , . . . , i n . Since the cyclic permutations of i 1 , . . . , i n are even for odd n , we get We can now use the cyclicity of c ii 1 ··· i n in its n +1 indices and the fact that n +1 is even to conclude that the right-hand side of (2.13) vanishes and thus These constraints generalize the conditions (1.7) indicated in the Introduction. In the differential form notation, they are equivalent to That the coefficients should satisfy such differential constraints is an interesting and quite unexpected feature of the general non-abelian Taylor expansion (2.2).", "pages": [ 6, 7 ] }, { "title": "2.3 General analysis", "content": "To work out the most general consequences of (2.8), it is convenient to decompose the cyclic tensors c i 1 ··· i n into irreducible components (see the Appendix for a general discussion of this method). Since the cyclic tensors c i 1 ··· i n are characterized by the constraint where is the idempotent associated with the cyclic subgroup Z n ⊂ S n generated by the cycle (12 · · · n ), this amounts to decomposing j Z n as a sum of primitive idempotents, One can then write where the irreducible pieces are given by Solving the constraints (2.8) is equivalent to deriving their consequences on the irreducible tensors c ( j a ) i 1 ··· i n . For example, the decomposition (2.20) always includes an irreducible completely symmetric tensor and, for odd n , an irreducible completely antisymmetric tensor. It is not difficult to show that (2.12) and (2.14) yield the most general constraints on these tensors that one can derive from (2.8) (that there cannot be any other constraint can be deduced from an ansatz like the one presented in 2.5). More generally, one can apply suitable primitive idempotents to (2.8) to isolate irreducible terms in the decomposition of c i 1 ··· i n +1 and express them in terms of the derivatives ∂ j 1 c j 2 ··· j n +1 of the lowest order coefficients. This is what we have done to derive (2.11) or (2.12). However, it is important to realize that some irreducible pieces in c i 1 ··· i n +1 do not appear on the right-hand side of (2.8), because they are projected out by j Z ' n . For this reason, the non-abelian Taylor expansion can contain new independent coefficients at various orders, which are not expressed in terms of the lower order coefficients. Moreover, one can note that the left-hand side of (2.8) is only constrained by the cyclicity in the indices i 1 , . . . , i n , or in other words belongs to the image of j Z ' n , whereas the right-hand side of (2.8) belongs to the a priori smaller image of j Z ' n j Z n +1 . In other words, the right-hand side of (2.6) contains a priori less irreducible pieces than the left-hand side. This implies the vanishing of the extra irreducible pieces of the left-hand side, which yields non-trivial differential constraints on the coefficients. The most salient examples of these constraints are the conditions (2.14). The other differential constraints that we have obtained in this way turn out to be consequences of the conditions discussed in the previous paragraph.", "pages": [ 7, 8 ] }, { "title": "2.4 The solution order by order", "content": "Let us now present the general solution of (2.8) along the lines explained in the previous subsection. We shall expand where S n is the action at order n , and work order by order up to n = 5. The calculations are tractable thank's to the techniques explained in the Appendix and their implementation in Mathematica. First and second order At first order we obtain (2.5). At second order, we get The right-hand side is symmetric and thus we get the first example of a differential constraint, ∂ [ i c j ] = 0. This is of course not a new constraint but a consequence of the first order constraint (2.5).", "pages": [ 8, 9 ] }, { "title": "Third order", "content": "The condition α ijk = α kji implies that α [ ijk ] = 0 and thus we have a decomposition in irreducible tensors of the form The coefficient α ( j 3 ) ijk = α ( ijk ) is fixed by (3.15), whereas is left unconstrained. If we define then one can show that the irreducible tensor u is characterized by the following symmetry properties, On the other hand, the condition β ijk = -β kji allows two irreducible components for the imaginary part coefficients. The completely antisymmetric irreducible piece β [ ijk ] is fixed by (3.18), and the other component, given by is fixed by the consistency condition (3.14) at n = 3 to be Note that we are using the same convenient notation as in Section 2 for the primitive idempotents, which are denoted according to the shape of the associated Young tableau. However, this notation is ambiguous, since two distinct idempotents can be associated with the same Young tableau. For example, the idempotent j 2 , 1 in (3.26) is clearly not the same as the one in (3.22), as the formulas for α ( j 2 , 1 ) and β ( j 2 , 1 ) show, but we are using the same notation because they are both associated with the same tableau .", "pages": [ 19, 20 ] }, { "title": "Fourth order", "content": "The condition α ijkl = α lkji allow six irreducible pieces, The consistency conditions fix four of the six irreducible tensors as The tensors α ( j 2 , 2 ) ijkl and α [ ijkl ] are unconstrained. If we define then one can show that the irreducible tensor r has precisely the same symmetries as the Riemann curvature tensor, Moreover, the four-form is simply the unconstrained A (4) in the notation (3.19). For the imaginary part coefficients, the condition β ijkl = -β lkji let four irreducible tensors, +2 β iljk + β ilkj +4 β jikl +3 β jilk +2 β jkil +2 β kijl + β kjil ) two of which are fixed by the consistency conditions, The unconstrained pieces β ( j 3 , 1 ) and β ( j 2 , 1 , 1 ) can be most easily described in terms of irreducible tensors s and t which are characterized by the following symmetry properties: One has where or equivalently, whereas For illustrative purposes, we have provided some details on the derivation of equations (3.41), (3.42) and (3.45) in the Appendix, Section A.3.", "pages": [ 20, 21, 22 ] }, { "title": "Fifth order", "content": "At fifth order, the cyclic coefficient has six irreducible pieces, according to the decomposition in which the Young tableau occurs twice. Acting on the cyclic coefficients with the primitive idempotents appearing in (2.34), we obtain the following explicit expressions for the irreducible tensors, The right-hand side of (2.38) contains eight additional terms obtained by circular permutations of the indices ijklm of the two terms that we have explicitly written down. It turns out that equation (2.8) for n = 4 fixes c ( j 5 ) as in (2.12) as well as c ( j 2 , 2 , 1 ), c ( j 3 , 2 ) and c ( j ' 3 , 1 , 1 ) in terms of the lower order coefficients. Explicitly, one finds The tensors c ( j 3 , 1 , 1 ) and c ( j 1 , 1 , 1 , 1 , 1 ) are the new tensors appearing at order five; the consistency conditions (2.8) do not relate them to lower order coefficients. Moreover, one can check explicitly that the differential constraints on the fifth order coefficients implied by (2.8) for n = 5 are all consequences of (2.12), (2.41) and (2.43), except for the condition (2.14) on c [ ijklm ] .", "pages": [ 10, 11 ] }, { "title": "Sixth and higher orders", "content": "At sixth order, we get twenty irreducible tensors, fourteen of which are fixed by (2.8) and six are new. And so on and so forth at higher and higher orders. It is possible to obtain explicit formulas using the computer, but they are complicated and not particularly useful. In particular, for most string theory applications, the knowledge of the action up to order five is sufficient, since it allows to derive the full supergravity background [1, 4], see Section 2.8. We now turn to a slightly less rigorous but possibly more illuminating discussion, based on a general ansatz for the solution of (2.8).", "pages": [ 11 ] }, { "title": "2.5 A convenient ansatz for the solution", "content": "It is actually very easy to write solutions to the conditions (2.8), based on the following two very simple remarks: is automatically invariant under the transformation (2.4). In particular, we can consider that it defines a function ˆ f ( X 1 , . . . , X d ) of the matrices X 1 , . . . , X d , with value in the space of Hermitian matrices. The most general solution of (2.8), up to order five, can then be reproduced by the following simple ansatz, which manifestly satisfies the consistency conditions of Section 2.1, The symmetry properties of the commutators in (2.45) allow us to constrain the coefficients s ij , s ijkl and s ijklm as By construction, the expansion of (2.45) in powers of glyph[epsilon1] , using (2.44), yields coefficients c i 1 ··· i n that automatically satisfy (2.8) and thus all the relations discussed in the previous subsection. Up to order two, only ˆ s contributes, with c = s , c i = ∂ i s , c ij = ∂ ij s . At order three, on top of ˆ s which yields the completely symmetric coefficient c ( ijk ) = ∂ ijk s , the term in ˆ s ij yields This formula automatically implements the differential constraint (2.14), the threeform being expressed as F (3) = 4d s (2) if s (2) is the two-form with components s [ ij ] . The fourth order action derived from (2.45) reads Using the cyclicity of the trace, it is easy to check that the tensor ∂ ij s kl actually enters only via the components ∂ i [ j s kl ] , consistently with (2.49), (2.33) and (2.28). Taking into account (2.47), the tensor s ijkl a priori contains three new irreducible components, but at order four only the cyclic combination appears, which yields the unique irreducible piece consistently with the discussion in 2.4. The fifth order action reads One can check that only the components of the form ∂ ij [ k s lm ] of ∂ ijk s lm contribute, consistently with (2.43) and (2.49). The term in ∂ i s jklm yields c ( j 2 , 2 , 1 ± j 3 , 2 ), consistently with (2.41), (2.42), (2.51) and the last equation in (2.30). It also yields the new irreducible tensor in a form that manifestly satisfies the constraint (2.14). Finally, the new tensor c ( j 3 , 1 , 1 ) picks contributions from ∂ ijk s lm , ∂ i s jklm and s ijklm ,", "pages": [ 11, 12, 13 ] }, { "title": "2.6 Reality condition", "content": "In some cases, for example if S ( X ) is the effective potential in a Minkowskian worldvolume action, it may be natural to impose a reality condition which is equivalent to on the coefficients. If we denote by σ the permutation such that σ ( k ) = n -k +1, it is not difficult to check that σj Z n = j Z n σ . The group algrebra elements are then Hermitian idempotents corresponding to orthogonal projectors on orthogonal subspaces. The decomposition (2.19) can be written as with σj ± a = ± j ± a . The associated irreducible tensors in (2.20) can thus be chosen such that The reality condition (2.56) is then equivalent to imposing that the irreducible tensors c ( j + a ) and c ( j -a ) are real and purely imaginary respectively. Let us note that the reality constraints obtained in this way are automatically consistent with the consistency conditions (2.8), since the invariance of S ( X ) under the shifts (2.4) implies the invariance of S ( X ) ∗ under the same shifts. In the order by order analysis performed in Section 2.4, one finds that the irreducible tensors are all real up to order two, together with c ( j 3 ), c ( j 4 ), c ( j 2 , 2 ), c ( j 5 ), c ( j 2 , 2 , 1 ), c ( j 3 , 2 ) and c ( j 1 , 1 , 1 , 1 , 1 ), whereas c ( j 1 , 1 , 1 ), c ( j 2 , 1 , 1 ), c ( j 3 , 1 , 1 ) and c ( j ' 3 , 1 , 1 ) are purely imaginary.", "pages": [ 13 ] }, { "title": "2.7 Summary of results", "content": "Through its Taylor expansion (1.2), a single-trace function of matrices S ( X ) can be characterized by an infinite set of irreducible tensors These tensors correspond to the irreducible pieces of the coefficients in the expansion that are not expressed as derivatives of lower order coefficients by the consistency conditions (2.8). Moreover, the completely antisymmetric tensors of odd order appearing in (2.60) must satisfy (2.14), i.e. they are associated with closed differential forms. If the function S ( X ) is real, the irreducible tensors must be real or purely imaginary according to their parity in (2.59).", "pages": [ 14 ] }, { "title": "2.8 The example of Myers D-instanton action", "content": "A very natural example of a single-trace matrix action like our S ( X ) is the effective action for D-instantons in type IIB string theory. Each Hermitian matrix X i is associated in this case with a Euclidean spacetime dimension and thus d = 10. The size k of the matrices is identified with the number of D-intantons. The single-trace structure of the action is a consequence of the small string coupling approximation, in which the action can be computed from open string diagrams having only one boundary. In this context, the tensors in (2.60) are naturally identified with the non-trivial closed string background fields in which the D-instantons can move. Myers [3] has proposed a general formula for S ( X ), using in particular constraints from T-duality. Myers' action is the sum of Dirac-Born-Infeld and Chern-Simons terms, The Dirac-Born-Infeld part is given by where φ , G ij and B ij are the usual dilaton, metric and Kalb-Ramond two-form of the Neveu-Schwarz sector and glyph[lscript] s is the string length. The fields are evaluated at X = x + glyph[epsilon1] . The determinant acts on the indices i, j (not on the U( k ) indices of the matrices glyph[epsilon1] ). The Str is an appropriate symmetrized trace on the U( k ) indices whose precise definition is given in [3] (and which should provide the correct ordering for the action up to order five in the expansion (1.2), but not beyond). The Chern-Simons part of the action is given by where we keep only the 0-form part in the right-hand side, the C 2 q are the RamondRamond forms and i glyph[epsilon1] the inner product. It is straightforward to check that, up to order five, (2.62) and (2.63) yield an action of the form (2.45), with The independent irreducible tensors entering into the expansion of S follow. We immediately get At order four, only the cyclic combination of the s ijkl enters, which eliminates the BB and antisymmetric terms in (2.66), yielding or equivalently from (2.30) At order five, the vanishing of s ijklm implies that there is only one new independent irreducible tensor at this order (instead of two for a generic matrix action), given by (2.53) as The would-be new independent tensor c ( j 3 , 1 , 1 ) is expressed in the present case in terms of lower order coefficients according to (2.54) for s ijklm = 0. Let us note that (2.68), (2.69) and (2.71) show that the full set of supergravity fields is encoded into the irreducible tensors appearing in the expansion (1.2) up to order five. The formulas (2.68), (2.70) and (2.71), derived from the Myers' action, correspond to the first terms in an infinite derivative expansion in powers of glyph[lscript] 2 s . This derivative expansion can in principle be obtained by computing open string disk diagrams. Terms of order n are generated when n open string vertex operators are inserted on the boundary of the disk, together with closed string vertex operators in the bulk. These calculations are of course extremely difficult, in particular at high orders, and only a few examples can be found in the literature, e.g. in [11]. However, the consistency conditions that we have studied above are exact and thus constrain the form of the action to all orders in glyph[lscript] 2 s . For example, from Myers', we find that the differential forms F (1) , F (3) and F (5) defined in (2.15) are given by The fact that these forms are locally exact is not an accident of the leading glyph[lscript] 2 s expansion, but instead a consequence of the general consistency conditions (2.14). This property will thus remain valid to all orders in glyph[lscript] 2 s and even at finite glyph[lscript] 2 s . Explicit examples are worked out in [4]. We could then use (2.72) to actually define what we mean by τ , B , C 2 and C 4 to all orders in glyph[lscript] 2 s . The general gauge transformations of the p -form potentials defined in this way will be discussed in Section 4. On the other hand, there are features of the Myers' action than will not remain valid to all orders in glyph[lscript] 2 s , because they are not protected by the general consistency conditions. For example, the fourth order coefficient c [ ij ][ kl ] factorizes in terms of a second rank metric tensor in (2.70). However, we have seen that the only general constraint on this coefficient is that it should have the same symmetries as the Riemann curvature tensor, including (2.31). This is not enough to ensure the existence of a factorized formula like (2.70). Such a special form for c [ ij ][ kl ] will thus be preserved only in exceptional situations, probably only when the glyph[lscript] 2 s corrections vanish, which occurs for the maximally supersymmetric AdS 5 × S 5 background [1].", "pages": [ 14, 15, 16 ] }, { "title": "3 The gauge group", "content": "In the abelian case, k = 1, two actions S and S ' related by a simple reparameterization of the spacetime coordinates, i.e. such that S ' ( x ' ) = S ( x ) for a diffeomorphism x ↦→ x ' , should of course be considered to be physically equivalent. In the non-abelian case, the d coordinates x i are promoted to k × k Hermitian matrices X i . One may then be tempted to consider the group of diffeomorphisms acting on the dk 2 real independent entries of the matrix coordinates X i . However, this huge group of transformations is not really interesting. It is much more fruitful to take into account basic properties of our matrix actions, which emerge naturally from string theory. First, there is a gauge group U( k ) acting on the matrices as X ↦→ UXU -1 . This gauge group is automatically present in the microscopic open string description. We wish to restrict the allowed transformations X ↦→ X ' to be compatible with the U( k ) action. Second, at small g s (which, in the microscopic gauge-theoretic description, corresponds to a large N limit), the effective actions are automatically single-trace. It is thus also very natural to restrict ourselves to transformations that respect the single-trace structure. These considerations yield the following definitions. Definition 1 : The quantum gauge group G of D-geometry is the subgroup of the group of diffeomorphisms acting on the dk 2 independent real entries of the matrix coordinates X i such that, for any F ∈ G , there exists a U ( k ) automorphism ρ such that for any U ∈ U ( k ) . The simple transformations for U 0 ∈ U( k ), belong to G , with associated (inner) automorphism ρ ( U ) = U 0 UU -1 0 . Another simple transformation belonging to G is complex conjugation, with associated automorphism ρ ( U ) = U ∗ . Since complex conjugation is actually the only outer automorphism of U( k ), we see that G is generated by (3.2), (3.3) and the transformations satisfying the simple constraint Definition 2 : The classical gauge group of D-geometry G D-geom is the subgroup of G preserving the single-trace structure of a matrix action, i.e., it corresponds to the transformations X ↦→ X ' such that, if S is single-trace, then S ' defined by S ' ( X ' ) = S ( X ) is also single-trace. Let us note that (3.2) and (3.3) belong to G D-geom and thus we can restrict ourselves without loss of generality to the transformations of G D-geom satisfying the simpler condition (3.4). Our aim in the present Section is to provide an explicit description of G D-geom and discuss some of its elementary structural properties.", "pages": [ 16, 17 ] }, { "title": "3.1 The consistency conditions", "content": "A transformation γ ∈ G D-geom satisfying (3.4) can be described by the set of coefficients γ i i 1 ··· i n that appear in the expansion Let us note that this is the most general form of the expansion that is compatible with both (3.4) and the single-trace restriction. In particular, if the expansion contained explicit traces, then multi-trace terms would be produced when acting on a singletrace matrix action, which is forbidden. The hermiticity of the matrix coordinates X i imply that the γ i i 1 ··· i n must satisfy a reality constraint Introducing the real and imaginary parts of the coefficients, this is equivalent to the conditions The most general consistent expansion (3.5) for γ ∈ G D-geom can be found by a rather straightforward generalization of the approach used in Section 2 to characterize the non-abelian Taylor expansions of single-trace functions. The fundamental consistency condition on the expansion (3.5) is the invariance under the shifts (2.4), which yields These equations are most conveniently written as in terms of the element of the group algebra C [S n ]. By taking the real and imaginary parts, we get Equations (3.13), (3.14) and (3.8), (3.9) are the analogues of the constraints (2.8) and (2.3) used in Section 2. Since the upper index in α , β or γ does not play any rˆole in the consistency conditions (3.8), (3.9), (3.13) and (3.14), we are going to suppress it in the following subsections in order to simplify the notation.", "pages": [ 17, 18 ] }, { "title": "3.2 Simple consequences", "content": "Let us symmetrize (3.13) and (3.14) with respect to the lower indices. We get which are the analogues of (2.12). Similarly, by antisymmetrizing with respect to the lower indices, we obtain the relations on the forms built from the totally antisymmetric coefficients. Equations (3.17) and (3.18) are the analogues of (2.16).", "pages": [ 19 ] }, { "title": "3.3 The solution order by order", "content": "Let us present the general solution up to order four. The derivations follow the same lines as in Section 2 and are based on the principles outlined in the Appendix. More details on an illustrative example can be found in Section A.3.", "pages": [ 19 ] }, { "title": "First and second order", "content": "For the real part coefficients, all the constraints at orders one and two are included in (3.15) and the conditions A (1) = d A (0) and A (2) = 0 from (3.17), which yield On the other hand, the zeroth and first order imaginary part coefficients vanish, β = β i = 0. At order two, since β ( ij ) = 0 we get one unconstrained irreducible tensor β [ ij ] corresponding to the form B (2) in (3.19). To summarize, up to order two, γ ∈ G D-geom is parametrized by an ordinary diffeomorphism α and a two-form b = B (2) , b ij = β [ ij ] = β ( j 1 , 1 ) ij .", "pages": [ 19 ] }, { "title": "3.4 Summary", "content": "Up to the transformations (3.2) and (3.3), an element γ of G D-geom can be parametrized by an ordinary diffeomorphism α and an infinite set of irreducible tensors that characterize the expansion (3.5), when the consistency conditions (3.6) and (3.10) are taken into account. Up to order four, this data can be conveniently described in terms of a set of tensors b, u, a, r, s, t ; b is a two-form, a is a four-form and the symmetry properties of u , r , s and t are given by (3.24), (3.39), (3.47) and (3.48). Introducing again the upper index, we can thus represent γ ∈ G D-geom as The explicit transformation associated with γ is where { A, B } = AB + BA and the · · · represent terms of higher order.", "pages": [ 22 ] }, { "title": "3.5 The composition law and the inverse element", "content": "Equipped with the explicit description of the elements γ ∈ G D-geom in terms of irreducible tensors, we can work out formulas for the composition law and the inverse element in terms of these tensors. The composition law γ ' · γ is straightforwardly obtained from the expansion (3.5). If X ' i = γ i ( X ), X '' i = γ ' i ( X ' ) and we find The simplest example corresponds to linear transformations γ L , for which ( γ L ) i j = L i j and ( γ L ) i i 1 ··· i n = 0 for n ≥ 2. Equation (3.57) then yields For more general transformations, it is important to realize that (3.57) contains a lot of redundant information. Indeed, the coefficients ( γ ' · γ ) i i 1 ··· i n automatically satisfy all the consistency conditions discussed in Section 3.1 if the coefficients γ i i 1 ··· i n and γ ' i i 1 ··· i n do. For example, the completely symmetrized version of (3.57) is simply equivalent to the standard composition law for multiple partial derivatives, consistently with (3.15), and thus contain no information beyond the fact that ( γ ' · γ ) i ( x ) = γ ' i ( γ ( x )). To obtain the non-trivial information coded in (3.57), we can act with suitable idempotents to isolate the irreducible tensors b , u , etc, in (3.54). Denoting γ ≡ ( α = γ, b [ γ ] , u [ γ ] , . . . ), x ' = γ ( x ), x '' = γ ' ( x ' ), γ p i = ∂ i γ p = ∂x ' p /∂x i , γ p ij = ∂ ij γ p = ∂ 2 x ' p /∂x i ∂x j , γ ' m p = ∂x '' m /∂x ' p , etc, we find, for example, and Formulas for the higher rank tensors are easy to obtain but they are complicated and not particularly illuminating. In the following, we shall only need (3.60) and (3.61). The inverse element γ -1 ≡ ( γ -1 , ¯ b, ¯ u, . . . ) can be computed from (3.57) by imposing ( γ -1 · γ ) i i 1 ··· i n = 0 for n ≥ 2. If x ' = γ ( x ), γ p i = ∂x ' p /∂x i , ¯ γ i p = ∂x i /∂x ' p , we get for example and", "pages": [ 23, 24 ] }, { "title": "3.6 The Lie algebra and the adjoint representation", "content": "It is useful to first briefly review the case k = 1 of ordinary diffeomorphisms. An infinitesimal diffeomorphism γ can be written as where ξ i is the infinitesinal generator. If we change the coordinate system from x to x ' , the same infinitesimal diffeomorphism γ will be expressed as with This shows that the Lie algebra of the diffeomorphism group is identified with the set of vector fields. By definition, the Lie bracket [ ξ 1 , ξ 2 ] between two generators ξ 1 and ξ 2 of infinitesimal diffeomorphisms γ 1 and γ 2 is the generator of the infinitesimal diffeomorphism γ 2 · γ 1 · γ -1 2 · γ -1 1 . A simple calculation then shows that where L ξ is the usual Lie derivative with respect to the vector field ξ . In particular, since [ ξ 1 , ξ 2 ] is by construction an infinitesimal generator, we know from (3.66) that it must transform as a vector field. This simple remark provides a calculation-free proof of the well-know fact that the Lie derivative (3.67) of a vector field is indeed a vector field. The transformation from (3.64) to (3.65) can be given a slightly different interpretation. Instead of considering a coordinate change, which is a passive transformation in the sense that it does not act on the points of the base manifold and does not change the diffeomorphism γ , we can consider the adjoint action of the diffeomorphism group on itself, γ ' = Γ · γ · Γ -1 for any diffeomorphism Γ. With this interpretation, the coordinate change is replaced by the active diffeomorphism Γ, with x ' i = Γ i ( x ). The diffeomorphisms γ and γ ' are then distinct and the formula (3.66) no longer relates the component of the same vector field in two coordinate systems but instead maps one vector field ξ , the generator of γ , to another vector field ξ ' = Γ ∗ ξ , the generator of γ ' . Of course, the two interpretations, active or passive, are equally valid. Let us now see how the above standard results generalize to the non-commutative case k > 1. An element γ of G D-geom is characterized by an ordinary diffeomorphism and by an infinite set of irreducible tensors ( b, u, a, r, s, t, . . . ), as in (3.54). An infinitesimal transformation will thus be parameterized by an infinitesimal vector field ξ together with infinitesimal tensors ( b , u , a , r , s , t , . . . ). In other words, an arbitrary element Ξ of the Lie algebra G D-geom is identified with a set of tensors, which have exactly the same symmetry properties as the corresponding tensors parametrizing the elements of G D-geom themselves. The adjoint action of G D-geom on itself, or on G D-geom , can be computed straightforwardly. For the simplest linear GL( d, R ) transformations (3.58), equation (3.59) implies that where L i k L k j = δ i j . This shows that the coefficients γ i i 1 ··· i n transform as tensors under GL( d, R ). Of course, the same is true for the irreducible pieces ( b, u, a, r, s, t, . . . ) in (3.54) or the ( b , u , a , r , s , t , . . . ) in (3.68). This property actually justifies the use of the terminology 'tensor' for these objects. The transformation laws under a general G D-geom transformation are much more complicated and interesting than simple tensor transformation laws. For example, the action of Γ ≡ (Γ i ( x ) = x ' i ( x ) , B ( x ) , . . . ) on (3.68) yields (3.66) and The first line in (3.70) is the standard tensor transformation law, whereas the second line represents a new term given in terms of a sort of bi-local Lie derivative of the tensor B . The fact that such bi-local terms enter is natural, since the transformation Γ really links the points x and x ' = Γ( x ), with the upper indices on the various tensors parametrizing Γ being associated with x ' and the lower indices being associated with x . This bi-locality is actually already visible in the tensor term, which involves both ∂x ' /∂x , which is naturally evaluated at x , and ∂x/∂x ' , which is naturally evaluated at x ' . Similar transformation laws can be straightforwardly derived for u and the other tensors. More interesting is the computation of the Lie algebra. The Lie algebra is automatically equipped with a bracket which provides a non-commutative, k > 1, generalization of the Lie derivative (3.67). If we find and more and more complicated formulas for the higher tensors. glyph[negationslash] A particularly interesting property of the generalized Lie bracket is its covariance with respect to the adjoint action, which generalizes in a rather non-trivial way the covariance of the ordinary Lie derivative. For example, if b 1 and b 2 transform as in (3.70), then b 3 given by (3.73) must also transform in the same way. If B = 0, this is the usual notion of covariance, which is manifest in formula (3.73) from the covariance of the Lie derivative. When B = 0, we obtain a non-trivial generalization of the notion of covariance, which can of course be checked explicitly by plugging the transformation laws (3.66) and (3.70) on the right-hand side of (3.73).", "pages": [ 24, 25, 26 ] }, { "title": "3.7 On the lift of ordinary diffeomorphisms", "content": "Let us now give a simple proof of an interesting result pointed out in [8]. First, equation (3.74) has an interesting consequence. Lemma : The set of elements of G D-geom of the form where f is an ordinary diffeomorphism, does not form a subgroup of G D-geom if k ≥ 2 . Proof : Let us first note that the transformation (3.75) does satisfy all the consistency conditions of Section 3.1 and thus does belong to G D-geom . In the representation (3.54), it has α i = f i and all the tensors b , u , etc, set to zero. However, the commutator glyph[negationslash] of two such transformations will have u = 0, because the terms in the third line of (3.74) are non-zero even when b 1 = b 2 = u 1 = u 2 = 0 (the other tensors do not enter in the formula for u 3 ). The same result can be obtained from the composition law (3.61), which shows that the product of two transformations of the form (3.75) will have u = 0. So we see that the simplest representation (3.75) of the usual group of diffeomorphism in the larger group G D-geom is actually inconsistent. With the machinery we have developed, it is actually very simple to prove the much more general result mentioned in [8]. Definition : A lift of the group of ordinary diffeomorphisms Diff into G D-geom is a group morphism Φ : Diff → G D-geom such that Φ( f ) ≡ ( f i , . . . ) . Theorem : There is no lift of Diff into G D-geom for k ≥ 2. Proof : The simplest proof is obtained by working at the level of the Lie algebra. Let us assume that a lift Φ does exist. If ξ is the generator of f , then the generator Ξ of Φ( f ) must be of the form (3.68), with the tensors b , u , etc, depending linearly on ξ . Equivalently, the infinitesimal coefficients γ i i 1 ··· i n , n ≥ 2, characterizing Φ( f ) must depend linearly on ξ . This linear dependence is strongly constrained by the tensorial transformation law (3.69) under the action of GL( d, R ): γ i i 1 ··· i n , for i ≥ 2, must be proportional to ∂ i 1 ··· i n ξ i . In particular, it must be completely symmetric in its lower indices. The consistency conditions (3.15) and (3.16) then imply that Φ( f ) must be a transformation of the form (3.75). We deduce from the lemma that Φ(Diff) is not a subgroup of G D-geom , which contradicts the fact that Φ is a group morphism. We conclude that the lift Φ cannot exist. A direct consequence of the above theorem is that there is no action of the group of diffeomorphisms on the space of matrix coordinates X i that respects the U ( k ) gauge symmetry, the single-trace structure and acts in the usual way on the diagonal matrices. This result might superficially suggest that there is an inconsistency with diffeomorphism invariance in string theory, but of course this is not so as we explain in the next Section.", "pages": [ 26, 27 ] }, { "title": "4.1 Closed strings gauge symmetries versus emergent gauge symmetries", "content": "The apparent paradox discussed in 3.7 can actually be solved in two different ways. One way of thinking is to assume the a priori existence of additional structures on top of the matrix coordinates X i . This is quite natural in the traditional point of view on string theory, where the closed string modes are on an equal footing glyph[negationslash] with the open string modes. The space of physical variables on which the group of diffeomorphisms has to act is thus no longer the space of matrices X i alone, but a bigger space including the X i alongside with all the supergravity fields, which we denote collectively by Σ. On this space of fields acts the full gauge group G SUGRA of supergravity gauge invariances, which includes the p -form gauge invariances on top of the diffeomorphisms. If f ∈ G SUGRA , let us denote the action by f · Σ = Σ f (Σ). It satisfies the consistency condition From the results on Section 3.7, we know that G SUGRA does not act on the space of the X s alone. However, this does not prevent us to define an action on ( X, Σ) of the form The crucial difference with the case discussed in 3.7 is that the transformation X f of the matrix coordinates is background dependent through its explicit dependence on Σ. The condition for a consistent group action now reads together with (4.1), and these conditions can a priori be solved. Indeed, this point of view was advocated in [8] and an explicit solution of (4.3) was constructed up to order four for f ∈ Diff ⊂ G SUGRA . The transformation X f built in [8] depends on an arbitrary background metric g . It can be expanded as in (3.5), with coefficients γ i i 1 ··· i n depending explicitly on g . This expansion must satisfy the conditions explained in 3.1 and thus can be parameterized as in (3.54), with α i = f i and tensors b , u , etc, depending on g . For example, the solution of [8] yields where the Γ k ij are the Christoffel symbols for the background metric g . One can easily check that the above definition of u is consistent with the constraints (3.24). If x ' = f 1 ( x ) and x '' = f 2 ( x ' ), the composition law (3.61) shows that the consistency condition (4.3) is equivalent to This equality can be straightforwardly checked from (4.4) and the well-known transformation properties of the metric and the Christoffel symbol under the action of f 1 ∈ Diff. It is plausible that the conditions (4.3) can be solved to all orders using appropriate formulas for a [ f, g ], r [ f, g ], s [ f, g ], t [ f, g ] and the higher tensors in (3.54). Unfortunately, the solutions to the consistency conditions will not be unique [8]. In particular, the metric g is arbitrary and is not clearly identified in terms of the supergravity fields (for instance, it could be the string frame metric, or the Einstein frame metric, or the metric seen by some particular D-brane, etc...). Moreover, there is no reason for the transformation X f to depend on the metric alone and more general possibilities may be found by including a dependence in other supergravity fields. Constraints on X f can be found by imposing diffeomorphism invariance, or more generally invariance under G SUGRA , on a particular D-brane action, if one knows the dependence of S on the supergravity fields Σ, for example by using Myers' results [3]. Even with this additional constraint, the solution is not unique. In [8], (4.6) was actually used the other way around, to put some constraints on the metric dependence of S , turning off all the other possible background fields. This is an interesting approach, since, beyond Myers' formulas, little is known about the general non-abelian D-brane actions in curved space. Unfortunately, but not surprisingly, the procedure is highly ambiguous and cannot fix the form of the action. Moreover, considering only the metric dependence might be misleading, since a fully consistent picture may require the closed string background to be on-shell. Because of all the above-mentioned difficulties, it may be more fruitful to use a different point of view, which is strongly favored if one interprets the closed string background as emerging from a microscopic, open-string like theory, as in the models studied in [1, 4, 5, 6]. In this point of view, the only natural gauge group is the group G D-geom discussed in Section 3. The closed string fields emerge from the coefficients c i 1 ··· i n in the expansion (2.2). Two sets of fields { c i 1 ··· i n , n ≥ 0 } and { c ' i 1 ··· i n , n ≥ 0 } will be physically equivalent if they correspond to the expansion of the same action in two different matrix coordinate systems X and X ' related to each other by a G D-geom transformation, Let us emphasize again the difference between (4.6) and (4.7). In equation (4.6), the background supergravity fields are given and one considers transformations under G SUGRA only. The transformation laws of the coefficients of the action, which are related to the supergravity background fields, are fixed a priori. The existence of a transformation law X f on the matrix coordinates such that (4.6) is valid is then required by consistency with the invariance under G SUGRA . On the other hand, equation (4.7) is not a consistency requirement, but the definition of the action of the group G D-geom on the coefficients c i 1 ··· i n and thus on the supergravity fields. Since S and S ' are physically equivalent, G D-geom is the group of gauge transformations. How can we see the usual gauge group G SUGRA emerge and how is the 'paradox' discussed in Section 3.7 solved in this picture? The point is that, even though there is no lift of G SUGRA into G D-geom , the groups G SUGRA and G D-geom can act in the same way on a set of fields. For example, one can define the action of γ ∈ G D-geom on scalar fields φ as where α is the ordinary diffeomorphism parametrizing γ in (3.54). This of course coincides with the usual action of Diff on a scalar field. It is obviously a consistent action of Diff, but it is also a consistent action of G D-geom because of the form of the composition law in G D-geom , In other words, even though there is no good group morphism Φ : G SUGRA → G D-geom in the sense explained in 3.7, there do exist surjective group morphisms Ψ : G D-geom → G SUGRA . If we have an action of G D-geom for which the kernel of Ψ acts trivially, then we can use Ψ to find a corresponding action of G SUGRA . This is the mechanism by which the usual G SUGRA transformations can emerge consistently from G D-geom and the open-string description. A simple explicit example, for the case of the AdS 5 × S 5 background studied in [1], will be given in Section 4.4. It is also important to realize that, in general, the action of G D-geom will induce transformation laws that are more general than the standard G SUGRA gauge transformations. In the rest of this Section, we are going to derive the form of these general transformation laws and discuss some of their consequences.", "pages": [ 27, 28, 29, 30 ] }, { "title": "4.2 The gauge transformations", "content": "Finding the explicit relation between two sets of fields c i 1 ··· i n and c ' i 1 ··· i n related by a G D-geom gauge transformation is completely straightforward. The action S ( X ) is expanded as in (1.2), S ' ( X ' ) is expanded as X and X ' are related to each other as in (3.5) and we impose the equality (4.7). This yields a general relation of the form where The simplest example corresponds to the case where γ = γ L ∈ GL( d, R ) is a linear transformation, as in (3.58). Equation (4.12) then yields the ordinary tensorial transformation law, which actually justifies our use of the term 'tensor' for the coefficients c i 1 ··· i n or their associated irreducible pieces. For general G D-geom transformations, the transformation laws are much more involved. Let us note, however, that the formulas (4.11) and (4.12) contain a lot of redundant information, since the coefficients c i 1 ··· i n and c ' i 1 ··· i n must satisfy the consistency conditions discussed in Section 2. In particular, if the set of coefficients { c i 1 ··· i n } satisfy these conditions, then the set { c ' i 1 ··· i n } determined by (4.11) and (4.12) automatically satisfy these conditions as well, and vice versa. All the information is thus contained in the transformation rules for the independent irreducible tensors (2.60), expressed in terms of the independent irreducible tensors (3.54) parametrizing the transformation law itself. The calculations required to express the transformation laws in this way are rather involved. We focus on the tensors c , c [ ijk ] , c [ ij ][ kl ] and c [ ijklm ] which, as explained in 2.8, encode Myers' action, and whose transformation laws will be explicitly used in the applications presented in 4.3 and 4.4. We find The transformation rules for c [ ijk ] and c [ ijklm ] can be most conveniently rewritten in the form language, using the definitions (2.15) and Equations (4.15) and (4.17) are then equivalent to Let us note that standard tensorial transformation laws would correspond to F (3) = F ' (3) and F (5) = F ' (5) . The additional terms enter because of the non-commutative structure of the space of matrix coordinates. The fact that F ' (3) -F (3) and F ' (5) -F (5) turns out to be exact forms is perfectly consistent with the constraints (2.16). Similarly, the simple tensorial transformation law of c [ ij ][ kl ] , which would correspond to the first term on the right-hand side of (4.16), is supplemented by additional terms which, of course, are consistent with the symmetries (2.31). The form of the gauge transformations (4.14)-(4.20) are quite interesting and non standard. Their form is, to some extent, dictated by the non-trivial structure of the group G D-geom discussed in Section 3. We are now going to provide a few simple applications and clarify their physical meaning.", "pages": [ 30, 31, 32 ] }, { "title": "4.3 Application to p -form gauge transformations", "content": "As a first application, let us show how the p -form supergravity gauge transformations are generated from the G D-geom gauge transformations and thus naturally emerge from the open string description. We shall treat below the case of Myers' D-instanton action and in Section 5 the case of D-particles. In particular, we are going to check explicitly the consistency of Myers' action with the p -form gauge symmetries via equation (4.6). An interesting feature, first derived in [9], is that the B -field gauge transformations must act non-trivially on the matrix coordinates, with δX ∼ [ X,X ]. This means that the transformation X f in (4.6) is non-trivial when f ∈ G SUGRA corresponds to a B -field gauge transformation. This result may be surprising from the closed string perspective but, from the discussion in 4.1, it is perfectly natural from the emergent geometry, or open string, point of view. The references [9] focused on the Chern-Simons part of the action and on the leading order transformation law for the matrix coordinates. As we now discuss, the formalism that we have developed so far allows us to generalize effortlessly the analysis to the full non-abelian D-brane action, including the Dirac-Born-Infeld part, and to work out the matrix coordinates transformation laws up to the fourth order. Myers' action was discussed in 2.8 and its dependence on the supergravity p -forms is coded in the forms F (1) , F (3) and F (5) given in (2.72). The Ramond-Ramond two- and four-forms gauge transformations are parametrized by a one-form µ and a three-form ω and induce the following non-trivial variations on the form fields, where H = d B is the Neveu-Schwarz three-form field strength. It is immediate to check that F (1) , F (3) and F (5) do not change under these transformations and thus the D-brane action is trivially invariant. Much more interesting is the case of the B -field gauge transformations, which acts only on B as It yields The quadratic term on the right-hand side of (4.25) ensures that the composition of two gauge transformations associated with λ 1 and λ 2 yields a gauge transformation of the same type with λ = λ 1 + λ 2 . It is straightforward to check that the formulas (4.23)-(4.25) are special cases of the general G D-geom gauge transformations (4.19) and (4.20), for x ' = x (the associated standard diffeomorphism is trivial, as expected) and up to an exact one-form which can always be added to λ . There remains to check that the other supergravity fields do not vary. The condition ∆ τ = 0 follows from the first equation in (2.68), (4.14) and x ' = x . From (2.70), the condition ∆ G ij = 0 is then equivalent to ∆ c [ ij ][ kl ] = 0. On the other hand, ∆ c [ ij ][ kl ] is given by (4.16), in the special case for which x ' = x (and thus γ m i = δ m i ) and, from the first equation in (4.26), On can then immediately check that the term proportional to c ' [ mnp ] on the righthand side of (4.16) automatically vanish. On the other hand, the term proportional to ∂ ' mn c ' can be made to vanish by choosing and the term proportional to ∂ ' m c ' can then be made to vanish by choosing Of course, the tensors u m ijk and r m ijkl defined in this way satisfy the required symmetry properties (3.24) and (3.39). The transformation law on the matrix coordinates given by (4.26), (4.27), (4.28) and (4.29) turn out to be background independent . It is very natural to expect that a background-independent extension of the transformation law to all orders could be found. Let us also note that the analysis can be performed independently on the detailed form of Myers' action and in particular independently of the small glyph[lscript] 2 s approximation. Indeed, the background field transformation laws (4.23), (4.24) and (4.25) are consistent with the general constraints (2.16) discussed in Section 2 and thus well-defined for any matrix action.", "pages": [ 32, 33, 34 ] }, { "title": "4.4 Diffeomorphisms and the emergent AdS 5 × S 5 background", "content": "As a consequence of the theorem reviewed in 3.7, the discussion of the previous subsection cannot be generalized straightforwardly to the case of space-time diffeomorphisms, because background-independent transformation laws associated with diffeomorphisms do not exist for the matrix coordinates. However, consistency with diffeomorphism invariance can nevertheless be achieved, as explained in 4.1. Let us see explicitly how this works for the D-instanton action in the presence of D3-branes, which was derived from a microscopic calculation in [1]. The action S ( X ) turns out to be precisely of the form predicted by Myers, as in equations (2.68), (2.70) and (2.71). The axion-dilaton τ is a constant and is expressed in terms of the ϑ angle and 't Hooft coupling λ of the N = 4 gauge theory living on the D3 branes as The coefficient c [ ij ][ kl ] factorizes as in (2.70) in terms of the usual Euclidean AdS 5 × S 5 metric, where 1 ≤ µ ≤ 4, dΩ 2 5 is the metric for the unit round five-sphere and the radius R is given by The form coefficients (2.72) are found to be where ω AdS 5 and ω S 5 are the volume forms associated with the AdS 5 and S 5 factors of the metric (4.31). Formulas (4.30)-(4.33) reproduce precisely the AdS 5 × S 5 background of type IIB supergravity. In particular, the condition F (1) = 0 comes from the fact that the axion-dilaton is constant, F (3) = 0 is equivalent to B = C 2 = 0 and F (5) yields the correct Ramond-Ramond five-form field strength. The above solution is derived from the microscopic computation of S ( X ), not from solving the supergravity equations of motion. By construction, it is then only defined modulo the general G D-geom gauge transformations discussed in 4.2. Because c is constant and c [ ijk ] = 0, the complicated transformation laws (4.14)-(4.17) actually simplify, for example where γ m i = ∂x ' m /∂x i . We simply find the action of ordinary diffeomorphisms, emerging from the field redefinition redundancy in the open string point of view. This is perfectly in line with the emerging space philosophy and the discussion around equation (4.9). It is also interesting to find that tensorial quantities in ordinary spacetime, like a metric or a five-form, can emerge from a purely scalar function of non-commuting matrix coordinates. The mechanism at work is quite different from the usual coupling of the metric to a kinetic term or of a p -form to a p -dimensional worldvolume, for instance.", "pages": [ 34, 35 ] }, { "title": "4.5 Comments on the general case", "content": "The discussion of the previous subsection uses heavily the special properties of the AdS 5 × S 5 background. If c is constant, corresponding to a constant axion-dilaton, we could still implement the ordinary diffeomorphisms with a G D-geom gauge transformation for which b m = 0, or more generally of the form b m = d ϕ ∧ d x m , which ensures that the transformation laws (4.14)-(4.17) reduce to the standard tensor transformation laws. However, as we have emphasized again and again, this is not natural. One should really consider the general action of G D-geom and draw the general consequences of the associated transformation laws. glyph[negationslash] Actually, for a generic background, the action of G D-geom on the action S ( X ) is very drastic. In the abelian case, k = 1, this is well-known. If we assume that dRe c ( x ) ∧ dIm c ( x ) = 0 then, in the vicinity of x , we can always pick a coordinate system such that x ' 1 = Re c and x ' 2 = Im c . In this coordinate system, S ' ( x ' ) = x ' 1 + ix ' 2 is a simple linear function. In the non-commutative case, we would like to make a similar statement. glyph[negationslash] Claim : Let us assume that dRe c ( x ) ∧ dIm c ( x ) = 0 . Then it is always possible to gauge away all the coefficients c i 1 ··· i n ( x ) for n ≥ 2 by using a general G D-geom gauge transformation. Up to order five, this is straightforwardly proved from our explicit formulas (4.14)(4.20) and the elementary Lemma : If v m is a complex valued vector such that Re v m and Im v m are linearly independent and if ρ i 1 ··· i n are arbitrary complex-valued coefficients, then it is always possible to solve the equations for some real coefficients r m i 1 ··· i n . Using this lemma, we can choose the a priori arbitrary two- and four-forms c ' m b m and c ' m a m in (4.19) and (4.20) in such a way that the closed forms F ' (3) and F ' (5) vanish. Similarly, the term 3 4 c ' m r m ijkl in (4.16) can be adjusted to any tensor with the general symmetries of c [ ij ][ kl ] and can thus be used to make c ' [ mn ][ pq ] vanish. An all order analysis is beyond the scope of our work, but it is interesting to mention that it is essentially equivalent to the following very natural 'lift' theorem. We have defined the general notion of a single-trace function f ( X ) in the beginning of Section 2, via an expansion where the cyclic coefficients c i 1 ··· i n must satisfy the constraints (2.8). A similar notion of a no-trace matrix-valued function F ( X ) can be defined as well, via the expansion where the coefficients ρ i 1 ··· i n satisfy the constraints These constraints are the same as in (3.11) and ensure, as usual, the invariance under the shifts (2.4). Examples of no-trace matrix-valued functions are the γ i ( X ) defining an element γ ∈ G D-geom in (3.5) or ˆ f ( X ) defined by (2.44). The lift conjecture then states that any single-trace function is the trace of a no-trace matrix-valued function . In other words, given cyclic coefficients satisfying (2.8), it is always possible to find coefficients ρ i 1 ··· i n satisfying (4.38) and such that The action of j Z n is defined in (2.18) and takes the cyclic combination of the coefficients ρ i 1 ··· i n . This statement seems extremely natural, but the proof is not trivial. For example, it can be easily checked that the choice ρ i 1 ··· i n = c i 1 ··· i n is not consistent. glyph[negationslash] Assuming this result to be correct, we can then proceed as follow to trivialize S ( X ). First, we use d Re c ∧ dIm c = 0 to choose the ordinary diffeomorphism x ' ( x ) in γ such that x ' 1 = Re c ( x ) and x ' 2 = Im c ( x ). In other words, Next, we pick an arbitrary complex-valued tensor ρ i 1 ··· i n satisfying the constraints and we choose the coefficients γ m i 1 ··· i n for n ≥ 2 in such a way that This is always possible. Indeed, by taking the real and imaginary parts and using (4.40) on the one hand and (3.8), (3.9) on the other hand, we see that (4.42) is equivalent to This is consistent, because the only conditions on the coefficients α m i 1 ··· i n and β m i 1 ··· i n , which are the constraints (3.13) and (3.14), are automatically satisfied if (4.41) is satisfied, as can be checked straightforwardly. Now, using the expansions (2.2) and (3.5), the condition is equivalent to (4.39), which can be solved by the lift theorem. The above discussion is just the beginning of what could be a much more elaborate mathematical study of single-trace functions modulo the action of G D-geom . This study would correspond to an important generalization of the standard singularity theory of ordinary functions [12], which deals with the classification of the possible expansions around a point modulo the action of diffeomorphisms (or biholomorphisms in the complex case). In view of the many connexions between D-brane physics, single-trace actions and (super)potentials, (singular) Calabi-Yau spaces and matrix models (see e.g. [13] and references therein), we believe that the development of this theory could have far-reaching consequences.", "pages": [ 35, 36, 37 ] }, { "title": "5 Matrix quantum mechanics", "content": "The analysis of the previous Sections can be straightforwardly generalized to higher dimensional actions. We are going to discuss briefly the case of quantum mechanical single-trace actions, which is used in particular in [5]. We continue to work in Euclidean signature, if not explicitly stated otherwise, for consistency with the rest of the paper.", "pages": [ 37 ] }, { "title": "5.1.1 General discussion", "content": "In the commutative k = 1 case, it is always possible to choose a gauge in which the time coordinate x d is identified with the parameter λ along the worldline. In the general case k > 1, we assume that such a static gauge still makes sense and set The D-brane actions derived from string theory are naturally found in this gauge. The quantum mechanical actions we consider, are thus functionals of matrix worldlines given, in parametric form, by d -1 matrix coordinate functions X i ( λ ), 1 ≤ i ≤ d -1. The Lagrangian L is assumed to be a single-trace function of X i ( λ ) and its derivatives. We can expand where the term L ( p ) contains p derivatives of the coordinates. We shall limit ourselves to the two-derivative action, p ≤ 2, and study the expansion around arbitrary diagonal time-independent configurations, The potential term, p = 0, is like an ordinary single-trace function (2.2), with c (0) i 1 ··· i n = c (0) i n i 1 ··· i n -1 . As for the one-derivative term, it can be written, using the cyclicity of the trace, as where is the covariant derivative along the worldline and z the worldline gauge potential. The two-derivative Lagrangian contains only glyph[epsilon1] and ˙ glyph[epsilon1] , up to the addition of total derivative terms. To the fourth order, we can arrange the terms as We now impose the invariance under the shift symmetry (2.4). Since the shift parameter a i is λ -independent, this yields constraints on each term L ( p ) independently of each other. The constraints on the potential term L (0) are of course exactly the same as the ones studied in Section 2. The coefficients c (0) i 1 ··· i n are thus characterized by irreducible tensors as in (2.60). The constraints on the expansion (5.6) match the ones studied in Section 3 for the expansion (3.5), see in particular (3.54), since the c (1) i 1 ··· i n do not satisfy any cyclicity condition. For example, whereas c (1) [ ij ]; k is unconstrained. We may wish to impose an additional reality condition, if we work in the Minkowskian. The coefficients c (1) k and c (1) [ ij ]; k must then be real and purely imaginary respectively. A similar analysis can be performed on the second derivative Lagrangian (5.8). For example, we find that and other constraints of a similar type.", "pages": [ 38, 39 ] }, { "title": "5.1.2 The example of Myers D-particle action", "content": "A particularly interesting example is the D0-brane action in type IIA string theory. In the Euclidean, a single D0-brane has a Lagrangian of the form where φ , G µν and A µ are the dilaton, the string-frame metric and the RamondRamond one-form respectively. This is the action for an ordinary charged particule of equal mass and charge m = q = √ 2 π/glyph[lscript] s moving in the d = 10 dimensional metric Going to the static gauge (5.1), x d = x 10 = λ , and expanding as in (5.3), yield with 1 ≤ i, j ≤ 9, d = 10 and The non-abelian version of this action, valid for an arbitrary number k ≥ 1 of Dparticles, can be computed from Myers' formulas [3]. The Dirac-Born-Infeld part of Myers Lagrangian reads with The latin indices always run from 1 to 9 whereas the greek indices run from 1 to 10. In particular, ˙ glyph[epsilon1] 10 = 1 because of (5.1). The determinant in (5.16) acts on the indices i, j and not on the U( k ) indices of the matrices glyph[epsilon1] . The Str is the symmetrized trace on the U( k ) indices defined in [3]. It provides the correct ordering up to order five in the expansion in powers of glyph[epsilon1] but not beyond. The Chern-Simons part of the Lagrangian is given by √ The C 2 q +1 are the type IIA Ramond-Ramond forms, i glyph[epsilon1] is the inner product and we keep only the one-form part of the expression in the bracket. The P denotes the U( k )-covariant pull-back to the D-particle worldline, A rather tedious calculation then yields explicit expressions for the various irreducible tensors parameterizing the Lagrangian. For example, by noting the Ramond-Ramond one-, three- and five-form potentials respectively, we get + H ik B jd B ld - H jk B id B ld - H il B jd B kd + H jl B id B kd , +3 A d B jk B lm ] +12 A j B kd B lm ] ) . ] Let us note that C and ˜ C are not independent, since the associated field strengths are dual to each other. Explicitly, if H = d B as usual, we have, in the Euclidean, Similarly, the first independent coefficients in the one- and two-derivative terms are given by We have checked explicitly the consistency of the above formulas with the D-particle Lagrangian obtained from the D-instanton action discussed in Section 2.8 by performing a T-duality in the direction of x 10 . This method is actually quite efficient. For example, the fifth order coefficient (5.25) in the potential is obtained more easily from T-duality than from the explicit Myers action (5.16) and (5.19). The formulas (5.22)-(5.29) allow to read off the type IIA supergravity background from the D-particle Lagrangian. They are crucially used in [5] to derive the emergent supergravity background generated by a large number of D4-branes.", "pages": [ 39, 40, 41 ] }, { "title": "5.2 Gauge symmetries", "content": "Let us now discuss the action of the gauge group G D-geom . The general qualitative discussion of Section 4.1 applies to the present quantum mechanical case as well. Since we are in the static gauge (5.1), we limit ourselves to transformations acting on the transverse matrix coordinates X i , 1 ≤ i ≤ 9. Moreover, if we assume that the coefficients appearing in the expansion (3.5) do not depend on x d = λ , then the general gauge transformations for the coefficients c (0) i 1 ··· i n of the potential term are given by equations (4.14)-(4.20). Similarly, we can find the gauge transformations of the fields appearing in the higher derivative terms. For example, As a simple application, we can study the p -form supergravity gauge transformations and extend the results of [9] to the full Dirac-Born-Infeld plus Chern-Simons non-abelian D-particle action. The Ramond-Ramond one- and three-forms gauge transformations are parameterized by a function h and a two-form κ , with If the background fields and the gauge transformation parameters h and κ do not depend on x d and κ has components on the transverse coordinates x i only, then the invariance of the D-particle action follows from the invariance of the D-instanton action proven in Section 4.3 and T-duality. In the general case of time-dependent background and general gauge transformations, we have checked, up to order four in the glyph[epsilon1] expansion, that the Myers' Lagrangian transforms as a total time derivative and thus that the action is invariant, as required. The B -field gauge transformation δB = d λ is more interesting, because, as in Section 4.3, it must act on the spacetime matrix coordinates in the non-commutative, k ≥ 2, case. Again, if the background and the gauge transformation parameter λ do not depend on x d , and if λ has components on the transverse coordinates x i only, then consistency follows from the D-instanton case studied in 4.3 and T-duality. If we drop this assumption, then the required G D-geom transformations, acting on the transverse matrix coordinates X i , will have to depend on the time x d explicitly. The transformation rules (4.14)-(4.17) and (5.30)-(5.33) are then generalized, because terms with different number of derivatives in the expansion (5.3) mix under timedependent G D-geom transformations. Limiting our analysis to the third order in glyph[epsilon1] , we need the following generalizations of (4.15) (or (4.19)) and (5.31), for a G D-geom transformation associated with the trivial diffeomorphism x ' = x , The other relevant transformation laws are unchanged. As in Section 4.3, we have to choose the transformation of the coordinates X i such that glyph[negationslash] This goes a long way in generating the B -field gauge transformation, but we also have to take into account the T-dual version of the non-trivial transformation law of the matrix coordinate X d , supplemented by an additional term when λ d = 0. The T-dual of X d is the worldline gauge field z and it can be checked that the correct transformation law is given by The effect of the background-independent field redefinitions associated with (5.37) and (5.38) precisely match the effect of the supergravity gauge transformation δB = d λ .", "pages": [ 41, 42 ] }, { "title": "6 Acknowledgments", "content": "I would like to thank Antonin Rovai and Micha Moskovic for many useful discussions. This work is supported in part by the belgian FRFC (grant 2.4655.07) and IISN (grant 4.4511.06 and 4.4514.08).", "pages": [ 43 ] }, { "title": "A A short review on tensor symmetries", "content": "This Appendix is devoted to a very brief review on the classification of tensor symmetries. This yields very useful calculational techniques that we have implemented in Mathematica and used to perform most of the calculations presented in the main text.", "pages": [ 44 ] }, { "title": "A.1 Generalities", "content": "We consider the Hilbert space T n d of complex tensors of rank n in d dimensions, with the norm The symmetric group S n acts on T n d in the usual way, where ˜ σ is the linear operator associated with the permutation σ . The group algebra C [S n ], defined to be the set of formal complex linear combinations of the elements of S n , also acts on T n d , by extending the action (A.2) by linearity. The group algebra is endowed with a Hilbert space structure, for which the n ! elements of S n form an orthonormal basis. It is also convenient to associate, to each element x ∈ C [S n ], a linear operator ˆ x acting on C [S n ] by left multiplication, ˆ x ( y ) = xy . If x ∈ S n , then both ˆ x (which acts on C [S n ]) and ˜ x (which acts on T n d ) are unitary. Moreover, the Hermitian conjugates ˆ x † and ˜ x † are both associated with the same element x † of C [S n ]. Explicitly, x and x † can be expanded as A right-ideal of C [S n ], or simply an ideal , is a subspace of C [S n ] stable under right multiplication (there is also a similar notion of left ideals, but for our purposes right ideals are more natural). An ideal I is called minimal if any ideal J ⊂ I is either the trivial ideal J = { 0 } or equal to I . It is known that I is minimal if and only if the representation of S n induced on I by the right multiplication is irreducible. The same irreducible representation can be associated with distinct ideals; a given irreducible representation of dimension δ actually occurs with multiplicity δ in C [S n ]. If I and I ' correspond to inequivalent representations, then xx ' = 0 if x ∈ I and x ' ∈ I ' . The ideals of C [S n ] have a simple explicit description. An idempotent is an element j ∈ C [S n ] such that j 2 = j . Then any ideal is of the form The idempotent generating a given ideal is not unique. One can easily show that I ( j ) = I ( j ' ) if and only if j ' = j -jx + jxj for some x ∈ C [S n ]. However, there is a unique Hermitian generating idempotent j I = j † I . This idempotent corresponds to the orthogonal projection of the identity element on the ideal I and as such it can be easily constructed algorithmically. The operator ˆ  I is the orthogonal projector on I . glyph[negationslash] Minimal ideals are generated by primitive idempotents. A primitive idempotent cannot be written as j = j 1 + j 2 for idempotents j 1 and j 2 satisfying j 1 j 2 = j 2 j 1 = 0, except if j 1 or j 2 is zero. A primitive idempotent is characterized by the fact that jxj = λ x j , λ x ∈ C , for any x ∈ C [S n ]. To a primitive idempotent is associated an irreducible representation of S n and thus a Young tableau. The representations associated to two primitive idempotents j and j ' are equivalent if and only if there exists x ∈ C [S n ] such that jxj ' = 0. A given ideal I can always be decomposed as a direct sum of minimal ideals, Moreover, if I = I ( j ), for j a ∈ I a , then j a j b = δ ab and I a = I ( j a ). The set of irreducible representations of S n and their multiplicities occurring in the decomposition of a given ideal I is unique. Minimal ideals associated with inequivalent irreducible representations are orthogonal. To compute the decomposition (A.5) algorithmically, we can proceed as follows. We start from an explicit decomposition of the algebra C [S n ], glyph[negationslash] For example, we can use for the j a the standard Young idempotents associated with Young tableaux (note that the Young idempotents are not Hermitian in general, but this is not a problem). If we find a j a 0 such that j I j a 0 = 0, then the ideal I a 0 = { j I j a 0 x, x ∈ C [S n ] } is minimal and enters into the decomposition of I . We compute j I a 0 and we iterate this process for the ideal generated by the idempotent j I -j I a 0 . This eventually yields a full decomposition of the form (A.5), with the additional bonus that the direct sum is automatically orthogonal.", "pages": [ 44, 45 ] }, { "title": "A.2 Tensor symmetries", "content": "The most general notion of a tensor symmetry is described by a set of linear equations that the components of tensors having the required symmetry must satisfy. Such equations are of course invariant under arbitrary changes of basis, corresponding to GL( d ) transformations. For example, given a subgroup G ⊂ S n and a linear character χ of G (i.e., a one-dimensional representation), one could consider tensors that satisfy σ · t = χ ( σ ) t for any σ ∈ G . Such tensors have the 'symmetry type' ( G,χ ). More generally, one can impose a set of conditions of the form Consider the ideal I = { x ∈ C [S n ] | x i x = 0 , 1 ≤ i ≤ p } . Then one can show that the tensors satisfying (A.8) are simply the tensors of the form x · τ for an arbitrary tensor τ and x ∈ I . If I = I ( j ), these are equivalently the tensors of the form j · τ or, again equivalently, the tensors satisfying This means that the many conditions (A.8) are always equivalent to the unique condition (A.9), for a certain idempotent j . For example, in the case of a tensor of symmetry type ( G,χ ), one has where | G | denotes the cardinal of G . It is straightforward to check in this case that the unique condition j G,χ · t = t is equivalent to σ · t = χ ( σ ) t for any σ ∈ G . The equivalence between (A.8) and (A.9) remains valid in all cases. In conclusion, the symmetry types of tensors are in one-to-one correspondence with the ideals of C [S n ]. Let T n d ( I ) denotes the vector space of rank n tensors in dimension d with symmetry type given by the ideal I . To the decomposition (A.5) corresponds the decomposition If the ideals I a and I a ' are orthogonal in C [S n ], then T n d ( I a ) and T n d ( I a ' ) are orthogonal in T n d . Elements of T n d ( I a ), where I a is a minimal ideal, are called irreducible tensors . Decomposing tensors into irreducible pieces can be a very useful tool which we have used to analyse the various constraints discussed in the main text. A simple application of the above formalism is to compute the number of independent components of a tensor in T n d ( I ). This is also the dimension of T n d ( I ) or, from (A.9), the rank of ˜  . Since ˜  2 = ˜  , rk ˜  = tr ˜  . Writing j = ∑ σ j σ σ , the dimension can be computed by using the fact that tr ˜ σ = d c ( σ ) , where c ( σ ) is the number of distinct cycles (including cycles of length one) in the cycle decomposition of σ . Example : consider the Riemann tensor R ∈ T 4 d . Its symmetries are described by the equations The elements x i s in (A.8) are given by The corresponding ideal I R is generated by the Young idempotent associated with the Young tableau 1 3 2 4 . The Young idempotent is not Hermitian, but the Hermitian generating idempotent can be found by projecting the identity element onto I R , A tensor R has the symmetries (A.12) if and only if it satisfies j Y R · R = R or equivalently j I R · R = R . Computing the traces, one finds which is the well-known number of independent components of the Riemann tensor in d dimensions.", "pages": [ 45, 46, 47 ] }, { "title": "A.3 A sample calculation", "content": "To illustrate the use of the above formalism on a typical example, let us give details on the derivation of the equations (3.41), (3.42) and (3.45) in the main text. The first step in the calculation is to decompose the tensor β ijkl into irreducible components, taking into account the constraint β ijkl = -β lkji . This constraint tells us that β ijkl has the symmetry type of the ideal generated by the idempotent Using the algorithm described around equation (A.7), we find that with and similar formulas for ˜  2 , 1 , 1 and ˜  ' 2 , 1 , 1 . In a second step, we analyse the consequences of the equation (3.14) for n = 4, By computing the decomposition of the ideal generated by J 4 j , we find that it contains each Young tableau and only once. This means that half of the irreducible tensors in (A.18) are projected out by J 4 , one for each irreducible representation appearing in the decomposition, and thus only the other half will be fixed by (A.21). To find out precisely which pieces are fixed by (A.21), we proceed as follows. Let N ( J 4 ) be the annihilating ideal associated with J 4 , i.e. and let Computing a basis for N ( J 4 ) and then for I is a simple problem of linear algebra. We can then apply our algorithm to compute the Hermitian generating idempotent j 3 , 1 of the ideal I , I = I ( j 3 , 1 ), and its orthogonal j ' 3 , 1 = ˜  3 , 1 +˜  ' 3 , 1 -j 3 , 1 . This yields Equations (3.41) and (3.42) are obtained by acting on the tensor β ijkl with j 3 , 1 and j ' 3 , 1 . By construction, β ( j 3 , 1 ) is left unconstrained by (A.21), since J 4 j 3 , 1 = 0. On the other hand, applying j ' 3 , 1 to both side of (A.21) and using the properties of the primitive idempotents listed in the paragraph between equations (A.4) and (A.5), we get noting β the tensor β ijkl and ∂β the tensor ∂ i β jkl . This yields β ( j ' 3 , 1 ) = 2 j ' 3 , 1 · ∂β which, by using the fact that β ijk is expressed in terms of β [ ij ] through equations (3.25) and (3.27), finally yields the equation (3.45). The same kind of reasoning allows to derive (3.43), (3.44) and (3.46) and many other results quoted in the main text.", "pages": [ 47, 48 ] } ]

[ { "title": "Quantum spins on star graphs and the Kondo model", "content": "N. Cramp'e a,b and A. Trombettoni c,d E-mail: [email protected], [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the XX model for quantum spins on the star graph with three legs (i.e., on a Y -junction). By performing a Jordan-Wigner transformation supplemented by the introduction of an auxiliary space we find a Kondo Hamiltonian of fermions, in the spin 1 representation of su (2), locally coupled with a magnetic impurity. In the continuum limit our model is shown to be equivalent to the 4-channel Kondo model coupling spin-1 / 2 fermions with a spin-1 / 2 impurity and exhibiting a non-Fermi liquid behavior. We also show that it is possible to find a XY model such that - after the Jordan-Wigner transformation - one obtains a quadratic fermionic Hamiltonian directly diagonalizable. Keywords: Quantum spin model; Kondo model; Jordan-Wigner transformation; Star graph", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Spatial inhomogeneities and their role in the emergence of coherent behaviors at mesoscopic length scales are the subject of a continuing interest. In general, spatial inhomogeneities may be random, due to the presence of disorder or noise, as well as non-random, as a result of an external control on the geometry of the system: in a broad sense, their formation can be dynamically generated or induced through a suitable engineering of the system. As a consequence the effects of spatial inhomogeneities have been investigated in a variety of systems, ranging from pattern formation in systems with competing interactions [1] to Josephson networks with non-random, yet non-translationally invariant architecture [2, 3]. A paradigmatic system in which the effects of spatial inhomogeneities can be studied is provided by spin models: on the one hand, not only do spin Hamiltonians directly describe many phenomena of magnetic systems [4], including the effects of frustration [5], but they are also routinely used to model physical properties of several condensed matter systems. On the other hand, spatial inhomogeneities can be straightforwardly included in spin models, to explore the consequences of the breaking of the translational invariance and the local properties on the length scales of the inhomogeneities [6]. As an example of the application of the study of spatial inhomogeneities in spin systems, we mention the spin chain Kondo effect. The standard Kondo effect arises from the interactions between magnetic impurities and the electrons in a metal and it is characterized by a net increase at low temperature of the resistance [7, 8, 9]. The Kondo effect has been initially observed for metals, like copper, in which magnetic atoms, like cobalt, are added: however, interest in the Kondo physics persisted also because it can be studied with quantum dots [10, 11]. The universal low-energy/long-distance physics of the Kondo model can be simulated and studied by a magnetic impurity coupled to a gapless antiferromagnetic one-dimensional chain having nearest- and next-nearest- neighbour couplings J 1 -J 2 [12], with the the correct scaling behavior of the single channel Kondo problem being exactly reproduced by this spin model only when J 2 equals a critical value [12]. The spin model reproducing the low-energy behaviour of the Kondo problem is defined on the half line, since the radial coordinate of the fermionic model as well varies in the half line: the rationale is that the free electron Kondo problem may be described by a one-dimensional model since only the s -wave part of the electronic wavefunction is affected by the Kondo coupling [13]. Another example in which the scaling behavior of a fermionic Kondo model may be well reproduced by a pertinently chosen spin model is discussed in [14]. Using the spin chain version of the Kondo problem, a characterization of the Kondo regime using negativity was recently presented [14, 15] and it was shown that long-range entanglement mediated by the Kondo cloud can be induced by a quantum quench [16]. It stands as a open and interesting line of research to introduce and study spin systems, eventually with suitably tailored spatial inhomogeneities, reproducing the scaling behaviour of more general systems of fermions coupled to magnetic impurities, as the general multichannel Kondo effect. Another reason of interest for introducing spatial inhomogeneities in spin models defined on networks is given by the study of the effects of the topology of the graph on the properties of the system and of the breaking of integrability. As a main example, consider a quantum (classical) spin model which is integrable in one (two) dimensions. Techniques have been developed to deal with open boundary conditions [17], as for free boundaries described by algebraic curves [18]. However, if some vertices of the graph on which the spins are located have a number of nearest neighbours larger than all others, then integrability is in general broken. One can see this by considering a one-dimensional quantum model which can be solved by a Jordan-Wigner (JW) transformation [19]: intersecting the chain at one site with a finite or infinite number of other chains the usual JW transformation on the spin variables will produce a fermionic model which is in general neither quadratic nor local. We recall that the two-dimensional classical Ising model at finite temperature can be solved by writing its partition functions in terms of a suitable quantum spin model on the chain which is solved by JW transformation [20, 21]. Therefore, finding an effective way of performing a JW transformation in non trivial graphs amounts to the possibility of studying and possibly solving the Ising model in some non trivial (non two-dimensional) lattices [22]. In this paper we study the XX model on a star graph obtained by merging three chains: the standard JW transformation cannot work for a star graph since there is no natural order on it (of course, this problem would generically appear for any graph, except for the circle and the segment where it works). We rather found convenient to supplement the application of the standard JW transformation with the introduction of an auxiliary space: the procedure of adding auxiliary sites to perform a JW transformation has been recently used to study higher-dimensional systems in [23, 24]. In our case, it is the use of this auxiliary space which allows us to get a Hamiltonian that is both quadratic and local in the JW fermions. Using such exact mapping, we show that the XX model on a star graph is equivalent to a generalized Kondo model, where the JW fermions enter locally and quadratically, and are coupled to a magnetic impurity. There are several reasons for our choice of the XX model on a star graph. On the one hand, we study the XX model since we are motivated by the need of emphasize the main point of our construction in the simplest case: for the XX model in a chain, the JW transformation gives rise to free fermions (our construction can be extended to other spin models solvable by JW transformations). On the other hand, we decide to restrict ourselves to the study of a star graph with three legs for a twofold reason: first, it is the simplest graph which can be constructed by merging a finite number of chains and having a finite number of vertices (three in our construction, see Figure 1) with coordination number z = 3 different and larger than the others (having z = 2, with the sites at the boundaries of the chain having z = 1). Second, the star graph (alias, the Y -junction) has been deeply studied in different contexts from different point of views: for three Tomonaga-Luttinger liquids (TLL) crossing at a point new attractive fixed points emerge [25, 26, 27]. Regular networks of TLL, with each node described by a unitary scattering matrix, were also studied [28], obtaining the same renormalization group equations derived for a single node coupled to several semi-infinite 1D wires [25]. The transport through one-dimensional TLL coupled together at a single point has been also studied [29]. Y -junctions of superconducting Josephson junctions were as well analyzed: for suitable values of the control parameters an attractive finite coupling fixed point is found [27], displaying an emerging two-level quantum system with enhanced coherence [30]. Star graphs were studied also in connection with bosonic models: properties of an ideal gas of bosons on a star graph were investigated in [31, 32] and the possible experimental realization with ultracold bosons was discussed in [32]. The dynamics of one-dimensional Bose liquids in Y -junctions and the reflection at the center of the star was studied, discussing the emergence of a repulsive fixed point [33]. Finally, we mention that the study of different theories on a graph and, particularly, on a star graph is a very active field of research: for example, for the Laplacian operator (also called quantum graphs) [34, 35, 36, 37, 38], for the Dirac operator [39, 40], for classical field theories and soliton theories [41, 42, 43] and for quantum field theories [44]. The plan of the paper is the following: in section 2, we introduce the XX model on a star graph, and we perform the JW transformation needed to obtain a fermionic Hamiltonian. The usefulness to add auxiliary sites is motivated, and the obtained Kondo Hamiltonian derived and discussed. In section 3, we show that it is possible to find an XY model such that after the JW transformation one obtains a quadratic fermionic Hamiltonian directly diagonalizable. Finally, our conclusions are presented in section 4.", "pages": [ 2, 3 ] }, { "title": "2 The XX model on a star graph", "content": "In this section, we want to obtain fermionic Hamiltonians from quantum spin models on a star graph by using a JW transformation. In particular, we point out the importance to add an auxiliary site to obtain a fermionic Hamiltonian: we show that to solve this model is equivalent to solve a generalized Kondo model. The treatment is explicitly done for the XX model to emphasize our construction in the simplest case, although the procedure can be used to study other models on the star graph.", "pages": [ 3 ] }, { "title": "2.1 The model and the Jordan-Wigner transformation", "content": "We introduce in this section the XX model on a three-leg star graph. The graph we consider is illustrated in Figure 1 and it is made of three chains of length L , each one having vertices labeled by 1 , · · · , L ; the sites 1 of each of the three chains are connected between them. In each vertex of the graph (having 3 L vertices) are defined the Pauli matrices σ x = ( 0 1 1 0 ) , σ y = ( 0 -i i 0 ) , σ z = ( 1 0 0 -1 ) . As usual we use the notation σ ± = 1 2 ( σ x ± iσ y ). The XX model is described by the following quantum Hamiltonian acting on the Hilbert space ( C 2 ) ⊗ 3 L : where σ ± α ( j ) stands for the matrix σ ± acting on the α th chain (with α = 1 , 2 , 3) and on the j th site from the vertex (the labeling of the sites is plotted in Fig. 1). In equation (2.1) we have used the convention σ ± 4 (1) := σ ± 1 (1). The parameter ρ (in general complex) entering in the definition of the Hamiltonian (2.1) is a free parameter allowing one to modify the coupling constant at the center of the star graph. In particular, for ρ = 0, one retrieves three independent XX models on segments with free (open) boundaries. At this point, we arrive at the main ingredient of our construction. To perform a JW transformation, we introduce, instead of ˜ H XX 3 , a slightly different Hamiltonian H XX 3 acting on the Hilbert space C 2 ⊗ ( C 2 ) ⊗ 3 L and defined by the following rules: H XX 3 acts as ˜ H XX 3 on the last 3 L C 2 -spaces and trivially on the first C 2 -space. The added space (in comparison with the Hilbert space of ˜ H XX 3 ) is denoted 0 and is called auxiliary space . We can write the link between both Hamiltonians as follows: where Id (0) is the 2 by 2 identity matrix acting on the auxiliary space. Notice that H XX 3 has exactly the same spectrum as ˜ H XX 3 but with a degeneracy multiplied by 2. Although the addition of this auxiliary site is trivial for the quantum spin model, we will see that it allows one to perform the JW transformation to get a fermionic model (see also the discussion at the end of this section to motivate why this auxiliary space seems necessary). The use of auxiliary sites to perform a JW transformation has been recently used in multidimensional spin systems [23, 24]. The JW transformation we use is defined, for j = 1 , 2 , . . . , L , by where we introduced the following operators: The last two factors in the r.h.s. of each of the equations (2.3) are the usual JW transformations [19] and give the anti-commutation between terms in the same leg. The anti-commutation between different legs is provided by the first factor, i.e., by the operators η a , with a = x, y, z , defined by equations (2.4). The choice (2.4) for the operators η a is due to the need to satisfy the three following requests: i) the operators c α ( j ) have to be fermionic; ii) the operator η a has to be a -th component of a spin operator; iii) the operators c α ( j ) and the operators η a have to commute. Defining as usual c α ( j ) † as the conjugate transpose of c α ( j ) (for α = 1 , 2 , 3 and j = 1 , 2 , . . . , L ), one can indeed show that c α ( j ) and c α ( j ) † are fermionic operators [property i )] and that they satisfy for α, β = 1 , 2 , 3 and j, k = 1 , 2 , . . . , L the following anti-commutation relations: (where { . , . } stands for the anti-commutator). Furthermore the operators η a share the same relations of the the Pauli matrices [property ii )], since they satisfy, for a, b = x, y, z , An important point is that η x commutes with ∏ j -1 k =1 σ z 1 ( k ) σ -1 ( j ) but anti-commutes with ∏ j -1 k =1 σ z 2 ( k ) σ -2 ( j ). Finally, we observe for a = x, y, z , β = 1 , 2 , 3 and j = 1 , 2 , . . . , L , the following relations hold: according to the requested property iii ). The factor η a in equations (2.3) may be viewed as a Klein factor, which has been used extensively in literature: it allows one to define correctly the bosonization [45] (see also [46, 47, 48]) and it has been used in different contexts, including the 2-channel Kondo model [49], quantum wire junctions described by coupled TLL [50, 26] or the free quantum field theory on a star graph [51]. We conclude this subsection by emphasizing that the introduction of the auxiliary site and the JW transformation (2.3) do not depend on the explicit form of the Hamiltonian. Therefore, the construction proposed here may be applied to other models as the anisotropic XY model with a transverse magnetic field.", "pages": [ 4, 5 ] }, { "title": "2.2 The Kondo model", "content": "By using the result of Section 2.1, it is possible to construct a model equivalent to H XX 3 expressed in terms of fermions. Indeed, by using relations (2.3), we can express the Hamiltonian H XX 3 in terms of the operators c α ( j ), c α ( j ) † and η a as follows To write more compactly the Hamiltonian (2.8) we introduce { S x , S y , S z } , the su (2) generators in the 3-dimensional representation, as Then, for ρ ∈ R , the Hamiltonian (2.8) becomes Finally, by introducing the vectorial notation the Hamiltonian (2.10) may be rewritten in a more compact way as The expression (2.10) is valid for three legs and it allows us to interpret the Hamiltonian H XX 3 as the Hamiltonian of free fermions coupled with a magnetic impurity. More precisely, it is a su (2) Kondo model with free fermions in the spin 1 representation and a magnetic impurity in the fundamental representation. The historical Kondo model [7] - studied using, for example, perturbation theory [52], numerical renormalization group [53] or exact methods [54, 55] - corresponds to spin 1 / 2 free fermions coupled with a spin 1 / 2 impurity. Different generalizations have been introduced and studied: spin S impurities [56], the su ( N ) version, so-called the Coqblin-Schrieffer model [57], the multi-channel Kondo models [58] or the multi-channel su ( N ) fermions in the fundamental representation with a spin S impurity [59, 60]. The most relevant results for our case are given in [61, 62]. These papers showed that the dynamics of the spin sector of the single channel Kondo model coupling spin j fermions with a spin S impurity is similar to the ones of the k ( j ) = 2 j ( j + 1)(2 j + 1) / 3 channel Kondo model coupling spin 1 / 2 fermions with a spin S impurity. Then, using the results for the multi-channel Kondo model obtained previously by the Bethe ansatz method [63, 64] and by the conformal field theory [65], the universal exponents for the thermodynamic quantities (e.g. the susceptibility or the specific heat) can be obtained. A general discussion of the mapping between different multichannel exchange models with spin j fermions, impurity spin S and channel number N f is presented in [66]. These results applied to our case leads to the fact that our model (2.10), in the continuum limit, is equivalent to the 4-channel Kondo model coupling spin 1 / 2 fermions with a spin 1 / 2 impurity. Then, since the number of channels is larger than twice the impurity spin, we deduce that our system shows a non-Fermi liquid behavior [58] and that the impurity contributes, for example, to the susceptibility and the specific, respectively, as following: We conclude this section by further commenting on the JW transformation we used, to more clearly show that the standard JW transformation does not generally give for a star graph a local and quadratic fermionic Hamiltonian and that the JW transformation (2.3)-(2.4) based on the introduction of the space of an auxiliary, fictitious site is functional to have the desired commutation relations between the fermionic operators and the spin η . We start by observing that one may think of using the JW transformation (3.2) instead of (2.3), replacing η a by σ a (0). However, in this case, although we get a Hamiltonian similar to (2.8), the fermionic operators obtained do not satisfy the commutation relations (2.7) with the η 's. Therefore, we cannot anymore to directly interpret the model as a Kondo model. Furthermore one could also think of different JW transformations without adding an auxiliary space. However, such transformations have their drawbacks: for example, if one performs the transformations (2.3) without the first factor (the η 's), one obtains a quadratic Hamiltonian, but quadratic in hardcore bosons, not fermions. Alternatively, one could consider a JW transformations following a 'spiral' according the operators c j are fermionic, however the Hamiltonian finally obtained is not quadratic in these operators.", "pages": [ 6, 7 ] }, { "title": "3 Free fermions on a star graph and associated spin chains", "content": "In this section we investigate if it is possible to find a XY model on a star graph which, after a JW transformation gives only a quadratic fermionic Hamiltonian. In comparison with the previous section, we allow ourselves to modify the interaction between the spins near the vertex.", "pages": [ 7 ] }, { "title": "3.1 Link between Hamiltonians", "content": "Since it would be very cumbersome to explore all the interactions between spins at the vertex to find the ones providing a quadratic fermionic Hamiltonian, we proceed in the following way: we start from a quadratic fermionic Hamiltonian on the star graph and we perform a JW transformation, obtaining a quantum spin model on the star graph. In this section, we restrict ourselves to the three-leg star graph (one may extend easily the obtained results). We start from the following quadratic fermionic Hamiltonian on a three-leg star graph where γ , a α and b α are coupling constants, d α ( j ) and d α ( j ) † are fermionic operators and we have used the conventions d 4 (1) := d 1 (1) , d † 4 (1) := d † 1 (1). As in the previous section, instead of ˜ H QF 3 we consider the Hamiltonian H QF 3 = Id (0) ⊗ ˜ H QF 3 which trivially acts on a supplementary C 2 -space, the auxiliary space denoted by 0. Then, we perform the following JW transformation: with j = 1 , 2 , . . . , L . A straightforward computation shows that the R.H.S. of (3.2) are fermionic operators (notice the differences with the previous Jordan-Wigner transformations (2.3)). By using the transformations (3.2), the quadratic fermionic Hamiltonian becomes the following quantum spin chain where This shows that it is possible to obtain a XY model on a three-leg star graph which is equivalent to a quadratic fermionic Hamiltonians. Notice, however, that the interactions between spins at the center of the star graph are not of the XY type and involve three spins.", "pages": [ 7, 8 ] }, { "title": "3.2 Solution for a α = a and b α = γ = 0", "content": "The Hamiltonian ˜ H QF 3 given in relation (3.1) is a quadratic fermionic Hamiltonian. Therefore, it can be diagonalized by usual procedures [67, 68]. Evidently, this result provides also the spectrum of the Hamiltonian (3.3) since it is the same spectrum with all the degeneracies multiplied by two. To present a specific example, we consider in the following the case a α = a ∈ R , b α = 0 and γ = 0 (for α = 1 , 2 , 3). We can rewrite the Hamiltonian (3.1) as where we have changed the numeration d α ( j ) → d ( α -1) L + j and the entries of the matrix A are 0 everywhere except To diagonalize ˜ H QF 3 one has to diagonalize the 3 L times 3 L matrix A : it is possible to show that the eigenvalues of A are the roots of the following three polynomials where U L ( x ) is the Chebyshev polynomials of the second kind of degree L . Let us remark that, for a = 0, we get three times the same equation which is expected since the system becomes three identical decoupled systems. The Hamiltonian then becomes where ξ k are fermionic operators and { λ k } is the set of the 3 L solutions of (3.8). Therefore, the spectrum of ˜ H QF 3 is the following set of 2 3 L elements and the eigenvalue of the ground state is -1 2 ∑ 3 L k =1 | λ k | . Finally, the problem is solved when the 3 L solutions of (3.8) are given. It is easy to find them numerically or even analytically. Indeed, for the last equation, its roots are 2 cos ( kπ/ ( L +1)) with k = 1 , 2 , . . . , L : we present them in Figure 2 for a = 1 and L = 150 as a dispersion relation. We remark that the three sets of solutions are very similar (in Figure 2, these three sets of roots are superimposed). The main difference relies on the presence of one isolated point for each of the first two equations. As shown in subsection 3.1, this result provides the spectrum for the quantum spin model (3.3).", "pages": [ 8, 9 ] }, { "title": "4 Conclusions", "content": "In this paper we studied the XX model for quantum spin model on a three-leg star graph: we showed that by introducing an auxiliary space and performing a Jordan-Wigner transformation, the model is equivalent to a generalized Kondo Hamiltonian in which the free fermions, in the spin 1 representation of su (2), are coupled with a magnetic impurity. Using previous results, we deduce that it is also equivalent to a 4-channel Kondo model with spin 1/2 fermions coupled with spin 1/2 impurity and conclude that it shows a non-Fermi liquid behavior. We also showed that it is possible to find a XY model such that - after the Jordan-Wigner transformation - one obtains a quadratic fermionic Hamiltonian directly diagonalizable. We observe that we may think of different generalizations of our method. Indeed, our method based on the Jordan-Wigner transformation (2.3) could be used for the Hamiltonian obtained replacing the XX Hamiltonian (2.1) by the anisotropic XY model with a transverse magnetic field. In perspective, one can also think to investigate more complicated graphs as the star graph with a number of legs p > 3 or as comb-like graphs: in order to get a Kondo-like Hamiltonian, i.e., an Hamiltonian of fermions coupled with magnetic impurities, one should identify the correct Klein factors, which in the present case p = 3 are given by equations (2.4). Furthermore, the properties of a quantum Ising model in a transverse magnetic field on a graph G may be related to the partition function of the classical Ising model in a corresponding higher-dimensional graph. Namely, the partition function of the classical Ising model on a graph made up of n copies of the 'base' graph G with couplings between corresponding sites of the adjacent copies can be written as the trace of the transfer matrix V to the power n where V is written as exponentials of terms proportional to σ z σ z and σ x . By organizing the factors in the exponentials, we may recognize the exponential of a quantum Ising model. For example, for a square (resp. cubic) lattice, the 'base' graph G is a line (resp. square): e.g., for the square, V can be written in terms of the quantum Ising model in a transverse magnetic field on the segment [20]. Therefore getting results for Kondo Hamiltonians of type (2.12) obtained from quantum Ising models on star-like graphs may be relevant to study the classical Ising model in non trivial geometries. The mapping presented in this paper between the XX quantum spin model on the star graph and the Kondo model illustrates that the introduction of a non trivial topology, even locally, can provide new interesting physical phenomena in comparison to models on the line or on the circle. At the same time, our results show that one may also think to use the XX model on a star graph to realize (or simulate) a Kondo model: to this respect we mention that a similar Hamiltonian, describing Majorana fermions, can be realized in a superconductor, coupled to conduction electrons [69]. Acknowledgments: We would like to thank V. Caudrelier, D. Giuliano, M. Fabrizio, P. Sodano and P.B. Wiegmann for very useful discussions. Useful correspondences with B. Beri, P. Lecheminant, D.C. Mattis and A.M. Tsvelik are also gratefully acknowledged. Note Added: After this paper was submitted, several very interesting papers on Y -systems appeared on the arXiv. In [70] the problem of an Ising model in a transverse field has been studied on the star graph: in the continuum limit, close to the quantum phase transition point and for coupling ρ << 1, the effective Lagrangian was worked out and the model shown to be equivalent to the overscreened two-channel Kondo model [70]. As previously mentioned, the approach presented in our paper can be used for the general case of an anisotropic XY in a transverse field: it would then very interesting to study the Kondo problem in such more general model, determining in particular how the low-energy physics varies across the parameter space (i.e, varying the anisotropy and the magnetic field). A discussion of the coupling of Majorana fermions to external leads was presented in [71]: the Klein factors of bosonization appear as extra Majoranas hybridizing with the physical ones and a SO ( M ) Kondo problem was shown to arise [71]. In [72] it was studied a setup with nanowires in proximity to a common mesoscopic superconducting island, showing that a weak finite charging energy of the center island may considerably affect the low-energy behavior of the system. Finally, in [73] a general Majorana junction was considered and the conditions for even-odd parity effects in the tunnel conductance for various junction topologies were examined.", "pages": [ 9, 10 ] } ]

[ { "title": "ABSTRACT", "content": "Nuclear Physics B Proceedings Supplement 00 (2021) 1-6", "pages": [ 1 ] }, { "title": "The strange case of HESS J0632 + 057 and the GLYPH<13> -ray High Mass X-ray Binaries", "content": "G. A. Caliandro[1] and A. B. Hill[2][3], on behalf of the Fermi -LAT collaboration a Institut de Ciencies de l'Espai (IEEC-CSIC), Campus UAB, 08193 Barcelona, Spain b W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA c School of Physics and Astronomy, University of Southampton, Highfield, Southampton, SO17 1BJ, UK", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the last decade Cherenkov telescopes on the ground and space-based GLYPH<13> -ray instruments have identified a new class of high mass X-ray binaries (HMXB), whose emission is dominated by GLYPH<13> rays. To date only five of these systems are known. All of them are detected by Cherenkov telescopes in the TeV energy range, while at GeV energies there is still one (HESS J0632 + 057) that has no reported detection with the Fermi -LAT. A deep search for GLYPH<13> -ray emission of HESS J0632 + 057 has been performed using more than 3.5 years of Fermi -LAT data. We discuss the results of this search and compare it to other GLYPH<13> -ray binary systems. Keywords: binaries: general, Gamma-rays: observations, binaries: individual(HESS J0632 + 057)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The High Mass X-ray Binaries (HMXBs) are relatively young ( < 10 8 year) Galactic sources composed of a massive OB or Be type star and a compact object, either a neutron star or a black hole. These systems are bright X-rays emitters, generally (but not always) due to matter from the massive star accreting onto the compact object. In the last decade a handful of HMXBs have been detected at high (HE; 0.1-100 GeV) or very highenergies (VHE; > 100 GeV). They are LS I + 61 GLYPH<14> 303 [14, 7, 2], LS 5039 [11, 3], PSR B1259 GLYPH<0> 63 [10, 6, 44], 1FGL J1018.6 GLYPH<0> 5856 [9], Cyg X GLYPH<0> 3 [4], and Cyg X GLYPH<0> 1 [15, 41]. With the exception of the last two sources, for the others the power of their emission is dominant at high energies, rather than at longer wave-lengths. For this reason they are some times mentioned with the name of GLYPH<13> -ray HMXBs. Recently a new source has been claimed to belong to this class of sources. It is HESSJ0632 + 057. In this proceeding we summarize the path covered from the discovery of HESS J0632 + 057 as unidentified source, to the claiming of its binarity. Finally, we describe the deep search for its emission in the GeV energy range, and we discuss the analogies and the di GLYPH<11> erences that this particular source shows with respects the others GLYPH<13> -ray HMXBs.", "pages": [ 1 ] }, { "title": "2. Discovery, and Cherenkov monitoring", "content": "HESS J0632 + 057 was discovered at VHE energies by the High Energy Spectroscopic System (H.E.S.S.) on 2007 [13]. It is in the Monoceros Loop region, very close to the Rosette Nebulae. It was classified as a point-like unidentified source. It is interesting to note that all the VHE Galactic sources are extended, with the exception of the HMXBs. Therefore, the pointlike feature of HESS J0632 + 057 was a strong indication of its binary nature. Further more, of many tens of unidentified sources discovered in the Galactic Plane H.E.S.S. survey, HESS J0632 + 057 is one of the only two unidentified VHE sources which appears to be point-like within the experimental resolution of the instrument. The other (HESS J1745-290) is coincident with the gravitational center of the Milky Way [12]. When HESS J0632 + 057 was discovered, several plausible association with sources in other wavelengths were discussed. A posteriori, the most relevant Nuclear Physics B Proceedings Supplement was the association with a massive emission-line star of spectral type B0pe (MWC 148, see section 4). The flux and the spectral index of this source measured by H.E.S.S. are F ( E > 1TeV) = 6 : 4 GLYPH<6> 1 : 5 GLYPH<2> 10 GLYPH<0> 13 cm GLYPH<0> 2 s GLYPH<0> 1 , GLYPH<0> = 2 : 53 GLYPH<6> 0 : 26, respectively. Subsiquently to the discovery, VERITAS and MAGIC started to monitor HESS J0632 + 057. It was observed by VERITAS for 31 hr in 2006, 2008, and 2009. The first observation was pointed toward the center of the Monoceros region, (at an angular distance of GLYPH<24> 0 GLYPH<14> : 5 from the HESS J0632 + 057). The other observations were targeted around the reported position of HESS J0632 + 057. During these observations, no significant signal in at energies above 1 TeV was detected, and a flux upper limit of 4 : 2 GLYPH<2> 10 GLYPH<0> 13 cm GLYPH<0> 2 s GLYPH<0> 1 at 99% confidence level [8]. The non detection by VERITAS during these observations provides variability in the VHE flux of HESS J0632 + 057. This was in favor with the binary interpretation of this source. Afterwards, the source was detected by MAGIC and VERITAS in February 2011, when TeV observation were triggered by an outburst in X-rays [24, 31, 16]. The spectrum measured by MAGIC is compatible with the one measured by H.E.S.S. Further observations with VERITAS have been performed until February 2012 [32]. In the right panel of figure 1 are plotted the integral GLYPH<13> -ray fluxes above 1 TeV of HESS J0632 + 057 measured by VERITAS in the di GLYPH<11> erent observations.", "pages": [ 1, 2 ] }, { "title": "3. X-ray follow up, and orbital period detection", "content": "Since its discovery, HESS J0632 + 057 was followed by several X-ray observatories. As first, an observation taken on the 2007 September 17 with the EPIC camera of XMM-Newton lead to the detection of a new Xray source (XMMU J063259.3 + 054801) within the error box of HESS J0632 + 057, and coincident with the massive star MWC 148. This source exhibits a hard spectrum, consistent with an absorbed power law with GLYPH<0> = 1 : 26 GLYPH<6> 0 : 04, and the unabsorbed 1 GLYPH<0> 10 keV flux is (5 : 3 GLYPH<6> 0 : 4) GLYPH<2> 10 GLYPH<0> 13 erg cm GLYPH<0> 2 s GLYPH<0> 1 . A long monitoring of this source was planned with the Swift X-ray telescope. A first set of data covering 108 days was analyzed by [23]. They measured flux increases by a factor GLYPH<24> 3 : , and found that X-ray variability was present on multiple timescales including days to months. The X-ray outburst on February 2011 that was announced by [24] triggered the observation of VERITAS and MAGIC (see section 2), as well as an observation with Chandra requested by [40]. Comparing the results of the Chandra data analysis with those of the first observation of XMM-Newton [40] found also a a spectral variability in addition to the flux one. Indeed, the spectral index measured was 1 : 61 GLYPH<6> 0 : 03. The flux and spectral variability are both characteristic features of the HMXBs, but in contrast with what observed for the other TeV binaries, in this source the higher the flux the softer the X-ray spectrum. Finally, the detection of the orbital periodicity of 321 GLYPH<6> 5 days was claimed by [19]. The periodical Xray flux modulation was the result of the analysis of the long set of Swift observations of the source from 2009 January to 2011 March. In figure 1 (left plot) is shown the folded X-ray light curve of HESS J0632 + 057. It shows a narrow high peak at phase 0.3 followed by a dip feature, and centered at phase 0.75 there is a moderate broad peak. The XMM-Newton observation fall in the dip feature, where the spectrum is harder, while the Chandra observation was taken during the high peak. Phase 0.0 in this plot is arbitrary set at the beginning of the Swift observations.", "pages": [ 2, 3 ] }, { "title": "4. Optical and Radio counterparts", "content": "MWC 148 is the companion massive star in the binary system HESS J0632 + 056. It is a Be star, as in the case of the HMXB systems LS I + 61 GLYPH<14> 303, and PSR B1259-63. Before the detection of the orbital period in X-rays, [17] analyzed optical spectra of the star acquired to investigate the stellar parameters, the properties of the Be star disk, and evidence of the binarity. They derived a mass of 13.2-19.0 M GLYPH<12> , and a radius of 6.0-9.0 R GLYPH<12> . Fitting the spectral energy distribution they found a distance between 1.1 and 1.7 kpc. Since their dataset was too short ( GLYPH<24> 1 month) it was not possible to detect the orbital period of the system. On the other hand, [21] collected optical spectra observations spanning for 4 years from 2008 October to 2011 May. They calculated the orbital parameters of the binary systems, finding that it is highly eccentric with e = 0 : 83 GLYPH<6> 0 : 08. Furthermore, the orbital solution implies that the high peak in the X-ray light curve is delayed GLYPH<24> 0 : 3 orbital phases after the periastron passage, similar to the case of LS I + 61 GLYPH<14> 303. In the radio regime [43] detected with the Giant Metrewave Radio Telescope (GMRT) and the Very Large Array (VLA) observations at 1.28 and 5 GHz, respectively, a point-like, variable radio source at the position of MCW 148. [34] observed HESS J0632 + 057 with the European VLBI Network (EVN) at 1.6 GHz in two epochs: during the January / February 2011 Xray outburst and 30 days later. In the first epoch the source appears point-like, whereas in the second one it appears extended with a projected size of GLYPH<24> 75 AU assuming a distance to the system of 1.5 kpc. The brightness temperature of 2 GLYPH<2> 10 6 K at 1.6 GHz together with the negative spectral index around GLYPH<0> 0.6 point to nonthermal synchrotron radiation producing the radio emission [43, 34]. The detected projected displacement of the peak of the emission of 21 AU in 30 days, the morphology and the size of HESS J0632 + 057 are similar to the ones observed in the well-established GLYPH<13> -ray binaries PSR B1259 GLYPH<0> 63, LS 5039, LS I + 61 GLYPH<14> 303.", "pages": [ 3 ] }, { "title": "5. Search for GeV emission", "content": "All the GLYPH<13> -ray HMXBs have a bright emission in the GeV energy range, but HESS J0632 + 057 is not present in the Second Fermi -LAT Source Catalog [35] (2FGL hereafter). Therefore, we performed a fine analysis with the Fermi -LAT data in order to deeply search for GeV emission. A dataset of GLYPH<24> 3 : 5 years from 4 th August 2008 to 3 rd March 2012 has been analyzed, that is almost twice the dataset used for the 2FGL. HESS J0632 + 057 lies at b = GLYPH<0> 1 : 44 GLYPH<14> from the Galactic plane, in the active region of the Monoceros loop. The region at GeV energies appears as shown in Figure 2, where the sources from the 2FGL catalog are labeled with a green cross, the yellow one in the center marks the position of HESS J0632 + 057, while in cyan is marked the pulsar PSR J0633 + 0632. It is evident that the pulsar emission dominates the region aroung HESS J0632 + 057. Their angular distance is only 0.78 GLYPH<14> .", "pages": [ 3 ] }, { "title": "5.1. PSR J0633 + 0632", "content": "PSR J0633 + 0632 is a bright radio-quiet GLYPH<13> -ray pulsar discovered in the first six months of the Fermi mission [1]. Indeed, it is listed in the first Fermi -LAT catalog of GLYPH<13> -ray pulsars [5]. It has a flux of 8 : 4 GLYPH<6> 1 : 2 GLYPH<2> 10 GLYPH<0> 8 ph cm GLYPH<0> 2 s GLYPH<0> 1 , and the pulse profile has the characteristic two narrow peaks separated GLYPH<24> 0 : 5 in phase. The weighted pulse profile of PSR J0633 + 0632 is shown in figure 3. A weight was associated to each event, corresponding to the probability that the event is emitted by the pulsar, rather than by a nearby source or by the diffuse emission. We defined the o GLYPH<11> -pulse phases of PSR J0633 + 0632 (0.66 - 0.98) applying a Bayesian block algorithm adapted for weighted-counts light curves [20]. In figure 3 the green line shows the calculated Bayesian blocks. The o GLYPH<11> -pulse is defined as the lowest block reduced by 10% of its total length at each edge. As well, we defined the intra-peaks or bridge phases (0.21 - 0.49). In order to avoid the contamination from the strong emission of PSR J0633 + 0632, we carried on the analysis of HESS J0632 + 057 selecting the o GLYPH<11> -pulse and the intra-peaks phases of PSR J0633 + 0632. With this phase selection the regioin around HESS J0632 + 057 appears quite crowded, and not perfectly modeled by the 2FGL sources. This is mainly due to the fact that the 2FGL catalog is built with 2 years of Fermi -LAT data, while in this analysis the dataset is almost doubled ( GLYPH<24> 3 : 5 years). Figure 4 shows the region out of the pulsar peaks. The 2FGL sources are labeled with a cross, and HESS J0632 + 057 by the yellow circle. Before to analyze directly HESS J0632 + 057 we as first applied an iterative method to reach a good modeling of the region around it. The method developed is described in detail in [20]. It accounts for the excesses in the map adding new point-like sources in the model of the region.", "pages": [ 3, 4 ] }, { "title": "5.2. Upper limits of HESS J0632 + 057", "content": "Once we obtained a good model for the region, we came back to focus specifically on the analysis of HESS J0632 + 057. We found no significant detection of the GeV emission of HESS J0632 + 057. Then, we derived the 95% flux upper limit for this source. The Helene's Bayesian method [25] was adopted to evaluate them. We calculated the flux upper limit for energies E > 100 MeV of F 100 < 3 : 0 GLYPH<2> 10 GLYPH<0> 8 cm GLYPH<0> 2 s GLYPH<0> 1 .", "pages": [ 4 ] }, { "title": "5.3. Folded analysis and variability", "content": "In the analysis described so far we have considered HESS J0632 + 057 as a steady source, and we concluded that its average GLYPH<13> -ray emission is below the sensitivity of Fermi -LAT. But it is still possible that the system have a brighter emission during a specific part of the orbit, like for the case of the HMXB system PSR B1259-63 ([6], [44]). In order to investigate this hypothesis, a further analysis was performed folding the Fermi -LAT data with the orbital period of 321 days, and setting the phase 0.0 to the epoch 54857 MJD [19]. We subdivided the orbit in 8 equally spaced phase bins, and for each of them we calculated the significance of the GeV emission. The results are showed in figure 5. We found that none of the bins have a significacnce above the detection threshold. Finally, we also searched for flares or high flux states of HESS J0632 + 057, calculating light curves with different time binning scales, from days to months. For none of the time scales investigated we got a significant signal.", "pages": [ 4 ] }, { "title": "6. Conclusions", "content": "In this proceeding we have summarized all the steps leading to the claim that HESS J0632 + 057 is a GLYPH<13> -ray HMXB system. Fermi -LAT data has been analyzed with the aim to search for the emission in the GeV energy range of HESS J0632 + 057. The analysis performed needed particular attention especially for the presence of a very nearby GLYPH<13> -ray pulsar PSR J0633 + 0632 (section 5.1). As first, we selected the events out of the peaks of the pulsars. Then, the region around the source was modeled adding new point-like sources in correspondence of unmodeled excesses with an iterative method. Despite the good modeling of the region, we did not detected any significant signal of GLYPH<13> -ray emission from HESS J0632 + 057. We calculated the upper limit for its permanent GeV emission, F 100 < 3 : 0 GLYPH<2> 10 GLYPH<0> 8 cm GLYPH<0> 2 s GLYPH<0> 1 . We also searched for GeV emission from a restricted portion of the orbit phases, as well as for plausible flares, or changes of state in its flux. The lack of significant signal from all these searches is di GLYPH<14> cult to understand if compared with the behavior of the other GLYPH<13> -ray HMXBs in the same energy range. Similar to HESS J0632 + 057, the massive stars in LS I + 61 GLYPH<14> 303 and PSR B1259-63 systems are Be star, that are characterized by the presence of a dense circumstellar disk. The distances of these three HMXBs are comparable: GLYPH<24> 1 : 5 kpc for HESS J0632 + 057, GLYPH<24> 1 : 9 kpc for LS I + 61 GLYPH<14> 303, and GLYPH<24> 2 : 3 kpc for PSR B1259-63. In contrast their orbital periods are significantly di GLYPH<11> erent. LS I + 61 GLYPH<14> 303 has the shortest orbital period of about one month (26.5 days). As consequence, the systems is also the most compact among the three. In the GeV energy range, LS I + 61 GLYPH<14> 303 has a continuous emission modulated by its orbital motion. Most probably the GeV emission is present in every phase of the orbit because it is so small that the compact object is always interacting with the circumstellar disk, or with the stellar wind. A di GLYPH<11> erent case is that of PSR B1259-63 with its very large orbital period of about 3.5 years. The orbit of this system is highly eccentric, with eccentricity e GLYPH<24> 0 : 87. This makes that when the compact object is close to the periastron, it pass through the circumstellar disk. Only during this passage PSR B1259-63 emits GeV and TeV GLYPH<13> -rays. HESS J0632 + 057 has an orbital period of about 1 year, that is in between those of LS I + 61 GLYPH<14> 303 and PSR B1259 + 63. The eccentricity of the orbit of HESS J0632 + 057 is also very high, with e GLYPH<24> 0 : 83. We expected a GeV emission at least during the periastron passage, but we did not detected it. This can be attributed to di GLYPH<11> erent factors. An important role can be played by the features of the circumstellar disk, like its size, its density, and its inclination respect to both the observer line of sight and the plane of the orbit. On the other hand, the di GLYPH<11> erences among LS I + 61 GLYPH<14> 303 PSR B1259 + 63, and HESS J0632 + 057, can also be attributed to the features of the compact objects, like the intensity of their relativistic winds. Or they could be caused by the di GLYPH<11> erent nature of the compact object itself, either a pulsars (as for PSR B1259 + 63) or a black hole. All these hypothesis are currently under investigation. Acknowledgments: The Fermi -LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT as well as scientific data analysis. These include NASA and DOE in the United States, CEA / Irfu and IN2P3 / CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the National Space Board in Sweden. Additional support from INAF in Italy and CNES in France for science analysis during the operations phase is also gratefully acknowledged. We thank the Spanish MICINN for additional support.", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "Nuclear Physics B Proceedings Supplement 00 (2022) 1-6", "pages": [ 1 ] }, { "title": "Paolo Giommi", "content": "ASI Science Data Center (ASDC), Via del Politecnico snc, I-00133 Roma, Italy", "pages": [ 1 ] }, { "title": "Abstract", "content": "Contemporary astronomy benefits of very large and rapidly growing amounts of data in all bands of the electromagnetic spectrum, from long-wavelength radio waves to high energy gamma-rays. Astronomers normally specialize in data taken in one particular energy window, however the advent of data centers world-wide and of the Virtual Observatory, which provide simple and open access to quality data in all energy bands taken at di GLYPH<11> erent epochs, is making multi-frequency and multi-epoch astronomy much more a GLYPH<11> ordable than in the past. New tools designed to combine and analyze these data sets are being developed with the aim of visualizing observational results and extracting information about the physical processes powering cosmic sources in ways that were not possible before. In this contribution blazars, a type of cosmic sources that emit highly variable radiation at all frequencies, are used as an example to describe the possibilities of this type of astronomy today, and the discovery potential for the near future. Keywords: astronomy, multi-frequency, timing analysis", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Modern astronomy rests upon highly technological ground and space-based observatories that are capable of probing the sky with high sensitivity in almost all bands of the electromagnetic spectrum. As a consequence, extremely large and rapidly growing amounts of high-quality digital data are being accumulated. Archive data centers, that often openly provide ready-to-use data products based on consolidated data format like FITS, together with the rapid increase of computing power and network communication speed, and the existence of world-wide initiatives like the Virtual Observatory (VO) [see e.g. 1] are providing unprecedented opportunities to obtain high quality multifrequency data. A new era of scientific discovery, based on large amounts of archival and fresh data covering the entire electromagnetic spectrum and accumulated over a very wide time interval, has started. Existing digital archives typically include astronomical data of one of the following types In this contribution I use blazars, a special type of extragalactic sources that emit highly variable radiation across the electromagnetic spectrum, to illustrate how this new opportunity of accessing large amounts of spectral and timing data is currently exploited in terms of techniques for visualization and analysis. I also briefly describe some new software tools, developed within the VO or related activities, that can be used to e GLYPH<14> ciently retrieve and analyze multi-frequency multi-temporal archival data. Nuclear Physics B Proceedings Supplement", "pages": [ 1 ] }, { "title": "2. Blazars as an example of multi-frequency multitemporal data analysis", "content": "Blazars are a special type of Active Galactic Nuclei (AGN) that are known to be strong emitters in all bands of the electromagnetic spectrum. These peculiar sources, known since the discovery of AGN fifty years ago [2], display very unusual properties like superluminal motion and are the most variable persistent sources in the extragalactic sky. The extreme properties of blazars are thought to be the result of emission from charged particles interacting with a magnetic field in a jet of plasma that moves at relativistic speeds and happens to point very close to the line of sight [see 3, for a review]. These are conditions that can happen only rarely, and that is why only about 3,000 blazars are known [4], compared to over one million AGN (http: // quasars.org / milliquas.htm). Over the past several years blazars have been observed, often repeatedly, at all frequencies; in some cases, especially in the radio and optical bands, some of the brighter ones have been monitored for long periods. Consequently there are many databases and catalogs that include measurements of blazars at all frequencies (e.g. radio, mm, IR, optical, UV, X-ray and gamma-ray). The broad-band emission in blazars is traditionally represented as Spectral Energy Distributions (SED), that is plots of intensity (usually flux density multiplied by frequency, GLYPH<23> f ( GLYPH<23> ), (or luminosity, GLYPH<23> L ( GLYPH<23> )) versus energy or, equivalently, frequency GLYPH<23> , of the emitted radiation. As an example, figure 1 shows the SED of the blazar 3C454.3, currently one of the most densely populated existing SED as it includes approximately 30,000 independent flux measurements collected over a time period of more than thirty years. A large fraction of the data shown in Fig. 1 comes from monitoring programs and from on-line databases like UMRAO (dept.astro.lsa.umich.edu / datasets / umrao.php) at 5, 8 and 14.5 GHz, OVRO (www.astro.caltech.edu / ovroblazars)[5] at 15GHz, Metsahovi (metsahovi.aalto.fi / en / research / projects / quasar / ) at 37 GHz, SMARTS (www.astro.yale.edu / smarts)[6] in the optical and infrared bands, WEBT (www.aoto.inaf.it / blazars / webt) [7] at optical, IR and radio frequencies, the BeppoSAX and Swift data bases in the X-rays, and Fermi in the gamma-ray band (www.asdc.asi.it / mmia). The 1GeV light-curve (that appears as a vertical line at 2.4 10 23 Hz in Fig. 1) was built with Fermi-LAT data using the adaptive-bin method developed by [8] Another important example of multi-frequency data acquisition is the organization of campaigns of simultaneous observations of one or more sources involving several di GLYPH<11> erent facilities. The data collected in these cases are more homogeneous. An example of this approach is the Planck, Swift, Fermi and ground-based simultaneous observations of a large sample of blazars, including 175 sources selected according to four di GLYPH<11> erent criteria in the radio, X-ray and gamma-ray band [9]. Figure 2 shows the SED of the source 3C279 taken from [9] which includes simultaneous data covering a spectral range of 15 orders of magnitudes. The simultaneous measurements from the instruments of Planck (LFI, HFI) [10], Swift (UVOT, XRT) [11] and Fermi (LAT) [12] are shown as red points, while quasisimultaneous data (i.e. observations carried our within two months of each other) are plotted as orange points. Archival data taken at di GLYPH<11> erent random times appear as gray points. Simultaneous data is clearly crucial for measuring the parameters related the emission process, like the energy where the emitted power peaks and the intensity level of the peak. As Figs. 1 and 2 demonstrate, the amplitude of variability in blazars is a strong function of the energy where the emission occurs, ranging from a factor of a few in the radio band, and up to a factor 10,000! at 1GeV. This dependence requires that the time scale of variability in each energy band must be properly taken into consideration when dealing with multi-frequency data that is not simultaneous. Clearly, observation times must be visualized with methods that go beyond simple spectral distributions like that of Fig. 1.", "pages": [ 2, 3 ] }, { "title": "3. SED software tools", "content": "As described above the remarkable number of measurements now accessible and the very large intensity variations observed in several well-known sources require that the interpretation of the physical emission processes is carried out by analyzing the SEDs in the time domain. Until recently, however, existing software packages for building SEDs did not take into account of time. This is rapidly changing and new tools capable of handling the time variable are appearing or are planned for the near future. Figure 3 is an example of how the time dependance of the energy distribution of a source (3C454.3 in this case) can be visualized as a 3D plot, illustrating in a single picture how time scales, flare details, variability amplitudes and time lags vary across the electromagnetic spectrum. This plot was generated using the TOPCAT application (http: // www.star.bris.ac.uk / GLYPH<24> mbt / topcat), which is a widely used interactive graphical viewer developed as part of some UK and Euro-VO projects. In the following I briefly describe two new software tools that can be used to download multi-frequency measurements from many di GLYPH<11> erent catalogs, databases and sky surveys, and build and analyze blazar SEDs.", "pages": [ 3 ] }, { "title": "3.1. The IRIS SED analysis tool", "content": "IRIS is a JAVA application developed as part of the activities of the Virtual Astronomical Observatory (VAO), the US contribution to the world-wide VO initiative. IRIS can retrieve data using VO protocols from the National Extragalactic Database (NED) and from the ASI Science Data Center (ASDC). It can be used to plot and fit Spectral Energy Distributions in a number of ways. As an example Fig. 5 shows a session of IRIS showing the SED of the blazar MKN421 displayed as a GLYPH<23> f( GLYPH<23> ) vs GLYPH<23> plot. The current version of this desktop application (V2.0) only allows limited control of the time variable, and visualization must be done using energy or frequency (in various units) on the X-axis. The application can be downloaded from the VAO web pages at the following link http: // www.usvao.org / science-toolsservices / iris-sed-analysis-tool /", "pages": [ 3, 4 ] }, { "title": "3.2. The ASDC SED builder", "content": "The ASDC SED builder is a web-based application developed at the ASDC (www.asdc.asi.it). The current version (V3.0, see Fig. 6) allows users to build SEDs using data from a large number of catalogs, on-line services, also in combination with personal data. This version of the tool can handle time resolved SEDs and multi-frequency light-curves. The service can be accessed at tools.asdc.asi.it / SED.", "pages": [ 4 ] }, { "title": "3.3. SEDs and the time domain", "content": "One important question in the analysis of the multifrequency emission in cosmic sources is whether the emission in di GLYPH<11> erent energy bands is correlated. In almost the totality of cases the measurements available are sparse and not uniformly sampled. An e GLYPH<14> cient method of measuring the amount of correlation between the emission in two energy bands with sparse data is the Ztransformed Discrete Correlation Function [ZDCF, see e.g. 13]. Fig. 7 shows the ZDCF of the emission form 3C454.3 at 1mm, 37GHz and 15GHz compared to the gammaray emission at 1GeV. The fluxes are clearly correlated but with time lags that range from approximately 0, for the case of the mm band, to several weeks depending on the frequency in the radio band. Note from Fig. 4 that, although the black (1GeV) and green (1mm) lightcurves show the same overall behavior in terms of peaks and minima occurring approximately at the same time, the emission at 1GeV displays more structured variability, reflecting di GLYPH<11> erent details in the emission mechanism. The optical light curve (red points) follows the 1GeV light curve also in the fine detail, although data in this band is certainly more sparse than at other frequencies. Figure 8 gives the discrete correlation function of the X-ray (1 keV from Swift-XRT data) and gamma-ray (1 GeV from Fermi-LAT data) emission from MKN421 recorded between the summer 2008 and spring 2012. No correlation is observed in this case. This is interesting since MKN 421 is a blazar of the HBL type, that is a blazar that radiates up to the TeV band, and the radiation from this type of sources is often interpreted as due to a single homogeneous component. The complete lack of correlation between the flux emitted in the X-ray and gamma-ray band challenges this simple interpretation. An e GLYPH<14> cient and novel way of representing fast and large variations in the energy distribution of cosmic sources is to run in a sequence a set of frames, each representing the status of the SED in a particular time interval, like in a movie. This, of course, is possible only if a su GLYPH<14> ciently large number of measurements across the electromagnetic spectrum are available at all times. This requirement is already satisfied for a small number of bright sources today; the very rapid increase in data production that we are experiencing ensure that many others will follow in the future.", "pages": [ 4, 5 ] }, { "title": "4. Conclusions and prospects for the future", "content": "Significant progress in the visualization and analysis of multi-frequency multi-temporal astronomical data has been made recently, mostly as part of the VO worldwide initiative and of related activities. This is true both in terms of availability of data from catalogs, databases and surveys, as well as in the development of new software tools. The discovery potential o GLYPH<11> ered by the exponential increase of available data, computer power and communications speed is extremely large. In this contribution I described some examples of the multifrequency data sets and tools that are available today. Of course there is ample room for improvements and much more progress is expected in the near future. With this motivation, and in an attempt of implementing new methods of visualizing variability of the emission across the electromagnetic spectrum, I developed a prototype software tool to run in sequence SEDs corresponding to di GLYPH<11> erent time slices, that is to produce SED movies. Figures 9 and 10 show two frames taken(stills) from the SED movie of 3C454.3 made with this prototype, corresponding to a period of approximately three days in late 2010 when the blazar underwent a very large gamma-ray flare. The data taken during the frame period is shown in red color, while all the remaining data is plotted in light gold color. The top panel shows the intensity of 3C454.3 in the gamma-ray band (1GeV) as a function of time and is used as a way to illustrate the passing of time. The bottom panel shows the full SED of the source using the same color coding for the data. This prototype will be further developed in the near future, likely as a collaborative e GLYPH<11> ort among international institutions, and made openly accessible within the ASDC SED builder. A sample of a full SED movie of the blazar 3C454.3 is currently available on-line and can be seen at the main page of the current version of the ASDC SED tool (V3.0). Usually blazar SED data is compared to theoretical models that are based on the current best understanding of the physical processes responsible for the emission of the multi-frequency radiation. When possible, this is done using simultaneous data gathered through observational campaigns involving ground-based and satellite observatories. However, as shown in Figs. 4 and 7 variability in di GLYPH<11> erent bands follow di GLYPH<11> erent dynamical timescales and may show significant time lags implying that fitting of simultaneous data may not be enough for a full comprehension of the physics at work in the source. New methods of analyzing data, that go beyond fitting simultaneous SEDs and fully take into account of the dynamical time-scales of the emission processes in di GLYPH<11> erent energy bands, require the development of new analysis tools. This is a challenge both for theoretician and scientific software developers to be addressed in the near future. The advent of advanced visualization and fitting techniques such as SED movies and dynamical model fitting will likely happen soon, providing more diagnostic and interpretation power.", "pages": [ 5, 6 ] }, { "title": "Acknowledgments", "content": "I acknowledge the use of archival data and software tools from the ASDC, a facility managed by the Italian Space Agency (ASI). Part of this work is based on archival data from the NASA / IPAC Extragalactic Database (NED).", "pages": [ 6 ] } ]

[ { "title": "Evidence for Recombining Plasma in the Supernova Remnant G346.6 -0.2", "content": "Shigeo Yamauchi 1 , Masayoshi Nobukawa 2 , Katsuji Koyama 2,3 , and Manami Yonemori 3 1 Department of Physics, Faculty of Science, Nara Women's University, Kitauoyanishi-machi, Nara 630-8506 [email protected] 2 Department of Physics, Graduate School of Science, Kyoto University, Kitashirakawa-oiwake-cho, Sakyo-ku, Kyoto 606-8502 3 Department of Earth and Space Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043 (Received 2012 May 29; accepted 2012 August 23)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We present the Suzaku results of the supernova remnant (SNR) G346.6 -0.2. The X-ray emission has a center-filled morphology with the size of 6 ' × 8 ' within the radio shell. Neither an ionization equilibrium nor non-equilibrium (ionizing) plasma can reproduce the spectra remaining shoulder-like residuals in the 2-4 keV band. These structures are possibly due to recombination of free electrons to the K-shell of He-like Si and S. The X-ray spectra are well fitted with a plasma model in a recombination dominant phase. We propose that the plasma was in nearly full ionized state at high temperature of ∼ 5 keV, then the plasma changed to a recombining phase due to selective cooling of electrons to lower temperature of ∼ 0.3 keV. G346.6 -0.2 would be in an epoch of the recombining phase. Key words: ISM: individual (G346.6 -0.2) - ISM: supernova remnants - X-rays: ISM - X-rays: spectra", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "G346.6 -0.2 is a supernova remnant (SNR) discovered in the radio band (Clark et al. 1975). The radio image shows a shell structure with the size of ∼ 8 ' (Clark et al. 1975; Dubner et al. 1993; Whiteoak & Green 1996). Flux densities at 408 MHz, 843 MHz, 1.47 GHz, and 5 GHz were measured to be 14.9 Jy, 8.7 Jy, 8.1 Jy, and 4.3 Jy, respectively (Green 2009 and references therein), then the spectral index was estimated to be 0.5 (Clark et al. 1975; Green 2009). The SNR would be interacted with a molecular cloud because an OH maser was found (Green et al. 1997). The Galactic plane survey project with ASCA discovered a faint X-ray emission from G346.6 -0.2 for the first time (Yamauchi et al. 2008). The ASCA GIS image showed the diffuse X-ray morphology in the radio shell. The X-ray spectrum was represented by either a thermal plasma model with a temperature of ∼ 1.6 keV or a power-law model with a photon index of ∼ 3.7. The absorption was as large as N H =(2-3) × 10 22 cm -2 (Yamauchi et al. 2008), which suggested that the SNR is located at a long distance, possibly in the Galactic inner disk or further. No further quantitative constraint was available with the ASCA data due to the limited photon statistics. G346.6 -0.2 was then observed with Suzaku. Sezer et al. (2011) found strong emission lines of Si and S, and fitted the X-ray spectrum with a model of a power-law (photon index ∼ 0.6) plus a non-equilibrium ionization (NEI) (ionizing) plasma. They predicted that the strong Si and S lines are due to an ejecta-dominated plasma which origi- rom a Type Ia supernova explosion, and the powerlaw component is regarded as synchrotron emission. Recently, strong radiative recombination continua (RRCs) have been discovered in the X-ray spectra of five mixed-morphology (MM) SNRs (Yamaguchi et al. 2009; Ozawa et al. 2009; Ohnishi et al. 2011; Sawada & Koyama 2012; Uchida et al. 2012). The RRC originates from radiative transitions of free electrons to the K-shell of ions, a sign of a recombination dominant plasma (RP). In the residuals of the NEI model fit for G346.6 -0.2 in Sezer et al. (2011), we see a similar structure to the RRC. G346.6 -0.2 is located on the Galactic ridge, where a strong X-ray emission, called the Galactic Ridge X-ray Emission (GRXE), is prevailing. However, the previous data reduction and analyses did not properly take account of the GRXE as a major background for the faint and diffuse source. We, therefore, revisited the Suzaku data and performed the data reduction, spectral construction, and spectral analysis paying particular concerns on the subtraction of the GRXE. We then discovered evidence for the recombining plasma from this MM SNR for the first time.", "pages": [ 1 ] }, { "title": "2. Observation and Data Reduction", "content": "Suzaku (Mitsuda et al. 2007) carried out the Galactic center/plane mapping project with the CCD cameras (XIS, Koyama et al. 2007) placed at the focal planes of the thin foil X-ray Telescopes (XRT, Serlemitsos et al. 2007). The SNR G346.6 -0.2 was observed on 2009 October 79 (Obs. ID 504096010). The pointing position was ( l , b )=(346 · .63, -0 · .22). G346.6 -0.2 is a faint X-ray SNR, located toward the inner Galactic disk. Therefore, the contribution of the GRXE, particular in the hard X-ray band above 5 keV, has a large impact on the source spectrum. Although the surface brightness of the GRXE is nearly constant along the Galactic plane within the range of a few degree of the Galactic longitude, the latitudinal variation is large, which is given by an exponential function with the scale height of ∼ 0 . · 5 (e.g., Koyama et al. 1986; Yamauchi & Koyama 1993; Kaneda et al. 1997). We, therefore, selected two background regions for the spectrum of G346.6 -0.2; one is the surrounding region of G346.6 -0.2 in the same field of view (FOV) (hereafter BGD-a) and the other is a nearby region at ( l , b )=(344 · .26, -0 · .22) (Obs. ID 502049010; hereafter BGD-b), at the same Galactic latitude but 1 · away from G346.6 -0.2 in longitude. XIS sensor-1 (XIS 1) is a back-side illuminated (BI) CCD, while the other three XIS sensors (XIS 0, 2, and 3) are front-side illuminated (FI) CCDs. The FOV of the XIS is 17.8 ' × 17.8 ' . Since one of the FIs (XIS 2) turned dysfunctional in 2006 November 1 , we used the data obtained with the other CCD cameras (XIS 0, 1, and 3). A small fraction of the XIS0 area was not used because of the data damage due possibly to an impact of micrometeorite on 2009 June 23 2 . The XIS was operated in the normal clocking mode. The degraded spectral resolution due to the radiation of cosmic particles 4 years after the launch was restored by the spaced-row charge injection (SCI) technique. Details of the SCI technique are given in Nakajima et al. (2008) and Uchiyama et al. (2009). Data reduction and analysis were made with the HEAsoft version 6.11, SPEX (Kaastra et al. 1996) version 2.02.04, and the processed data version 2.4. The XIS pulse-height data for each X-ray event were converted to Pulse Invariant (PI) channels using the xispi software and the calibration database version 2011-11-09. We re- jected the data taken at the South Atlantic Anomaly, during the earth occultation, and at the low elevation angle from the earth rim of < 5 · (night earth) and < 20 · (day earth). The exposure times after these screenings were 56.8 and 215.7 ks for G346.6 -0.2/BGD-a and BGDb fields, respectively. In the following data analysis, we subtracted the nonX-ray background (NXB), which was constructed from the night earth data generated by xisnxbgen (Tawa et al. 2008) in the HEAsoft package.", "pages": [ 1, 2, 3 ] }, { "title": "3.1. X-Ray Image", "content": "Figure 1 shows the vignetting corrected X-ray images for the G346.6 -0.2/BGD-a (1a-1c) and BGD-b (1d-1f) fields, in the 0.5-1, 1-5, and 5-8 keV energy bands. The solid contours in figures 1a and 1b are the radio band image at the 843 MHz (Whiteoak & Green 1996). In order to increase X-ray photon statistics, we co-added all the data of XIS 0, 1, and 3. The images contain all the X-rays including the GRXE and the cosmic X-ray background (CXB). Above the X-ray fluxes of these backgrounds, diffuse X-rays from G346.6 -0.2 are found only in the 1-5 keV band image (figure 1b); no significant X-ray is found in the 0.5-1 and 5-8 keV bands (figures 1a and 1c). We see a center-filled X-ray emission within the shell of the radio emission. In addition, four and three faint point-like sources are found in the G346.6 -0.2 (Src 1-4) and BGD-b field (Src 5-7), respectively.", "pages": [ 3 ] }, { "title": "3.2. X-Ray Spectra", "content": "The XIS spectra of G346.6 -0.2, BGD-a, and BGDb were extracted from the solid ellipse in figure 1c, the dashed lines in figures 1c, and 1f, respectively. For these data, we excluded regions of the point-like sources (Src 1-7) and the calibration sources at the field corners. The source and background spectra are shown in figure 2. The hard X-ray flux above ∼ 5 keV is essentially comparable with each other, as is also noted in the hard X-ray band image of figure 1. The hard X-rays are likely due to the sum of the CXB and the GRXE. The Fe K-line emission at 6.7 keV is due to the GRXE, and the line intensities are also comparable between the source and two background spectra. We, therefore, conclude that after the subtraction of the proper background of the GRXE and the CXB, G346.6 -0.2 should have no significant X-rays above ∼ 5 keV. The BGD-a spectrum shows an X-ray excess over BGDb in the low energy band below ∼ 3 keV. This implies that a local diffuse emission is prevailing around G346.6 -0.2. We discuss this local diffuse emission in section 4.2. BGDa should be a better background for G346.6 -0.2 because of the close vicinity, but photon statistics are limited due to less exposure time of 56.8 ks and off-axis vignetting effect. BGD-b, on the other hand, provides better photon statistics due to longer exposure of 215.7 ks and larger correcting area than BGD-a. We, therefore, made two source spectra by the subtraction of each background, BGD-a or BGD-b. The exposure times and vignetting effects of these backgrounds were corrected by the method shown by Hyodo et al. (2008). For the spectral analysis, the Response files, Redistribution Matrix Files (RMFs) and Ancillary Response Files (ARFs), were made using xisrmfgen and xissimarfgen , respectively, in the HEAsoft package. The ARFs were created assuming the observed image to the photon distribution.", "pages": [ 3 ] }, { "title": "3.2.1. G346.6 -0.2", "content": "The spectra of G346.6 -0.2 were made by subtracting BGD-a or BGD-b (here spectra-a and spectra-b, respectively). These spectra were separately given in figures 3a and 3f, respectively. For these spectra, we applied a thin thermal plasma model in collisional ionization equilibrium (CIE) ( vapec model in XSPEC) modified by interstellar absorption ( wabs model in XSPEC). The cross-sections of the photoelectric absorption were taken from Morrison & McCammon (1983), while the abundance data were taken from Anders & Grevesse (1989). The abundances of Mg, Si, S, Ar, and Fe were set to be free and those of Ca and Ni were assumed to be the same as Ar and Fe, respectively. The others were fixed to the solar values. The XIS0, 1 and 3 spectra were simultaneously fitted. The 1-CIE model was rejected with the large χ 2 values of 354 and 423 (d.o.f.=193) for spectra-a and spectra-b, respectively. The best-fit parameters are listed in table 1, while the residuals from the best-fit model are plotted in figures 3b and 3g. We found systematic residuals (shoulder-like structures) in the 1.5-4 keV energy band. We further examined a plasma model in an NEI state ( vnei model in XSPEC) to the source spectra and confirmed that the model was also rejected with the χ 2 values of 309 and 428 (d.o.f.=192) for spectra-a and spectra-b, respectively. The residuals were essentially similar to those of 1-CIE model. We note that similar shoulder-like structures are found in the NEI model fit of Sezer et al. (2011) (figure 3), although the data reduction process was different from the present paper. We then applied a 2-component CIE plasma (2-CIE) model (2vapec model), assuming the abundances of the two plasma components to be equal. This model was also rejected with the χ 2 values of 249 and 373 (d.o.f.=191) for spectra-a and spectra-b, respectively (table 1). The residuals from the best-fit model are plotted in figures 3c and 3h. The residuals in the 1.5-2 keV band disappeared, but those at the energy of 2-4 keV remained. These residuals show shoulder-like shapes with the leading edge energies of ∼ 2.4 keV and ∼ 3.2 keV which correspond the ionization energies of He-like Si and He-like S ions, respectively. The same features were also reported from IC443, G359.1 -0.5, W28, and W44 (Yamaguchi et al. 2009; Ohnishi et al. 2011; Sawada & Koyama 2012; Uchida et al. 2012). These authors predicted that the residuals are due to the RRC which is a sign of the RP. We, therefore, applied a cie model in the SPEX package (Kaastra et al. 1996) applying the absm model for the interstellar absorption. The cie model treats an electron temperature ( kT e ) and an ionization temperature ( kT z ) independently (here 1-RP model). The best-fit param- eters are listed in table 1. Although the χ 2 value for spectra-b was significantly improved from 373 (d.o.f=191) to 306 (d.o.f=192), that for spectra-a was not improved (248 of d.o.f=192). Furthermore, the best-fit abundances of Ar and Ca are 4-5 solar, which is far higher than any other elements of solar or sub-solar abundances. The residuals from the best-fit 1-RP model are plotted in figures 3d and 3i. The residuals at 2-4 keV are partly reduced, but a new residual was appeared at the energy of ∼ 1.35 keV corresponding to the He-like Mg K α line. The additional excess from the He-like Mg K α and unreasonable high abundances of Ar and Ca suggest that the single ionization-temperature (1kT z ) plasma in the cie model is inadequate. In fact, Sawada & Koyama (2012) and Uchida et al. (2012) found that the spectra of MM SNRs, W28 and W44, have multi-ionization temperatures. They successfully predicted the multi-ionization temperature structure with a scenario that the X-ray emitting plasma is in a transition phase of recombining process. In order to examine a possibility of a multikT z plasma for G346.6 -0.2, we fitted the spectra with the neij model in SPEX. The neij model describes a plasma state when the initial plasma was in CIE with the temperature of kT e1 , then only the electron temperature dropped to kT e2 by a rapid electron cooling in the past. In the initial phase, the plasma was in 1kT z plasma ( kT z = kT e 1 ), then forced recombination as a function of n e t , where n e and t are an electron density and an elapsed time after the electron cool-down epoch, respectively. Since the time sale of recombination is element-dependent, the time evolution realizes a multikT z plasma. The neij model gave better reduced χ 2 value than the simple 1-RP ( cie ) model with ∆ χ 2 = 22 (spectra-a) and 38 (spectra-b), but no constraint on kT e1 was obtained with the error ranges of > 4 keV and > 3 keV (90% con- nce level) for spectra-a and spectra-b, respectively. These error ranges mean that the initial plasma of kT e1 is in nearly full ionized state for all the relevant elements except for Fe (Fe would be in between full ionized and H-like states). Taking account of the initial plasma condition, we assumed kT e1 =5 keV as a physically reasonable value and tried the final neij fitting. The best-fit parameters and the χ 2 values are listed in table 1. The best-fit model is plotted in figures 3a and 3f, while the residuals are shown in figures 3e and 3j. On the contrary to the simple 1-RP model, the neij model gave reasonable abundances of Ar and Ca similar to the other elements. Thus, we conclude that the overall spectra of G346.6 -0.2 were nicely fitted with the neij model. 3.2.2. Local Emission around G346.6 -0.2 In the background estimation process, we found a local excess in soft X-rays around G346.6 -0.2. The X-ray spectrum of this local excess emission was made by subtracting BGD-b from the BGD-a data (figure 2), which is given in figure 4. The X-ray spectrum shows a weak Si K α line, which means a thin thermal origin. The spectrum was nicely fitted by a thin thermal plasma model, the apec in XSPEC, with χ 2 /d.o.f. =102/104. The results are listed in table 2 and the best-fit model is plotted in figure 4.", "pages": [ 3, 4, 5, 6 ] }, { "title": "4.1. G346.6 -0.2", "content": "For the background estimation, we took account of the difference of vignetting effects between the source and background regions. Furthermore, we tried two different backgrounds of BGD-a and BGD-b (spectra-a and spectra-b). Although the fluxes of these backgrounds are different, the spectral shapes are very similar. As the results, these two spectra, spectra-a and spectra-b, gave essentially the same best-fit parameters except for the ab- solute luminosity. Since BGD-a and G346.6 -0.2 are likely in a local soft X-ray excess around G346.6 -0.2, we discuss based on the results of spectra-a, the BGD-a subtracted spectra. The spectrum was nicely described by the RP in a transition epoch of its recombining phase ( neij model). The best-fit electron temperature was about 0.3 keV. These conclusions are inconsistent with those of Yamauchi et al. (2008) and Sezer et al. (2011). In the thermal plasma fitting, their results were in CIE or NEI with the electron temperatures of ∼ 1.6 keV and 0.97-1.3 keV in Yamauchi et al. (2008) and Sezer et al. (2011), respectively. These apparent inconsistencies are mainly due to the background subtraction. Sezer et al. (2011) used the background from very small region (near the corner of the FOV), while Yamauchi et al. (2008) used the annulus region around the source. No vignetting effect was corrected in Yamauchi et al. (2008) and Sezer et al. (2011). Therefore, the background-subtracted spectra provide less statistics and should contain some fractions of the GRXE emission, particularly in the hard X-ray band. In fact, they reported that the X-ray spectrum has excess flux above ∼ 5 keV. This may artificially predict higher electron temperatures and/or power-law component. Assuming the mean density of 1H cm -3 , the derived N H value of (2.3 ± 0 . 1) × 10 22 cm -2 corresponds to the distance of 7-8 kpc, which is well consistent with the distance of 8.2 kpc estimated from ΣD relation (Case & Bhattacharya 1998). If we assume the distance of 8 kpc, the luminosity was calculated to be 3.5 × 10 35 erg s -1 in the 0.5-10 keV energy band. The abundances are sub-solar to solar, which is consistent with those of Sezer et al. (2011). They predicted that the remnant may originate from a Type Ia supernova (SN) explosion, and the solar/sub-solar abundances of heavy elements are explained by the scenario that the SNR is young, and the reverse shock does not reach yet to the interior of the ejecta. Our best-fit n e t is (4.8 +0 . 1 -0 . 4 ) × 10 11 cm -3 s. Then, assuming the mean density of 1H cm -3 , the age of G346.6 -0.2 is estimated to be 1.4 × 10 4 -1.6 × 10 4 yr. This means that G346.6 -0.2 is not a young SNR, and hence no evidence for the young Type Ia SN scenario is obtained. The solar/sub-solar abundances plasma would be mainly due to interstellar matter. G346.6 -0.2 is the sixth sample of the SNRs with the RP, after IC443 (Yamaguchi et al. 2009), W49B (Ozawa et al. 2009), G359.1 -0.5 (Ohnishi et al. 2011), W28 (Sawada & Koyama 2012), and W44 (Uchida et al. 2012). Among these SNRs with the RP, the plasma structures in W28 and W44 were studied in detail. The results were that the RP is in a recombining phase of ∼ 10 11 ( n e / 1H cm -3 ) -1 s after the production of the initial RP. We found that G346.6 -0.2 is explained with a similar scenario as these SNRs. We propose that the other SNRs, IC443, W49B, and G359.1 -0.5, can also be explained by the same scenario, a plasma in an epoch of recombining process. All the previous RP-detected SNRs share some common characteristics: (1) they are classified to MM SNRs (Rho & Petre 1998), (2) an OH maser is detected, suggesting the interaction with molecular clouds, and (3) TeV/GeV γ -ray emission is detected. For possible origins of the RP based on these common features, one can refer the discussions in Yamaguchi et al. (2009), Sawada & Koyama (2012), and Uchida et al. (2012). The center-filled thermal X-ray emission within the radio shell suggests that G346.6 -0.2 is a MM SNR. An OH maser has been found (Green et al. 1997), but no TeV/GeV γ -ray emission has been found from G346.6 -0.2. Therefore, future TeV/GeV γ -ray search from this SNR is encouraged.", "pages": [ 6, 7 ] }, { "title": "4.2. Local Emission", "content": "The excess emission around G346.6 -0.2 was shown by the optically thin thermal plasma with the temperature of 0.79 keV, 0.14 solar abundance, and N H value of 1.2 × 10 22 cm -2 . The ASCA Galactic plane survey revealed that the GRXE spectra observed in various regions are well represented by an optically thin thermal plasma model with two temperatures of < 1 keV and ∼ 7 keV (Kaneda 1997; Kaneda et al. 1997). The two-temperature structure was confirmed by Chandra (Ebisawa et al. 2005) and Suzaku (Ryu et al. 2009). The spectral parameters of the excess emission in the G346.6 -0.2 field are similar to those of the soft component of the GRXE. The intensity distribution of the hard component along the Galactic plane is symmetric with respect to the Galactic center, while that of the soft component is more asymmetric: a local peak and a local minimum were found at l ∼ 347 · and l ∼ 355 · , respectively (Kaneda 1997); the longitudinal distribution of the GRXE soft component shows that the intensity near the BGD-a region is about ∼ 1.3 times higher than that of the BGD-b region (Kaneda 1997). Our present result of BGD-a is ∼ 2 times larger than BGD-b in the soft Xray band, and hence the excess would not be due to the fluctuation of the GRXE, but would be a local plasma. The N H value of the local plasma is smaller than that of G346.6 -0.2 in the GRXE, hence this emission is located at the near side of G346.6 -0.2 and the GRXE. The local plasma is near to the direction of the non-thermal SNR RX J1713.7 -3946 and the N H value is similar to that of RX J1713.7 -3946 (Koyama et al. 1997). Therefore, the local plasma is located near RX J1713.7 -3946 at the distance of 1-2 kpc. Whether this thermal plasma is physically associated with RX J1713.7 -3946 or not is unclear. To clarify this, we encourage further observations around RX J1713.7 -3946. We would like to express our thanks to all of the Suzaku team. We thank Dr. M. Sawada for his useful comments. This work was supported by the Japan Society for the Promotion of Science (JSPS); the Grant-in-Aid for Scientific Research (C) 21540234 (SY), 24540232 (SY), and 24540229 (KK), Young Scientists (B) 24740123 (MN), Challenging Exploratory Research program 20654019 (KK), and Specially Promoted Research 23000004 (KK).", "pages": [ 7 ] }, { "title": "References", "content": "Dubner, G. M., Moffett, D. A., Goss, W. M., & Winkler, P. F. 1993, AJ, 105, 2251 Kaneda, H., Makishima, K., Yamauchi, S., Koyama, K., Matsuzaki, K., & Yamasaki, N. Y. 2007, ApJ, 491, 638 Rho, J., & Petre, R. 1998, ApJ, 503, L167 Ryu, S., Koyama, K., Nobukawa, M., Fukuoka, R., & Tsuru,", "pages": [ 7 ] } ]

[ { "title": "Annual Parallax Distance and Secular Motion of the Water Fountain Source IRAS 18286 -0959", "content": "Hiroshi Imai 1,2 , Tomoharu Kurayama 3 , Mareki Honma 4 , and Takeshi Miyaji 4 1 Department of Physics and Astronomy, Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima 890-0065 [email protected] 2 International Centre for Radio Astronomy Research, M468, The University of Western Australia, 35 Stirling Hwy, Crawley, Western Australia, 6009, Australia 3 Center for Fundamental Education, Teikyo University of Science, 2525 Yatsusawa, Uenohara, Yamanashi 409-0193 4 Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588 (Received 2012 April 28; accepted 2012 October 16)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We report on results of astrometric observations of H 2 O masers in the 'water fountain' source IRAS 18286 -0959 (I18286) with the VLBI Exploration of Radio Astrometry (VERA). These observations yielded an annual parallax of IRAS 18286 -0959, π = 0 . 277 ± 0 . 041 mas, corresponding to a heliocentric distance of D =3 . 61 +0 . 63 -0 . 47 kpc. The maser feature, whose annual parallax was measured, showed the absolute proper motion of ( µ α , µ δ ) = ( -3 . 2 ± 0 . 3 , -7 . 2 ± 0 . 2)[mas yr -1 ]. The intrinsic motion of the maser feature in the internal motions of the cluster of features in I18286 does not seem to trace the motion of the bipolar jet of I18286. Taking into account this intrinsic motion, the derived motion of the maser feature is roughly equal to that of the maser source I18286 itself. The proximity of I18286 to the Galactic midplane ( z ≈ 10 pc) suggests that the parental star of the water fountain source in I18286 should be intermediatemass AGB/post-AGB star, but the origin of a large deviation of the systemic source motion from that expected from the Galactic rotation curve is still unclear. Key words: masers - stars: AGB and post-AGB - stars: individual (IRAS 18286 - 0959)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Energetic mass loss (rate up to 10 -3 ˙ M /circledot yr -1 ) has been observed from dying stars such as asymptotic giant branch (AGB) and post-AGB stars. The spatio-kinematics of the mass-loss flows have been investigated in great detail by observations of maser emission such as SiO, H 2 O, and OH using very long baseline interferometry (VLBI). These maser sources are associated with circumstellar envelopes (CSEs) of AGB stars, but in rare cases with highly collimated bipolar jets, so called 'water fountains,' and CSE remnants around post-AGB stars or central objects of planetary nebulae (PNe). In the latest decade, high precision astrometry in very long baseline interferomery (VLBI) has enabled the measurement of trigonometric parallax distances and proper motions of maser sources. Even in the rare cases mentioned above, the heliocentric distances and three-dimensional secular motions of the water fountains, pre-PNe, and PNe have been measured, in those cases where H 2 O masers could be detected (Imai et al. 2007c; Imai et al. 2011b; Tafoya et al. 2011). These kinematic approaches have contributed to estimating the physical parameters of these parental stars and their evolutionary properties. Taking into account their locations and 3D motions in the Milky Way, it has been suggested that these stars should be intermediate-mass AGB/post- AGB stars (e.g., Imai 2007). Here we report on the measurement of an annual parallax of H 2 O masers in IRAS 18286 -0959 (hereafter abbreviated as I18286) with the VLBI Exploration of Radio Astrometry (VERA). The VERA astrometry for I18286 was conducted in one of the VERA key science projects, which focuses on H 2 O maser sources at intermediate distances (2-5 kpc from the Sun) in order to extend the distance scale for the VERA parallax measurements. The result of the I18286 astrometry could be used as one of the data points for exploring the Galactic dynamics if the motion of I18286 closely follows that of the Galactic rotation (i.e. Reid et al. 2009; Honma et al. 2012). I18286 is a member of the class of water fountains, whose spatiokinematics of H 2 O masers have been investigated in detail. Yung et al. (2011) suggest that most of the H 2 O masers in I18286 are associated with 'double helices' of highly collimated bipolar jets with speeds of ∼ 100 km s -1 . The formation of the double helices is expected from discontinuous mass ejection from a moving star on a period of ∼ 30 yr. The annual parallax measurement for I18286 is the second case of parallax measurements for the water fountains after that for IRAS 19134+2131 (Imai et al. 2007c). This paper discusses the galactic kinematical properties of I18286 in Sect. 3.2. However, the existence of a low-velocity 'equatorial flow' should be taken into account when one interprets the observed secular motion of the maser feature in the cluster of features in I18286 (Imai 2007). The argument of the equatorial flow will be described in a separate paper in more detail.", "pages": [ 1, 2 ] }, { "title": "2. Observations and data reduction", "content": "The VERA observations of the I18286 H 2 O ( J K -K + = 6 12 → 5 23 , 22.235080 GHz) masers were conducted at 16 epochs from 2007 October to 2009 September. Table 1 gives a summary of these observations, maser source mapping, and astrometry. Out of 16 epochs, 14 produced scientifically meaningful output. At each epoch, the observation was made for 6.7 hr in total. I18286 was observed together with the fringe-phase and position reference source, ICRF J183220.8 -103511 (hereafter abbreviated as J1832), separated by 0 · .67 from I18286, simultaneously using VERA's dual-beam system. They were tracked for 30-35 min out of every 40 min, followed by scans on other band-pass calibrator sources. The received signals were digitized in four quantization levels, then divided into 16 base band channels (BBCs) with a bandwidth of 16 MHz each. One of the BBCs collected data from I18286 at the frequency band including the H 2 O maser emission, while other BBCs from J1832 spanning a total frequency band range of 480 MHz. The BBC outputs had a recording data rate of 1024 Mbits s -1 . The data correlation was made with the Mitaka FX correlator with a correlation accumulation period of 1 s. The correlation outputs consisted of 512 and 32 spectral channels for the H 2 O maser and reference continuum emission, respectively. The former corresponds to a velocity spacing of 0.42 km s -1 , which is narrow enough to resolve an H 2 O maser feature (corresponding to a maser gas clump), which consists of two or more spectral channel components called maser spots . Data reduction was mainly made with the National Radio Astronomy Observatory (NRAO) Astronomical Image Processing System (AIPS) package. For astrometry, we need special procedures described as follows (see also e.g., Honma et al. 2007; Imai et al. 2007b). Firstly, delay-tracking was repeated for the correlated data using better delay-tracking models calculated with the software equivalent to the CALC9 package developed by the Goddard Space Flight Center/NASA VLBI group. Throughout the whole data analysis, we adopted the coordinates of the delay-tracking center: α J 2000 =18 h 31 m 22 s .934, δ J 2000 = -09 · 57 ' 21 '' .70 for I18286 and α J 2000 =18 h 32 m 20 s .836521, δ J 2000 = -10 · 35 ' 11 '' .20055 for the reference source J1832. The delay-tracking solutions include residual delay contributions from the atmosphere, which were estimated using the global positioning system (GPS) data (Honma et al. 2008b). Secondly, differences in instrumental delays between two signal paths in the dual beam system were calibrated using the differential delays, which were measured using artificial noise signals injected into the two receivers at the same time. The measurement accuracy has been improved for observations since 2007 by installing one to four artificial noise sources (Honma et al. 2008a). Thirdly, fringe-fitting and self-calibration were performed using the continuum source data, whose solutions were applied in the data analysis of maser emission. Only the solutions in the BBC at the same frequency as that for the H 2 O maser emission were valid for the fringe-phase calibration. The accuracy of the phase-fluctuation compensation and the coherence of the integration is dependent on weather condition, particularly the humidity, which is seasonally variable. These effects affect the final astrometric accuracy. Finally, image cubes of the maser source were obtained by deconvolution through the CLEAN algorithm with a typical synthesized beam of 0.9 × 2.5 in milliarcseconds (mas) in the case of full operation of VERA's four antennas. Because the CLEAN deconvolution was performed by automatically selecting local peaks of brightness in each of channel maps without smaller CLEAN boxes, side lobes of the synthesized beam pattern affected the final CLEAN image cube 1 . Each maser spot (or velocity component) was identified as a Gaussian brightness component using the AIPS task SAD. In order to obtain higher quality maser maps, we also adopted a normal procedures for maser source mapping. Fringe-fitting and self-calibration were performed using a spectral channel that contain bright maser emission. Column 4 in table 1 gives the local-standard-of-rest (LSR) velocity of the spectral channel selected as phase- and position-reference. The obtained solutions of calibration were applied to the data in all spectral channels.", "pages": [ 2 ] }, { "title": "3. Results", "content": "3.1. The distribution and proper motions of H 2 O maser features in IRAS 18286 -0959 Column 6 of table 1 gives the numbers of maser features identified in the individual epochs. Figure 1 shows the distributions of H 2 O maser features at six out of 14 epochs. The maser feature distribution was highly variable, with only a small fraction of the maser features having lifetimes longer than half year. Note that the VLBA observations conducted by Yung et al. (2011) and the VERA observations reported in this paper, respectively, lasted for similar seasons, 2008 April-2009 May and 2007 October-2009 September, respectively. The latter observations identified most of the same maser features as found in Yung et al. (2011). However, there exists some discrepancy of maser feature detections between these observations because of the different image sensitivities 2 and, more im- portantly, different epochs of observations. Although our VERAobservations could not trace the maser feature that was selected as position-reference in the VLBA observations (at an LSR velocity V LSR /similarequal 52 km s -1 in the eastern side of the maser distribution) over a full year, we newly identify another maser feature that survived over one year in the western side of the maser distribution. This maser feature (IRAS 18286 -0959:I201314 , see table 2), has a similar LSR velocity and is denoted by a black opened circle in figure 1. The maser spots included in this maser feature was used for the annual parallax measurement as described in subsection 3.2. The relative proper motions of maser features with respect to the position-reference feature were identified for the features detected at three or more epochs. Some proper motions were also removed from consideration because their LSR velocity drifts were higher than 10 km s -1 yr -1 . Thus only 50 maser proper motions were measured although there existed a few hundred identified features. Because there was no single maser feature that was detected at all epochs, different maser features were selected as position reference in the first nine and final five epochs. Table 2 gives the parameters of the maser features whose proper motions were measured. Figure 2 shows the spatial distribution of the relative maser proper motions. The spatio-kinematical structure roughly reproduces a fast bipolar outflow found by Yung et al. (2011), but it is difficult to recognize the double helix jets as seen by Yung et al. (2011). The VLBA identification of maser proper motions seems to have biases to high velocity motions. It is noteworthy that, in the present result, there exist a group of maser features moving in slow velocities ( V < ∼ 30 km s -1 ) near the center of the maser feature distribution. They were moving perpendicularly to the high velocity ( V > ∼ 100 km s -1 ) motions of maser features in the north-south direction. This implies the existence of an 'equatorial flow' (Imai 2007), different from the high velocity flows ( V ∼ 100 km s -1 ) that are associated with highly collimated bipolar jets described by Yung et al. (2011). This equatorial flow will be described in a separate paper. Nevertheless it is still necessary to estimate the motion of the star itself in I18286 (or the systemic motion) with respect to the maser proper motion in order to discuss the systemic motion of I18286 in the Milky Way (see subsection 4). The maser motions shown in figure 2 are well approximated to an outflow with a point-symmetric velocity field. We performed the least-squares fitting method for the maser motion data using a radially expanding flow model. This method has been repeatedly described in our previous papers (e.g. Imai et al. 2011a; ? ). Note that the model fitting assumes independent radial-expansion velocities of the maser features rather than a common radial velocity field as a function of distance from the dyamical center of the flow. This is, however, sufficient for estimat- ing the systemic motion of the outflow origin as one of the free parameters. Table 3 gives the derived fitting parameters. Figure 2a displays the measured maser motion vectors after subtracting the derived systemic motion vector. The model-fitting result suggests that position-reference feature (IRAS 18286 -0959:I2013-14) itself may belong to the equatorial flow mentioned above.", "pages": [ 2, 3 ] }, { "title": "3.2. The annual parallax distance to IRAS 18286 -0959", "content": "Column 7 of table 1 shows validity of astrometry for the annual parallax measurement in the individual epochs (Y/N/S). As described in subsection 3.1 and table 1, the maser feature IRAS 18286 -0959:I201314 was identified in the observations between 2007 October and 2009 March in the maser image cubes synthesized through selfcalibration and/or phase-referencing procedures and two maser spots in this feature were used for the parallax measurement. Figure 3 shows the motions of these maser spots. The spot motions can be fitted to combination of an annual parallax and a constant-velocity motion as described later. The uncertainty of spot position presented in figure 3 indicates only the contribution from thermal noise, which is underestimated if the spot is extended. We note that Yung et al. (2011) and the data of Imai (2007) suggest that the H 2 O maser spots in I18286 were extended. Thus, deviations of the data points from the modeled motions (up to 0.5 mas) are larger than the position errors expected from combination of contributions from thermal noise, instrumental, and atmospheric phase-delay residuals (0.1 mas level, Honma et al. 2010, see also general formulation in Asaki et al. 2007). They should be mainly attributed to the variation of maser spot structures. Note that the temporal variation of maser spot/feature structure is not necessarily random as assumed in Honma et al. (2010) if the feature is associated with some physical feature (e.g. a shock front, Imai et al. 2002) changing on a specific time scale. Nevertheless, random variation of a spot position around its ballistic motion as shown in figure 3 is expected when the maser is highly spatially resolved with a VLBI synthesized beam whose shape may change from one epoch to another as small intrinsic variation of the maser structure will be highly enhanced (c.f. Imai et al. 2007b). We attempted the least-square method for the spot motions to fit the modeled motions each of which is composed of an annual parallax, a constant secular motion, and a position offset at the reference epoch (J2000.0) 3 . In order to obtain the common annual parallax of these spots, iterative procedures were adopted, which is similar to the approach by Sanna et al. (2012). Firstly, the model fitting was performed independently for the individual spot motions (independent fitting). Secondly, a mean annual parallactic motion was subtracted from the spot motions to estimate only the position offsets and secular motions of spots (proper motion fitting). Thirdly, the systemic motions estimated from the derived parameters were sub- tracted from the original spot motions. Finally, the position residuals were used to estimate a common parallax and position offset and proper motion residuals (combined fitting). The proper motion fitting was again performed using the estimated common parallax. Then the procedures from the proper motion to combined fitting were iterated until the estimated parameters seemed converged. Table 4 gives the derived parameters after the final iteration. Although the two maser spots were associated with the same maser feature and their motions are not completely independent, the data combination may mitigate the error contribution from statistical (large) errors expected from the signal-to-noise ratios of maser spot detection and the random variation of spot structures. In the model fitting, weighting with position accuracy was adopted as performed in our previous analysis (Imai et al. 2011b). Thus we obtained an annual parallax of I18286, π = 0 . 277 ± 0 . 041 mas, corresponding to a distance value of D = 3 . 61 +0 . 63 -0 . 47 kpc. The kinematic distance to I18286 of 3.1 kpc (Deguchi et al. 2007) is roughly consistent with the annual parallax distance. The absolute proper motion of the maser feature IRAS 18286 -0959:I201314 was derived to be ( µ α , µ δ ) = ( -3 . 2 ± 0 . 3 , -7 . 2 ± 0 . 2)[mas yr -1 ] from the mean motions of the two spots, whose parameters are given in table 4. Here we estimate the location and the three-dimensional motion of the I18286 system in the Milky Way. Table 5 gives the derived parameters. I18286 is located very close to the Galactic midplane ( z /similarequal 7 pc). On the other hand, the systemic secular motion of I18286 has some uncertainty, described as follows, which is dependent on the relative motion of the feature IRAS 18286 -0959:I201314 with respect to the central star. If the motion of this maser feature is the same as that of the star (Case 1 in table 5), the systemic motion has a very large deviation (up to 100 km s -1 ) from that expected from the Galactic rotation curve. This is a similar case to that in which one adopts the relative motion of the feature estimated from the model fitting as described in subsection 3.1 (Case 2). The systemic motion only follows the Galactic rotation curve closely if one assumes a relative proper motion of the feature ( -3 , -5) [mas yr -1 ], or ( -51, -85) [km s -1 ] with respect to the star, as indicated with a magenta arrow in figure 2 (Case 3).", "pages": [ 3, 4 ] }, { "title": "4. Discussion", "content": "Through the present work, we can obtain some constraint on the properties of the central star in the water fountain I18286. The annual parallax distance to I18286 gives the luminosity value of I18286, ∼ 1 . 2 × 10 4 L /circledot , a factor of 1.4 higher than the value previously derived (8 . 7 × 10 3 L /circledot , Deguchi et al. 2007). This suggests that I18286 has a luminosity higher than the typical value of AGB stars ( ∼ 6 000 L /circledot ), supporting the hypothesis of I18286 to be a higher mass AGB/post-AGB star (e.g., Imai 2007; Imai et al. 2007c). Note that the high mass population of evolved stars such as red supergiants follows closely the Galactic rotation within 30 km s -1 (e.g. Reid et al. 2009; Asaki et al. 2010; ? ) while lower mass evolved stars may have the large kinematical deviations due to the dynamical relaxation in the Milky Way during their long lifetime. In the case of I18286, the large deviation of its motion from the Galactic rotation favors the hypothesis of an AGB star over that of a red supergiant. However, its close proximitiy to the Galactic mid-plane does not rule out the possibility that the large kinematical deviation is attributed to a binary motion in I18286. SiO maser or millimeter continuum emission can pinpoint the location of the star (Imai et al. 2005; Imai 2007), but has not yet been detected toward I18286. Detection of a circumstellar envelope in molecular line emission with a new facility such as the Atacama Large Millimeter/submillimeter Array (ALMA) will shed light on the property of of the central star. In practice, it is difficult to distinguish the intrinsic emission of I18286 from interstellar foreground/background emission (Imai et al. 2009) although this object is clearly identified in the Spitzer /GLIMPSE image (Deguchi et al. 2007). In addition, more intensive monitoring VLBI observations of the I18286 H 2 O masers are necessary in order to trace a greater number of proper motions of maser features including shorter-lived (a few months) features, enabling to elucidate the maser spatiokinematics in more detail, including a possible equatorial flow or an AGB envelope. VERA/Mizusawa VLBI observatory is a branch of the National Astronomical Observatory of Japan, an interuniversity research institute supported by the Ministry of Education, Culture, Sports, Science and Technology. We acknowledge all staff members and students who have helped in array operation and in data correlation of the VERA. We also thank Richard Dodson for carefully reading the manuscript and give us useful comments. HI was financially supported by Grant-in-Aid for Young Scientists from the Ministry of Education, Culture, Sports, Science, and Technology (18740109) as well as by Grant-in-Aid for Scientific Research from Japan Society for Promotion Science (JSPS) (20540234 and 22-00022). HI also acknowledge for the support for his stay at ICRAR in the Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation funded by JSPS.", "pages": [ 4 ] }, { "title": "References", "content": "Asaki, Y., Deguchi, S., Imai, H., Hachisuka, K., Miyoshi, M., & Honma, M. 2010, ApJ, 721,267 Asaki, Y., et al. 2007, PASJ, 59, 397 Claussen, M., Sahai, R., & Morris, M. R. 2009, ApJ, 691, 219 Deguchi, S., Nakashima, J, Kwok, S., & Koning, N. 2007, ApJ, 664, 1130 Dehnen, W., & Binney, J. 1998, MNRAS, 294, 429 Francis, C., & Anderson, E. 2009, New Astron., 14, 615 Hammersley, P. L., Garz'on, F., Mahoney, T., & Calbet, X. 1995, MNRAS, 273, 206 Honma, M., et al. 2012, PASJ, 64, 136 Honma, M., et al. 2010, PNAOJ, 13, 57 Honma, M., et al. 2007, PASJ, 59, 889 Honma, M., et al. 2008a, PASJ, 60, 935 Honma, M., Tamura, Y., & Reid, M. J. 2008b, PASJ, 60, 951 Imai, H., Sakai, N., Nakanishi, H., Sakanoue, H., Honma, M., & Miyaji, T. 2012, PASJ, 64, 142 Imai, H., Omi, R., Kurayama, T., Nagayama, T., Hirota, T., Miyaji, T., & Omodaka, T. 2011a, PASJ, 63, 1293 Imai, H., Tafoya, D., Honma, M., Hirota, T., & Miyaji, T. 2011b, PASJ, 63, 81 Imai, H., He, J.-H., Nakashima, J., Ukita, N., Deguchi, S., & Koning, N. 2009, PASJ, 61, 1365 Imai, H. 2007, in: IAU Symposium 242, Astrophysical Masers and their Environments, Baan, W., & Chapman, J. (Cambridge University Press: Cambridge), p279 Imai, H., Sahai, R., & Morris, M. 2007b, PASJ, 59, 1107 Imai, H., et al. 2007c, ApJ, 669, 424 Imai, H., Nakashima, J., Diamond, P. J., Miyazaki, A., & Deguchi, S. 2005, ApJ, 622, L125 Imai, H., Deguchi, S., & Sasao, T. 2002a, ApJ, 567, 971 Reid, M. J., et al. 2009, ApJ, 700, 137 Sanna, A., et al. 2012, ApJ, 745, 82 Soubiran, C., Bienaym'e, O., & Siebert, A. 2003, ARA&A, 398, 141 Tafoya, D., et al. 2011, PASJ, 63, 71, 745, 82 Vlemmings, W. H. T., Diamond, P. J., & Imai, H. 2006, Nature, 440, 58 Yung, B. H. K., Nakashima, J., Imai, H., Deguchi, S., Diamond, P. J., & Kwok, S. 2011, ApJ, 741, 94 a Telescope whose data were valid for phase-referencing maser imaging. M: Mizusawa, R: Iriki, O: Ogasawara, S: Ishigakijima. The station with parentheses had some problem during the observations and affected the annual parallax measurements. g Peak intensity, in units of Jy beam -1 , of the phase-referencing-based image of the maser feature IRAS 18286 -0959:I201314 that contained the maser spots selected for the astrometry. h A too long fringe-fitting solution interval (3 min.) was adopted. The maser image cube obtained through self-calibration did not have better quality than that through phase-referencing calibration, causing some of the maser features found in the latter image cube missing in the former one. Year since J2000.0", "pages": [ 4, 5, 6, 11 ] } ]

[ { "title": "VLBI Imagings of Kilo-parsec Knot in 3C 380", "content": "Shoko Koyama 1,2 , Motoki Kino 3 , Hiroshi Nagai 2 , Kazuhiro Hada 4 , Seiji Kameno 5 , and Hideyuki Kobayashi 1,2 1 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan [email protected] 4 INAF Istituto di Radioastronomia, via Gobetti 101, 40129 Bologna, Italy 5 Department of Physics, Faculty of Science, Kagoshima University, 1-21-35 Korimoto, Kagoshima 890-0065, Japan (Received 2012 August 22; accepted 2012 October 19)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate observational properties of a kilo-parsec scale knot in radio-loud quasar 3C 380 by using two epoch archival data obtained by Very Long Baseline Interferometry (VLBI) at 5 GHz on 1998 July and 2001 April. We succeed in obtaining the highest spatial resolution image of the bright knot K1 located at 732 milliarcseconds, or > = 20 kpc de-projected, downstream from the nucleus three times better than previously obtained highest resolution image by Papageorgiou et al. (2006). Our images reveal, with new clarity, 'inverted bow-shock' structure in K1 facing the nucleus and its morphology resembles a conical shock wave. By comparing the two epoch images directly, we explore the kinematics of K1 and obtain the upper limit of apparent velocity, 0 . 25 mas yr -1 or 9 . 8 c of K1 for the first time. The upper limit of apparent velocity is marginally smaller than superluminal motions seen in the core region. Further new epoch VLBI observations are necessary to measure the proper motion at K1. Key words: galaxies: active - galaxies: jets- galaxies: quasars: indi vidual (3C 380 = 1828+487) - radio continuum: galaxies - techniques: interferometric", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Thanks to recent progress in the field of radio interferometry, the properties of knots at large scales (down to ∼ 100 pc-10 kpc from the nucleus) in several nearby radio galaxies have been investigated. For example, VLBI images clarify the detail complex internal structures such as knot HST-1 in M87 ( z =0 . 0036) (e.g., Cheung et al. 2007;Chang et al. 2010;Giroletti et al. 2012) and knot C80 in 3C 120 ( z = 0 . 033) (e.g., Roca-Sogorb et al. 2010; Agudo et al. 2012), located 50-100 pc away from their nucleus (details and other examples are summarized in § 5.1.1). HST-1 complex, and knots D and E located at kpc order from the nucleus of M87, display superluminal motions up to 6 c by Very Large Array (VLA), Hubble space telescope (HST) and VLBI (e.g., Biretta et al. 1995; Biretta et al. 1999; Cheung et al. 2007;Chang et al. 2010; Giroletti et al. 2012). The kinematics of knots in nearby broad line radio galaxy 3C 120 are also studied out to 3 kpc, but the none of the superluminal motion originally claimed by Walker et al. (1988) was found in further VLBI observations (Muxlow & Wilkinson 1991; Walker 1997). As for radio-loud quasars, VLBI images at large scales, let alone their kinematics are hardly studied due to their locations and lack of spatial resolutions. Although the majority of the cores showing superluminal motions are quasars (e.g., Kellermann et al. 2004; Lister et al. 2009), it is not clear where jet deceleration happens. To develop a detailed understanding of the jet deceleration process and shock dissipation process at large scales, it is crucial to obtain direct images of kpc scale knots with sufficient spatial resolution. There is only one attempt to image kpc scale knots and constrain on their motions in radio-loud quasars. With a global VLBI network of 16 radio telescopes, Davis et al. (1991) conducted the observations of 3C 273 ( z =0 . 158) at 1.7 GHz. By comparing with an earlier image, they indicate a possible superluminal motion about 2-5 c on 100 pc scales, but it is difficult to confirm because different components emerged. To overcome the above shown difficulty and explore observational properties of large scale knots in radio-loud quasars, we select 3C 380 which is known as a compact steep spectrum (CSS) radio source ( z =0 . 692) with VLBI. As having a steep spectrum, this source is considered to be associated with a misaligned jet (Fanti et al. 1990). Since the position angle of each inner parsec scale jet, ranging from 284 · to 352 · , are almost parallel to their motion vectors, it is suggested that the jet was ejected ballistically from the core. There are two distant bright knots, K1 and K2, located ∼ 0.73 and 1 arcseconds at position angle around 308 · , which is approximately in the direction of continuation of the inner jet (Kameno et al. 2000). The distance between K1 and the core corresponds to more than 20 kpc, using viewing angle < = 15 · (e.g., Wilkinson et al. 1984; Kameno et al. 2000). Detection of the linear polarization by Multi-Element Radio Linked Interferometer Network (MERLIN) (Flatters 1987) and the optical emission by HST (deVries et al. 1997) at K1 and K2 implies the presence of strong interaction between the knots and ambient medium. K1 in 3C 380 is the best target to explore the observational properties of quasar kpc scale knot, because the knot is sufficiently bright and the source entire angular size is sufficiently compact for VLBI observations. We attempt to image the kpc scale knot K1 in 3C 380 with a high resolution VLBI. The organization of this paper is as follows. The observation data and data reduction is described in § 2 and § 3, respectively. The results are presented in § 4 and the discussions are given in § 5. Throughout this paper, we adopt the following cosmological parameters : H 0 = 71 km s -1 Mpc -1 , Ω M = 0 . 27, and Ω Λ = 0 . 73 (Komatsu et al. 2009), or 1 mas = 7 . 11 pc and 0 . 1 mas yr -1 =3 . 92 c .", "pages": [ 1, 2, 3 ] }, { "title": "2. Archival Radio Data", "content": "We analyzed two VLBI archival data of quasar 3C 380 at 4.815 GHz in left hand circular polarization, observations of which were made on 1998 July 4 and 2001 April 24. In Table 1, we summarize the details of the VLBI data. All the ten antennas of VLBA (Very Long Baseline Array) are used in the observations. The first epoch is our data as a part of VSOP (VLBI Space Observatory Programme) observation, including the Effelsberg telescope. The second epoch data is obtained from VLBA archival data services. All the correlation processes were performed by the National Radio Astronomy Observatory (NRAO) VLBA correlator in Socorro, NM, USA.", "pages": [ 3 ] }, { "title": "3. Data Reduction", "content": "We used the Astronomical Image Processing System (AIPS) software package developed by the NRAO for a priori amplitude calibration, fringe fitting, and passband calibration process. For the first epoch data, we did not use the data of spacecraft baselines. Since the distance between the core and K1 corresponds to around 700 beamwidth, time and frequency averaging would cause time and bandwidth smearing in the K1 region (Thompson et al. 2001). To minimize the smearing effect on K1 to make wide field of view images, the frequency channels were averaged within each IF, and these IFs were kept separate during imaging process. We did not use any time averaging, but the error on each visibility data point was also adopted as the standard deviation within ten seconds by using AIPS task FIXWT (see Table 1). Imaging was performed using CLEAN and self-calibration algorithm. This was performed with the Difmap software package (Shepherd et al. 1994). We started imaging the core and the inner jet to eliminate the sidelobes from the core over K1, because we expected the K1 flux was around 10% of the integrated flux density of the inner jets, which was ∼ 2.0 Jy at 5 GHz. We adopted both uniform and natural weighting, and performed phaseonly self-calibration several times. After converging visibility phase model and observed phase, we imaged the inner jet and K1 together by applying natural weighting and uv -tapering and performed phase and amplitude self-calibration. Since all the data had the shortest uv distance much longer than 143 k λ , which corresponded to half of the beam size ∼ 720 mas which covered both core and K1 at 5 GHz, missing flux would be caused at K1. Therefore we only obtained the lower limit of the K1 flux.", "pages": [ 3 ] }, { "title": "4.1. Inverted Bow-shock Structure of K1", "content": "In Fig. 1, we show the overall images of 3C 380 in total intensities. In the left panel, the entire image of 3C 380 and zoom-in core image are shown, while the K1 image is shown in the right panel with the beamsize of 1.66 × 1.10 mas resolution at beam position angle -32 . 5 · . K1 in Fig. 1, located at around 0.73 arcsec away from the core, is detected with signal-to-noise ratio over eight. From Fig. 1, we find, with new clarity, the inverted bow-shock shaped structure in K1 edge-brightened region as is previously described (Simon et al. 1990; Wilkinson et al. 1991; Papageorgiou et al. 2006). This finding helps to confirm the original suggestion of inverted bow-shock structure in K1 by Cawthorne (2006) and Papageorgiou et al. (2006), with three times better resolution than these of their images. The width of K1 is about 280 pc (40 mas) measured perpendicular to the jet direction, which is the same size as the K1 diameter suggested by (Simon et al. 1990), and the length of K1 is 140 pc (20 mas) in our images. Compared with the previously obtained VLBI images of K1 (Fig. 12 in Papageorgiou et al. 2006; top left of Fig. 1 in Kameno et al. 2000), we have attained the highest spatial resolution image by adding outer five VLBA and Effelsberg telescopes. The spatial resolution of obtained K1 image is three times higher than that of the previously obtained highest resolution image of K1 at 1.6 GHz (the beam size of Fig. 12 in Papageorgiou et al. 2006 is 5.0 × 3.7 mas).", "pages": [ 4 ] }, { "title": "4.2. Kinematics of K1", "content": "We further attempt to explore the kinematics of K1 by comparing Gaussian-fitted peak positions of K1 in these two epoch images. In Fig. 2, we present the two epoch images of K1, using only VLBA ten antennas, with the common restored beam size 2.54 × 1.81 mas resolution at beam position angle -1 . 95 · with natural weighting. From Fig. 2, we find that the inverted bow-shock structure seen in the first epoch image is also detected in the second epoch. This structure would be maintained between these two observations, that is, over 2.82 years. To compare the position of K1 between 1998 July and 2001 April, we overlay two images of Fig. 2 to produce Fig. 3 left with reference to the core peak position measured by AIPS task JMFIT. Previous studies support that core brightness peak position converges to a stable point within 0.2 mas order (e.g., O'Sullivan & Gabuzda 2009; Hada et al. 2011). Since it is sufficiently small compared with our beam size, we can regard the core as stable in the present work. To measure the peak position and estimate the position accuracy of K1, we fit a single Gaussian model to each slice profile (Fig. 3 right) by using the task SLFIT in AIPS. From Fig. 3 right, we can find the core-facing edge of K1 in each epoch is located at the same position, that is, at ∼ 725 mas distance from the core. The full width of half maximum (FWHM) of a single Gaussian fitted to slice profile of K1 is almost identical to the slight change of each image due to the method of self-calibration. The position accuracy of K1 is derived as a ratio of FWHM of the fitted Gaussian to signal-to-noise ratio (SNR) at K1 (e.g., Walker 1997), which is conservatively estimated less than ∼ 0.79 mas. The derived peak position and their accuracy is summarized in Table 3. Finally we estimate the maximum apparent velocity of K1, β app , max , as peak position displacement over ∆ t =2.82 years with propagation of uncertainty. The peak position displacement is ∆ R = R 1 -R 2 = -0 . 27 mas, where R 1 and R 2 are the peak position at first epoch and second epoch, respectively. The error in the displacement is estimated to be 0 . 97 mas by using propagation of uncertainty, or the root-mean-square (r.m.s.) of the position uncertainty at each epoch. Therefore, the upper limit of the peak position displacement is ∆ R max = -0 . 27 + 0 . 97=0 . 70 mas and the maximum apparent proper motion is estimated to be ∆ R max / ∆ t =0 . 25 mas yr -1 . Thus, we obtain β app , max =9 . 8.", "pages": [ 4, 5 ] }, { "title": "5.1.1. Classification of previously known cases", "content": "As shown in the Introduction, VLBI observations of large-scale knots are quite limited and only a handful of sources are explored. Here, we classify them. Below, we attempt to categorize the internal structures into three typical ones. We do not include some known sources which are difficult to categorize because of their peculiarities (e.g., HST-1 in M87 by Giroletti et al. 2012; northern hot spot of broad line radio galaxy PKS 1421 -490 by Godfrey et al. 2009). Inverted bow-shock typeAs shown in the previous section, the apex of K1 edgebrightened region in 3C 380 faces towards the core. In this work, we call this feature inverted bow-shock structure. The same structure as K1 in 3C 380 is found at C80 in 3C 120, which is the stationary jet feature located 140 pc (80 mas) away from the core with 35 pc in size (Agudo et al. 2012). The key common property between 3C 380 and 3C 120 is their viewing angle. They are classified as misaligned AGN (Abdo et al. 2010), since their viewing angles are larger than those of blazars but smaller than those of radio galaxies. Bow-shock type3C 205 is known as a high-redshift quasar with large viewing angle because there exists a pair of strong hot spots. In the pioneer work of Lonsdale & Barthel (1998), VLBA images of the primary hot spot A in 3C 205 at 1.4 GHz are shown. The VLBI hot spot, located more than 40 kpc away from the core, has the overall size 1400 pc and the jet width around 250 pc. The apex of edge-bright region in hot spot A of 3C 205 against the core face the opposite direction of that of 3C 380. Therefore, here we call this feature bow-shock type structure to tell contrast to inverted bow-shock type. 1 Multi-spots typeThere are several knots and spots having multi-spots in a hot spot. Pictor A is a representative of this. Tingay et al. (2008) reveals that the northwest hot spot in Pictor A at 3.5 kpc scale contains five compact pc-scale components in the spot. The sizes of these components are 30-170 pc. One of the other examples is the southern hot spot of FRI/FRII radio galaxy PKS 2153 -69, which is 200 pc in diameter and contains three components as small as 50 pc Young et al. (2005). The hot spot is located 5 kpc away from the core and would trace the varying position of the precessing jet interaction region with clouds.", "pages": [ 5, 6 ] }, { "title": "5.1.2. Origin of various internal structures", "content": "Bearing the above brief summary in mind, let us discuss possible origins of apparently different internal structures in large scale knots. Viewing angle effectThe inverted bow-shock can be observed in broad-line radio galaxies (BLRGs) and CSS-QSOs, both of which have relatively narrow viewing angles. It is known that 3C 120 is identified as a BLRG and its viewing angle is estimated as θ < = 19 · (e.g., G'omez et al. 2000) and CSS-QSO 3C 380 with inclination angle θ < = 15 · (e.g., Wilkinson et al. 1984; Kameno et al. 2000). On the other hand, the viewing angle of quasar 3C 205 is suggested to be around 40 · , which is the upper end of the quasar/radio galaxy unification according to low lobe flux density ratio (Bridle et al. 1994). Therefore, we speculate that the difference of viewing angles divide images into bow-shock and inverted bow-shock. This point has been already suggested by Cawthorne (2006), modeling the edge-bright region in K1 as a conical-shock seen with small viewing angle. Our work contributes to offer the highest resolution image of K1 structure with new clarity and to show the inverted bow-shock structure supporting the Cawthorne's model. Regarding the physical origin of bow-shock and inverted bow-shock, Lind & Blandford (1985) suggest that the bow-shock is caused by a fast stream moving at relativistic speed up the center of the jet, while for example, Norman et al. (1982) indicate the inverted bow-shock is triggered by Kelvin-Helmholtz instability inside the unshocked jet. In the case of K1 in 3C 380, the inverted bow-shock might be interpreted as the bent backflow (reverse shock) at the jet termination point (Mizuta et al. 2010), since (Wilkinson et al. 1991) mention that K1 is similar to a hot spot seen in the lobes of some Fanaroff-Riley class II sources seen approximately pole-on. Precession effectThe jet precession effect, or we may say jet-jittering effect, is explored and modeled by Scheuer (1982) and Cox et al. (1991) and are known as the 'dentist drill' model. We consider that when the direction of the straight jet changes, causing the termination point to vary over a large-scale spot larger than the cross section of the jet, dynamically young (or long-lived) relic components can be seen as multi-spots. The multi-spots seen in Pictor A can be explained by the dynamically young (or we may say the long-lived) relic components produced by the precessing jets (Tingay et al. 2008). They estimate that a typical synchrotron cooling time scale of these regions from 100 to 700 years is much longer than the dynamical (Alfvenic crossing) time scale of a few decades and indicate that these are dynamically young regions.", "pages": [ 6, 7 ] }, { "title": "5.2. Kinematics of kpc knot K1", "content": "First of all, we stress that the present work is the first attempt to constrain the upper limit on possible proper motion at kpc scales in radio-loud quasars. By comparing Gaussian peak position of K1 slice profiles in 1998 July and 2001 April as reference to the core peak position ( § 4.2), we constrain the resolution of K1 apparent proper motion up to 0.25 mas yr -1 , corresponding to apparent velocity β app , max =9 . 8. In the core region, proper motions of several components are measured by Kameno et al. (2000) and Lister et al. (2009), ranging from 1 . 2 c to 15 c , from sub-mas to 30 mas away from the core, respectively. Our constraint is marginally slower than the fastest and outermost apparent motions measured in the core region, which is the apparent motion of component F, 0.38 mas yr -1 or 15 c , labeled by Kameno et al. (2000) 2 . This implies the jet deceleration or bending occurs between inner jet and K1, or the ejection angle (viewing angle) of K1 has changed from those of the inner jets assuming straight ballistic jets. To confirm jet proper motion at large scales with the maximum resolution of apparent velocity less than 2 c , further new epoch VLBI observation more than 14 years interval from the first epoch observation is required. In the case of jet bending, the apparent position angle difference φ pos ∼ 13 · between F and K1 would be magnified by projection with fixed small viewing angle ( θ view < = 15 · ). Intrinsic jet bending angle φ bend is estimated to be < = 3 . 3 · , where tan φ bend =tan φ pos × sin θ view (Kameno et al. 2000). As for the changes of jet ejection angle, if we assume the same intrinsic velocity β =0 . 9978 for F and K1, the viewing angle should be 3 . 8 · for component F and 1 . 4 · or 10 . 2 · for K1.", "pages": [ 7 ] }, { "title": "5.3. Future prospect", "content": "As a first step, we deal with only 5 GHz VLBI data in this paper. Here we mention future prospects to investigate the properties of K1.", "pages": [ 7 ] }, { "title": "5.3.1. Low frequency spectrum turnover", "content": "Low frequency spectrum turnover can constrain the jet component properties such as magnetic field strength (e.g., PKS 1421 -490: Godfrey et al. 2009). Regarding the case of K1 in 3C 380, previous work of Megn et al. (2006) suggests spectral flattening below ∼ 100 MHz. However discussions in Megn et al. (2006) are based on flux values collected from literatures derived from various different interferometers and in which K1 is smaller than the beam size of each interferometer. Therefore, it seems difficult to determine fluxes accurately. Square Kilometer Array (SKA) 3 will, in future, tell us the real turnover frequency with sufficiently high resolution.", "pages": [ 7 ] }, { "title": "5.3.2. Polarization properties", "content": "Polarization properties are crucial to explore magnetic field geometries. Only Papageorgiou et al. (2006) report the resolved distribution of magnetic vector polarization angle (MVPA) in K1. The MVPA distribution appears tangential to the inverted bow-shock. In order to clarify a change of shock structure in K1, time-variation of MVPA is one of the key quantities for future observations because the sudden change of MVPA strongly suggest the existence of magnetohydrodynamical fast/slow mode waves (e.g., Nakamura et al. 2011). To clarify polarization properties of synchrotron emission is also substantial (Nalewajko & Sikora 2012) for testing reconfinement shock models (e.g., Komissarov & Falle 1997; Stawarz et al. 2006; Bromberg & Levinson 2009).", "pages": [ 8 ] }, { "title": "6. Summary", "content": "To explore the properties of kpc scale knots in radio-loud quasars, we produced the pc scale images of distant knot K1 in a bright CSS quasar 3C 380 with VLBI. Below we summarize the main results obtained in this work. We are grateful to K. Asada and A. Doi for constructive discussions. We thank the anonymous referee for useful comments and suggestions. S.K. acknowledges this research grant provided by the Global COE program of University of Tokyo. This work was partially supported by Grant-in-Aid for Scientific Researches, KAKENHI 2450240 (MK) from the Japan Society for the Promotion of Science (JSPS). This research has made use of data from National Radio Astronomy Observatory (NRAO) archive. The NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.", "pages": [ 8 ] }, { "title": "References", "content": "Thompson, A. R., Moran, J. M., & Swenson, G. W., Jr. 2001, Interferometry and synthesis in radio Young, A. J., Wilson, A. S., Tingay, S. J., & Heinz, S. 2005, ApJ, 622, 830 ✏ ✗ ✑ ✖ ✍ ✕ ✔ ✓ ✒ ✑ ✎ ✎ ✍ ✌ ☞ ☛ ✡ ✠ ✟ ❇", "pages": [ 10, 15 ] } ]

[ { "title": "ABSTRACT", "content": "We present 100 year light curves of Kepler planet-candidate host stars from the Digital Access to a Sky Century at Harvard (DASCH) project. 261 out of 997 host stars have at least 10 good measurements on DASCH scans of the Harvard plates. 109 of them have at least 100 good measurements, including 70% (73 out of 104) of all host stars with g ≤ 13 mag, and 44% (100 out of 228) of all host stars with g ≤ 14 mag. Our typical photometric uncertainty is ∼ 0 . 1 -0 . 15 mag. No variation is found at 3 σ level for these host stars, including 21 confirmed or candidate hot Jupiter systems which might be expected to show enhanced flares from magnetic interactions between dwarf primaries and their close and relatively massive planet companions. Subject headings: Planetary Systems - Stars: general", "pages": [ 1 ] }, { "title": "100-year DASCH Light Curves of Kepler Planet-Candidate Host Stars", "content": "Sumin Tang 1 , 2 , 3 , Dimitar Sasselov 1 , Jonathan Grindlay 1 , Edward Los 1 , Mathieu Servillat 1 , 4", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In 2011 the Kepler Mission announced 997 stars with a total of 1235 planetary candidates that show transit-like signatures (Borucki et al. 2011). It is estimated that more than 95% of these candidates are planets (Morton & Johnson 2011), although a much higher false positive rate of 35% for Kepler close-in giant candidates is reported by Santerne et al. (2012). Kepler has completed 3 years of observations and with a 4-year extension will accumulate 7.5 years of data for these host stars. Searching the host stars for significant very long-term photometric variations on timescales of decades and for rare flare events is important for understanding fully the environments of their planets and for constraining the habitability of exoplanets in general. For example, a recent study by Lecavelier des Etangs et al. (2012) found increased evaporating atmosphere of hot Jupiter HD 189733b during a transit, which is probably related to a X-ray flare 8h before the transit. Here we present the 100-year light curves of Kepler planet-candidate host stars from the Digital Access to a Sky Century at Harvard (DASCH) project (Grindlay et al. 2009, 2012). The photometric variability of about 150,000 Kepler target stars, including the sample with planetary candidates, shows low-level variability due to modulation by spots and other manifestations of magnetic activity, e.g., white light flares (Basri et al. 2011). None of these variabilities have amplitudes that would be detectable ( > 0 . 3 mag if only a single observation) in the DASCH data. Here we aim to complement the short-timescale and high-precision photometry by Kepler with the much longer timescales available from DASCH. The DASCH project can reveal stars with rare flare events or extremely slow changes in luminosity. It is of interest to search for both kinds of variability with DASCH. Extremely large ( > 1mag) flares could drive significant mass loss from the atmospheres of hot Jupiters, and long term changes in host star luminosity could drive changes in planetary winds and spectral composition. This paper provides DASCH light curves for the initial sample of Kepler planets presented by Borucki et al. (2011).", "pages": [ 1 ] }, { "title": "2. DASCH light curves", "content": "In order to take advantage of the unprecedented Kepler data on short timescales (Borucki et al. 2010), which complements DASCH data on long timescales, we have scanned and processed 3735 plates taken from the 1880s to 1990 in or covering part of the Kepler field. Each plate covers 5 -40 degrees on a side, and most of them are bluesensitive (close to Johnson B). Most plates have limiting magnitudes 12 -14 mag, while ∼ 9% of them (mc series) are down to ∼ 17 mag. More details on the coverage and limiting magnitudes of the plates in the Kepler field are described in Tang et al. (2013). We used the Kepler Input Catalog (KIC; Brown et al. 2011) for photometric calibration. The typical relative photometric uncertainty is ∼ 0 . 1 -0 . 13 mag, as measured by the median light curve RMS of stars (Laycock et al. 2010; Tang et al. 2013; see also Table 1 and Figure 3 in this paper). Given that most stars are constant at the 0 . 1 mag level, their light curve RMS values are dominated by photometric uncertainty, and thus the median light curve RMS represents the relative photometric uncertainty. The typical absolute photometric uncertainty, as measured by the difference between our measurements and KIC g band magnitudes, is ∼ 0 . 2 mag, and up to ∼ 0 . 35 mag for bright stars ( g < 11 mag). The larger absolute photometric uncertainty is caused by additional contributions from the uncertainties in KIC g mag (especially at the bright end), and the difference between the effective color of the plates (close to Johnson B) and g band. The magnitude uncertainties we provide in DASCH light curves, are defined as the scattering of ( m DASCH -g KIC ) for stars in local spatial bins with similar magnitudes, and thus represent the absolute photometric uncertainty (Tang et al. 2013). The typical absolute astrometric uncertainty (per data point) is ∼ 0 . 8 -5 '' , depending on plate scale (Laycock et al. 2010; Los et al. 2011; Servillat et al. 2011). Example DASCH light curves of 3 Kepler hot Jupiter host stars, as well as a binary are shown in Figure 1. Fainter stars have fewer DASCH measurements, due to smaller number of deeper plates available. For a given star, every measurement is typically separated by days to months, and could be up to years. Therefore, one stellar flare, if hap- pened during the plate observation, is expected to appear on a single plate only. Some fields do have multiple exposures which our pipeline is able to process individually (see Los et al 2011), and would allow stellar flares to be confirmed by successive measurements; no such flares were seen. Among the 4 stars plotted in Fig. 1, K10666592 (HAT-P-7; P'al et al. 2008), K10264660 (Kepler14; Buchhave et al. 2011) and K8191672 (Kepler5; Koch et al. 2010) are confirmed hot Jupiter host stars. K5122112 (KOI 552; Borucki et al. 2011) was listed as a hot Jupiter candidate, but later has been unveiled as a binary (Bouchy et al. 2011). All three planets have R ≥ R Jupiter , a < 0 . 1 AU, and equilibrium temperature > 1300 K. No variation is detected. All the light curve data and plots are available at the DASCH website 1 . Only good measurements are included. We excluded blended images, measurements within 0 . 75 mag of the limiting magnitude which are more likely to be contaminated by noise, images within the outer border of the plates whose width is 10% of the plate's minor-axis length (annular bin 9; see Laycock et al 2010), and dubious points with image profiles different from neighbor stars and thus are suspected to be emulsion defects or dust. Stars with strong correlation between magnitude measurements and plate limiting magnitudes, or between magnitude measurements and plate astrometry uncertainties, are also excluded, which are very likely to be polluted by noise or blends. More detailed descriptions can be found at Tang et al. (2013). Note that these Kepler planet host stars have variations at the 0 . 01% -0 . 1% level in the Kepler light curves, which are much smaller than the plotting symbols in Figure 1 and our photometric accuracy. For comparison, the Kepler light curves of the 4 example Kepler planet-candidate host stars (from Q0 or Q1 to Q6) are also shown in Figure 1 as green dots. We used the PDC corrected flux, which are converted to magnitudes and shifted to the mean magnitudes of DASCH light curves (See http://keplergo.arc.nasa.gov/CalibrationSN.shtml). Among 997 host stars, 261 stars have at least 10 good measurements on DASCH plates, and 109 stars have at least 100 good measurements. Distributions of g band magnitudes for all the host stars and host stars with at least 10 or 100 DASCH measurements, are shown in Figure 2. We have at least 100 measurements for 70% (73 out of 104) of all host stars with g ≤ 13 mag, and 44% (100 out of 228) of all host stars with g ≤ 14 mag. Most stars brighter than g = 13 mag we lost (i.e. with less than 100 good measurements), were due to blending with neighbor stars, as expected in such a crowded field, with 74% of them (23 out of 31) having bright neighbor stars with g < 14 mag within 1 arcmin. For comparison, for g ≤ 13 stars with at least 100 good measurements, only 14% (10 out of 73) of them have g < 14 mag neighbor stars within 1 arcmin. We have carefully examined light curves of the 261 planet candidate host stars, and none of them showed variations (flares or dips) at the > 3 σ level, where σ here is our absolute photometric uncertainty with typical value of ∼ 0 . 2 mag for g > 13 objects, and up to 0 . 35 mag for g < 12 objects. Nor did we see any long-term photometric trends. We also compare the light curve RMS of these host stars vs. other stars with similar magnitudes in the Kepler field, as shown in Figure 3. The raw light curve RMS vs. g band magnitude for the 261 host stars with at least 10 DASCH measurements are shown as blue dots. The black solid line shows the median light curve RMS of all the stars with at least 10 DASCH measurements in the Kepler field, and the red dashed lines and green dash-dotted lines show the 1 σ and 2 σ distributions, respectively. The median, 1 σ , and 2 σ distributions of light curve RMS are calculated in 0.5 magnitude bins, after three iterations of 4 σ -clipping, to exclude variable stars and dubious light curves contaminated by blending or plate defects. Stars with bright neighbors ( ≥ 30% the flux of the star in KIC g band) within 30' are also excluded in the calculation of median, 1 σ , and 2 σ distributions, to avoid contamination from blending. Note the drop of RMS for stars fainter than 14 mag is due to the fact that in general deeper plates are of better quality. Our plates are not a homogeneous sample. Stars with different magnitudes, even in the same region of the sky, are actually covered by different plates. For example, a 12th mag star is detected on all the plates deeper than 12th mag, while a 15th mag star is only detected on plates deeper than 15th mag. Most of the deeper plates, such as mc series, are of much better quality compared with the shallow plates, which leads to the decrease of typical (median, etc.) light curve rms of fainter stars (g > 14 mag). None of the planet candidate host stars has a light curve RMS more than 2 σ greater than the median RMS of stars of similar magnitudes. There is one planet host star, i.e. Kepler-21 (KIC 3632418, g = 8 . 74; Howell et al. 2012), with light curve RMS close to the 2 σ distribution. Further examination shows that some images are marginally contaminated by a g = 11 . 43 mag neighbor star located at 51' away from the star, and thus its RMS excess is dubious. It has been suggested that the magnetic interaction between hot Jupiters and their host stars should enhance stellar activity, and may lead to phase shifts of hot spots on the stellar chromosphere (Cuntz & Shkolnik 2002; Shkolnik et al. 2003, 2005; Kopp et al. 2011; Poppenhaeger & Schmitt 2011). Given the extremely long timescale covered by DASCH, it is interesting to examine the light curves of stars hosting hot Jupiters. Among the 261 planet candidate host stars with at least 10 DASCH measurements, 21 of them host hot Jupiter planet candidates ( R ≥ R Jupiter and a < 0 . 4 AU), and 9 of them have at least 100 DASCH measurements (4 are shown in Figure 1 as examples). DASCH light curve properties of these Jupiter planet candidate host stars, including number of measurement, median magnitude and RMS in the light curves, are listed in Table 1. None of them showed variations at the > 3 σ level.", "pages": [ 2, 3, 4 ] }, { "title": "3. Conclusion", "content": "The Kepler mission has now discovered more than 2000 planetary candidates (only the first 1235 considered here) and provided unparalleled precision light curves for their host stars. Here we complement that database with much longer timescale 100-year light curves from the DASCH project. Despite their inferior photometric accuracy, the DASCH light curves sample such an extended period of time (e.g., tens of solar-like cycles of activity), that rare or very slow phenomena can be studied. From the statistical sample of the Kepler planet host stars, limits on their longterm variations and rare flare events, such as the X-ray/EUV flare from HD189733 (Lecavelier des Etangs et al. 2012), help us understand the planetary environments around main sequence stars and the habitability of exoplanets in general. We note that the HD189733 system is not within the Kepler field and so has not yet been scanned by DASCH. However, given the luminosity of the flare (7 × 10 28 erg s -1 ; Lecavelier des Etangs et al. 2012), and the relatively luminous (K1.5V) host star, such a flare is beyond the detection limit of DASCH. Relatively larger optical flares would be expected for cooler stars with hot Jupiter companions. Because Kepler is mostly targeting on Sun like stars, none of M dwarf and hot Jupiter system is in the sample of the 261 host stars we studied. We have scanned and processed 3735 plates taken from the 1880s to 1990 in or covering part of the Kepler field, and studied the light curves of 261 planet host stars that have at least 10 good measurements on DASCH plates. We find no photometric variations at the 3 σ level. All the light curve data and plots of planet hosts in this study are available at the DASCH website 2 . Besides, we have released all the DASCH data in the Kepler field to the public in DASCH Data Release 1 3 . DASCH light curves over ∼ 100 year timescales will continue to provide unique constraints for planet host stars, as well as any other interesting objects in the Kepler field. We thank the anonymous referee for suggestions that have helped improve this paper. We thank Alison Doane, Jaime Pepper, David Sliski and Robert J. Simcoe at CfA for their work on DASCH, and many volunteers who have helped digitize logbooks, clean and scan plates (http://hea-www.harvard.edu/DASCH/team.php). This work was supported in part by NSF grants AST0407380 and AST0909073 and now also the Cornel and Cynthia K. Sarosdy Fund for DASCH .", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Basri, G., Walkowicz, L. M., Batalha, N., et al. 2011, AJ, 141, 20 Bonomo, A.S., et al. 2012, A&A, 538, 96 Borucki, W. J., et al. 2010, Science, 327, 977 Borucki, W. J., et al. 2011, ApJ, 736, 19 Bouchy, E., et al. 2011, A&A, 533, 83 Brown, T. M., Latham, D. W., Everett, M. E., & Esquerdo, G. A. 2011, AJ, 142, 112 Buchhave, L.A., et al. 2011, ApJS, 197, 3 Cuntz, M., & Shkolnik, E. 2002, Astron. Nachr., 323, 387 D'esert, J-M., et al. 2011, ApJS, 197, 14 Endl, M., et al. 2011, ApJS, 197, 13 Fortney, J. J., et al. 2011, ApJS, 197, 9 Grindlay, J., Tang, S., Los, E., & Servillat, M. 2012, Proc. IAUS285, Camb. Univ. 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[ { "title": "Using Energy Peaks to Count Dark Matter Particles in Decays", "content": "Kaustubh Agashe a , Roberto Franceschini a , Doojin Kim a , and Kyle Wardlow a a Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, U.S.A.", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the determination of the symmetry that stabilizes a dark matter (DM) candidate produced at colliders. Our question is motivated per se , and by several alternative symmetries that appear in models that provide a DM particle. To this end, we devise a strategy to determine whether a heavy mother particle decays into one visible massless particle and one or two DM particles. The counting of DM particles in these decays is relevant to distinguish the minimal choice of Z 2 from a Z 3 stabilization symmetry, under which the heavy particle and the DM are charged and the visible particle is not. Our method is novel in that it chiefly uses the peak of the energy spectrum of the visible particle and only secondarily uses the M T 2 endpoint of events in which the heavy mother particles are pair-produced. We present new theoretical results concerning the energy distribution of the decay products of a three-body decay, which are crucial for our method. To demonstrate the feasibility of our method in investigating the stabilization symmetry, we apply it in distinguishing the decay of a bottom quark partner into a b quark and one or two DM particles. The method can be applied generally to distinguish two- and three-body decays, irrespective of DM.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Extensions to the Standard Model (SM) of particle physics are motivated for various reasons; perhaps the most important among these is the necessity of a fundamental mechanism for electroweak symmetry breaking (EWSB). Additionally, the related Planck-weak hierarchy problem of the SM must also be addressed. In such extensions of the SM, there generally exists a new particle at or below the TeV scale which cancels the quadratic divergence of the Higgs mass from the top quark loop in the SM. Such a particle is typically a color triplet with a significant coupling to the SM top quark, and has an electric charge of +2 / 3. Following the literature, we will generically call such particles 'top partners' and denote them by T ' 1 . These top partners often come along with bottom partners, which we similarly denote as B ' . The typical reason for this is that the left-handed (LH) top quark is in a doublet of SU (2) L with the LH bottom quark. We then expect top and bottom quark-rich events from the production and decay of these new particles at the LHC. Another seemingly unrelated motivation for new physics at the TeV scale is the evidence for the existence of dark matter (DM) in the Universe, combined with the absence of a viable DM candidate in the SM [1]. A well-motivated candidate for this DM is found in a stable weakly interacting massive particle (WIMP), especially one that arises as part of an extension to the SM at the TeV scale. The motivation for this new physics becomes even stronger when the extension to the SM solves other problems inherent in the SM. These scenarios often involve heavier new particles that are charged under both the symmetry that keeps the DM stable and the SM gauge group. These new particles should then be copiously produced at the LHC and must decay into DM particles and SM states, given that the latter are not charged under the DM stabilization symmetry. Thus we expect this new physics to give rise to events at the LHC with large missing energy, in association with jets, leptons, and photons. Combining the above two lines of argument, we realize that the most attractive scenarios are those extensions of the SM which not only solve the Planck-weak hierarchy problem, but also have a WIMP DM candidate. In this case, it is likely that the top and bottom partners are also charged under the DM stabilization symmetry. These extensions will then result in top and bottom quarkrich events at the LHC in which the new particles give rise to missing energy . The classic example of such an extension is SUSY, where R -parity stabilizes the DM [2]. The associated signals from the scalar top and bottom partners have been studied in great detail. A more recent example is little Higgs models [3] with T -parity [4]. Like SUSY, the signals from the fermionic partners of the top and other quarks in these models have been thoroughly studied. In short, we find that a search for events with top or bottom quarks and missing energy should be a top priority of the LHC. Once the existence of new physics has been established, the most urgent issue that will then have to be addressed is the determination of the details of the dynamics underlying this new physics. In particular, it will be crucial to determine the properties of the top and bottom partners using as model-independent an approach as possible. This detailed study would also offer major hints regarding the resolution of the Planck-weak hierarchy problem. For largely model-independent work on fermionic bottom and top partners' discovery potential at the LHC see Refs. [5, 6] and for the determination of generic partners' spin and mass see Refs. [7]. However, we remark that in these works it has been assumed that the top or bottom partner decays into only one DM particle, which is expected when the DM is stabilized by a Z 2 symmetry. While Z 2 is perhaps the simplest DM stabilization symmetry, it is by no means the only possibility: see references [8, 9]. The point, especially in the case of such nonZ 2 symmetries, is that more than one DM can appear in the decays of top and bottom (and other SM) partners: for example, two DM are allowed with Z 3 as in [8], but not with Z 2 . We believe that a truly model-independent approach to the determination of the top and bottom partners' properties should include this possibility of multiple DM in addition to different spins for the top and bottom partners. With this goal in mind, we aim to devise a strategy that uses experimental data to determine the number of DM in these decays and accordingly to identify the stabilization symmetry of the dark matter. Below, we outline a general strategy and then apply it to the specific case of bottom partner decays. We concentrate on the distinction between two general decay topologies: where b is a (single) SM visible particle, X and Y are two potentially different invisible particles and A is a heavier particle that belongs to the new physics sector. In the context of the models that we have discussed, A is the heavy particle charged under the DM stabilization symmetry and the particles labeled X and Y are the DM particles. In particular, we focus on scenarios where the two decays are mutually exclusive, i.e. where the stabilization symmetry and the charges of the involved particles are such that one decay can happen and not the other. This mutual exclusivity can be the case with both Z 2 and Z 3 as the stabilization symmetry. To wit, if the SM particle b is not charged under the stabilization symmetry and all the new particles A,X,Y are, then the Z 2 symmetry allows only for two-body decays of A . On the other hand, both the two and three-body decays of A are allowed by the Z 3 symmetry by itself. However, we assume that other considerations forbid (or suppress) the two-body decay in this model. We choose to concentrate on this realization of the Z 3 -symmetric model in part because this is the case that cannot be resolved using the results of previous work on the DM stabilization symmetry. This is the case, for instance, in Ref. [10], where purely two-body decays of A could be distinguished from mixed two- and three-body decays, but not from the purely three-body decays that we are now taking into consideration. In this paper, we develop a method based primarily on the features of the energy distribution of the visible final state b to differentiate between the cases of purely two- and three-body decays. We remark that this is the first work to use the energy distribution of the the decay products to study the stabilization symmetry of the DM. In fact, other work has typically focused on using Lorentz invariant quantities or quantities that are invariant under boosts along the beam direction of the collider. This is the case for Refs. [10, 11, 12, 13]. In particular, Refs. [10, 11, 12] used the endpoints of kinematic distributions to probe the stabilization symmetry of the DM, whereas our method relies quite directly on peak measurements and only marginally on endpoint measurements. Additionally, we note that the methods developed in Refs. [12, 13] apply only to the case where there are more than one visible particle per decay. Therefore, our result for cases where there is only one visible particle per decay is complementary to the results of the above references. Our basic strategy is explained in the following. It relies on a new result: assuming massless visible decay products and the unpolarized production of the mother particles, we will show that in a three-body decay the peak of the observed energy of a massless decay product is smaller than its maximum energy in the rest frame of the mother. This observation can be used in conjunction with a previously observed kinematic characteristic of the two-body decay to distinguish the stabilization symmetry of the DM. Specifically, it was shown in Ref. [14, 15] that for an unpolarized mother particle, the peak of the laboratory frame energy distribution of a massless daughter from a twobody decay coincides with its (fixed) energy in the rest -frame of the mother. Clearly, to make use of these observations in distinguishing two from three-body decays, we need to measure the 'reference' values of the energy that are involved in these comparisons. Moreover, the procedure that is to be used to obtain this reference value from the experimental data should be applicable to both two and three-body decays. To this end, we find that when the mother particles are pair produced, as happens in hadronic collisions, the M T 2 variable can be used. Thus, these observations make counting the number of invisible decay products possible by looking only at the properties of the single detectable particle produced in the decay. However, it is worth noting that our proof of the above assertion regarding the kinematics of two- and three-body decays is only valid with a massless visible daughter and an unpolarized mother. Therefore, care must be taken when discussing cases with a massive daughter or a polarized mother. To illustrate the proposed technique, we will study how to distinguish between pair-produced bottom partners each decaying into a b quark and one DMfrom pair-produced bottom partners each decaying into a b quark and two DM particles at the LHC 2 . As discussed above, a bottom partner appears in many motivated extensions to the SM, so we posit that this is a relevant example. Furthermore, we remark that the b quark is relatively light compared to the expected mass of the bottom partner, so that our theoretical observation for massless visible particles is expected to apply. Additionally, the production of bottom partners proceeds dominantly via QCD and is thus unpolarized. In this sense, the example of a bottom partner is well-suited to illustrate our technique. Finally, it is known that the backgrounds to the production of bottom partners may be rendered more easily manageable than for those of top partners [5], which would be a well-motivated alternative example. Specializing to the example of bottom partners, our goal then is to distinguish the two processes illustrated in Figure 1 at the collider where χ is an invisible particle and a bar denote anti-particles. In these processes, we assume that there are no on-shell intermediate states. We consider the case where the decay into two χ can happen only if the stabilization symmetry of the DM is Z 3 , while the decay into one χ is characteristic of the Z 2 case. As said before, we focus on this scenario because it has thus far been left uninvestigated by previous studies on the experimental determination of the stabilization symmetry of the dark matter [10, 12]. From here, we organize our findings as follows: In Section 2, we review the current theory and we derive new results about the energy spectrum of the decay products of two- and threebody decays. These are then the foundation of the general technique presented in Section 3 for differentiating decays into one DM particle from those into two DM particles. In Section 4, we apply this technique to the specific case of bottom partners at the LHC. We conclude in Section 5.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 Theoretical observations on kinematics", "content": "We begin first by reviewing the relevant theoretical observations about the kinematics of twobody and three-body decays. Specifically, we review the remarks on two-body decays described in [14]. We then generalize this result to three-body decay kinematics and study the features that distinguish it from two-body decay kinematics. We also briefly review applications of the kinematic variable M T 2 to two-body and three-body decays and discuss the distinct features of the two different decay processes [10, 16]. For the two-body decay, we assume that a heavy particle A decays into a massless visible daughter b 3 and another daughter X which can be massive and invisible: On the other hand, for a three-body decay the heavy particle A decays into particles b , X and another particle Y Like particle X , particle Y can also be massive and invisible, but it is not necessarily the same species as particle X .", "pages": [ 5, 6 ] }, { "title": "2.1.1 Two-body decay", "content": "It is well-known that the energy of particle b in the rest frame of its mother particle A is fixed, which implies a δ function-like distribution, and the simple analytic expression for this energy can be written in terms of the two mass parameters m A and m X : Typically, the mother particle is produced in the laboratory frame at colliders with a boost that varies with each event. Since the energy is not an invariant quantity, it is clear that the δ functionlike distribution for the energy as described in the rest frame of the mother is smeared as we go to the laboratory frame. Thus, naively it seems that the information encoded in eq. (6) might be lost or at least not easily accessed in the laboratory frame. Nevertheless, it turns out that such information is retained. We denote the energy of the visible particle b as measured in the laboratory frame as E b . Remarkably, the location of the peak of the laboratory frame energy distribution is the same as the fixed rest-frame energy given in eq. (6): as was shown in [14, 15]. Let us briefly review the proof of this result while looking ahead to the discussion of the threebody case. As mentioned before, the rest-frame energy of particle b must be Lorentz-transformed. The energy in the laboratory frame is given by where γ is the Lorentz boost factor of the mother in the laboratory frame and θ ∗ defines the angle between the emission direction of the particle b in the rest frame of the mother and the direction of the boost glyph[vector] β , and where we have used the relationship γβ = √ γ 2 -1. If the mother particle is produced un polarized, i.e., it is either a scalar particle or a particle with spin produced with equal likelihood in all possible polarization states, the probability distribution of cos θ ∗ is flat, and thus so is that of E b . Since cos θ ∗ varies between -1 and +1 for any given γ , the shape of the distribution in E b is simply given by a rectangle spanning the range It is crucial to note that the lower and upper bounds of the above-given range are always smaller and greater, respectively, than E b = E ∗ b for any given γ , so that E ∗ b is covered by every single rectangle. As long as the distribution of the mother particle boost is non-vanishing in a small region near γ = 1, E ∗ is the only value of E b to have this feature. Furthermore, because the energy distribution is flat for any boost factor γ , no other energy value has a larger contribution to the distribution than E ∗ b . Thus, the peak in the energy distribution of particle b is unambiguously located at E b = E ∗ b . The existence of this peak can be understood formally. From the fact that the differential decay width in cos θ ∗ is constant, we can derive the differential decay width in E b for a fixed γ as follows: where the two Θ( E b ) are the usual Heaviside step functions, which here merely define the range of E b . To obtain the full expression for any given E b , one should integrate over all γ factors contributing to this E b . Letting g ( γ ) denote the probability distribution of the boost factor γ of the mother particles, the normalized energy distribution f 2-body ( E b ) can be expressed as the following integral The lower limit in the integral can be computed by solving the following equation for γ : with the positive (negative) signature being relevant for E b ≥ E ∗ b ( E b < E ∗ b ). We can also calculate the first derivative of eq. (11) with respect to E b as follows: glyph[negationslash] The solutions of f ' 2-body ( E b ) = 0 give the extrema of f 2-body ( E b ), and given the expression f ' 2-body ( E b ) in eq. (13), these zeros originate from those of g ( γ ). For practical purposes, one can take g ( γ ) to be non-vanishing for particles produced at colliders for any finite value of γ greater than 1 4 . As far as zeros are concerned, two possible cases arise for g (1) (corresponding to E b = E ∗ b ). If it vanishes, f ' 2-body ( E b = E ∗ b ) ∝ g (1) = 0, which implies that the distribution has a unique extremum at E b = E ∗ b . If g (1) = 0, f ' 2-body ( E b ) has an overall sign change at E b = E ∗ b . As a result, the distribution has a cusp and is concave-down at E b = E ∗ b . Moreover, the function f 2-body ( E b ) has to be positive to be physical, and has to vanish as E b approaches either 0 or ∞ , which is manifest from the fact that in those two limits the definite integral in eq. (11) is trivial. Combining all of these considerations, one can easily see that the point E b = E ∗ b is necessarily the peak value of the distribution in both cases.", "pages": [ 6, 7, 8 ] }, { "title": "2.1.2 Three-body decay", "content": "glyph[negationslash] We now generalize the above argument to three-body decays. We denote the energy of the visible particle b measured in the rest frame of the mother particle A as ¯ E b . We also denote the normalized rest-frame energy distribution of particle b as h ( ¯ E b ). In the two-body decay, this rest-frame energy is single-valued (see eq. (6)), and so the corresponding distribution h ( ¯ E b ) was trivially given by a δ -function. However, when another decay product is introduced, for instance, particle Y in eq. (5), then the energy of particle b is no longer fixed, even in the mother's rest frame: h ( ¯ E b ) = δ ( ¯ E b -E ∗ b ) . Although the detailed shape of this rest-frame energy distribution is model-dependent, the kinematic upper and lower endpoints are model-independent. Since particle b is assumed massless, the lower endpoint corresponds to the case where energy-momentum conservation is satisfied by particles X and Y alone. On the other hand, the upper endpoint is obtained when the invariant mass of X and Y equals m X + m Y ,which corresponds to the situation where X and Y are produced at rest in their overall center-of-mass frame. Thus, we have For any fixed γ , the differential decay width in the energy of particle b in the laboratory frame is no longer a simple rectangle due to non-trivial h ( ¯ E b ). For any specific laboratory frame energy E b , contributions should be taken from all relevant values of ¯ E b and weighted by h ( ¯ E b ). This can be written as where with E b running from 0 to ¯ E max b ( γ + √ γ 2 -1 ) . Again, since the visible particle is assumed massless, ¯ E min b is zero and so the second equality in eq. (17) holds trivially. Finding an analytic expression for the location of the peak is difficult because of the modeldependence of h ( ¯ E b ), and it follows that the precise location of the peak is also model-dependent. Nevertheless, we can still obtain a bound on the position of the peak for fixed γ . Suppose that we are interested in the functional value of the energy distribution at a certain value of E b in the laboratory frame; according to the integral representation given above, the relevant contributions to this E b come from a range of center of mass energies which go from ¯ E ' b to ¯ E '' b , where these are defined by Each energy contributes with weight described by h ( ¯ E b ), as implied by eq. (16). Let us assume that ¯ E '' b = ¯ E max b and denote the corresponding energy in the laboratory frame as E limit b , given by From these considerations, it follows that all rest-frame energies in the range from ¯ E ' b = E limit b ( γ + √ γ 2 -1) to ¯ E '' b = ¯ E max b contribute to a chosen energy in the laboratory frame, E limit b . On the other hand, any laboratory frame energy greater than E limit b has contributions from ¯ E ' b > E limit b ( γ + √ γ 2 -1) to ¯ E '' b = ¯ E max b ; the relevant range of the rest-frame energy values will shrink so that the peak cannot exceed E limit b : In order to ensure that the first inequality holds even for γ = 1, we assume in the last equation that h ( ¯ E max b ) = 0, which is typically the case for a three-body decay. In order to obtain the shape of the energy distribution of particle b in the laboratory frame, all relevant values of γ should be integrated over as with the two-body kinematics in the previous section. Hence, the laboratory frame distribution reads Since the argument leading to eq. (22) holds for every γ , the superposition of contributions from all relevant boost factors does not alter this observation. Therefore, we can see that irrespective of g ( γ ) and h ( ¯ E b ), the peak position of the energy distribution of particle b in the laboratory frame is always less than the maximum rest-frame energy: To gain intuition on the magnitude of the typical difference between the peak of the energy distribution in the laboratory frame and the maximum rest frame energy, we show the ratio of the two as a function of γ in Fig. 2. From the figure, it is clear that as the typical γ increases beyond γ = 1, i.e., as the system becomes more boosted, the location of the peak in the energy distribution becomes smaller. An appreciable shift of order 10% is achieved for a modest boost of order γ -1 glyph[similarequal] 10 -2 . It should be noted that all results here for both two-body and three-body decays are valid to leading order in perturbation theory. The presence of extra radiation in the decay will effectively add extra bodies to the relevant kinematics. Specifically, extra radiation can turn a two-body decay into a three-body one, which for our investigation would constitute a fake signal of two DM particles being produced in the decay of a heavy new physics particle. Therefore, we have to remark that in some cases, for instance, when the heavy new physics is typically produced with very small boost, the differences between the two scenarios of DM stabilization may be tiny and a study beyond leading order may be necessary. From Fig. 2 it seems, however, that the typical effect of the presence of two dark matter particles per decay of the heavy new particle is to easily induce an order one effect on the peak position. Therefore, we anticipate that such an effect would be much larger than the expected uncertainty from higher order corrections, which we estimate to be of order 10%. GLYPH<144> Before closing this section, we emphasize that we shall use the right-hand sides of eqs. (7) and (24) as 'reference' values to which the measurements of their respective left-hand side values (extracted from the energy distribution) are to be compared. In the next section, we show that such a reference value can, in fact, be extracted from an analysis of M T 2 .", "pages": [ 8, 9, 10, 11 ] }, { "title": "2.2 The kinematic endpoint of the M T 2 distribution", "content": "In this section, we review how the M T 2 variable is implemented for the two- and three-body decays of heavy particles produced at a collider. For our M T 2 analysis, we make further assumptions as follow: for the two-body decay and the three-body decay, respectively. The last assumption is especially relevant to make contact with the problem of distinguishing the Z 2 and the Z 3 dark matter interactions, as detailed in the introduction.", "pages": [ 11 ] }, { "title": "2.2.1 Two-body decay, one visible and one invisible", "content": "The M T 2 variable generalizes the transverse mass to the cases where pair-produced mother particles each decay into visible particles along with missing particles (see Ref. [16] and references therein for a detailed review). Specifically, it can be evaluated for each event by a minimization of the two transverse masses in each decay chain, under the constraint that the sum of all the transverse momenta of the visible and invisible particles vanishes. By construction, each of the transverse masses in both decay chains involve the mass of the invisible particle(s), and thus so does M T 2 . Since a priori we are not aware of the invisible particles' masses, we are required to introduce a trial mass parameter into the definition of M T 2 . We denote this trial mass by ˜ m . The dependence of the definition of M T 2 on the trial mass makes it a function of ˜ m . This function has been shown in Ref. [16] to have a kinematic endpoint where the C parameter is given by This C parameter can be deduced from eq. (26) by substituting the experimental value of the kinematic endpoint and the chosen trial DM mass.", "pages": [ 11, 12 ] }, { "title": "2.2.2 Three-body decay, one visible and two invisibles", "content": "As previously mentioned, for three-body decays we assume that the extra particle Y is also invisible. Therefore, as far as the detectable final state is concerned, the three-body decay looks like a twobody process. Since we are not a priori aware of the number of invisible particles involved in the decay process, a natural assumption is to hypothesize a single invisible particle per decay chain as in a two-body decay. In this context, we shall refer to this supposition as the 'na¨ıve' M T 2 method (for three-body decay) [10]. In each event, this three-body decay can be understood as a two-body decay process where the two invisible particles X and Y behave like a single invisible particle with an effective mass equal to the invariant mass of the system formed by particles X and Y . As is well-known, the invariant mass of the particles X and Y follows a distribution and ranges from m X + m Y to m A . Therefore, the overall kinematic endpoint in the corresponding M T 2 distribution arises when the invariant mass of the X -Y system is minimized [10]. The theoretical expectation for M max T 2 , 3 -body is similar to that of the two-body decay: where the C parameter is given by When comparing to the two-body case, two different features should be noted. First, given the same mother particle, visible state, and trial DM mass, the kinematic endpoint of the M T 2 distribution for the three-body process is expected to be smaller than that of the two-body process. This is because for the three-body decay, one more invisible particle, Y , is involved (see and compare eqs. (27) and (29), i.e., m X + m Y ≥ m X ). Second, the fall-off of the distribution of the three-body process at the endpoint is faster than in the two-body process. This is because in the three-body case more kinematic constraints need to be satisfied to reach the kinematic endpoint [10, 11]. Before closing the Section, a further critical observation is in order. According to eqs. (26) and (28), we see that the observed values of M max T 2 as a function of the various chosen trial DM masses ( ˜ m ) can be fitted with the same equation in both the two- and three-body cases: where the parameter C can be extracted from the fit. This will be used in the following to extract the C parameter without making any assumption on the number of invisible products in the decay. The fact that the M T 2 endpoint can be described with the same parametrization in terms of a generic C parameter, as in eq. (30), is not surprising. In fact, for the two-body case in events near the endpoint each mother needs to have its decay products ( b and X ) emitted at the same rapidity (although the two mothers A can be at different rapidities) [16]. Analogously for the three-body case, the two invisible decay products ( X and Y ) and the particle b produced at the same interaction vertex all need to share the same rapidity. In such a situation, the two invisible particles are kinematically equivalent to a single invisible particle, and so the decay can still be effectively reduced to a two-body decay. In this sense, M max T 2 for the three-body case corresponds to the same kinematic configuration that gives the endpoint for the two-body case. However, it must be noted that the C parameter actually provides different information in the two cases. For two-body decays, the C parameter in eq. (27) is the same as the rest-frame energy of particle b in eq. (6), whereas for three-body decays, the C parameter in eq. (29) is the same as the maximum energy of particle b in the rest frame in eq. (15) 5 : This observation puts us in the position to extract the C parameter from the M T 2 distribution and compare it with the peak value in the energy distribution of the visible particle so as to test the nature of the decay.", "pages": [ 12, 13 ] }, { "title": "3 General Strategy to distinguish Z 2 and Z 3", "content": "We now apply the above theoretical observation to the determination of the underlying DM stabilization symmetry. To pinpoint this stabilization symmetry, we study the energy distribution of the particle b from the process defined in eq. (25). In particular, we exploit relation between this energy distribution and the distribution of the M T 2 variable in the same process. As will be clear from the following analysis, the correlation between features of the distribution of these two observables will allow us to make a much firmer statement than merely utilizing one of them. In point of fact, the M T 2 distribution of the process eq. (25) could itself in principle be a good discriminator between Z 2 and Z 3 models. Indeed, as discussed in Section 2.2.2, the kinematic endpoint in the M T 2 distribution of the visible particles from a duplicate three-body decay, which is realized under Z 3 symmetry, develops a longer tail than that of two-body decays, the latter being realized under Z 2 symmetry. Therefore, a less sharp fall-off near the endpoint could be a sign of more than one invisible particle in the decay [10, 11]. However, shape analyses of the tail of the M T 2 distribution are rather delicate, especially in the presence of a background. Besides the issues raised by the backgrounds, there are also some inherent complications in using only the shape of the M T 2 distribution to determine the underlying stabilization symmetry. For example, the effects of spin correlation could change the shape of the M T 2 distribution, particularly the behavior near the upper endpoint of the distribution. In other words, a certain 'choice' of spin correlation could alter the sharp edge of the M T 2 distribution in Z 2 models, mimicking the typical distribution shape characteristic of Z 3 models, and vice versa. Alternatively, one could try to use the energy distribution of the b particles in events from the process eq. (25). Recall that the distribution of the visible particle energy in their mother particle's rest frame is δ function-like in Z 2 models, whereas the distribution in Z 3 models is nontrivial. Therefore, once the decay products are boosted to the laboratory frame from their mother particle's rest frame, the energy distribution for Z 3 physics is expected to be relatively broader for a given mother particle. However, it is very hard to quantify the width of the resulting energy distributions in both Z 2 and Z 3 models because it is strongly model-dependent. In particular, the shape of the energy distribution in the laboratory frame is governed by the boost distributions of the mother particles, which are subject to uncertainties. Such uncertainties come from the fact that we are not a priori aware of the underlying dynamics governing the new physics involved in the process eq. (25), which affects, for instance, the production mechanism of the mother particles. In order to overcome the difficulties described above, we propose here a combined analysis of the two distributions. The goal is to obtain a more robust technique that is sensitive to the differences between the Z 2 and the Z 3 models but largely independent of the other details of the models. Also, we aim at formulating a method that is less demanding from an experimental standpoint and more stable against the inclusion of experimental errors. The analysis proceeds in two steps as explained in the following. From the data, one first produces the M T 2 distribution using a trial DM mass and extracts the kinematic endpoint M max T 2 , obs . . Then, by substituting the measured endpoint into the function given in eq. (30), one obtains the C parameter. As illustrated in eq. (31), the C parameter has different physical implications depending on the stabilization symmetry of the DM. For the Z 2 case, it is the energy of the visible particle in the rest frame of its mother particle, and by virtue of [14, 15], it is expected to be the value of the peak of the energy distribution in the laboratory frame. Alternatively, for a Z 3 model the C parameter is an upper bound to the peak of the energy distribution in the laboratory frame. Therefore, the comparison between the extracted C parameter and the peak position in the b particle energy distribution enables us to determine whether the relevant physics is Z 2 or Z 3 . This observation can be summarized as follows: Some remarks must be made about our proposal. First, the use of the distribution of M T 2 is needed only to the extent that this is useful to extract the C parameter. In fact, in order to find the reference value needed for the comparison of eq. (32), any other observable that is sensitive to the relevant combination of masses could be used. Second, spin correlation effects do not change the location of the peak in the energy distribution of the b particle as long as the bottom partners are produced unpolarized, as discussed earlier. Additionally, although the overall shape near the endpoint of the M T 2 distribution could be affected by non-trivial spin correlation effects, the endpoint value is not. Furthermore, substantial errors in the determination of the M T 2 endpoint can be tolerated. In fact, as shown in Fig. 2, the difference between the reference value and the typical peak of the energy distribution in a three-body decay is quite large. For the above reasons, we believe that compared with other methods which utilize only M T 2 , the method presented here is more general and more robust in highlighting the different kinematic behavior inherent to the two different stabilization symmetries. In order to demonstrate the feasibility of the proposed analysis, we work out in detail an application of our method to the case of pair production of partners of the b quark that decay into a b quark and one or two invisible particles in the next section.", "pages": [ 13, 14, 15 ] }, { "title": "4 Application to b quark partner decays", "content": "In this Section, we study in detail the production of b quark partners, B ' , and their subsequent decay into b quarks and one or two DM particles. As mentioned in the introduction, b quark partners occur in many well-motivated extensions to the SM. In the following, we apply the results of Sections 2 and 3 with the underlying goal of 'counting' the number of DM particles in the above decay process. Although we employ DM and a b quark partner with specific spin for the purpose of illustrating our technique, we emphasize that our method can be applied for any appropriate choice of spins for the involved particles. In fact, the choice of spins does not alter our results so long as the mother particles are produced unpolarized. Because the b quark partners are charged under QCD, the dominant production channel at hadron colliders would be via color gauge interactions, which guarantee that the b quark partners would be produced unpolarized and in pairs. Due to the fact that these particles are produced in pairs, the above results given for M T 2 are in force. Furthermore, the unpolarized production guarantees that the results of Section 2 can be applied to the energy distribution. In what follows, we consider the QCD pair production of heavy b quark partners at the LHC running at a center-of-mass energy √ s = 14 TeV, and we take as signal processes: where χ is the DM particle. Once produced, we assume that each B ' decays into a b quark and either one or two stable neutral weakly-interacting particles (see also Fig. 1). These processes will appear in the detector as jets from the two b quarks and missing transverse energy Note that our program is meant to be carried out only after the discovery of heavy b quark partner. In fact, our focus is not on discovery, but on determining what type of symmetry governs the associated decays of such a particle once the discovery is made, specifically in the b ¯ b + E/ T channel. In order to achieve this goal, a high integrated luminosity would be required to make a definitive determination of the underlying symmetry. Likewise, compared with the criteria necessary to claim the discovery of such a resonance, a different set of event selection conditions would be likely have to be used in order to make a definitive determination of the underlying stabilization symmetry. For our proof-of-concept example, we take m B ' = 800 GeV and m χ = 100 GeV while noting that searches for scalar b quark partners such as Ref. [17] are in principle sensitive to our final state. Unfortunately, there is no available interpretation of this search in terms of a fermionic partner; a naive rescaling of the current limits on a scalar partner with mass of about 650 GeV shows that our choice of mass parameters might be on the verge of exclusion. However, we remark that our choice is only for the purpose of illustrating our technique, and can just as easily be applied to a heavier B ' . There are several SM backgrounds that are also able to give the same detector signature as our signal. Since we require a double b -tagging, the main backgrounds to our signal are the following three processes: i) Z + b ¯ b , where Z decays into two neutrinos, ii) W ± + b ¯ b , where the W decay products are not detected, and iii) t ¯ t where again the two W 's from the top decay go undetected 6 . The first background is irreducible, while the latter two are reducible. To reduce these backgrounds to a level that allows clear extraction of the features of the b -jet energy and M T 2 distribution, we put constraints on the following observables: In general, the mismeasurement of the momenta of the observable objects used to compute E/ T can produce an instrumental source of E/ T , as opposed to a 'physical' source of E/ T which originates from invisible particles carrying away momentum. The mismeasurement of E/ T can grow as objects of larger p T are found in an event, and it is therefore useful to compare the measured missing transverse energy with some measure of the global transverse momentum of the event. For this reason, we introduce the quantity 7 which is expected to be small for events where the E/ T comes from mismeasurements, but should be large for events where invisible particles carry away momentum. Furthermore, when the instrumental E/ T originates mostly from the mismeasurement of a single object, the E/ T is expected to point approximately in the direction of one of the visible momenta. Therefore, the events where the E/ T is purely instrumental are expected to have a small which is the angle between the direction of the missing transverse momentum and any glyph[vector] p T j . To select signal events and reject background events, we choose the following set of cuts: (36c) Note that the our cuts are of the same sort used in experimental searches for new physics in final states with large E/ T , 0 leptons and jets including 1 or more b -jets (see, for instance, [19]). However, notice that in our analysis, we privilege the strength of the signal over the statistical significance of the observation. As already mentioned, we imagine this investigation being carried out after the initial discovery of a B ' has taken place. Hence, we favor enhancing the signal to better study the detailed properties of the interaction(s) of B ' . For this reason, we cut more aggressively on E/ T and S T than in experimental searches and other phenomenological papers focusing on the discovery of B ' s (see, for example, [5]). We consider quarks separated by ∆ R > 0 . 7 as jets. With this as our condition on jet reconstruction, the cuts of eq. (36) can be readily applied to the signals and to the Z + b ¯ b background; the resulting cross-sections are shown in Table 1. These cross-sections are computed from samples of events obtained using the Monte Carlo event generator MadGraph5 v1.4.7 [20] and parton distribution functions CTEQ6L1 [21]. For the sake of completeness, we specify that in generating these event samples we assumed a fermionic B ' and a weakly interacting scalar χ . However, as already stressed, we anticipate that different choices of spin for these particles will not significantly affect our final result because the production via QCD gives rise to an effectively unpolarized sample of b quark partners. The estimate of the reducible backgrounds requires more work, as it is particularly important to accurately model the possible causes that make a background to our 2 b + E/ T signal. In fact, these processes have larger cross sections than Z + b ¯ b . However, they also typically give rise to extra leptons or extra jets with respect to our selection criteria in eq. (36). Therefore, in order for us to consider them as background events, it is necessary for the extra leptons or jets to fail our selection criteria. Accordingly, the relevant cross-section for these processes is significantly reduced compared to the total. In fact, we find that t ¯ t and W ± b ¯ b are subdominant background sources compared to Z + b ¯ b . In what follows, we describe how we estimated the background rate from t ¯ t and W ± b ¯ b . An accurate determination of the proportion of t ¯ t and W ± b ¯ b background events that pass the cuts in eq. (36) depends on the finer details of the detector used to observe these events. However, the most important causes for the extra jets and leptons in the reducible backgrounds to fail our jet and lepton identification criteria can be understood at the matrix element level. We estimate the rate of the reducible backgrounds by requiring that at the matrix element level, a suitable number of final states from the t ¯ t and W + b ¯ b production fail the selections of eq. (36) for one of the following reasons: Additionally, when any quark or lepton is too close to a b quark, we consider them as having been merged by the detector, and the resulting object is counted as a b quark (i.e., ∆ R bl < 0 . 7, ∆ R bj < 0 . 7), or if any light quark or lepton is too close to a light jet, they are likewise merged, and the resulting object is counted as a light quark (i.e., ∆ R jl < 0 . 7, ∆ R jj < 0 . 7). In the latter case, the light 'jet' resulting from a merger must then also satisfy the p T and η criteria given above for going undetected. Using our method to estimate the results on the backgrounds in Ref. [5], the analysis of which was carried out with objects reconstructed at the detector level, we find that our estimates agree with Ref. [5] within a factor of two. Because we successfully captured the leading effect, we did not feel the necessity of pursuing detector simulations in our analysis. Estimating the reducible background after the selections in eq. (36), we find that t ¯ t and W + b ¯ b are subdominant compared to Z + b ¯ b . The suppression of the reducible backgrounds, and in particular, of t ¯ t , comes especially from the combination of the S T and E/ T cuts. This is shown in Fig. 3, where we plot the E/ T distributions of the three backgrounds under different S T cuts: S T > 0, S T > 0 . 2, and the cut S T > 0 . 4, which is used in our final analysis. Clearly, one can see that for a E/ T as large as our requirement in eq. (36), the dominant background is Z + b ¯ b , and that in particular, the t ¯ t is significantly suppressed by simultaneously requiring a large E/ T and moderate S T cut (rightmost panel in the figure). As the first step in our analysis, we compute the M T 2 distributions expected at the LHC for our two potential cases of new physics interactions, Z 2 and Z 3 . The distributions for the two cases are shown in Fig. 4. Since we found that with selections of eq. (36), the Z + b ¯ b process is the dominant background, as seen in the figure, we consider it the only background process. The two distributions have been computed assuming a trial mass ˜ m = 0 GeV and have an endpoint at 787.5 GeV and 750 GeV for the Z 2 and the Z 3 cases, respectively. Interpreting the distributions under the na¨ıve assumption of one invisible particle per decay of the B ' , we obtain from eq. (30) a C parameter that is 383.75 GeV and 375 GeV for Z 2 and Z 3 , respectively. These are the reference values that we need for the analysis of the energy distributions 8 . As the final step in our analysis, we need to compare the obtained reference values with the peaks of the energy distributions. These distributions are shown in Fig. 5. We clearly see that the location of the peak in the energy distribution the Z 2 case coincides with the associated reference value, whereas for the Z 3 case the peak is, as expected, at an energy less than the associated reference value. We remark that in the Z 3 case, the peak of the energy distribution is significantly displaced with respect to the reference value. Therefore, we expect our test of the Z 2 nature of the interactions of the B ' to be quite robust under the inclusion of both experimental and theoretical uncertainties, such as the smearing of the peak due to the resolution on the jet energy, the errors on the extraction of the reference value obtained from the M T 2 analysis, and the shift of the peak that is expected due to radiative corrections to the leading order of the decay of the B ' .", "pages": [ 15, 16, 17, 18, 19, 20 ] }, { "title": "5 Conclusions", "content": "In this treatise, we studied the problem of the experimental determination of the general structure of the interactions of an extension to the SM that hosts collider-stable WIMPs. If these new particles are charged under a new symmetry and the SM particles are not, then the lightest such WIMP is stable and is concomitantly a candidate for the DM of the universe. In the context of such H L H L H L H L DM models, our work is thus relevant for the determination of the stabilization symmetry of this DM. In more detail, such models typically have heavier new particles that are charged under both the SM gauge group and the DM stabilization symmetry. Thus, these particles can be produced via the collision of SM particles, and will decay into DM plus SM particles. The number of DM particles in such a decay depends on the DM stabilization symmetry. Our goal was to devise a strategy to count this number of DM and thus probe the nature of this symmetry, based only on the visible part of the decays. To illustrate the technique, we studied models with fermionic b quark partners, i.e. colored fermions with electric charge -1 / 3 with sizable coupling to the b quark. In our example, we considered the case of b quark partners with mass at or below the TeV scale. The possibility of such is motivated by extensions to the SM that solve the Planck-weak hierarchy problem, since they contain top partners and, thus by SU (2) L symmetry, bottom partners. In the same model, it is also possible to have a WIMP DM. The b quark partners, as the typical states of the new physics sector, are charged under this stabilization symmetry and will then decay into a bottom quark, plus DM. Furthermore, thanks to their color gauge interactions, the b quark partners have a large production cross-section at hadronic colliders. Therefore the study of b quark partners is very well-suited to illustrate our technique. The literature on b quark partners thus far has only considered single DM in each decay chain, as would be the case in models where the DM is stabilized by a Z 2 symmetry. However, in general, there can be more than one DM in this decay chain; for example, two DM are allowed in the case of a Z 3 stabilization symmetry, albeit not in the case of a Z 2 symmetry. So, the question we posed is whether we can distinguish the hypothesis of one vs. (say) two DM particles appearing in each of these decay chains. As mentioned above, in this way we can probe the nature of the DM stabilization symmetry. The question is non-trivial, because in either case the detectable particles produced are the same, and so is the signal of the b quark partners' production, i.e. b ¯ b + E/ T . To distinguish between one and two DM in each b quark partner decay chain, the first result we used is that the measured M T 2 endpoints can be fitted by the formula eq. (30) irrespectively of how many DM particles are produced. The value of the free parameter obtained by fitting eq. (30) to the data is used in the next step of our analysis as follows. The second theoretical observation is that the peak of the distribution of the b quark energy in the laboratory frame is the same as the mother rest frame value for the two-body decay, but is smaller than the maximum value in the mother rest frame for the three-body decay. The crux is that the rest frame energy that is used as a reference value in this comparison is precisely the parameter obtained in the above M T 2 analysis. Combining the above two facts, we showed that the peak of observed bottom-jet energy being smaller than (vs. same as) the reference value obtained from the M T 2 endpoint provides evidence for two (vs. one) DM particles in the decay of a b quark partner, and thus a Z 3 symmetry can be distinguished from Z 2 . We verified our theoretical observations in B ' pair production and decay at the LHC. To assess the feasibility of the determination of the stabilization symmetry with our method, we simulated the signal and the dominant SM backgrounds. Using suitable cuts, we showed that the background in this case is due mostly to Z + b ¯ b . We studied in detail the case where the b quark partner has a mass m B ' = 800 GeV and the invisible particles have a mass m χ = 100 GeV. In this case, the background can be made small compared to the signal using the cuts of eq. (36). In Figures 4 and 5, we show the resulting M T 2 and b quark energy distributions relevant to our analysis. We observed that the peak in the b quark energy distribution for Z 2 models is consistent with the reference value from the M T 2 endpoint, while that of Z 3 models is apparently less than the corresponding reference value. The determinations of the peak of the energy distribution and of the reference value needed for our analysis are subject to uncertainties, e.g. those that propagate from the error in the determination of the M T 2 endpoint. However, the evidence for a Z 3 stabilization symmetry comes from a difference between the peak of the energy distribution and the reference value. The theoretical prediction for this difference is large enough compared to the relevant uncertainties so that the proposed method seems to be quite robust, and should allow a clear discrimination of the stabilization symmetry of the DM. In future work we plan to extend the theory of Section 2 to deal with massive visible decay products. Thus, we shall be able devise a strategy to tell apart Z 2 and Z 3 stabilization symmetry in top quark partners decays into a top quark and invisible particles, which arise in the same scenario that we studied here. We also expect that our theoretical observation can be relevant in other applications, such as distinguishing two-body from three-body decays independently of the issue of DM.", "pages": [ 20, 21, 22, 23 ] }, { "title": "Acknowledgments", "content": "We would like to thank Johan Alwall and Shufang Su for discussions. This work was supported in part by NSF Grant No. PHY-0968854. D. K. also acknowledges the support from the LHC Theory Initiative graduate fellowship that is funded through NSF Grant No. PHY-0969510. The work of R. F. is also supported by NSF Grant No. PHY-0910467, and by the Maryland Center for Fundamental Physics.", "pages": [ 23 ] } ]

[ { "title": "Gwenael Giacinti", "content": "University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK E-mail: [email protected] Abstract. The propagation of TeV-PeV cosmic rays (CR) in our Galaxy can be described as a diffusive process. We discuss here two effects, with important observational consequences, that cannot be predicted by the diffusion approximation (DA) in its usual form. First, we present an explanation for the CR anisotropies observed at small angular scales on the sky. We show that the local magnetic field configuration within a CR mean free path from Earth naturally results in CR flux anisotropies at small and medium scales [1]. Second, we point out that TeVPeV CRs should be expected to diffuse strongly anisotropically in the interstellar medium on scales smaller than the maximum scale of spatial fluctuations of the field, ∼ 100pc [2, 3]. This notably questions the usual assumptions on CR diffusion around sources.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The turbulent magnetic field that permeates the interstellar medium (ISM) of our Galaxy contains fluctuations on spatial scales ranging from L max ∼ 100 pc down to a damping scale L min , smaller than ∼ 1 AU [4]. Galactic cosmic rays (CR) resonantly scatter on modes with fluctuation scales that match their Larmor radius r L : 2 π/k ∼ r L . In turn, individual trajectories of CRs may be modeled as random walks. CR propagation in the disk and halo can be globally described as a diffusive process, by the diffusion approximation (DA) -see for instance Refs. [5, 6] for more exhaustive discussions. Streaming CRs generate Alfv'en waves [7, 8]. However, above energies of a few tens of GeVs, CRs should not be expected to be self-confined, and, hence, do not stream in the ISM to only ∼ the Alfv'en speed [8, 9]. The exact impact of higher energy CRs on the magnetic fields they diffuse in is still an open question. The authors of Refs. [10] argue that the spectral inflection detected by PAMELA and CREAM experiments [11] at rigidities E/Z ∼ 200 GeV could mark the limit between CR diffusion in self-generated turbulence and diffusion in the pre-existing turbulence. In Galactic CR propagation codes [6, 12], this can be implemented by assuming a break in the diffusion coefficient at that rigidity, but it does not change the global effective picture of CR diffusion in the ISM. We discuss in this work two topical subjects that cannot be predicted by the DA in its usual form. Several observatories have detected anisotropies in the TeV-PeV CR arrival directions on the sky at Earth [13, 14]. These anisotropies have been observed both at large (dipole) and, more surprisingly, small angular scales (higher order multipoles). The amplitude of the dipole is /similarequal 0 . 1%, and that of the smaller scale anisotropies (SSA) is lower by a factor of a few to ten. SSA have been detected down to /similarequal 10 · scales, with narrow regions receiving a flux larger or smaller than would be expected if only a dipole anisotropy were present. The DA cannot explain the SSA. We show in Section 2 that it is so because the DA is not designed to predict phenomena arising on spatial scales smaller than the CR mean free path (MFP) : Anisotropies naturally appear on small angular scales and reflect the local configuration of the turbulent magnetic field within a CR MFP from Earth. In Section 3, we study the diffusion of high-energy Galactic CRs around their sources. We show that the simplifying hypothesis of isotropic diffusion is usually not acceptable. CR densities around sources should be expected to be irregular and, sometimes, to display filamentary structures.", "pages": [ 1, 2 ] }, { "title": "2. Origin of TeV-PeV cosmic ray anisotropies", "content": "The DA only predicts a dipole anisotropy. The direction and amplitude of the dipole δ are directly related to the local relative gradient of CR density ∇ n/n by δ = 3 D ( E ) /c · ∇ n/n , where D ( E ) is the CR diffusion coefficient. ∇ n/n is mostly determined by one or a few nearby recent sources [15]. Refs. [16], amongst others, put constraints on the amplitude of D ( E ). Taking values of D in the preferred range, and | ∇ n/n | ∼ 1 kpc -1 , one finds | δ | ∼ 0 . 1% [1], which is compatible with observations. See Refs. [15] for more detailed calculations. However, the DA does not predict the SSA. Previous works have assumed additional effects so as to explain the SSA, see for example Refs. [17]. We demonstrate here that no additional effect is needed, and that rigidity-dependent SSA automatically arise from the local realization of the random magnetic field within about a CR MFP from Earth. This statement is valid even in isotropic magnetic turbulence. In other words, the DA does not predict SSA because it loses its validity on distances smaller than the CR MFP. Let us schematically model CR trajectories as random walks. CRs can then be regarded as performing jumps on straight lines with MFP lengths, l mfp . They have an equal probability to go in any direction after every jump because the process is Markovian. In this simplified picture, CRs arriving at Earth during a time interval l mfp /c are located within a sphere of radius l mfp . Assuming a gradient of CR density ∇ n at Earth, one then only finds a dipole anisotropy in the direction of ∇ n , with the amplitude predicted by the DA. However, in reality, the magnetic turbulence has a given configuration, and CRs do not travel on straight lines within this sphere of radius ≈ l mfp . The magnetic field within ≈ l mfp from Earth does not vary significantly during the CR crossing time ∼ l mfp /c . This gives rise to anisotropies at medium and small scales by reshuffling pieces of the dipole to smaller scales, on different parts of the sky. Equivalently, one can say that the DA cannot predict the SSA because it averages over all possible magnetic field configurations. This averaging notably causes the problem to be artificially symmetric around the axis defined by ∇ n and containing the Earth.", "pages": [ 2, 3 ] }, { "title": "Earth.", "content": "Figure 2 shows the CR flux smoothed over 90 · circles on the sky (upper row) and the remaining SSA after subtracting the dipole and smoothing over 20 · circles (lower row), for two different realizations of the magnetic turbulence (one for each column). For computing time reasons, we take CRs with E/Z = 10PeV and a relative gradient of CR density | ∇ n/n | = (290 pc) -1 , which gives a dipole amplitude of /similarequal 6%. More realistic values would give an amplitude in line with those observed. The amplitude of SSA is about the same. In practice, CRs with different rigidities are mixed (different charges and broad energy ranges), which leads to a smaller amplitude because SSA are rigidity-dependent and values for different E/Z add non-constructively. Taking a CR distribution with a median energy of 10 PeV and a relative width ∆ E/E equal to that inferred for IceCube measurements at 20 TeV (see Fig. 3 of Ref. [13]), we find SSA with an amplitude ≈ 10 times lower than the dipole. This is in line with observations. In Fig. 2, both dipoles should point towards (180 · ,0 · ), which is the direction of ∇ n here. Results are not far from the DA prediction, but slight deviations are visible, because of the local realization of the turbulence. Concerning the SSA, the two lower panels are completely different from one another. This illustrates the full dependence of SSA on the field realization.", "pages": [ 4 ] }, { "title": "3. Anisotropic diffusion of cosmic rays in the ISM", "content": "Observations are consistent with either a Kolmogorov or a Kraichnan spectrum for the power spectrum of the turbulent Galactic magnetic field [6]. They suggest that the coherence length of the field, L c , is about a few tens of parsecs [4, 20]. This implies that the Larmor radius of TeV-PeV CRs should be /lessmuch L c , L max , and that most of the power is contained in fluctuations on spatial scales larger than r L . Since the ratio of power in modes with 2 π/k ∼ r L to modes with 2 π/k /greatermuch r L is expected to be small, CRs should diffuse strongly anisotropically locally, because the latter modes are seen as local regular fields. As in Section 2, we propagate individual CRs in turbulent fields. To quantify the anisotropy of CR distributions around sources, we compute the eigenvalues d ( b ) 1 < d ( b ) 2 < d ( b ) 3 of D ( b ) ij = 1 2 nt ∑ n a =1 x ( a ) i x ( a ) j for n = 10 4 /greatermuch 1 CRs with E/Z = 1PeV, injected at (0,0,0) and t = 0 in one given realization b of the isotropic Kolmogorov turbulence described previously. The effects of variation over time of the field are negligible during the time scales we consider ( t ∼ 10 kyr) : The velocity of ISM fluid parcels and the Alfv'en speed are ∼ 10 km/s, and 10 km/s × 10 kyr /lessmuch √ d i × 10 kyr. The degree of anisotropy of CR distributions vary from one configuration b to another. Therefore, we study the averages of d ( b ) 1 , 2 , 3 over N b /greatermuch 1 different configurations of the field, d k = 1 N b ∑ N b b =1 d ( b ) k . Figure 3 (left panel) shows the evolution over time of d 1 , 2 , 3 averaged over N b = 10 magnetic field realizations. N b = 10 is sufficient for the purpose of this work. At early times, the ratio between extreme eigenvalues reaches about a few tens. Then, all eigenvalues tend towards the same value for t ∼ t ∗ ∼ 10 kyr. t ∗ corresponds to the time when the bulk of spreading CRs reach a distance ≈ L max from the source, and therefore start to experience other 'cells' of size L 3 max . Schematically, CRs are initially contained in a more or less narrow flux tube containing the source ( t ∼ t ∗ / 10). The same effect can explain solar energetic particle dropouts in the solar wind, see [21]. Then, CRs start to be more isotropized in space ( t ∼ t ∗ ), though their radial distributions from their sources still differ at large radii r from the predictions of isotropic diffusion, see Fig. 3 (right panel). The excess in the tail of the distribution at large r becomes unnoticeable by t ∼ 10 t ∗ . We plot in Fig. 4 the projection on a plane of the CR distribution around a given source for different times and energies. In the upper row, E/Z = 1PeV and t = 0 . 5, 2, 7 kyr. Such results are in line with those of Fig. 3 (left panel): Diffusion is initially strongly anisotropic -if not filamentary, and then slowly tends towards the predictions of isotropic diffusion. Deviations, in particular from isotropy have also been found in simulations of energetic protons propagating in more or less anisotropic turbulence, see Ref. [22]. The two lower rows of Fig. 4 present CR distributions at lower rigidities ( E/Z = 100 and 10 TeV). For decreasing E/Z , t ∗ increases. From the similar shapes of distributions in panels on diagonals, one can see that the expected scaling t ∗ ∝ 1 /D ( E ) ∝ E -1 / 3 is approximately satisfied. We also find that t ∗ approximately grows as L 2 max . We estimate t ∗ ∼ 10 kyr ( L max / 150 pc) β (( E/Z ) / PeV) -γ ( B rms / 4 µG ) γ , where β /similarequal 2 and γ = 0 . 25-0 . 5. Secondary gamma-rays from CRs somehow map the CR distribution around the sources. For a uniform density of thermal protons in the surrounding ISM, the gamma-ray images would be similar to those of the CR column density, such as in Fig. 4. The above similarities between CR distributions at different energies, through the scaling t after escape ∝ 1 /D ( E ), have been found when considering CRs as test particles. However, CR-driven instabilities may modify interstellar magnetic fields. CRs amplify and modify magnetic fields just ahead of supernova remnant shocks, see [23] for a recent study. However, CR currents are significantly lower in the case studied here. Whether CRs would still have a non-negligible impact on fields within ∼ l c from their sources depends on several parameters, such as the amount of escaping CRs. If so, the anisotropy of the CR distribution would be lowered, and this should be expected to happen preferentially at lower energies where the CR current is larger. Our orderof-magnitude estimate below shows that the above test particle calculations should be sufficient for, at least, CRs with energies larger than ∼ a few tens of TeV, and therefore for photons with energies above a few TeV. Deviations would be larger in regions with strong cosmic ray currents j . Largest j are mainly expected at t < t ∗ when CRs are still close to their sources, and contained in well collimated flux tubes. For a source that has channeled 10 50 erg in CRs with E = 1GeV -1 PeV and with an E -2 spectrum, 10 49 erg would be present in each of the 10 bins in energy with logarithmic widths. Let us assume the extreme case where CRs are contained in a collimated tube of radius 3 pc and length 2 L c = 2 L max / 5 = 60 pc, around the source. This yields a CR density U ∼ 100 eV/cm 3 . CRs propagate inside with a speed roughly ∼ D/L c where D = D 0 E 1 / 3 and D 0 ∼ 10 29 cm 2 /(s · PeV 1 / 3 ) [2]. The non-resonant hybrid (NRH) instability uncovered by Bell [24] dominates over the Alfv'en instability when Bjr L /ρv 2 A > 1, where ρ /similarequal 1 m p /cm 3 is the density of the ISM. For the above values, we are at the limit where it may play a role. The respective growth rates of these instabilities are [24] : γ NRH = 0 . 5 j √ µ 0 /ρ and γ A ≈ 0 . 3 j √ µ 0 /ρ . Our parameters lead to a typical growth time of /similarequal 5 γ -1 NRH ≈ 10 √ ρ/µ 0 E 2 / 3 L c /UeD 0 . 5 γ -1 NRH ≈ 3 . 1 and 67 kyr for CRs with E = 10TeV and 1PeV, respectively. The bulk of CRs roughly spend a time ∼ L 2 c /D in the collimated flux tube. Then, instabilities cannot grow sufficiently for CRs with roughly E > √ µ 0 /ρUeL c / 10 ≈ 40 TeV. This energy may be further lowered if waves are damped sufficiently quickly. See for example [25, 26, 10] for sources of damping. If the impact of low energy CRs on magnetic fields is nonnegligible, we expect the parallel diffusion coefficient along the filament to be suppressed, and CRs to diffuse more isotropically. j and the growth rate 5 γ -1 NRH would become lower. Hence, some anisotropies in the CR distribution should be expected to remain at TeV energies even in such a case. Assuming a given template for CR escape from the source, one may check in the future if CR-driven instabilities have time to grow at low E and have an impact on the surrounding fields, by 'comparing' gamma-ray images at low and high energies. At high energies, γ -ray observations will improve our knowledge of the structure of interstellar magnetic fields.", "pages": [ 4, 5, 6, 7 ] }, { "title": "4. Conclusions and perspectives", "content": "We have discussed here two effects that are not predicted by the DA in its usual form. First, we have proposed a natural explanation for the TeV-PeV CR anisotropies observed at small scales on the sky. The DA cannot predict them because it is not applicable on spatial scales below the CR mean free path. We have demonstrated in Sec. 2 that SSA must automatically appear due to the given local configuration of the magnetic field within ≈ l mfp from the observer, provided a dipole anisotropy exists. In the future, TeV-PeV CR anisotropies should become a convenient way to probe the structure of interstellar magnetic fields within a few tens of parsecs from Earth. The argument presented here holds for any field within ≈ l mfp from the observer. At E ∼ 1 -10 TeV, the SSA may start to probe heliospheric fields, see also [27]. Let us mention that electric fields in the heliosphere may result in SSA too, see [28]. In the future, a thorough analysis can determine the relative contributions of both effects. Second, we have shown in Sec. 3 that diffusion of TeV-PeV CRs in the ISM should be expected to be non-negligibly anisotropic on scales smaller than L max ∼ 100 pc. Therefore, CR distributions around recent sources should look anisotropic and irregular. This has important implications for γ -ray astronomy, and some first observations may hint at our findings, see [3]. Detailed comparisons at high and low photon energies of the extended γ -ray emissions around CR sources may give insights into the potential impact at low energies of CR-driven instabilities on the surrounding ISM. In general, deviations from standard diffusion are expected to have an impact in several other situations. For example, Refs. [29] consider, in particular, their implications for particles accelerated at the solar wind termination shock.", "pages": [ 8 ] }, { "title": "Acknowledgments", "content": "The author acknowledges funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007 -2013)/ERC grant agreement no. 247039.", "pages": [ 8 ] } ]

[ { "title": "Constraint on Pulsar Wind Properties from Induced Compton Scattering off Radio Pulses", "content": "Shuta J. Tanaka 1 and Fumio Takahara 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulsar winds have longstanding problems in energy conversion and pair cascade processes which determine the magnetization σ , the pair multiplicity κ and the bulk Lorentz factor γ of the wind. We study induced Compton scattering by a relativistically moving cold plasma to constrain wind properties by imposing that radio pulses from the pulsar itself are not scattered by the wind as was first studied by Wilson & Rees. We find that relativistic effects cause a significant increase or decrease of the scattering coefficient depending on scattering geometry. Applying to the Crab, we consider uncertainties of an inclination angle of the wind velocity with respect to the radio beam θ pl and the emission region size r e which determines an opening angle of the radio beam. We obtain the lower limit γ /greaterorsimilar 10 1 . 7 r 1 / 2 e , 3 θ -1 pl (1 + σ ) -1 / 4 ( r e = 10 3 r e , 3 cm) at the light cylinder r LC for an inclined wind θ pl > 10 -2 . 7 . For an aligned wind θ pl < 10 -2 . 7 , we require γ > 10 2 . 7 at r LC and an additional constraint γ > 10 3 . 4 r 1 / 5 e , 3 (1 + σ ) -1 / 10 at the characteristic scattering radius r c = 10 9 . 6 r 2 / 5 e , 3 cm within which the 'lack of time' effect prevents scattering. Considering the lower limit κ /greaterorsimilar 10 6 . 6 suggested by recent studies of the Crab Nebula, for r e = 10 3 cm, we obtain the most optimistic constraint 10 1 . 7 /lessorsimilar γ /lessorsimilar 10 3 . 9 and 10 6 . 6 /lessorsimilar κ /lessorsimilar 10 8 . 8 which are independent of r when θ pl ∼ 1 and 1 + σ ∼ 1 at r LC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index xxxx, xxx", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Pulsar magnetospheres create pulsar winds through pair creation and particle acceleration [1]. Because pulsar winds are radiatively inefficient, it is difficult to constrain their properties. However, their properties are inferred from observations of surrounding pulsar wind nebula (PWN) and pulsed emissions of the pulsar itself. Interestingly, a secular increase of their pulse period tells us their total energy output L spin . Because most of the spin-down power is converted into the pulsar wind, L spin constrains its properties as (see also Equation (22)) where κ is the pair multiplicity ( e ± number flux normalized by the Goldreich-Julian number flux ˙ N GJ ), γ is the bulk Lorentz factor, and σ is the magnetization parameter (the ratio of the Poynting to the kinetic energy fluxes) of the pulsar wind, respectively. We used ˙ N GJ ≡ 2 πr 2 pc cn GJ ( r pc ) = √ 6 cL spin /e , where r pc is the polar cap radius, n GJ ( r pc ) is the GoldreichJulian density at an magnetic pole and the numerical factor two comes from the north and south magnetic poles. Pair cascade models within the magnetosphere of the Crab pulsar ( L spin = 4 . 6 × 10 38 erg s -1 ) predict κ ∼ 10 4 with σ ∼ 10 4 and γ ∼ 10 2 in the vicinity of the light cylinder r LC [e.g., 2-4]. On the other hand, magnetohydrodynamic (MHD) models of the Crab Nebula reproduce its non-thermal emission from optical to γ -ray with κ ∼ 10 4 , σ ∼ 10 -3 -10 -2 and γ ∼ 10 6 [5-8]. Although κ ∼ 10 4 in both models is consistent with particle number conservation, σ (and also γ ) differs by many orders of magnitude, which is called the ' σ -problem' [c.f., 9]. It is noted that there is an additional problem of the pulsar wind properties [c.f., 10, 11]. Because the MHD models of the Crab Nebula do not explicitly account for the origin of radio emitting particles, they may underestimate the pair multiplicity. Recent studies of spectral evolution of PWNe showed κ > 10 6 for the Crab Nebula and κ > 10 5 for other PWNe [e.g, 11-14]. Although the origin of the low energy particles that are responsible for the radio emission of PWNe is still an open problem, they originate most likely from the pulsar because of the continuity of the broadband spectrum and because of the radio structures apparently originating from the pulsar [15-17]. Thus there arises another problem on κ besides the σ -problem, while only the combination of κγ (1 + σ ) in Equation (1) is firm. In view of the σ - and κ -problems, it is interesting to consider other independent constraints on the physical conditions of pulsar winds. Wilson & Rees (1978, hereafter WR78) [18] considered induced Compton scattering off radio pulses by a pulsar wind. So far, it is thought that we have not observed a signature of scattering in radio spectra of pulsars, although we do not fully understand how scattering changes the radio spectrum (e.g., scattering by a nonrelativistic plasma was studied by [19, 20]). Observations suggest that the optical depth to induced Compton scattering is less than unity, and the radio spectrum is not changed. Based on this consideration, WR78 obtained the lower limit of the bulk Lorentz factor of the Crab pulsar wind γ > 10 4 at 10 3 r LC ∼ 10 11 cm away from the pulsar. Substituting Equation (1), only for (1 + σ ) ∼ 1 at 10 3 r LC , their conclusion is marginally consistent with the conclusion of κ /greaterorsimilar 10 6 . 6 ≡ κ PWN obtained from the study of the Crab Nebula spectrum by Tanaka & Takahara (2010, 2011) [11, 13]. Induced Compton scattering process has been studied for the application to high brightness temperature radio sources, such as the pulsars [e.g., 18, 21-23], active galactic nuclei [e.g., 19, 24, 25] and other sources [e.g., 26-28]. Induced Compton scattering is about a factor of θ 4 bm k B T b ( ν ) /m e c 2 times effective compared with spontaneous one in the rest frame of the plasma, where θ bm ( < 1) and T b are a half-opening angle and a brightness temperature of a radio beam, respectively (see Equation (15)). Note that the value of k B T b ( ν ) /m e c 2 can be larger than 10 15 for the Crab pulsar (see Equation (24)). However, for scattering by relativistically moving electrons, the scattering coefficient is modified by relativistic effects and, as we will see below, either an increase or a decrease is possible depending on situations considered, e.g., the velocity u = γ β of the electrons and an inclination between an electron motion u and a radio beam k , where k is the wavenumber vector. In this paper, we reconsider induced Compton scattering by a relativistically moving plasma and reevaluate a lower limit of the bulk Lorentz factor. Despite strong dependence on scattering geometry, WR78 considered a specific scattering geometry where the pulsar wind is completely aligned with respect to the radio pulse beam and where θ bm of the radio beam is the widest value inferred from the observations. We consider rather general geometries of the system, such as the direction of the wind being inclined with respect to the radio pulse beam. Even if the direction of pulsed radio emission is almost radial, the pulsar wind is likely to have a significant toroidal velocity just outside r LC , or its motion in the meridional plane is not strictly radial. As already noted by WR78, the scattering coefficient may be significantly reduced if the pulsar wind inclines with respect to the radio beam. For θ bm , the scattering coefficient is reduced when the radio beam is narrow in the rest frame of the plasma. If this is the case, the lower limit of the bulk Lorentz factor of the pulsar wind may be reduced so as to be consistent with recent studies of the Crab Nebula spectrum. While we focus on geometrical effects in this paper, we ignore effects of the magnetic field and background photons following WR78. The magnetic field effect may be important when the frequency of the photon at the plasma rest frame ν ' is smaller than the electron cyclotron frequency ν ce [e.g., 21, 29]. For the Crab pulsar, although the magnetic field in the observer frame is about B obs ∼ 10 6 G at the light cylinder ( ν ce = 5 . 8 × 10 12 Hz for the magnetic field of B ' = 10 6 G in the plasma rest frame), ν ce strongly depends on the magnetic field configuration and a direction of plasma motion in the observer frame. For example, if B obs ⊥ u , we find B ' = B obs /γ and ν ' = ν/δ D where δ D is the Doppler factor. Basically, the magnetic field effect reduces the scattering cross section, i.e., smaller γ would be allowed. For the effect of background photons, Lyubarsky & Petrova (1996) [21] discussed that scattering off the background photons induced by the beam photons may be important. They discussed that the occupation number of the background photons increases exponentially, i.e., the beam photons may decrease accordingly, when the scattering optical depth to the background photons well exceeds unity, say 10 2 . In this paper, we ignore background photons ( θ bm < θ ≤ π ) assuming that the occupation number of the beam photons is much larger than that of the background photons. If scattering off the background photons is efficient, scattering would be more efficient and larger γ would be required. These processes will be discussed in a separated paper. In Section 2, we describe the scattering coefficient of induced Compton scattering by a relativistically moving plasma in a general geometry. We also show simple analytic forms of the scattering coefficient in some specific geometries. In general geometry, the scattering coefficient is written in an integral form and is obtained numerically in Appendix A. In Section 3, we consider induced Compton scattering at pulsar wind regions, specifically applying to the Crab pulsar. We show the resultant lower limits of γ and also discuss the corresponding upper limits of the pair multiplicity κ . We summarize the present results in Section 4.", "pages": [ 1, 2, 3 ] }, { "title": "2. INDUCED COMPTON SCATTERING OFF A PHOTON BEAM", "content": "Here, we express the scattering coefficient at a certain position x and see that the scattering coefficient strongly depends on geometry of scattering. The kinetic equation for a photon occupation number n ( x , k , t ) is expressed as [e.g., 21, 30] where Ω = k /k , f ( p ) is the distribution function of plasma and dσ/d Ω is the differential scattering cross section, respectively. Note that when the electron is initially at rest, the recoil g is expressed as g ( k, ξ ) = k/ (1 + kλ e (1 -cos ξ )), where λ e = /planckover2pi1 /m e c represents the Compton wavelength for an electron and ξ is the angle between incident and scattered photons. We omit arguments x and t in Equation (2) and in this section. The terms 1 + n represent spontaneous and induced scattering terms, and we only consider the induced process below, assuming n /greatermuch 1.", "pages": [ 3, 4 ] }, { "title": "2.1. Scattering Coefficient", "content": "The scattering coefficient of induced Compton scattering is the right-hand side of Equation (2) divided by n ( k ) [e.g., 31]. Equation (2) is simplified by following three approximations. (I) Plasma is cold, and moves with the velocity u = γ β (the bulk Lorentz factor γ = (1 -β 2 ) -1 / 2 ). (II) The magnetic field is weak enough to satisfy the condition ν ce < ν ' , where ν ce and ν ' are the electron cyclotron frequency and the frequency of an incident photon in the plasma rest frame, respectively [e.g., 21]. (III) Photons are in the Thomson regime, i.e., kλ e /lessmuch 1 [c.f., 30]. The condition (III) is a good approximation for scattering off radio photons by plasma of γ /lessmuch 10 10 . In the observer frame, Equation (2) then becomes, [e.g., 18, 21], where and n pl is a number density of plasma. R ( Ω , Ω 1 , u ) is order unity (1 ≤ R ≤ 2) and σ T is the Thomson scattering cross section. The scattering coefficient contains the integral which depends on the occupation number itself and on scattering geometry at x , i.e., directions of photons ( Ω and Ω 1 ) and a velocity of the plasma u . While WR78 performed this integral on a specific scattering geometry, we reevaluate it in more general geometries.", "pages": [ 4 ] }, { "title": "2.2. Geometry", "content": "Scattering geometry at a certain position x in the observer frame is depicted in Figure 1. The photon beam with a half-opening angle θ bm directs to an observer on z -axis. An inclination angle of the plasma velocity is θ pl . Note that the plasma should be depicted as a line rather than a cone on Figure 1, i.e., zero opening angle, because we assume that the plasma is cold. However, we will see that there is the characteristic angle γ -1 around the plasma velocity and then we associate the plasma with the cone of its half-opening angle γ -1 in the figures in this paper. For the plasma, we express the velocity u as GLYPH<1> GLYPH<1> We assume that the occupation number of photons is uniform inside the beam and is expressed as where H is the Heaviside's step function. The spectrum n ( ν ) is assumed to be a broken power-law form where p 1 and p 2 are power-law indices of low and high frequency parts and n 0 is the occupation number at a break frequency ν 0 , respectively. Observed pulsar radio spectra correspond to -7 /lessorsimilar p 2 /lessorsimilar -3, and we require p 1 > -3 for the number density of photons to be finite at ν → 0. For the application in Section 3, we take p 2 = -5 and ν 0 = 10 MHz considering the radio observations. Adopting p 1 = 3, the brightness temperature k B T b ( ν ) = hνn ( ν ) to be maximum at ν 0 . We consider scattering off photons toward the observer, i.e., Ω = e z . The scattering coefficient χ at x is expressed as As is the conventional definition of the optical depth dτ = χdl for a path l along z -axis, we include a minus sign, where the occupation number decreases along the path for a positive value of χ and vice versa. The sign of χ can change with the sign of the function", "pages": [ 4, 5, 6 ] }, { "title": "2.3. Analytic Estimates", "content": "It is convenient to rewrite Equation (11) by introducing the normalization The scattering coefficient becomes where the integral I ( ν, θ bm , θ pl , γ ) represents a geometrical effect. Note that χ contains a factor of γ -3 which is independent of scattering geometries. The value of I ( ν, θ bm , θ pl , γ ) is obtained numerically in general and can take a wide range of values even for a fixed frequency. The numerical results of the integral I ( ν ) for different parameter sets ( θ bm , θ pl , γ ) are described in Appendix A and are also shortly summarized in the last paragraph of this section. Below, we describe simple analytic forms of the integral I ( ν ) for some special cases. They help understanding of dependence on ( θ bm , θ pl , γ ) and turn out to be useful for applications in the next section. We first see the non-relativistic limit β /lessmuch 1 ( D,D 1 ∼ 1) where the θ pl -dependence can be neglected. Considering θ bm < 1, we obtain where we use R ( e z , Ω 1 , u ) ≈ 1 + (1 -θ 2 1 / 2) 2 ≈ 2. When the photon beam is narrow ( θ bm /lessmuch 1), the scattering coefficient can be small. This is because the number of photons which stimulate the scattering process decreases with θ 2 bm and another factor θ 2 bm comes from the recoil term ∝ 1 -µ ≈ θ 2 1 / 2. For typical values of p 1 and p 2 , | I NR ( ν ) | (i.e., | χ NR ( ν ) | ) has a peak and changes sign at ν ≈ ν 0 . To see relativistic effects, we expand sin θ , cos θ and β to second-order in θ 1 , θ pl and γ -1 , i.e., we concern the situations 0 ≤ ( θ pl , θ bm ) /lessorsimilar 1 and γ /greatermuch 1. The integrand is composed of following three factors. (I) The solid angle (and the recoil) factor originates from the solid angle element d Ω 1 and from the recoil term 1 -µ , and is expressed as This factor already appeared in the non-relativistic case (Equation (15)). (II) The aberration factor originates from the Lorentz transformation of a solid angle element from the plasma rest frame to the observer frame, and is expressed as where we introduced an angle ψ 1 between β and Ω 1 , given by the approximation ψ 2 1 = θ 2 1 -2 θ 1 θ pl cos φ 1 + θ 2 pl . (III) The frequency shift factor also originates from the Lorentz GLYPH<1> GLYPH<1> transformation of a frequency, and is expressed as Analytic forms of the integral I ( ν ) presented below are explained by a simple combination of these three factors. We also show numerical results of the integral I ( ν ) for these cases in Figures 2 -4, where we adopt p 1 = 3, p 2 = -5 and γ = 10 2 . Introducing normalized angles Θ bm ≡ γθ bm and Θ pl ≡ γθ pl , it is easy to find that the integral I ( ν ) depends on (Θ bm , Θ pl ) rather than separately on θ bm , θ pl and γ . We first consider the case 1 > Θ 2 bm +Θ 2 pl where the narrow photon beam and Ω = e z are well inside the γ -1 cone associated with the plasma as shown in the right panel of Figure 2. We call this case 'Narrow'. In this case, we obtain D -2 1 ≈ 4 γ 4 and D/D 1 ≈ 1, and then the integral I ( ν ) is approximated as where we use R ( e z , Ω 1 , u ) ≈ 1 + (1 -2Θ 2 1 ) 2 ≈ 2 (Θ 1 ≡ γθ 1 ). This expression with γ → 1 (Θ bm → θ bm ) is almost the same as that of the non-relativistic case (Equation (15)). For the 'Narrow' case, the aberration factor increases the integral I ( ν ) by a factor of D -2 1 ≈ 4 γ 4 compared with I NR ( ν ) because the opening angle increases by a factor of ∼ γ in the plasma rest frame, while the frequency shift is negligible ( D/D 1 ≈ 1). Note that χ Narrow ( ν ) is a factor of γ larger than χ NR ( ν ) accounting for the factor of γ -3 in Equation (14). In the left GLYPH<1> GLYPH<1> panel of Figure 2, we plot numerical results of absolute values of the integral I ( ν ) (Equation (14)) as a function of ν . | I ( ν ) | has a discontinuity because S ( ν ) changes sign at ν ∼ ν 0 , where I ( ν ) > 0 (i.e., χ ( ν ) > 0) for ν > ν 0 and vice versa. Next case is Θ 2 pl > Θ 2 bm +1 where u is inclined with respect to Ω and the associated cones do not overlap with Ω as shown in the right panel of Figure 3. We call this case 'Inclined'. The integral I ( ν ) also suffers from little frequency shift ( D/D 1 ≈ 1) and the aberration factor is approximated as D -2 1 ≈ 4 θ -4 pl . We obtain an approximated form of where we use R ( e z , Ω 1 , u ) ≈ 1 + (1 -2Θ 2 1 Θ -4 pl ) 2 ≈ 2. In the left panel of Figure 3, we show numerical results for the 'Inclined' case. The aberration factor decreases the integral I ( ν ) by a factor of Θ -4 pl compared with I Narrow ( ν ). Note that χ Inclined ( ν ) can be smaller than χ NR ( ν ), as χ Inclined ( ν ) /χ NR ( ν ) ∼ γ -3 θ -4 pl . For example, we find χ Inclined ( ν ) ∼ γ -3 χ NR ( ν ) for θ pl ∼ 1, while χ Inclined ( ν ) ∼ γχ NR ( ν ) for Θ pl ∼ 1. /negationslash The scattering geometry satisfying Θ bm > 1 > Θ pl is sketched in the right panel of Figure 4 where the γ -1 cone of plasma contains Ω and is well within the photon beam. We call this case 'Wide'. Note that although we take θ pl = 0 in Figure 4 and in Equation (21), we will find that the integral I ( ν ) behaves in a similar way for Θ bm > 1 > Θ pl = 0 in Appendix A. For θ pl = 0, the frequency shift factor is approximated as D/D 1 ≈ (1 + Θ 2 1 ) -1 ≤ 1. The aberration factor behave as D -2 1 ≈ 4 γ 4 / (1 + Θ 2 1 ) 2 and makes the angular distribution of the photon beam almost isotropic in the plasma rest frame. Simple analytic form is found for the frequency range ν > (1 + Θ 2 bm ) ν 0 ≈ Θ 2 bm ν 0 , where we use the expressions T b ( ν 1 ) ≈ GLYPH<1> GLYPH<1> T b ( ν 0 )( ν/ (1 + Θ 2 1 ) ν 0 ) p 2 +1 and S ( ν 1 ) ≈ p 2 +2. We obtain an approximated form where we take R ( e z , Ω 1 , u ) ≈ 1 + (1 -2Θ 2 1 (1 + Θ 2 1 ) -1 ) 2 ≈ 1 because the value varies in the range between 1 ≤ R ( e z , Ω 1 , u ) ≤ 2 for 0 ≤ θ 1 ≤ θ bm . I Wide ( ν ) is order unity at ν ∼ Θ 2 bm ν 0 . Numerical results are shown in Figure 4. Figure 4 shows that I Wide ( ν ) is approximated as -1 (order unity) even for ν 0 < ν < Θ 2 bm ν 0 . I Wide ( ν < ν 0 ) is approximated as ( T ( ν ) /T ( ν 0 )) S ( ν ) corresponding to Equation (15) with θ bm ∼ 1, i.e., almost isotropic. It is important to note that I Wide ∼ -1 can be used for applications in Section 3 rather than Equation (21). Note that χ Wide ( ν ) can also be smaller than χ NR ( ν ) depending on p 2 and θ bm in somewhat complex way because of the frequency shift. There remains the geometry Θ bm > Θ pl > 1 where the cone of plasma does not contain Ω but is within the photon beam. We do not find an analytic form of the integral I ( ν ) in this case. The numerical calculation in Appendix A shows that | I ( ν ) | takes between | I Inclined ( ν ) | and | I Wide ( ν ) | for the frequency range ν > ν 0 in which we are interested in Section 3. Note that | I Inclined ( ν ) | gives the smallest value and | I Wide ( ν ) | gives the largest value in any geometries (Θ bm , Θ pl ) for ν > ν 0 . We give a detailed discussion including this exceptional geometry in Appendix A.", "pages": [ 6, 7, 8, 9 ] }, { "title": "3. APPLICATION TO THE CRAB PULSAR", "content": "We evaluate the optical depth to induced Compton scattering applying to the Crab pulsar. We require that the optical depth | τ ( ν ) | is less than unity and then we constrain the Crab pulsar wind properties κ , γ , and σ .", "pages": [ 9 ] }, { "title": "3.1. Setup", "content": "We describe assumptions to estimate the normalization χ 0 for the Crab pulsar. For a pulsar wind, three assumptions are made. (I) Almost all of the spin-down power L spin goes to the pulsar wind. (II) The pulsar wind is a cold magnetized e ± flow whose bulk Lorentz factor is γ . (III) The number density of the pulsar wind decreases with r -2 , and we ignore structures in the pulsar wind, such as the current sheet [e.g., 32]. Now, the number density of the pulsar wind in the observer frame is where we assume the radial velocity β r ∼ 1. Note that we obtain Equation (1) from Equation (22) by normalizing 4 πr 2 cβ r n pl ( r ) with ˙ N GJ . Note also that a product γ (1 + σ ) does not depend on r because we expect no particle production outside the light cylinder r LC , i.e., n pl ∝ r -2 . For radio pulses, uncertainty of the brightness temperature arises from an opening angle of the radio emission θ bm . Following WR78, we assume that the emission is isotropic at r = r e where r e is an emission region size. The opening angle θ bm ( r ) is written as We adopt Equation (23) for the opening angle of the radio pulse throughout this paper. The brightness temperature is expressed as [e.g., 34] where F ν and d are a flux density at a frequency ν and a distance to the object, respectively. WR78 adopted r e = 10 7 cm which is estimated from the integrated pulse width W 50 = 3 msec [33, 34]. We study dependence on r e in Section 3.5. In Section 3.5, we will take r e = 10 3 cm considering the 'microbursts' of which individual pulses from the Crab pulsar show nano -microsecond duration structures [35]. Note that r e = 10 3 cm would also be considered as almost the minimum size of plasma to emit the coherent electromagnetic wave of the frequency ν = 100 MHz ( c/ν = 3 × 10 2 cm). Figure 5 shows the radio spectrum of the Crab pulsar. We assume F ν ∼ 50 ( ν/ 100 MHz) p 2 +3 Jy for ν 0 ≤ ν ≤ 100 MHz with ν 0 = 10 MHz and p 2 = -5. Adopting d = 2 kpc, L spin = 4 . 6 × 10 38 erg s -1 and the light cylinder radius r LC = 1 . 6 × 10 8 cm for the Crab pulsar, we obtain the normalization Although we used ν 0 = 10 MHz, we require | τ ( ν ) | < 1 at ν = 100 MHz because the Crab pulsar spectrum (Figure 5) is obviously unaffected by scattering in a range ν ≥ 100 MHz. On the assumptions made in this section, the scattering coefficient χ ( ν, r ) is considered to be a rapidly decreasing function of r . We introduce the exponents a and b (( a, b ) > 0) characterizing the r -dependence of the velocity u ( r ) as γ ∝ r a and θ pl ∝ r -b . Now, the r -dependence of χ ( ν, r ) (Equation (14)) is expressed as where I Wide ( ν ) ≈ -1 is used in this section because ν 0 /lessorsimilar ν < Θ 2 bm ν 0 ( ν 0 = 10 MHz and ν = 100 MHz) is mostly attainable for the 'Wide' case (Θ bm > 1). In Equation (26), b -a < 1 . 25 is sufficient for χ ( ν, r ) to be considered as a rapidly decreasing function of r . Otherwise we consider moderate values of a and b , say, 0 < ( a, b ) /lessorsimilar 1 . 25 below. Therefore, the choice of the innermost scattering radius is important to evaluate the optical depth. Here, we consider scattering beyond the light cylinder r ≥ r LC , because we do not know where the electron-positron plasma and the radio emission are produced inside the magnetosphere and because we do not take into account magnetic field effects which may be important close to the pulsar. We evaluate the optical depth as where r in and ∆ r are the innermost scattering radius and the path length, respectively. In Equation (27), we should not simply put r in = ∆ r = r LC because the path length ∆ r has a lower limit originating from the 'lack of time' effect which we will discuss in the next subsection.", "pages": [ 10, 11 ] }, { "title": "3.2. Characteristic Scattering Length", "content": "The 'lack of time' effect introduced by WR78 should be taken into account for the evaluation of r in and ∆ r in Equation (27). This is similar to the concept of the 'coherence radiation length' [e.g., 39, 40]. The normal treatment of scattering breaks down when an electron does not see one cycle of the electric field oscillation of radio waves. We determine this characteristic length l c as follows. A cycle of the incident and scattered photons in the plasma rest frame is described as ∆ t ' = δ D /ν where δ D = ( γD ) -1 or ( γD 1 ) -1 is the Doppler factor. The characteristic length l c is the speed of light multiplied by the time interval ∆ t = γ ∆ t ' in the observer frame. Using D -1 ≈ 2 γ 2 / (1 + Θ 2 pl ) and D -1 1 ≈ 2 γ 2 / (1 + Ψ 2 1 ) (Ψ 2 1 ≡ γ 2 ψ 2 1 ), we obtain l c is considered as a function of only r through γ ( r ) or θ pl ( r ) for the given frequency ν = 100 MHz. On the other hand, for the geometry Θ bm > Θ pl > 1, we obtain We find l c for this case is equal to or larger than that for the 'Inclined' case. Because l c depends on Ω 1 , we cannot separate integrals over Ω 1 and r in Equations (14) and (27). In this subsection, we limit the discussion about the 'Narrow', 'Inclined' and 'Wide' cases. Now, we describe how we determine r in and ∆ r taking into account the r -dependence of l c ( r ). Although we describe only for the 'Narrow' and 'Wide' cases ( l c ∝ γ 2 ), the same discussion is applicable to the 'Inclined' case ( l c ∝ θ -2 pl ) by replacing γ with θ -1 pl . We set γ ( r ) = γ LC ( r/r LC ) a where γ LC is the Lorentz factor at r LC . Substituting it into Equation (28), we obtain In Figure 6, we show the l c ( r ) -r diagram. We do not consider the region r < r LC . The region r > r LC is divided into two regions by the line l c ( r ) = r which corresponds to γ LC ≈ 10 2 . 7 and a = 0 . 5. Scattering off the radio pulse should be considered when l c ( r ) < r so that three different choices of r in are possible for different values of γ LC and the exponent a , corresponding to points 'A', 'B' and 'C' in Figure 6. Point 'A' corresponds to γ LC < 10 2 . 7 with any values of the exponent a . Since l c ( r LC ) < r LC in this case, we take r in = ∆ r = r LC . Point 'B' corresponds to γ LC > 10 2 . 7 with a < 0 . 5. The radio pulse is not scattered at r LC but beyond r LC . Here, we introduce the characteristic scattering radius r c which satisfies r c = l c ( r c ) > r LC so that we take r in = ∆ r = r c = (10 2 . 8 γ 2 LC r -2 a LC ) 1 / (1 -2 a ) cm. For γ LC > 10 2 . 7 with a ≥ 0 . 5, we obtain l c ( r ) > r everywhere beyond r LC , i.e., the electron never sees one cycle of radio waves (dot-dashed line: red in color). However, γ ( r ) cannot be infinitely large so that there should exist the radius satisfying r in = l c ( r in ) > r LC corresponding to point 'C'. In this case, we also take r in = ∆ r = r c whose expression is different from that for a < 0 . 5. Therefore, γ LC = 10 2 . 7 or θ pl , LC = 10 -2 . 7 is a critical value in determining which to adopt as r in . We consider whether the radio pulse can escape from scattering at the two radii r LC and r c . Rather than using the exponents a and/or b , it is convenient to introduce γ c ≡ γ ( r c ) and θ pl , c ≡ θ pl ( r c ). We evaluate the optical depth by treating the velocities u LC and u c , i.e., ( γ LC , θ pl , LC ) and ( γ c , θ pl , c ), as free parameters. Relation between the exponent a ( b ) and γ c ( θ pl , c ) will be discussed shortly in Section 3.3.3. Note that we indirectly obtain the characteristic scattering radius r c from Equation (28) once γ c or θ pl , c is obtained.", "pages": [ 12, 13 ] }, { "title": "3.3. Constrains on Lorentz Factor", "content": "Lower limits of γ are obtained from the condition | τ ( ν ) | < 1 for a given θ pl . We evaluate the optical depth, at ν = 100 MHz. τ (100 MHz) strongly depends on u LC or u c (Tables 1 and 2). Below, we search allowable region on γ -θ pl planes for r in = r LC (Figure 7) and for r in = r c (Figure 8), respectively. The results will be combined in Section 3.3.3. For a given r e , i.e., θ bm ( r in ) (Equation (23)), scattering geometry is classified into four cases on the γ -θ pl plane corresponding to the 'Narrow' (1 > Θ 2 bm +Θ 2 pl ), 'Inclined' (Θ 2 pl > Θ 2 bm +1) and 'Wide' (Θ bm > 1 > Θ pl ) cases, and the geometry satisfying Θ bm > Θ pl > 1. The first three geometries are studied in section 2.3 and the expressions of τ (100 MHz) for them are obtained in Tables 1 and 2. For Θ bm > Θ pl > 1, τ (100 MHz) is not expressed by Equation (31) because l c depends on Ω 1 as already discussed in Equation (29). Here, we infer the optical depth for Θ bm > Θ pl > 1 from the resuls of other three cases. Thus, the | τ (100 MHz) | = 1 lines at the Θ bm > Θ pl > 1 area in Figures 7 and 8 (thick dashed lines) are not calculated but inferred ones. We adopt r e = 10 7 r e , 7 cm to evaluate θ bm ( r in ) and will study when r e = 10 3 cm in Section 3.5 ( r e -dependence is already included explicitly in Tables 1 and 2). We consider customarily used values of σ ( σ LC and σ c ) in a range of 1 < 1 + σ /lessorsimilar 10 4 . We take ν 0 = 10 MHz, ν = 100 MHz and p 2 = -5, i.e., T b ( ν ) /T b ( ν 0 ) ∼ 10 -4 in the integrals I Narrow ( ν ) and I Inclined ( ν ) While I Wide ( ν ) ∼ -1 is used as the same reason discussed in Equation (26). Again, only the pulsar wind velocities u LC and u c are remaining parameters, i.e., we take ( γ LC , θ pl , LC ) and ( γ c , θ pl , c ) as the free parameters. 3.3.1. Escape from scattering at the light cylinder. Here, we are interested in whether the radio pulse can escape from scattering at r LC . Figure 7 shows the resultant γ -θ pl diagram which tells us whether the radio pulses can escape from scattering or not at a given point on the diagram, i.e., a given velocity u LC of the pulsar wind (see also Table 1). Since we obtain θ bm ( r LC ) ≈ 10 -1 . 2 from Equation (23), the scattering geometries are divided by the lines γ = 10 1 . 2 (Θ bm , LC = 1), θ pl = 10 -1 . 2 (Θ pl = Θ bm , LC ) and γ = θ -1 pl (Θ pl = 1). Areas above the thick lines | τ LC | = 1 correspond to the pulsar wind structures which allow the radio pulses to escape, where τ LC is the optical depth for r in = r LC . At the upper left corner on the diagram, the region satisfies l c ( r LC ) > r LC and the radio pulses also escape from scattering at r LC due to the 'lack of time' effect. The lines | τ LC | = 1 and l c ( r LC ) = r LC are different for different scattering geometries as described below and summarized in Table 1. First, we consider the 'Narrow' case (1 > Θ 2 bm +Θ 2 pl ) corresponding to the lowermost area on the diagram. The optical depth of τ LC ∼ 10 14 . 9 (1 + σ LC ) -1 obtained from Equations (19), (25) and (31) is independent of both γ LC and θ pl , LC . Therefore, a region | τ LC | < 1 does not appear for 1 + σ LC /lessorsimilar 10 4 and then we conclude that this case is not realized for the Crab pulsar. Next, we consider the 'Inclined' case (Θ 2 pl > Θ 2 bm +1) corresponding to the rightmost area on the diagram. In this case, the optical depth is expressed as τ LC ∼ 10 14 . 9 γ -4 LC θ -4 pl , LC (1 + σ LC ) -1 . The condition for | τ LC | < 1 is equivalent to γ LC /greaterorsimilar 10 3 . 7 θ -1 pl , LC (1 + σ LC ) -1 / 4 with θ pl , LC /greaterorsimilar 10 -1 . 2 where the painted area above | τ LC | = 1 line in the 'Inclined' area on the diagram. We find that the radio pulses can escape for reasonable parameters when the pulsar wind has a significant non-radial motion. For example, the pulsar wind of γ LC > 10 2 . 7 with θ pl , LC ∼ 1 and 1 + σ LC ≈ 10 4 can escape from scattering at r LC . The 'Wide' case (Θ bm > 1 > Θ pl ) corresponds to the left triangle area on the diagram. For | τ LC | ∼ 10 23 . 3 γ -4 LC (1 + σ LC ) -1 to be less than unity, we require γ LC > 10 5 . 8 (1 + σ LC ) -1 / 4 where the | τ LC | = 1 line in the 'Wide' area on the diagram. However, because the line is already above γ LC > 10 2 . 7 for 1 < 1 + σ LC /lessorsimilar 10 4 , therefore, γ LC > 10 2 . 7 (the 'lack of time' effect) is the condition for the radio pulses to escaping from scattering at r LC in this case. Lastly, we mention the geometry of Θ bm > Θ pl > 1 which appears in the upper triangle area on the diagram. The l c ( r LC ) = r LC and | τ LC | = 1 lines (dashed lines) are not calculated but interpolated ones. For escaping by the 'lack of time' effect ( l c ( r LC ) > r LC ), we obtain at least θ pl , LC < 10 -2 . 7 from Equation (29). The | τ LC | = 1 line is expected to be continuous at the boundaries on the γ LC = θ -1 pl , LC and θ pl , LC = 10 -1 . 2 lines because these boundaries just divide the approximated forms of Equation (14). On the other hand, the | τ LC | = 1 line would have at least one singular point because τ LC changes the sign at the left and right boundaries and a singular line (or curve) which satisfies τ LC = 0 would be drawn on the diagram. Although a significantly small value of γ LC might be allowed on the sides of the singular line, such a region on the γ -θ pl diagram would be as small as the dip around the discontinuity of I ( ν ) in Figures 2 -4 because S ( ν 1 ) which appears in Equation (14) controls the singularity τ LC = 0. When we neglect such a singular region, the allowed region would be above the thick dashed line and the lower limit of γ LC is clearly larger than the 'Inclined' case. 3.3.2. Escape from scattering beyond the light cylinder. We investigate whether the radio pulse can escape from scattering at r c further than r LC . Because r c > r LC , we have only to consider a region of γ > 10 2 . 7 and θ pl < 10 -2 . 7 . The behaviors of γ ( r ) and θ pl ( r ) at r LC < r < r c will be discussed in Section 3.3.3. Figure 8 shows the resultant γ -θ pl diagram at r c . We set θ bm ( r c ) ≈ 10 4 . 2 γ -2 c for the 'Narrow' and 'Wide' cases or θ bm ( r c ) ≈ 10 4 . 2 θ 2 pl , c for the 'Inclined' case from Equations (23) and (28). The scattering geometries are divided by the lines γ = 10 4 . 2 (Θ bm , c = 1), θ pl = 10 -4 . 2 (Θ pl = Θ bm , c ) and γ = θ -1 pl (Θ pl = 1) (see Table 2). It should be noted that each scattering geometry appears in a different layout on the γ -θ pl diagram compared with Figure 7 because θ bm ( r c ) depends on γ c or θ pl , c . The pulsar wind velocity u c which allows the radio pulses to escape corresponds to the area satisfying γ c ≥ 10 4 . 2 and θ pl , c ≤ 10 -4 . 2 corresponding to the 'Narrow' or 'Inclined' cases. Except for the extrapolated line in the geometry Θ bm > Θ pl > 1 (thick dashed line), the | τ c | = 1 line is not drawn on the diagram as described below, where τ c is the optical depth for r in = r c . The 'Narrow' case (1 > Θ 2 bm +Θ 2 pl ) corresponds to the left triangle area on the diagram. In this case, the optical depth is written as τ c ≈ 10 42 . 0 γ -10 c (1 + σ c ) -1 , i.e., we require γ c /greaterorsimilar 10 4 . 2 (1 + σ c ) -1 / 10 to be | τ c | < 1. The | τ c | = 1 line is degenerate to or a bit lower than the γ c = 10 4 . 2 line for 1 + σ c > 1. Therefore, whole of the 'Narrow' geometry area γ c ≥ 10 4 . 2 is allowed for radio pulses to escape. The corresponding characteristic scattering radius is r c /greaterorsimilar 10 11 . 2 cm ∼ 10 3 r LC . Next, we consider the 'Inclined' case (Θ 2 pl > Θ 2 bm +1) corresponding to the right triangle area on the diagram. For the optical depth, we require | τ c | ∼ 10 42 . 0 γ -4 c θ 6 pl , c (1 + σ c ) -1 = 10 42 . 0 γ -10 c Θ 6 pl , c (1 + σ c ) -1 < 1 at r c . The | τ c | = 1 line satisfies γ c = 10 4 . 2 Θ 3 / 5 pl , c (1 + σ c ) -1 / 10 which has slope γ ∝ θ 3 / 2 pl and is continuous with the | τ c | = 1 line for the 'Narrow' case on GLYPH<1> the boundary line γ = θ -1 pl . Note that large θ pl , c does not reduce | τ c | as | τ LC | is reduced by large θ pl , LC (see the 'Inclined' area in Figure 7) because r c is a rapidly decreasing function of θ pl , c . Therefore, whole of the 'Inclined' geometry area θ pl , c ≤ 10 -4 . 2 is allowed for radio pulses to escape and we obtain r c /greaterorsimilar 10 11 . 2 cm again. The 'Wide' case (Θ bm > 1 > Θ pl ) corresponding to the lowermost area on the diagram. The condition to be | τ c | < 1 is γ c /greaterorsimilar 10 4 . 8 (1 + σ c ) -1 / 6 . In this case, a region | τ c | < 1 does not appear in the 'Wide' area for 1 + σ c < 10 4 and then we conclude that this case is not realized for the Crab pulsar. For the geometry of Θ bm > Θ pl > 1 corresponding to the rightmost area on the diagram, we do not draw the | τ c | = 1 line in the same manner as Figure 7 because no | τ c | = 1 line appears in Figure 8 for other geometries. One possibility is that the | τ c | = 1 line emerges from the boundary θ pl = 10 -4 . 2 , such as the thick dashed line on the diagram. As implied from the | τ c | = 1 line for the 'Inclined' case, the line has slope γ ∝ θ q pl with q ≥ 3 / 2 because r c rapidly decreases with increase θ pl , c . 3.3.3. Summary. There exist two possible cases of u LC where the radio pulses are not scattered at r LC . First, when u LC is significantly inclined with respect to the radio pulses 10 -1 . 2 < θ pl , LC /lessorsimilar 1 and has the Lorentz factor satisfying γ LC θ pl , LC (1 + σ LC ) 1 / 4 /greaterorsimilar 10 3 . 7 , we obtain τ LC < 1. In this case, the radio pulses reach the observer without scattering because χ ( ν, r ) decreases rapidly with r for 0 < ( a, b ) /lessorsimilar 1 . 25 as discussed in Equation (26). The second corresponds to the 'lack of time' effect, i.e., u LC is almost aligned with respect to the radio pulses θ pl , LC < 10 -2 . 7 with γ LC > 10 2 . 7 . In this case, r in = ∆ r = r c , we require | τ c | < 1 when an electron reaches r c and also require l c ( r ) > r at r LC < r < r c . Using the result γ c > 10 4 . 2 and θ pl , c < 10 -4 . 2 for | τ c | < 1 ( r c /greaterorsimilar 10 11 . 2 cm ≈ 10 3 r LC ), γ ( r ) at the range of r LC < r < 10 11 . 2 cm should be changed with r as follows (see also Equation (30) and Figure 6). For the 'Narrow' and 'Wide' cases, we require that the point 'B' ( a < 0 . 5) or point 'C' ( a ≥ 0 . 5) in Figure 6 is more distant than 10 11 . 2 cm. For example, if γ has a constant value ( a = 0), we require γ > 10 4 . 2 at r LC . On the other hand, if a ≥ 0 . 5 with γ LC > 10 2 . 7 , γ should have a terminal value of γ > 10 4 . 2 . Although the 'Inclined' case is a bit complicated, we can constrain the behavior of γ by replacing γ with θ -1 pl in the above discussion and using the condition γ > θ -1 pl (Θ pl > 1) for the 'Inclined' case. Required values of the exponents a and b change with the value of u LC , σ LC and σ c . Lastly, we mention the result obtained by WR78. Essentially, the 'Wide' geometry with scattering at r c ∼ 10 11 . 2 cm of ours corresponds to the situation which they considered, although their setup is not exactly the same as ours in the radial variations of γ ( r ) and n pl ( r ). Our result of γ c /greaterorsimilar 10 4 . 8 (1 + σ c ) -1 / 6 obtained in Section 3.3.2 is close to their result of γ > 10 4 . 4 (see their Equation (16)). Note that we did not consider the 'Wide' case with scattering at r c because γ c < 10 4 . 2 is also required for the geometry to be 'Wide'. Also note that they did not account for the constraint at r LC , although we require γ LC > 10 2 . 7 and θ pl , LC < 10 -2 . 7 for r c > r LC .", "pages": [ 13, 14, 15, 16, 17, 18 ] }, { "title": "3.4. Constraints on Pair Multiplicity", "content": "In the last section, we obtain lower limits of γ for a given inclination angle θ pl and a magnetization σ of the pulsar wind. Here, we consider corresponding upper limits of κ using Equation (1). Note that the combination of κγ (1 + σ ) = 10 10 . 5 is independent of r from energy conservation law and that κ alone is also expected to be independent of r from the law of conservation of particle number. Below, we consider the upper limits of κ for the two possible u LC of the pulsar wind and we do not consider constraint for the geometry Θ bm > Θ pl > 1 for simplicity. When the pulsar wind is inclined with respect to the radio pulses at r LC (10 -1 . 2 < θ pl , LC /lessorsimilar 1), we obtain an upper limit of κ by eliminating γ LC from γ LC θ pl , LC (1 + σ LC ) 1 / 4 /greaterorsimilar 10 3 . 7 with the use of Equation (1) ( κγ ( r )(1 + σ ( r )) = 10 10 . 5 ). We obtain The upper limit is κ < 10 6 . 8 for both 1 + σ LC ∼ 1 and θ pl , LC ∼ 1. This upper limit of the pair multiplicity can satisfy κ /greaterorsimilar κ PWN = 10 6 . 6 obtained by Tanaka & Takahara (2010, 2011) [11, 13]. However, for σ LC ∼ 10 4 , an upper limit becomes κ /lessorsimilar 10 3 . 8 θ pl , LC and γ LC /greaterorsimilar 10 2 . 7 θ -1 pl , LC which can be close to the customarily believed picture of the pulsar wind at the light cylinder [2, 3]. In other words, 1 + σ LC /lessorsimilar 10 0 . 2 θ 4 / 3 pl , LC is required for κ ≥ κ PWN . For the second case when the pulsar wind is aligned with respect to the radio pulse at r LC , we require both γ LC > 10 2 . 7 ( θ pl , LC < 10 -2 . 7 ) and γ c > 10 4 . 2 ( θ pl , c < 10 -4 . 2 ). Using κγ ( r )(1 + σ ( r )) = 10 10 . 5 , we require both Because κ conserves along the flow, κ should satisfy both of the two inequalities. Even for 1 + σ c ∼ 1, κ /lessorsimilar 10 6 . 3 at r c ∼ 10 3 r LC is marginal for κ > κ PWN . For customarily used magnetization σ LC ∼ 10 4 , an upper limit is κ /lessorsimilar 10 3 . 8 /lessmuch κ PWN . The results are summarized in Table 3. A little bit larger κ is allowed for the inclined u LC ( θ pl , LC ∼ 1) than for the aligned u LC with respect to the radio pulse beam.", "pages": [ 18, 19 ] }, { "title": "3.5. Dependence on the Size of Emission Region", "content": "We assume r e = 10 7 cm in the above calculations. Here, we discuss the constraints on γ and κ assuming Equation (23) with r e = 10 3 cm for example. The dependence on r e (10 3 ≤ r e ≤ 10 7 cm) is described explicitly in Tables 1 and 2. When we take a different value of r e , the brightness temperature T b (Equation (24)) and the integrals I Narrow and I Inclined (Equations (19) and (20)) are changed. In Tables 1 and 2, we find that the optical depth for the 'Narrow' and 'Inclined' cases is proportional to r 2 e . This is because I Narrow and I Inclined are proportional to r 4 e and T b is proportional to r -2 e . On the other hand, for the 'Wide' case, the optical depth is proportional to r -2 e because I Wide ( ν ) ∼ -1 whose value does not depend on θ bm in the range of ν 0 /lessorsimilar ν < Θ 2 bm ν 0 . Note that the layout of scattering geometry on the γ -θ pl diagrams (Figure 9) is also changed where the 'Narrow' and 'Inclined' areas spread on the planes compared with those in Figures 7 and 8. We obtain the lower limits of γ and the upper limits of κ in the same manner as the case of r e = 10 7 cm. Figure 9 shows the resultant γ -θ pl diagrams both at r LC (left) and r c (right). Obtained lower limits of γ and upper limits of κ are summarized in Table 4. At r LC ( θ bm ( r LC ) ≈ 10 -5 . 2 ), we find two allowed regions on the diagram in the left panel of Figure 9. First is when the pulsar wind has a significant non-radial motion 10 -2 . 7 < θ pl , LC /lessorsimilar 1. We require γ LC θ pl , LC (1 + σ LC ) 1 / 4 /greaterorsimilar 10 1 . 7 r 1 / 2 e , 3 for | τ LC | < 1 and no scattering occurs beyond r LC for the moderate values of the exponents a and b . We also find that the non-relativistic pulsar wind β LC /lessmuch 1 is unfavorable even for such a small opening angle of the radio beam θ bm , LC = 10 -5 . 2 with 1 + σ LC ≈ 10 4 . Secondly, the region which satisfies γ LC > 10 2 . 7 and θ pl , LC < 10 -2 . 7 is also allowed to escape from scattering at r LC due to the 'lack of time' effect. In this case, in addition, we require | τ c | < 1 at r c ( > r LC ). The right panel of Figure 9 shows the γ -θ pl diagram at r c . We do not find the 'Wide' and Θ bm > Θ pl > 1 geometries on the diagram because θ bm ( r c ) for r e = 10 3 cm is much smaller than that for r e = 10 7 cm. The region which satisfies | τ c | < 1 is γ c /greaterorsimilar 10 3 . 4 r 1 / 5 e , 3 (1 + σ c ) -1 / 10 for the 'Narrow' case and γ c /greaterorsimilar 10 3 . 4 r 1 / 5 e , 3 Θ 3 / 5 pl , c (1 + σ c ) -1 / 10 for the 'Inclined' case. Corresponding r c is larger than 10 9 . 6 r 2 / 5 e , 3 cm = 10 1 . 4 r 2 / 5 e , 3 r LC . It is important to note that the constraint at r c very weakly depends on r e as r 1 / 5 e . Accordingly, we obtain upper limits of κ with the help of Equation (1). When the pulsar wind is inclined with respect to the radio pulse at r LC (10 -2 . 7 < θ pl , LC /lessorsimilar 1), we obtain We require σ LC /lessorsimilar 10 3 /lessmuch 10 4 for κ > κ PWN . When the pulsar wind is aligned with respect to the radio pulse at r LC ( θ pl , LC < 10 -2 . 7 and γ LC > 10 2 . 7 ), we obtain κ > κ PWN is attainable for both the 'Narrow' and 'Inclined' cases again. We obtain the lower limits of γ and the upper limits of κ for different sizes of the emission region r e . Basically, as is found from Table 4, the smaller the emission region size becomes, the easier the radio pulses escape from scattering, i.e., small γ and large κ are allowed. We obtain the most optimistic constraint for large κ ( κ /lessorsimilar 10 8 . 8 at the uppermost row of Table 4), when θ pl , LC ∼ 1 (inclined u LC ), 1 + σ LC ∼ 1 and r e = 10 3 cm. Combined with κ /greaterorsimilar κ PWN = 10 6 . 6 , we can write the pulsar wind properties as 10 1 . 7 /lessorsimilar γ /lessorsimilar 10 3 . 9 and κ PWN /lessorsimilar κ /lessorsimilar 10 8 . 8 . Although all these constraints are at r LC , the radio pulse can escape from scattering and κ /greaterorsimilar κ PWN is satisfied beyond r LC because γ ( r )(1 + σ ( r )) ≈ γ ( r ) = constant beyond r LC for 1 + σ LC ∼ 1 from Equation (1) and conservation of particle number ( κ = constant). Note that we obtain 10 1 . 2 /lessorsimilar γ LC /lessorsimilar 10 1 . 9 and κ PWN /lessorsimilar κ /lessorsimilar 10 7 . 3 for 1 + σ LC ∼ 10 2 , and we require γ ( r )(1 + σ ( r )) = constant and also κ = constant beyond r LC .", "pages": [ 19, 20, 21 ] }, { "title": "4. Summary", "content": "To constrain the pulsar wind properties, we study induced Compton scattering by a relativistically moving cold plasma. Induced Compton scattering is θ 4 bm k B T b ( ν ) /m e c 2 times significant compared with spontaneous scattering for the non-relativistic case. However, for scattering by the relativistically moving plasma, scattering geometry of the system changes the scattering coefficient significantly. We consider fairly general geometries of scattering in the observer frame and obtain the scattering coefficient for induced Compton scattering off the photon beam. On the other hand, we do not take into account the magnetic field effects and the scattering off the background photons in this paper. We obtain approximate expressions of the scattering coefficient for three geometries corresponding to the 'Narrow' (1 > Θ 2 bm +Θ 2 pl ), 'Inclined' (Θ 2 pl > 1 + Θ 2 bm ) and 'Wide' (Θ bm > 1 > Θ pl ) cases, while the scattering coefficient for Θ bm > Θ pl > 1 is obtained numerically. Behavior of the scattering coefficient against a given scattering geometry is governed by a simple combination of four factors. In addition to the solid angle factor θ 4 bm appearing even for the non-relativistic case, there exist three relativistic effects; the factor independent of scattering geometry γ -3 and the other two factors depending on geometry, the aberration factor D -2 1 and the frequency shift factor D/D 1 . When the photon beam is inside the γ -1 cone of the plasma beam (the 'Narrow' case), the aberration factor increases the scattering coefficient by a factor of ∼ γ 4 (up to γθ bm ∼ 1). On the other hand, when the plasma velocity is significantly inclined with respect to the photon beam (the 'Inclined' case), this factor of γ 4 does not appear. The frequency shift factor is important when the photon beam is wider than the γ -1 cone of the plasma beam (the 'Wide' case) and is rather complex and mostly increases the absolute value of the scattering coefficient compared with the non-relativistic case. Basically, the 'Inclined' case gives the smallest and the 'Wide' case gives the largest scattering coefficient, i.e., the Θ bm > Θ pl > 1 case is in between. We apply induced Compton scattering to the Crab pulsar, where the high T b ( ν ) radio pulses go through the relativistic pulsar wind and constrain the pulsar wind properties by imposing the condition of the optical depth being smaller than unity. We introduce the characteristic scattering radius r c where the 'lack of time' effect prevents scattering at r < r c . We evaluate the scattering optical depth for both r in = r LC and r in = r c cases. We consider more general scattering geometries than WR78 and also study the dependence on the size of the emission region 10 3 ≤ r e ≤ 10 7 cm which directly affects the opening angle of the radio pulses θ bm ( r ). Allowable pulsar wind velocities at r LC ( u LC ) and at r c ( u c ) are explored assuming the canonical value of the magnetization 1 < 1 + σ /lessorsimilar 10 4 . The two pulsar wind velocities u LC are allowed for radio pulses to escape from scattering at r LC . One is that the plasma velocity is inclined with respect to the photon beam ( θ pl , LC ∼ 1). When γ LC /greaterorsimilar 10 1 . 7 r 1 / 2 e , 3 θ -1 pl , LC (1 + σ LC ) -1 / 4 is satisfied, the radio pulses reach the observer without scattering for moderate radial variation of γ ( r ) and θ pl ( r ) where γ ∝ r a and θ pl ∝ r -b with 0 < ( a, b ) /lessorsimilar 1 . 25. The other is when the plasma velocity is aligned with respect to the photon beam ( θ pl , LC < 10 -2 . 7 ). We require the lower limit γ LC /greaterorsimilar 10 2 . 7 for the 'lack of time' effect preventing scattering at r LC . In this case, we also require the optical depth at r c /greaterorsimilar 10 9 . 6 r 2 / 5 e , 3 cm = 10 1 . 4 r 2 / 5 e , 3 r LC to be less than unity, where r c (= l c ) depends on γ c or θ c (Equation (28)). For example, we require γ c /greaterorsimilar 10 3 . 4 r 1 / 5 e , 3 (1 + σ c ) -1 / 10 for the completely aligned case θ pl = 0. Basically, the smaller the emission region size and the larger the inclination angle of the pulsar wind become, the smaller γ is allowed. We discussed upper limits of the pair multiplicity using obtained constraints on the velocities of the Crab pulsar wind and Equation (1). In principle, κ /greaterorsimilar κ PWN ≡ 10 6 . 6 [11, 13] is possible although we require 1 + σ LC /lessmuch 10 4 , i.e., customarily used value 1 + σ LC ≈ 10 4 contradicts κ > κ PWN . The most optimistic constraint which allows large κ is obtained when θ pl , LC ∼ 1 and r e = 10 3 cm (Equation (34)). In this case with κ /greaterorsimilar κ PWN , we can write the pulsar wind properties as 10 1 . 7 /lessorsimilar γ /lessorsimilar 10 3 . 9 and κ PWN /lessorsimilar κ /lessorsimilar 10 8 . 8 for 1 + σ LC ∼ 1 and 10 1 . 2 /lessorsimilar γ /lessorsimilar 10 1 . 9 and κ PWN /lessorsimilar κ /lessorsimilar 10 7 . 3 for 1 + σ LC ∼ 10 2 . Note that all these constraints are at r LC and we also require moderate radial variation of θ pl ( r ) and γ ( r ) ( ∝ (1 + σ ( r )) -1 ) beyond r LC .", "pages": [ 21, 22 ] }, { "title": "Acknowledgment", "content": "S. J. T. would like to thank Y. Ohira, R. Yamazaki, T. Inoue and S. Kisaka for useful discussion. We would also like to thank the anonymous referees for a meticulous reading of the manuscript and very helpful comments. This work is supported by JSPS Research Fellowships for Young Scientists (S.J.T. 2510447).", "pages": [ 22 ] }, { "title": "A. Numerical Integration", "content": "We show results of numerical integration of I ( ν, γ, θ bm , θ pl ) (Equation (14)). We focus on the situation 0 ≤ ( θ pl , θ bm ) /lessorsimilar 1 and γ /greatermuch 1, and then the integral I ( ν ) depends on the GLYPH<1> GLYPH<1> normalized angles Θ bm ≡ γθ bm and Θ pl ≡ γθ pl rather than on θ bm , θ pl and γ , separately. As seen in Section 2.3, the behavior of I ( ν ) is very different for the value of Θ bm and Θ pl , i.e., different scattering geometries. To obtain the results of Figures A1 and A2, we set γ = 10 2 and adopt the broken power-law spectrum with p 1 = 3 and p 2 = -5 (Equation (10)). The figures show absolute values of the integral I ( ν ) versus frequency ν for different sets of parameters Θ bm and Θ pl . All the lines in these figures have a discontinuity where the sign of the integral I ( ν ) changes. The sign of the integral I ( ν ) is positive at high frequency side where the photon number decreases and vice versa. Before describing details of Figures A1 and A2, we mention that the approximated forms studied in Section 2.3 can describe behaviors of most of lines in the figures. Behaviors of lines with no frequency shift is described by I Narrow and I Inclined and behaviors of lines whose discontinuity point shifted to ν > ν 0 is described by I Wide . Only behaviors of 'line d' and 'line e' in the bottom-left panel in Figure A1 and of 'line e' in the bottom-left panel in Figure A2 are not explained by these three approximated forms corresponding to Θ bm > Θ pl > 1 which we will discuss later. GLYPH<1> GLYPH<1> Figure A1 shows how the integral I ( ν ) changes with Θ pl (0 ≤ Θ pl ≤ 10) for fixed Θ bm . Three panels in Figure A1 correspond to different fixed values of Θ bm and the bottom-right sketch describes scattering geometry when Θ bm = 10 corresponding to the bottom-left panel in Figure A1, for example. It is common for all the panels that 'line a' is very close to 'line b', i.e, we can approximate that the photon and plasma are completely aligned (Θ pl = 0) even for Θ pl < 1. It is also common for all the panels that 'line a' is larger than other lines for ν > ν 0 and | I ( ν ) | decreases in order from 'line a' to 'line e', i.e., | I ( ν ) | is large when the photons and the plasma are aligned at least the frequency range ν > ν 0 . The top-left panel (Θ bm = 0 . 1) shows the case when the photon beam is considered as narrow (compared with γ -1 cone associated with the plasma) and shows little frequency shift D/D 1 ≈ 1 corresponding to I Narrow and I Inclined studied in Section 2.3. The bottom-left panel in Figure A1 is the case when the photon beam is considered as wide (Θ bm = 10: the bottom-right sketch of Figure A1). In this case, the frequency shift effect is extreme and the absolute values | I ( ν ) | is almost unity at broad frequency range. sketch describes scattering geometry when Θ pl = 1 corresponding to the top-right panel in Figure A2, for example. Note that some lines are the same parameter set with Figure A1. It is common for all the panels that | I ( ν ) | decreases with the smaller values of Θ bm . 'Line d' and 'line e' on the top-left panel (Θ pl = 0) and top-right panel (Θ pl = 1) show I Wide studied in Section 2.3. Lastly, we discuss the behaviors of 'line d' and 'line e' in the bottom-left panel in Figure A1 and of 'line e' in the bottom-left panel in Figure A2. These lines satisfy Θ bm ≥ Θ pl > 1 and shows two notable features. One is the discontinuity point shifting toward ν < ν 0 ('feature one') and the other is | I ( ν ) | being significantly greater than unity at ν < ν 0 ('feature two'). We can discuss these features qualitatively. To simplify explanation, we take Θ 2 bm = Θ 2 pl /greatermuch 1 corresponding to 'line e' in the bottom-left panel both in Figures A1 and A2 (Θ bm = Θ pl = 10). For the 'feature one', we obtain from Equation (18) that the frequency shift factor has a peak value D/D 1 ∼ Θ 2 pl at Θ 1 = Θ bm and φ 1 = 0, this value corresponds to the frequency which gives the peak of | I ( ν ) | . For the 'feature two', we try to evaluate | I ( ν ≈ Θ -2 pl ν 0 ) | . For ν ≈ Θ -2 pl ν 0 , we obtain ν 1 = ( D/D 1 ) ν ≈ (Θ 2 pl / (1 + Ψ 2 1 )) ν ≈ ν 0 / (1 + Ψ 2 1 ) ≤ ν 0 so that we take S ( ν 1 ) ∼ p 1 +2 and T b ( ν 1 ) ≈ T b ( ν 0 )(1 + Ψ 2 1 ) -p 1 -1 . Assuming that R is a constant of order unity, we obtain, Although this integral cannot be performed analytically, we find that the integrand has a peak value Θ 3 pl at φ 1 = 0 and Θ 1 = Θ pl . A crude estimate may be obtained by taking a peak value of the integrand Θ 3 pl with ∫ 2 π 0 dφ ∼ 2 π and ∫ Θ pl 0 d Θ 1 ∼ Θ pl . This must be overestimate and gives 3 RS Θ 4 pl / 2 ∼ 10 4 for Θ pl = 10. Although the value does not fit to the numerical calculation ( I (Θ -2 pl ν 0 ) ∼ 10 2 from Figures A1 and A2), we find the I (Θ -2 ν 0 ) can be much greater than unity.", "pages": [ 22, 23, 24, 25 ] } ]

[ { "title": "MODULAR THEORY FOR OPERATOR ALGEBRA IN BOUNDED REGION OF SPACE-TIME AND QUANTUM ENTANGLEMENT", "content": "DAISUKE IDA, TAKAHIRO OKAMOTO, AND MIYUKI SAITO Abstract. We consider the quantum state seen by an observer in the diamondshaped region, which is a globally hyperbolic open submanifold of the Minkowski space-time. It is known from the operator-algebraic argument that the vacuum state of the quantum field transforming covariantly under the conformal group looks like a thermal state on the von Neumann algebra generated by the field operators on the diamond-shaped region of the Minkowski space-time. Here, we find, in the case of the free massless Hermitian scalar field in the 2dimensional Minkowski space-time, that such a state can in fact be identified with a certain entangled quantum state. By doing this, we obtain the thermodynamic quantities such as the Casimir energy and the von Neumann entropy of the thermal state in the diamond-shaped region, and show that the Bekenstein bound for the entropy-to-energy ratio is saturated. We further speculate on a possible information-theoretic interpretation of the entropy in terms of the probability density functions naturally determined from the Tomita-Takesaki modular flow in the diamond-shaped region.", "pages": [ 1 ] }, { "title": "1. Background and Motivation", "content": "We often regard the quantum state of a field on the space-time as being a pure state that has the zero von Neumann entropy. Of course, this does not imply that an observer always has a perfect knowledge of the quantum field. Rather, each observer would not be able to distinguish it from a certain mixed state, and the identified mixed state would in general depend on the observer's trajectory and the measuring means available. Thus, each observer has his own nonzero von Neumann entropy for the quantum states of the field. For example, let us consider an observer with a finite lifetime whose world-line is a timelike segment in the space-time bounded by future and past end points, and the measurements of the quantum field by him in terms of an apparatus located at each space-time point. When this observer sends a command to a remote measuring apparatus, the apparatus immediately performs a measurement of the quantum field on the corresponding space-time point and the result is returned to the observer. If this is the only way for the observer to measure the quantum state of the field, the set of points from which the observer can get the information is the intersection of the chronological future and the chronological past of the observer's world-line, which we call, for obvious reasons, the 'diamond region' associated with the observer. The limitation of the observed region would cause the loss of the information on the quantum state of the field. This can be heuristically understood from general considerations as follows. In general, a quantum measurement can be reduced to the evaluation of the expectation value of a non-negative self-adjoint operator belonging to a C ∗ -algebra A . In the quantum field theory, the corresponding C ∗ -algebra A might be regarded as the von Neumann algebra A ( M ) constructed from the field operators on the space-time M . (Though the polynomial ∗ -algebra generated by field operators is not a von Neumann algebra, for the field operators are unbounded, one can define a von Neumann algebra A ( O ) constructed from field operators on the open submanifold O of M , if O is M itself, a diamond region, a so-called Rindler wedge, or their image under a Poincar'e transformation [1, 2]. More precisely, the von Neumann algebra A ( O ) is the double commutant of the C ∗ -algebra generated by the projection operators composing the field operators smeared by test functions with support in O .) However, not all the projection operators in A ( M ) are available for every observer. Rather, the available projection operators, or more generally non-negative self-adjoint operators, generate a proper von Neumann subalgebra of A ( M ), which would be regarded as the algebra of physical quantities for the observer. For an observer with a finite lifetime, the corresponding von Neumann subalgebra of physical quantities would be A ( O ), where O is the diamond region associated with the observer. On the other hand, a quantum state ω : A → C on a C ∗ -algebra A is a pure state if and only if the GNS representation of A associated with the quantum state ω is irreducible. However, the GNS representation of its C ∗ -subalgebra A ' associated with the restriction of ω to A ' is not always irreducible. If it is reducible, the quantum state ω is indistinguishable from a certain mixed state in terms of any quantum measurements solely of the operators in A ' . Hence, an observer with a finite lifetime would perceive a certain mixed state. Then, how does the vacuum state in the Minkowski space-time look like for the observer with the finite lifetime? In the case of the conformally invariant Hermitian scalar field, Martinetti and Rovelli [3] conclude that such an observer will see a certain thermal state. Their reasoning is based on the conformal invariance of the vacuum state and the conformal equivalence between the diamond region and the Rindler wedge. The outline of their argument is as follows. Let W be the Rindler wedge, which is the open submanifold of the n -dimensional Minkowski space-time ( n ≥ 2) specified by x 1 > | x 0 | in terms of the standard time coordinate x 0 and one of the standard spatial coordinates x 1 in the Minkowski space-time. The Rindler wedge W is globally static in the sense that the Lorentz boost generated by the Killing vector field x 1 ∂ 0 + x 0 ∂ 1 acts isometrically on W . A uniformly accelerated observer following an orbit of the Lorentz boost in the Poincar'e invariant vacuum state would find himself apparently in a thermal bath with the temperature proportional to the proper acceleration. This is well known as the Unruh effect [4]. One of rigorous explanations of the Unruh effect is given by the BisognanoWichmann theorem [5]. This theorem shows that the von Neumann algebra A ( W ) gives in an essential way an example of the application of the Tomita-Takesaki modular theory [6] of operator algebras. According to the Tomita's fundamental theorem in the modular theory, given a von Neumann algebra A acting on a Hilbert space H , and a cyclic and separating vector | Ω 〉 ∈ H , there uniquely exists the one-parameter group of automorphism { σ s } acting on A , which is called the modular flow. Furthermore, the modular flow is subject to the Kubo-MartinSchwinger (KMS) condition with respect to the vector state corresponding to | Ω 〉 , which means that | Ω 〉 is identified with a thermal state. The Bisognano-Wichmann theorem states that in the case of A = A ( W ), | Ω 〉 corresponds to the Poincar'e invariant vacuum, and hence the vacuum is subject to the KMS condition, where the generator of the Lorentz boost plays a role of the Hamiltonian. Thus, the modular flow here can be seen as the geometric flow generating the time translation in W . A relativistic quantum field in the Minkowski space-time is often assumed to transform covariantly under the Poincar'e group [7]. If we further require the covariance under the conformal group, and the conformal invariance of the vacuum state, we can, in a sense, map the geometric modular flow in the Rindler wedge W to that in the conformal image of W . (Though in the case of n = 2, there is no vacuum state invariant under the conformal group, it is sufficient to consider a state invariant under the projective conformal group, which is the subgroup generated by the dilatations, the special conformal transformations and the Poincar'e transformations.) In fact, Hislop and Longo show that for the quantum field in the diamond region O , the conformally invariant vacuum is subject to the KMS condition [8], which relies on the conformal equivalence between the Rindler wedge and the diamond region. Martinetti and Rovelli interpret the modular flow as determining the 'thermal time' in O , and this leads to the notion of the 'diamond temperature' which is the proper temperature for the observer following the modular flow [3]. The relevant observer in O is the inertial observer or the uniformly accelerated observer with the finite lifetime. A remarkable point here is that even an inertial observer may perceive a nonzero temperature. Another feature of the diamond temperature is that it in general diverges around the future and past end points of the observer's world-line. It is not clear whether the behavior of the diamond temperature as above is universal one or whether it is peculiar to the operator-algebraic method. Hence, we would like to verify the diamond temperature in terms of the standard method [9] via the determination of the Bogoliubov transformation between different Fock representations. We will see that it gives the same temperature as that derived by Martinetti and Rovelli. Then, we discuss the thermodynamic quantities such as the Casimir energy and the quantum entanglement entropy for the observer with a finite lifetime based on the standard quantum field theory. We further introduce the probability density function naturally determined by the modular flow in the diamond region, and attempt to give the information-theoretic interpretation of the entropy of the diamond region. In this paper, we consider the free massless Hermitian scalar field in the 2dimensional Minkowski space-time M , which transforms covariantly under the projective conformal group. We use the natural unit system in which c = /planckover2pi1 = 1. The diamond region O is specified by | t | + | x | < L with a length parameter L , when the Lorentzian metric is written as ds 2 = -dt 2 + dx 2 (Fig. 1).", "pages": [ 1, 2, 3, 4 ] }, { "title": "2. Thermal State in Diamond Region", "content": "The modular flow in O coincides with the geometric flow generated by the conformal Killing vector field, which is timelike in O . This conformal Killing vector field naturally defines the positive frequency modes of the Hermitian scalar field for observers following the modular flow. In fact, we define the positive frequency modes as the conformal image of the positive frequency solutions defined on the Rindler wedge W , under the conformal diffeomorphism: W → O , which pushes forward the timelike Killing vector field in W to the conformal Killing vector field in O . More precisely, if the Lorentzian metric g O µν in O is conformally equivalent to the Lorentzian metric g W µν in W as g O µν = C 2 g W µν , and ξ µ is the timelike Killing vector field with respect to g W µν , then ξ µ is the conformal Killing vector field with respect to g O µν . The positive frequency mode χ O ω in O is required to satisfy the eigenvalue equation on O The null coordinates U ± ∈ R 1 covering O are defined by where u ± = t ± x are the Minkowski null coordinates. Then, the positive frequency modes in O are subject to Therefore, the positive frequency mode in O consists of For the later convenience, we introduce the mode functions as extension of χ O ± ω to M , where we call χ + ω the ingoing mode, and χ -ω the outgoing mode, and these mode functions are normalized with respect to the Klein-Gordon inner product. On the other hand, by continuing analytically the positive frequency modes χ O ± ω to M , we obtain where the null coordinates U ± ex are defined by for the regions: | u ± | > L . Although there are two options to extend χ O ± ω to | u ± | > L corresponding to the double signs in the relation log( -1) = ± iπ , we remove this ambiguity by requiring that ˜ χ ± ω correspond to the positive frequency modes with respect to the Poincar'e invariant vacuum. The positive frequency modes complement to { χ O ± ω } are determined as where we set the sign in the exponent to positive for U ± ex are past-directed. The analytic extension of χ ex ± ω from the regions: | u ± | > L to M is obtained in the form Now, let us derive the Bogoliubov transformation between the Poincar'e invariant vacuum and the vacuum defined by the conformal time flow in the diamond region O . We can expand the field operator in the form Then, the vacuum state in the diamond region | 0; O 〉 is defined in terms of the annihilation operators ( b ± ω , b ex ± ω ) as On the other hand, the Poincar'e invariant vacuum is defined by the set of modes ( ˜ χ ± ω , ˜ χ ex ± ω ). More precisely, by writing the mode expansion of the field operator as ˜ ˜ the Poincar'e invariant vacuum | 0; M 〉 is defined by From Eqs. (1) and (2), the transformation between the two sets of mode functions turns out to be This leads to the Bogoliubov transformation of the creation and annihilation operators as From this, we see that the vacuum | 0; M 〉 is also written formally as By taking the partial trace of the density operator over the subsystem generated by the operators b ex ±† ω , we obtain the Gibbs state with the inverse temperature β = πL as It should be noted, however, that there is an ambiguity in the normalization of the conformal Killing vector field, ξ µ ↦→ αξ µ , which affects the inverse temperature as β ↦→ α -1 β . The invariant inverse temperature is given by β O = β √ -ξ µ ξ µ , which is regarded as the local inverse temperature associated with the observer following the flow determined by the conformal Killing vector. In terms of the proper time τ of the observer, it becomes where a denotes the proper acceleration of the observer and 2 τ a is the proper length of his lifetime. This is identical with the diamond temperature of Martinetti and Rovelli [3]. For we have explicitly determined the quantum state in the diamond region, we would be able to verify this peculiar behavior of the proper temperature along the observer with a finite lifetime by constructing a concrete model of the particle detector. Note that this result does not immediately imply that a real thermometer with a finite lifetime in the Minkowski space-time indicates a nonzero temperature. In principle, if a thermometer is prepared such that it interacts only with the field modes in the diamond region, it would indicate a finite temperature. However, such a preparation would be extremely difficult, that is an arbitrarily prepared thermometer would in general be inevitably coupled with field modes outside the diamond region. Thus, we come to the same conclusion with different independent arguments, which is the evidence that the diamond temperature has the universal significance. We also note that the tunneling approach recently proposed by Banerjee and Majhi [10, 11], which is another independent method to obtain the temperature of the subsystem, also gives the same temperature, though we don't state details here. To promote a better understanding of the thermodynamics of an observer in O , we try to find the expression for the energy and the entropy in the diamond region. Firstly, we compute the expectation value of the stress-energy operator T O µν for the observer in O with respect to the Poincar'e invariant vacuum. The operator T O µν is defined in O by where the colons denote normal ordering with respect to the vacuum | 0; O 〉 , and the metric g O µν in O is written as Noting that the Hamiltonian H O defining the present thermal state can be written as the spatial integral of T O TT , we formally obtain its expectation value as Here it should not be interpreted this divergent result as that the infinite energy has been confined in a bounded region. This quantity is merely the expectation value of the ideal Hamiltonian operator such that the vacuum state projected onto the diamond region becomes the thermal state with respect to it, and hence it does not imply the instability of the Minkowski space-time. The divergent factor δ (0) here is controlled by introducing the cutoff scale /lscript in the following manner. Let the scalar field φ be confined within the interval -/lscript < X < /lscript , and let T O µν ( /lscript ) be the corresponding stress-energy operator. Then, the frequency of the scalar field is discretized as ω n = nπ//lscript ( n = 1 , 2 , · · · ), and the energy is regarded as the limit where has been defined. On the other hand, the von Neumann entropy S O ( ρ O ) of the Gibbs state ρ O can be formally computed as where the trace is taken over the Fock space of the creation and annihilation operators ( b ±† ω , b ± ω ). This may be called as the entanglement entropy of the diamond region O . The divergent factor δ (0) here also arises in the limit where In general, the amount of the entropy to energy ratio S/E contained within a given finite region is believed to be bounded from above by the typical length scale R of the region as This is known as the Bekenstein bound [12]. In the present case, we find the relationship (in the sense that S O ( ρ O ; /lscript ) = 2 πLE O ( /lscript )(1 + O ( L//lscript )) holds,) among the entropy S O ( ρ O ), the length scale L and the energy E O . This shows that the present system saturates the Bekenstein bound.", "pages": [ 4, 5, 6, 7, 8, 9 ] }, { "title": "3. Final Remarks", "content": "Finally, let us try to speculate on another interpretation of the entropy S O ( ρ O ) in terms of the information theory. The trajectory of the modular flow is the curve: X = const . , which corresponds to the uniformly accelerated motion with the proper acceleration a = -L -1 sinh(2 X/L ). In other words, each congruence class of trajectories of the modular flow under the action of the proper Poincar'e group is represented by the pair of parameters ( L, X ). Each trajectory of the modular flow defines a nonnegative function of the modular parameter T on the trajectory, which integrates to unity: We interpret this as determining a certain probability density associated with the modular flow. For example, if an observer ( L, X ) following the modular flow regards the increase of the Minkowski time u + as a probabilistic process, so that u + jumps from -L to L once in his history at the modular time T , he could expect that this jump occurs with the probability density P ( L, X ; T ). Given the family of probability density functions P ( L, X ; T ), the parameter space ( L, X ) inherits the structure of the Riemannian manifold. The Riemannian metric on the parameter space is given by the Fisher information metric where y i = ( L, X ) denotes the coordinates on the parameter space. In the present case, the parameter space ( L, X ) turns out to be the Poincar'e half plane. In fact, the Fisher information metric has the form The distance in the parameter space determined by the Fisher information metric gives an invariant measure of the difference between a pair of probability density functions. Applying this to the distant pair: P = ( L, -/lscript ) and Q = ( L, /lscript ) in the diamond region O of the fixed size L , we get Thus, this amount of information discrepancy is proportional to the entanglement entropy. We can also compute explicitly the Shannon entropy S Sh ( L ) as which is a function of L . This quantity can be also related with the distance between the point P = ( L, X ) and P ' = ( L + δL, X ) as Thus, the Shannon entropy is relevant to the entropy correction [13, 14] associated with the variation of the size of the diamond region O . In this way, the Riemannian structure of the parameter space of a certain kind of the probability density functions might have to do with the entanglement entropy of the subsystem and its corrections. We hope this viewpoint provides some insight into the better understanding of the information-theoretic origin of the BekensteinHawking entropy of black holes.", "pages": [ 9, 10, 11 ] }, { "title": "References", "content": "Department of Physics, Gakushuin University, Tokyo 171-8588, Japan.", "pages": [ 11 ] } ]

[ { "title": "Gravitational Waves and Inspiraling Compact Binaries in Alternative Theories of Gravity", "content": "A dissertation presented to the Graduate School of Art and Sciences of Washington University in St. Louis in partial fulfillment of the requirements for the degree arXiv:1308.5240v1 [gr-qc] 23 Aug 2013 of Doctor of Philosophy Dissertation Examination Committee: Prof. Clifford M. Will (chair), Prof. James H. Buckley, Prof. Ram Cowsik, Prof. Francesc Ferrer, Prof. Henric Krawczynski, Prof. Sándor J. Kovács, Prof. Xiang Tang St. Louis, Missouri August 2013", "pages": [ 1 ] }, { "title": "List of Tables", "content": "162", "pages": [ 13 ] }, { "title": "Acknowledgements", "content": "Since I started my journey in the field of theoretical physics until today, there have been many people from several universities and institutes who have had undeniable effects on my career and have made all the work presented in this dissertation possible. I want to take this opportunity to acknowledge some of them. I would like to thank the Physics Department at University of Tehran and Washington University in St. Louis for their great undergraduate and graduate programs, specially for their advanced courses on gravitation and astrophysics. I would also like to thank McDonnell Center for the Space Sciences in Washington University for their strong support. I am very thankful to the Massachusetts Institute of Technology , Institute d'astrophysique de Paris , and the University of Florida for their hospitality during working on parts of this dissertation. This research was supported in part by the National Science Foundation , Grant Nos. PHY 09-65133 and 12-60995. First and foremost, I would like to thank my PhD advisor for critically reading this dissertation and for his extremely helpful comments. It was a great pleasure for me to have Clifford Will as my advisor during my doctoral studies since Spring 2009. Here I could -and probably should- acknowledge him for all those many unique and precious lessons he gave me, but I keep it short by saying that people like Cliff, make me have a strong faith in a brighter future for science and humanity. I look at him as a role model and I hope some day I could contribute to the scientific community just as he does. I know this won't be easy, but that is my goal. I also have to greatly thank Nicolas Yunes , who has been a collaborator on a part of this dissertation. His advices and comments were always constructive and to the point. I would also like to thank Frances Ferrer and K. G. Arun for helpful discussions and acknowledge Leo Stein for his help in streamlining our Mathematica code. My thanks and appreciations also go to Emanuele Berti and Michael Horbatsch for their essential argument on a part of this work. Alongside my advisor, who taught me a great deal of what I know in the field of gravitation, I would like to thank all of my teachers and professors. My first thanks go to my high-school physics teacher H. Doroodian for an excellent first impression he gave me into physics. I still keep the Halliday-Resnick book translated in Farsi which I got from him as an award in one of his classes. The same gratitude goes to all my teachers and professors in University of Tehran and Washington University in St. Louis, which I was very fortunate to be a student of them; I would like to thank Amir M. Abbassi , Mark Alford , Carl Bender , Claude Bernard , Anders Carlsson , Ram Cowsik , Jonathan Katz , Hamid R. Moshfegh , and Wai-Mo Suen , to name a few. As a graduate student in Washington University in St. Louis I had the chance to visit several other universities and research institutes and interact with many excellent professors and researchers in my field of research. I would like to take this opportunity to thank Scott Hughes for his hospitality and the great conversation that we had during my visit at MIT in Spring 2011. I need to thank Luc Blanchet at Institut d'astrophysique de Paris for his hospitality and helpful discussions. From the same institute I thank Roya Mohayaee , and Jacques Colin for making me feel at home during my stay in Paris in the summer of 2012. I am thankful for invitations of Reza Mansouri at IPM, Mohammad Nouri-Zonoz at University of Tehran, and Sohrab Rahvar at Sharif University of Technology that gave me the opportunity to present parts of this work and interact with the experts in their institutes. I also have to thank Steven Detweiler for his support and hospitality at the University of Florida. During my doctoral studies in Dr. Will's research group I have been glad to work with several postdocs and graduate students in our group including K. G. Arun , Adamantios Stavridis , Ryan Lang , Dimitris Manolidis , Laleh Sadeghian , Alexandre Le Tiec , and Pierre Fromholz . I would like to thank all of them for creating such a good working environment. I would also like to thank all the people whom I have learnt 'new' things from them during my PhD time period inside and outside of the working environment, including James Bendert , Benjamin Burch , Steven Dorsher , Lauren Edge , Daniel Flanagan , Daniel Hunter , Joben Lewis , Matthew Lightman , Faraz Monifi , Ryan Murphy , Danial Sabri , Sarah Thibadeau , Kaveh Vejdani , Kasey Wagoner , and Shannon Kian Zare , to name a few. A special thanks goes to Sina Mossahebi , Morvarid Karimi , Javad Komijani , Morteza Shahriari-Nia , Mehdi Saremi , and Moojan Daneshmand for their warm hospitalities. Last but not least, I am grateful to my family and friends for all the support and positive energy that they always offer. In particular, I would like to thank my wife, Laleh Sadeghian , my parents, Gholam-Abbas Mirshekari and Goli Abedi , and my siblings, Fatemeh , Masoumeh , and Soroush . I have been very lucky to have so many valuable friends wherever I have lived so far: Tehran, St. Louis, Paris, and Gainesville. It is impossible to list all the names here but I would like them to know that I won't forget their support, help, friendship, brotherhood, and love. to Laleh", "pages": [ 14, 15, 16 ] }, { "title": "Saeed Mirshekari", "content": "Doctor of Philosophy in Physics Washington University in St. Louis, 2013 Professor Clifford Will, Chair This dissertation consists of four parts. In Part I, we briefly review fundamental theories of gravity, performed experimental tests, and gravitational waves. The framework and the methods that we use in our calculations are discussed in Part II. This part includes reviewing the methods of the Parametrized Post-Newtonian (PPN) framework, Direct Integration of Relaxed Einstein Equations (DIRE), and Matched Filtering. In Part III, we calculate the explicit equations of motion for non-spinning compact objects (neutron stars or black holes) to 2.5 post-Newtonian order, or O ( v/c ) 5 beyond Newtonian gravity, in a general class of alternative theories to general relativity known as scalar-tensor theories. For the conservative part of the motion, we obtain the two-body Lagrangian and conserved energy and momentum through second post-Newtonian order. We find the contributions to gravitational radiation reaction to 1.5 post-Newtonian and 2.5 post-Newtonian orders, the former corresponding to the effects of dipole gravitational radiation. For binary black holes we show that the motion through 2.5 post-Newtonian order is observationally identical to that predicted by general relativity. 1 In Part IV, we construct a parametrized dispersion relation that can produce a range of predictions of alternative theories of gravity for violations of Lorentz invariance in gravitation, and investigate their impact on the propagation of gravitational waves. We show how such corrections map to the waveform observable by a gravitational-wave detector, and to the 'parametrized post-Einsteinian framework', proposed to model a range of deviations from General Relativity. Given a gravitational-wave detection, the lack of evidence for such corrections could then be used to place a constraint on Lorentz violation. 2", "pages": [ 18 ] }, { "title": "PART I Foundations", "content": "This part includes introductory materials to the rest of the dissertation. In Chapter 1 the fundamental theory of gravity from its early days up to date is reviewed briefly. The status of tests of theories of gravity specially general relativity is discussed in Chapter 2. Chapter 3 is an introduction to gravitational waves. 'The fact that we live at the bottom of a deep gravity well, on the surface of a gas covered planet going around a nuclear fireball 90 million miles away and think this to be normal is obviously some indication of how skewed our perspective tends to be.' -Douglas Adams 1", "pages": [ 20, 22 ] }, { "title": "1.1 From Newtonian Gravity to Einstein's General Relativity", "content": "It is not a long time in the history of humanity that we know where we are in the Universe. Since ancient thinkers until the development of the heliocentric model by Nicolaus Copernicus in the 16th century, the accepted view about the Universe was that the Earth is at the center and the Sun and other planets orbit around it 1 . This popular belief was based on the Ptolemaic geocentric system. The publication of Copernicus' book proposing a heliocentric system, just before his death in 1543, is considered a major event in the history of science. Tycho Brahe (15461601) performed the most accurate and comprehensive astronomical and planetary observations until his time. Brahe's observational data helped his young colleague, Johannes Kepler (15711630), to develop his laws of planetary motion. These works eventually led to the first wellstablished theory of gravitation by Isaac Newton in 1679. We refer the interested readers to [162] for an interesting detailed history of astronomical science before Newton. Newtonian gravity was the dominant theory of gravity in celestial mechanics for almost two centuries. The first observed deviation from Newtonian gravity in the solar system was recognized in 1859 in the motion of Mercury [246]. Analysis of the best available timed observations of transits of Mercury over the Sun's disk shows that the actual rate of the precession of Mercury's perihelion (point of closest approach to the Sun) disagrees with that predicted from Newton's theory by 43\" (arc seconds) per century. All attempts failed to explain this deviation by Newtonian gravity until Einstein's theory of gravity in 1916 [104]. The basic concepts of this theory are briefly summarized in the next section. Einstein showed that general relativity agrees closely with the observed amount of perihelion shift of Mercury. This was a powerful factor motivating the further tests of general relativity. Although general relativity has successfully passed all the performed tests (see Chapter 2), we are still interested to continue testing general relativity and studying alternative theories, for three reasons: (1) Gravity is a fundamental interaction of nature; deeper understanding of gravity leads to deeper understanding of the Universe. (2) All attempts to quantize gravity and to unify it with other types of interaction (i.e. electroweak and strong interactions) suggest that standard general relativity is not likely to be the last word. (3) Since general relativity contains no adjustable parameter, its predictions are fixed and therefore every test of the theory is either a potentially deadly test or a possible probe for new physics [269].", "pages": [ 22, 23 ] }, { "title": "1.2 General Relativity in a Nutshell", "content": "The way general relativity describes the cause of motion is quite different from the Newtonian explanation. In general relativity, there is no need to define gravitational forces, as Newton did, to describe the motion of massive objects in gravitational fields. In general relativity, the distribution of matter (massive particles) changes the geometry of spacetime such that massive objects just follow their optimum natural paths through the spacetime (geodesics). Paraphrasing John Wheeler (1911-2008), spacetime tells matter how to move and matter tells spacetime how to curve. To briefly review the basic concepts of the theory of general relativity, we start from the key concept of the invariant, differential line element ds at spacetime point x as where g µν is a 4 × 4 symmetric tensor ( metric tensor ), and repeated indices imply summation. Two examples are (1) the Minkowski metric in a Cartesian-coordinate system i.e. ( t, x, y, z ) as g µν = diag ( -1 , 1 , 1 , 1) which has fixed values for its components and describes the flat spacetime in the absence of matter (or at very far distances from the gravitational source where the gravitational field is negligible), and (2) the Schwarzschild metric in a spherical-coordinate system i.e. ( t, r, θ, φ ) which at a distance r from the source mass M is given by describing the curved spacetime around a static, spherically symmetric mass distribution of total mass M , where G is Newton's gravitational constant and c is the speed of light. The geodesic equation of motion for a test particle is given by where τ is the proper time measured by a clock traveling with the particle, and Γ µ αβ are the Christoffel symbols (also known as connection coefficients) defined by where the comma followed by a subscript denotes a partial derivative with respect to that coordinate, and from which we can define the Riemann curvature tensor as The Ricci tensor and Ricci scalar can be defined by contracting two of the indices of the Riemann tensor, and then contracting again, the Ricci tensor and scalar appears in the famous Einstein's field equations in general relativity: where T µν is the energy-momentum tensor for the matter. The Einstein-Hilbert action in general relativity is the action that yields the Einstein field equations, given by Eq. (1.7), through the principle of least action. It is given by The stress-energy tensor can be regarded as having the following qualitative form: Specifically, T 00 ( x ) is the local energy density, T 0 i ( x ) and T i 0 ( x ) are, respectively, the flux of energy and the density of momentum both in the direction of x i (note T 0 i = T i 0 ). T ij is the i th component of the force per unit area exerted across a surface with normal in direction x j . The diagonal elements T ii (no summation over i ) represent pressure components, and the off-diagonal elements represent shear stresses. For more details about general relativity, we refer the interested readers to many published textbooks on this topic including those written by Wald [254], Weinberg [258], Misner, Thorne, and Wheeler [181], Schutz [225], Hughston and Tod [147], Stephani [238], d'Inverno [94], Carroll [63], Kopeikin, Efroimsky, and Kaplan [163], and Poisson and Will [198].", "pages": [ 23, 24, 25 ] }, { "title": "1.3 Alternative Theories of Gravity", "content": "Alternative theories of gravity are interesting because although general relativity has successfully passed all the tests performed to date, but there are some issues in which general relativity is not quite promising, such as to quantize and unify gravity. An alternative theory might be the solution such that it is compatible with general relativity in certain limits and also can explain the ambiguous sectors like quantum gravity and unifying gravity with other forces. However, so far no alternative theory has been completely successful. The space of possible alternative theories is infinite but the most desirable theories of gravity are those which satisfy a certain number of properties including [295]: Since Einstein (1916) many various feasible and unfeasible alternative theories of gravity have been proposed to modify or replace general relativity. In this section we shall introduce two classes out of many: (1) scalar-tensor theories and (2) massive graviton theories. In this dissertation, we only focus on these specific classes (see Part III,IV). To have a review of alternative theories of gravity specially those are testable via gravitational wave observations we refer the interested reader to [13, 68, 198, 265, 295].", "pages": [ 25, 26 ] }, { "title": "1.3.1 Scalar-Tensor Theories", "content": "One of the cornerstones of every theory of gravity is its action. Although the Einstein frame [111, 118] gives the simplest presentation of the scalar-tensor theory, the metric used in this frame is not the same as the physical metric g µν that governs clocks and rods. Through a conformal transformationone can recast the theory into the Jordan frame in which clocks and rods measure the physical values of time and distance. The action in the Jordan frame is given by where the non-gravitational, matter action S NG involves the matter fields m and the metric only. Applying the principle of the least action to Eq. (1.10) leads to the following field equations If the coupling ω ( φ ) = ω BD is constant, then the general scalar-tensor theory in Eqs. (1.10) reduces to the massless Brans-Dicke theory [58] which is the simplest scalar-tensor theory that one could construct. For more details and more complicated versions of this theory we refer the interested reader to [84, 118, 265, 269]. Like general relativity, scalar-tensor theories are among metric theories of gravity and predict gravitational waves. But they predict an extra scalar (spin-0) mode of polarization in addition to the two transverse-traceless (spin-2) modes of general relativity. The emission of dipolar radiation in scalar-tensor theories is not predicted by general relativity. The form of the action in Eqs. (1.10) suggests that in the weak-field limit one may consider scalar-tensor theories as modifying Newton's gravitational constant via G → G ( φ ) = G/φ . Scalar-tensor theories have a continuous limit to Einstein's theory such that in the limit of ω →∞ one recovers general relativity. Because of this, scalar-tensor theories have passed all the performed precision tests. The massless Brans-Dicke theory agrees with all known experimental tests provided ω BD > 4 × 10 4 , given by measurements of the time delay in tracking signals to the Cassini spacecraft, while observations of the Nordtvedt effect with Lunar Laser Ranging and observations of the orbital period derivative of white-dwarf/neutron-star binaries yield looser constraints [37]. Massive Brans-Dicke theory has been recently constrained to ω BD > 4 × 10 4 and m s < 2 . 5 × 10 -20 eV, with m s the mass of the scalar field, through the observations of Shapiro time delay [5]. Scalar-tensor theories have not only passed the precision tests but also are very wellmotivated by fundamental physics. Specially, they can be derived from the low-energy limit of certain string theories. The integration of string quantum fluctuations leads to a higherdimensional string theoretical action that reduces locally to a field theory similar to a scalartensor one [116, 124]. In addition, scalar-tensor theories can be mapped to the general class of f ( R ) theories which have been proposed as a way to account for the acceleration of the universe without resorting to dark energy. (see [88, 233, 234] for a review of f ( R ) theories and their correspondence to scalar-tensor theories). Black holes and stars continue to exist in scalar-tensor theories. Stellar configurations are modified from their general relativistic profile [4, 144, 276], while black holes are not. Hawking [134] has shown that stationary black holes in Brans-Dicke theory are identical to those in general relativity. Many extensions of Hawking's theorem have been carried out since then, including [24, 26, 35, 35]. In particular, Sotiriou and Faraoni [235] have generalized Hawking's proof from pure Branse-Dicke theory to a general class of scalar-tensor theories. Recently, Hawking's result has been extended even further to quasi-stationary black holes. These extensions have been done in general scalar-tensor theories, through the study of post-Newtonian comparable-mass inspirals [178], extreme-mass ratio inspirals [293] and numerical simulations of comparable-mass black hole mergers [135]. Post-Newtonian calculations, accurate to ( v/c ) 5 order beyond Newtonian limit, predict no measurable difference between the equations of motion of binary black holes in general relativity and in general scalar-tensor theories of gravity [178].", "pages": [ 26, 27 ] }, { "title": "1.3.2 Massive Graviton Theories and Lorentz Violation", "content": "glyph[negationslash] Einstein's theory of general relativity predicts massless gauge bosons i.e. gravitons for gravitational propagation which travel with the speed of light. In the other hand, in massive graviton theories, the gravitational interaction is propagated by a massive gauge boson i.e. a graviton with mass m g = 0 . The corresponding Compton wavelength is λ g ≡ h/ ( m g c ) < ∞ . For a detailed review of massive graviton theories see e.g. [140]. Like scalar-tensor theories, massive graviton theories are somewhat well-motivated by fundamental physics, especially by theories of quantum gravity. In the cosmological extension of loop quantum gravity i.e. loop quantum cosmology [15, 54], the graviton dispersion relation predicts massive gravitons [55]. Massive graviton models also arise in some alternative theories inspired by string theory such as Dvali's compact, extra-dimensional theory [97]. Other modified theories that imply massive gravitons include Rosen bimetric theory [206, 207], Visser's theory [248], TeVez [27], and Bigravity [192]. Massive graviton theories have a theoretical issue, the van Dam-Veltman-Zakharov (vDVZ) discontinuity [245, 298]. They do not quite satisfy the precision tests. In particular, certain predictions of massive graviton theories do not reduce to those of general relativity in the m g → 0 limit. Roughly speaking, this discontinuity is due to the fact that, in this limit the scalar mode in spin states does not decouple [295]. The vDVZ discontinuity, however, can be evaded by carefully including non-linearities in massive graviton theories [31, 89, 105]. Although the absence of any particular well-accepted action for massive graviton theories makes it very difficult to ascertain many of the properties of these theories, we can still consider certain phenomenological effects [295]. The two main consequences of massive graviton theories are modifications to (1) the Newtonian limit, and (2) gravitational wave propagation. The first class of modifications corresponds to the replacement of the Newtonian potential by a Yukawa-type potential. In the non-radiative, near-zone of mass M , the Yukawa potential is given by V = ( M/r ) exp( -r/λ g ) , where r is the distance to the massive body [267]. The proposed tests of Yukawa interactions include the observations of bound clusters, tidal interactions between galaxies [128], and weak lensing [66]. These proposed tests are all model-dependent. The second class of modifications can be clearly seen in a modified gravitational wave dispersion relation [179, 267]. Explicit forms of modifications are given in Eqs. (13.1, 13.2). Either modification to the dispersion relation has the net effect of slowing gravitons down, such that for the same observable event the arrival times of photons and gravitons are different (see Fig. 12.1). We will discuss this issue in more detail in Chapter 12. Although it is extremely difficult (if not impossible) to measure the mass of a single graviton [98], many authors have tried to put an upper limit on the graviton's mass via different methods including the data analysis of binary pulsars and gravitational waves [80, 112, 161, 179, 267]. Table 12.1 shows a list of obtained upper limits on the mass of the graviton by a recent matched filtering analysis. Although massive graviton theories unavoidably lead to a modification to the graviton dispersion relation, the converse is not necessarily true. A modification of the dispersion relation is usually accompanied by a modification to either the Lorentz group or its action in real or momentum space [295]. Such Lorentz-violating effects are commonly found in quantum gravitational theories, including loop quantum gravity [55] and string theory [67, 239], as well as other effective models [29, 30]. In Doubly Special Relativity [6-8, 177], the graviton dispersion relation is modified at high energies by modifying the law of transformation of inertial observers. Modified graviton dispersion relations have also been shown to arise in generic extra-dimensional models [226], in Hořava-Lifshitz theory [53, 142, 143, 244] and in theories with non-commutative geometries [121-123]. None of these theories necessarily requires a massive graviton, but rather the modification to the dispersion relation is introduced due to Lorentz-violating effects.", "pages": [ 27, 28, 29 ] }, { "title": "1.4 Parametrized Post-Newtonian Theory as a Powerful Tool", "content": "In the 1970's, Nordtvedt and Will [185, 187, 265, 273] developed a general Parametrized PostNewtonian theory (PPN) of gravity in which general relativity and many viable alternative theories of gravity such as scalar-tensor theories can be described by choosing proper values for 10 independent parameters. The PPN parameters and their physical significance are shown in Table 1.1. The values of the PPN parameters differ for different theories (e.g. see Table 4.1). In general relativity, all the PPN parameters vanish except γ = β = 1 . In the next chapter we use the PPN framework as a powerful tool to study the tests of gravitational theories, and leave more details until Chapter 4 where we discuss the PPN framework. 'No amount of experimentation can ever prove me right; a single experiment can prove me wrong.' -Albert Einstein 2", "pages": [ 29, 30 ] }, { "title": "Tests of Gravitational Theories", "content": "This chapter is devoted to reviewing tests of gravitation theory. It is important to know that what kind of experiments and observations have been done so far primarily in the weak-field slow-motion regime, the regime covered by the PPN framework. Our results in Part III and Part IV are among the next steps toward providing new tools and abilities to test alternative theories of gravity, using future observations by gravitational-wave detectors. For a review of possible tests of gravitational theories with gravitational-wave detectors see [13, 119, 180]. In this chapter we briefly review the classical tests and tests of the Strong Equivalence Principle. Then we discuss the gravitational-wave's properties that we can use to put alternative theories of gravity to the test. We finish up this chapter with a list of performed tests and a summary of all the obtained bounds on the PPN parameters via various tests.", "pages": [ 30 ] }, { "title": "2.1 The Classical and SEP Tests", "content": "In this section we focus on three ket tests of relativistic gravity, including: (1) the perihelion advance of Mercury, (2) the deflection of light, and (3) the time delay of light. Strong Equivalence Principle (SEP) tests make up another class of tests for gravitational theories, that we discuss in this section.", "pages": [ 30 ] }, { "title": "2.1.1.1 The Perihelion Advance of Mercury's Orbit", "content": "An anomalous rate of precession of the perihelion of Mercury's orbit had been a puzzle since 1859 [246]. taking all the possible Newtonian effects into account, the observational results still showed a deviation as big as 43 '' per century in the perihelion shift of Mercury. This remaining precession can be explained accurately by Einstein's general relativity, and the predicted value agrees closely with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity. Based on recent measurements of the perihelion advance of Mercury's orbit and using the PPN formalism for fully conservative theories of gravity ( α 1 ≡ α 2 ≡ α 3 ≡ ζ 2 ≡ 0 ) it is possible to place a bound on the PPN parameters γ and β (Eq.7.55 in TEGP). The results agree with general relativity. Using 24 years of observing the perihelion shift of Mercury (1966-1990), Shapiro and his collaborators have estimated the following constrains on the PPN parameter combination [227, 229]: Analysis of data taken since 1990 could improve the accuracy.", "pages": [ 30, 31 ] }, { "title": "2.1.1.2 The Deflection of Light", "content": "Accurate measurements of the deflection of light near massive bodies like our Sun can test gravitational theories in the PPN formalism by bounding the value of the PPN parameter γ . A straightforward calculation in the PPN formalism, based on the equations of motion for photons i.e. Eqs. (6.14, 6.15) of [265], shows that the deflection angle of a light ray coming from a very distant source which is passing nearby a massive object on its way to our detectors on the Earth is given by where γ is the PPN parameter, m is the mass of the body which causes the deflection, d is the closest distance between the light ray and the mass m , and θ 0 indicates the angle between the undeflected ray and the direction to the source star (see Fig. 2.1). For the Sun, the deflection is maximum for a grazing ray i.e. for θ 0 glyph[similarequal] 0 , d glyph[similarequal] R glyph[circledot] glyph[similarequal] 6 . 96 × 10 5 km, m = m glyph[circledot] = 1 . 476 km. For light in the visible band, the effect is detectable from the Earth only at the time of total solar eclipses. In this case The light deflection phenomenon had been predicted as a Newtonian effect [64, 231] many years before Einstein's general relativity in 1915. The first observational test to measure this effect was performed by Arthur Eddington in 1919 [99]. The level of accuracy was not very high in the first experiment but clearly enough to reject the Newtonian prediction for the deflection angle which is half of what general relativity predicts. Figure 2.2 illustrates the prediction of these theories for the path of a light ray from a far star passing near the Sun. After Eddington, many other groups measured the deflection of light via different methods and techniques such as very-long-baseline radio-interferometric techniques (VLBI). A complete list of performed measurements of light deflection has been presented in Fig. 7.2 of TEGP [265]. A recent VLBI analysis [230] yieldes which is much more accurate that earlier measurements in the 1970's (see [241], for example).", "pages": [ 31, 32 ] }, { "title": "2.1.1.3 The Time-Delay of Light", "content": "The spacetime path of a light ray is affected by the gravitational field that it travels through, in two ways: (1) non-uniform gravitational fields cause the optimal path of the light rays to be curved, not straight (2) for a given distance, general relativity predicts a longer time travel for photon compared to what Newtonian gravity predicts. Here we concentrate on the second aspect i.e. the time delay of light. For a radar signal, we can measure the time travel of a round trip by sending it toward a far planet such that it passes close to the Sun. The additional time delay δt caused by the gravitational field of the Sun is a maximum when the reflector planet is on the far side of the Sun from the earth (superior conjunction); Fig. 2.3 shows this configuration. It is straightforward to show that [265] where R glyph[circledot] is the radius of the Sun, d is the closest distance between the radar beam and the Sun, r p is the distance between the Sun and the target planet, and a is an astronomical unit. Many different tests have been done so far to measure the time delay of light. With a high level of accuracy all of the tests confirm general relativity. A complete list of the performed radar time-delay experiments is presented in Fig.7.3 of TEGP [265]. Compared to earlier experiment in the 70's, such as Viking experiment [203], a significant improvement was reported in 2003 in measuring the parameter γ using Doppler tracking data for the Cassini spacecraft [38]. Most of the theories shown in Table.5.1 of TEGP can select their adjustable parameters or cosmological boundary conditions with sufficient freedom to meet this constraint. From the results of the Cassini experiment, we can conclude that the coefficient 1 2 (1 + γ ) must be within at most 0.0012 percent of unity. Scalar-tensor theories must have ω > 40 , 000 to be compatible with this constraint.", "pages": [ 33, 34 ] }, { "title": "2.1.2.1 Weak, Strong, and Einstein Equivalence Principles", "content": "Besides the classical tests of gravity, there is another class of solar-system experiments that tests the Strong Equivalence Principle (SEP). SEP contains the Einstein Equivalence principle (EEP) as a special case in which local gravitational forces are ignored. EEP is the cornerstone of all metric theories of gravity including general relativity, scalar-tensor gravity, etc. In metric theories of gravity, matter and non-gravitational fields respond only to the spacetime metric g µν . The only theories of gravity that have a hope of being viable are metric theories, or possibly theories that are metric apart from very weak or short-range non-metric couplings (such as string theory ). In all metric theories of gravity: Here we list all the conditions (sub-principles) that are required for a gravitational theory to satisfy EEP : Every metric theory of gravitation satisfies the conditions of EEP, yet does not necessarily satisfy SEP. SEP contains the same principles as EEP but with stronger conditions. SEP is satisfied if and only if the following conditions are satisfied:", "pages": [ 34, 35 ] }, { "title": "2.1.2.2 Nordtvedt Effect and Other SEP tests", "content": "It has been pointed out [265] that many metric theories of gravity (perhaps all except general relativity) can be expected to violate one or more aspects of SEP (for example see the following equations in TEGP: 6.33, 6.40, 6.75, 6.88). The breakdown in SEP has some observable consequences that many experiments have tested. The Lunar Eötvös experiment to test the Nordtvedt effect is one in which the breakdown in GWEP is the target. The Nordtvedt effect is a prediction of many gravitational theories in which the Earth and the Moon fall toward the Sun with different accelerations. Considering the inertial mass as m i and passive gravitational mass as m p we have m i a = m p ∇ U and from [265], we find that many theories predict where η N (Nordtvedt parameter) is a linear combination of PPN parameters as and E g is the gravitational self-energy of the body. Since for laboratory-sized objects the value of E g /m i is extremely small ( E g /m i ≤ 10 -27 ) the existence of the Nordtvedt effect does not violate the results of laboratory Eötvös experiments [107]. This is far below the sensitivity of current and future Eötvös-type experiments. On the other hand, for the Sun, Earth, and the Moon, E g /m i is respectively 3 . 6 × 10 -6 , 4 . 6 × 10 -10 , and 0 . 2 × 10 -10 . Measuring the Nordtvedt effect for the Earth-Moon-Sun system via Lunar Laser Ranging gives General relativity does not violate SEP and therefore there is no Nordtvedt effect in general relativity ( η N = 0 ), but this effect is certainly present in general scalar-tensor theories 1 such that η N = 1 / (1 + 2 ω ) + 4 ζλ 1 where λ 1 and ζ are defined in Eqs. (4.38, 4.37). In scalar-tensor theories of gravity, the internal structure of bodies clearly affects the dynamics of motion and therefore violates the SEP. Besides the Nordtvedt effect and Lunar Eötvös experiments there are many other SEP experiments that test preferred-frame effects, preferred-location effects, and constancy of the Newtonian gravitational constant. Preferred-frame and preferred-location effects can be tested via two type of experiments: (1) geophysical tests (2) orbital tests. Interested readers can see lots of details in sections 8.1-8.4 of TEGP [265]. These SEP experiments can measure the PPN parameters and therefore put additional bounds on some of them.", "pages": [ 35, 36 ] }, { "title": "2.2 Gravitational-Wave Tests", "content": "In the previous section we showed that a variety of tests of gravity in the solar system confirm general relativity. However the post-Newtonian limit of any other alternative metric theories of gravity, within a small margin of error (ranging from 1% to parts in 10 -7 ) must agree with that in general relativity. Most currently viable theories of gravity, such as scalar-tensor theories, can accommodate these constraints by choosing appropriate values for their arbitrary, intrinsic parameters and functions. Of course, no such adjustments are needed for general relativity. This fact makes general relativity the simplest and the most favorable one. In the other hand, because general relativity contains no adjustable parameter, any deviation from the fixed general relativistic predictions would kill the theory. In addition to the post-Newtonian tests that we discussed in Section 2.1, new testing grounds where the differences among competing theories may appear in observable ways are also possible. Measuring the properties of gravitational waves, observing binary pulsars, and cosmological tests are new arenas for testing theories of gravity besides the classical and SEP tests. In this section we focus on gravitational radiation as a tool for testing relativistic gravity. Although Einstein's theory of relativity had predicted the existence of gravitational waves as ripples of spacetime, Eddington [103] suggested that they might represent merely ripples of the coordinates of spacetime and as such would not be observable. Forty years later, Bondi and his collaborators [56] showed in invariant, coordinate-free terms that gravitational radiation is physically observable. They explicitly showed that gravitational waves carry energy and momentum away from systems, and that the mass of systems that radiate gravitational waves must decrease. The existence of gravitational radiation is not particularly strong evidence for or against any proposed theories of gravity, because almost all viable alternative metric theories of gravity predict gravitational waves as well as general relativity. Therefore it is not the existence of gravitational waves that will concern us here to test gravity but the detailed properties of these waves, including speed, polarization, and radiation back-reaction. In the weak-field, slow-motion, and far-zone limit, the predictions of various viable metric theories of gravity might be different from each other and from the predictions of general relativity at least in three important ways. They may predict: (1) different values for the speed of radiated gravitational waves which might not be necessarily equal to the speed of light, (2) different polarization states for generic gravitational waves, and (3) different multi-polarities (monopole, dipole, quadrupole, etc.) of gravitational radiation. Although the detection of gravitational waves is required for tests of speed and polarization, the tests of multi-polarities do not necessarily require direct gravitational-wave detection. The multi-polarities of gravitational waves can be studied by analyzing the back influence of the emission of radiation on the source (radiation reaction) for different multipoles. For instance, the emission of gravitational radiation changes the period of a two-body orbit, such as a binary pulsar. This is because the system loses energy via radiation of gravitational waves.", "pages": [ 36, 37 ] }, { "title": "2.2.1 Speed of Gravitational Waves", "content": "General relativity and scalar-tensor theories of gravity both predict that gravitational waves propagate along null geodesics with a speed equal to the speed of light, v g = c (in the limit in which the wavelength of gravitational waves is small compared to radius of curvature of the background spacetime). On the other hand, if gravitation propagates by a massive field (a massive graviton), the speed of gravitational waves could differ from c (see more details in Section 13.1). Vector-tensor theories [137, 272], Rosen's bimetric theories [206, 207], and Rastall's theory [202] predict different values for the speed of gravitational radiation depending on the parameters of the theory (see section 10.1 of [265] for details). The most obvious way to measure (or bound) the speed of gravitational waves is by comparing the arrival times of a gravitational-wave signal and of an electromagnetic-wave signal from the same event, for example a supernova. For an event at a distance D from our detector, the speed of gravitational radiation can be bounded by measuring the time interval between emission and arrival of an electromagnetic and gravitational signal from the same source. According to [269] where ∆ t ≡ ∆ t a -(1 + Z )∆ t e is the time difference , where ∆ t a and ∆ t e are the differences in arrival time and emission time of the two signals, respectively, and Z is the redshift of the source. The value of ∆ t e is considered to be unknown in many cases, so that the best one can do is to employ an upper bound on ∆ t e based on observation or modeling. If the frequency of the gravitational-waves is such that hf glyph[greatermuch] m g c 2 , where h is Planck's constant, then where λ g = h/ ( m g c ) is the graviton Compton wavelength, and the bound on v g at Eq. (2.9) can be converted to a bound on λ g as In the above analysis we have assumed that the source emits both gravitational and electromagnetic signals and we are able to detect them accurately enough. We have also assumed that the relative time of emission, ∆ t e , is either very small or measurable to sufficient accuracy. Instead of using both electromagnetic and gravitational signals from the same source, Will [267] proposed a method in which a bound on the graviton mass can be set by studying gravitational radiation alone. This has been shown specifically in the case of inspiralling compact binary systems. Roughly speaking, by using Will's method the phase interval f ∆ t in Eq. (2.11) can be measured to an accuracy 1 /ρ , where ρ is the signal-to-noise ratio. Thus, one can estimate the bounds on λ g achievable for various compact inspiral systems, and for various detectors. In part IV we will discuss this method and a generalized version of it in detail. Other possible gravitational-wave based methods include (1) using binary pulsar data to bound modifications of gravitational radiation damping by a massive graviton [112], and (2) using LISA-like observations of the phasing of waves from compact white-dwarf binaries, eccentric galactic binaries, and eccentric inspiral binaries [80, 157].", "pages": [ 37, 38 ] }, { "title": "2.2.2 Polarization of Gravitational Waves", "content": "In principle, a well-designed gravitational-wave antenna, for example AdLIGO, can measure the local components of a symmetric 3 × 3 tensor which is composed of the electric components of the Riemann curvature tensor, R 0 i 0 j , via the equation of geodesic deviation [101, 102]. If we show the spatial separation distance between two freely falling test masses by x i , based on general relativity, the equation of geodesic deviation is x i = -R 0 i 0 j x j . The symmetric R 0 i 0 j has six independent components, which can be expressed in terms of six modes of polarization. Figure 2.4 shows these six possible independent polarization modes. This figure indicates how a ring of freely falling test particles can be distorted due to each of these polarization modes. Three of these six generic polarization modes represent transverse waves and the other three y y represent longitudinal waves. Four of them (a), (b), (e), (f) are quadruple modes in different planes while there is one monopolar breathing mode (c) and one axially symmetric stretching mode in the propagation direction (d). In general relativity only two transverse quadrupole modes (a), (b) are present, independent of the source. Modes (a) and (b) correspond to the waveforms h + and h × , respectively. A suitable array of gravitational-wave antennas could describe or limit the number of polarization modes present in a given wave. Any observational evidence for other modes, besides (a) and (b), will be disastrous for general relativity. Massless scalar-tensor theories differ from general relativity by prediction of an extra polarization mode beside the general-relativistic polarization modes, namely a monopolar breathing mode (c). Notice that the absence of a breathing mode in future observational data would not necessarily rule out scalar-tensor gravity, because the strength of that mode depends on the nature of the source. In massive scalar-tensor theories the longitudinal stretching mode (d) is also possible, in addition to (a), (b), and (c), but it is suppressed relative to breathing mode (c) by a factor of ( λ/λ c ) 2 , where λ is wavelength of the radiation, and λ c is the Compton wavelength of the massive scalar. More general metric theories predict additional longitudinal modes, up to the full complement of six (see chapter 10 of [265] for details). Implementing polarization observations has been studied in detail [174, 249, 265]. One important question is whether the current and future interferometric gravitational-wave detectors (ground-based and space-based, or a combination of both types) could perform interesting polarization measurements [60, 125, 175, 250, 259]. The two LIGO observatories (in Washington and Louisiana states) have been constructed to have their arms as parallel as possible, apart from the curvature of Earth. Although this maximizes the joint sensitivity of the two detectors to gravitational-waves, unfortunately it minimizes their ability to detect the two modes of polarizations. Installing the INDIGO detector [284] in India will be a major help in this regard.", "pages": [ 38, 39, 40 ] }, { "title": "2.2.3 Gravitational Radiation Back-Reaction", "content": "In addition to measuring the speed and polarization of gravitational-waves, gravitational-wavebased tests of gravity are also possible via studying radiation reaction effects in compact binary sources. In the case of binary pulsars, the first derivative of the binary frequency ˙ f b is measured using radio signals from the orbiting pulsar to measure the orbit precisely, while in the case of inspiralling compact binaries, we are able to measure the full nonlinear variation of f b as a function of time via gravitational-wave signals. Broad-band laser interferometers are especially sensitive to the phase evolution of the gravitational waves. To extract gravitational-wave signals from noisy outputs of the detectors, we need to have an ensemble of theoretical template waveforms which depend on the intrinsic parameters of the inspiralling binary, such as the component masses, spins, and so on, and on its inspiral evolution. Data analysis involves some matched filtering of the noisy detector output against this ensemble of templates. For this purpose we need templates, accurate to an appropriate post-Newtonian order. The evolution of the gravitational-wave frequency f = 2 f b has been calculated up to the accuracy of 3.5PN order (see [47] for a review). To avoid lengthy expressions at higher orders, here we only show the expression until 2PN order, calculated by Blanchet and his collaborators", "pages": [ 40 ] }, { "title": "[48, 49, 274]:", "content": "where m , M , η are total mass, chirp mass, and mass-ratio parameters, respectively, given by Eqs. (6.22, 6.23). This rate of change in the frequency is related to the rate of orbital energy loss by Kepler's third law via In a generic metric theory of gravity the rate of energy loss from an inspiralling compact binary system can be parametrized to leading order in a post-Newtonian expansion, as [265]: where r is orbital separation, and v is relative velocity. S is the difference in the self-gravitational binding energy per unit mass between the two bodies. κ 1 and κ 2 are known as PM parameters , because of the pioneering work of Peters and Mathews [194], and their values depend on the theory (see Table 2.1). While κ 1 and κ 2 represent quadruple radiation, κ D represents dipole radiation. There is no dipole radiation in general relativity and therefore κ D = 0 , but scalartensor theories predict a dipolar contribution in the energy rate. In general relativity ( κ 1 = 12 , κ 2 = 11 ), the orbital frequency change induced by Eq. (2.14) corresponds to the leading term -the factor unity in the square brackets- in Eq. (2.12). Based on above discussion, there are three possibilities that can be suggested to use radiation reaction effects to test gravity: so-called tail-effects (the third term in Eq. (2.12)) could be possible by observing a source with a sufficiently strong signal [51, 52].", "pages": [ 41, 42 ] }, { "title": "2.3 Other Tests and Summarizing the Experimental Results", "content": "In addition to classical tests, tests of SEP, and gravitational-wave based tests, there remains a number of tests of post-Newtonian gravitational effects that do not fit into any of these mentioned categories. In some cases, the prior constrains on the parameters are tighter than the best limit these experiments could hope to achieve. Obviously, one might ask why we should bother performing any other test when we already have obtained stronger bounds on the PPN parameters? The answer is that in spite of previous tests, for the following reasons it is important to carry out such experiments: (1) each new test provides independent, though potentially weaker, checks of the values of the PPN parameters and therefore is an independent test of gravitation theory, (2) we should not treat the PPN formalism in a prejudicial way; it reduces the importance of experiments that have independent, compelling justifications for their performance, (3) any result which shows any disagreement with general relativity would be very interesting. Remaining tests of general relativity and alternative theories of gravity include: the Gravity Probe-B gyroscope experiment [109, 110, 173, 219, 221, 265, 271], laboratory tests of postNewtonian gravity [57, 126, 166, 183], tests of post-Newtonian conservation laws [263, 265], stellar system tests which include: internal structure dependance [4, 169, 265] and the binary pulsars [82, 165, 236, 265, 296], cosmological tests [69, 70, 87, 212, 265]. Table 2.2 summarizes the tightest bounds on the PPN parameters, obtained by different experiments. Notice that no feasible experiment or observation has ever been proposed that would set direct limits on the parameters ζ 1 or ζ 4 . However, these parameters do appear in combination with other PPN parameters in observable effects, for example in the Nordtvedt effect. A resource letter by Will [270] provides an introduction to some of the main current topics in experimental tests of general relativity as well as to some of the historical literature. 'It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong.' -Richard P. Feynman 3", "pages": [ 42, 43, 44 ] }, { "title": "Gravitational Waves: Sources and Detection", "content": "The existence of gravitational waves is one of the direct predictions of general relativity (and of almost all other alternative theories of gravity), produced by the acceleration of mass. No gravitational-wave signal has been detected directly to date. What are gravitational waves? What sources can generate these waves? How do they propagate and how can we detect them? These issues will be discussed in this chapter. In summary, gravitational-waves can be thought of as ripples in the curvature of spacetime. Why are we interested in their direct detection? First, that would be another verification of general relativity and it would be a major upset if gravitational waves did not exist! Second, and more importantly, the detection of gravitational waves will open a new window to the Universe, as a new branch of astronomy. Gravitational-wave astronomy will provide powerful tools for looking into the heart of some of the most violent events in the Universe in a way that is totally different from electromagnetic astronomy. It is believed that the reason for not detecting any gravitational-wave signal so far with the first generation of detectors such as initial-LIGO/VIRGO is the lack of strong-enough astronomical sources in the sensitive range of the detectors. In addition to the references cited in this chapter for specific topics on gravitational-waves, there exist many informative books and review articles including those written by Saulson [216], Maggiore [176], Creighton and Anderson [76], Jaranowski and Krolák [155], Hartle [133], Collins [72] Misner, Thorne, and Wheeler [181], Schutz [225], Sathyaprakash and Schutz (2009) [214], Freise and Strain (2010) [117], Pitkin et al. (2011) [195], and more recently by Blair et al. (2012) [41], and Riles (2013) [204].", "pages": [ 44 ] }, { "title": "3.1 Generation and Propagation", "content": "According to general relativity, the presence of any matter will curve the spacetime around it. The proper distance between two neighboring points is given by ds 2 = g µν dx µ dx ν where g µν is the metric tensor. In the absence of matter (or at very far distances from the matter), spacetime is flat (asymptotically flat) and the metric tensor is the Minkowski metric i.e. η µν = ( -1 , +1 , +1 , +1) in a Cartesian coordinate system. The origin of gravitational waves is implicit in the tensorial field equations of the theory (for general relativity and scalar-tensor theories of gravity see Eq. (1.7) and Eq. (1.11), respectively). To see why, consider a region far from a source, a nearly-flat region where the gravitationalwave perturbs a flat Cartesian metric η µν by only a small amount h µν , i.e. g µν = η µν + h µν where h µν glyph[lessmuch] 1 . Choosing an appropriate gauge condition, it can be shown that this linearized gravity yields simple wave equations for the components of tensor h such that in vacuum we have glyph[square] h µν = 0 . The amplitude of the wave h is related to the perturbation of the metric which is in turn related to the curvature of spacetime. In addition, h can be interpreted as a physical strain in space or more precisely h ∼ δL/L where δL is the change in separation of two masses a distance L apart. From elementary electrodynamics, we know that the acceleration of charged particles generates electromagnetic waves. In the same way, we expect accelerating gravitationally charged particles (masses) to generate waves. However, the existence of only one sign of mass (not two positive/negative types of charge as in electrodynamics) together with the conservation law of linear momentum implies that there is neither monopolar nor dipolar gravitational radiation. Gravitational radiation starts from quadrupole radiation and continues up to higher multipoles. In general relativity, gravitational waves propagate with the speed of light and there are two possible polarization modes: h + and h × . The effect of these polarization modes on a ring of test particles is shown in Fig. 3.1. From this, the principle of most gravitational-wave detectors looking for changes in the length of mechanical systems such as bars of aluminum or the arms of Michelson-Morley-type interferometric detectors - can be clearly seen. We will discuss more details about interferometric detection in Section 3.3.1. The magnitude of the components of a perturbing gravitational signal h ij produced at a distance r from a source at time t is proportional to the second time derivative of the quadrupole moment of the source (at earlier time t -r/c ) and inversely proportional to r [133], Notice that in the above formula the extremely small value of coefficient G/c 4 clearly shows why the gravitational-wave signals are very hard to detect. The energy luminosity of the source is proportional to the square of the third time derivative of the quadrupole moment [133] i.e. where 〈 〉 represents an average over several cycles; I ij is the moment of inertia defined as and I ij is the symmetric trace-free (STF) moment of inertia or quadrupole tensor : The energy flux of gravitational waves can be very large. For example, the energy flux of a sinusoidal, linearly polarized wave of amplitude h + and angular frequency ω is [133] which for a 100 -Hz sinusoidal wave of amplitude h + = 10 -21 , one obtains a flux of 1 . 6 mW.m -2 . A simple comparison shows that during a short time when the waves of a coalescing binary neutron-star system in Virgo cluster pass the Earth, the implicit energy flux is more than a millionth that from the Sun! As we will see, however, detecting the passage of this energy flux is a very difficult task. In a sense, spacetime is extremely stiff, in that the 'ripples' may be exceedingly small, yet can transmit considerable energy. Before moving on to likely sources of detectable gravitational waves, it is useful to make a comparison between gravitational and electromagnetic waves:", "pages": [ 45, 46, 47 ] }, { "title": "3.2 Sources of Gravitational Waves", "content": "Studying sources of gravitational waves by current and future detectors will uncover dark sectors of the Universe in extreme physical conditions including strong, non-linear gravity in relativistic motion and extremely high density, temperature and magnetic fields. Sources of gravitational waves are expected to emit in a wide range of frequency, from 10 -7 Hz in the case of ripples in the cosmological background to 10 3 Hz for the birth of neutron-stars in supernova explosions. Fig. 3.2 shows the signal strength at the Earth, integrated over appropriate time intervals, for a number of sources. This figure also illustrates the estimated frequency range for different types of sources. The ground- and space-based detectors are sensitive to high and low frequency ranges, respectively. In section 3.3 we will discuss different types of detectors together with their abilities and limitations. There are many sources of great astrophysical interest including the interaction and coalescences of black-holes and neutron-stars, low-mass X-ray binaries such as Sco-X1 1 , supernova explosions, rotating asymmetric neutron stars such as pulsars, and processes in the early Universe. In this dissertation we focus on the inspiralling compact binary sources, which are crucial for our work in parts III and IV, and refer the reader to recent reviews [81, 129, 214, 224] for further reading on other types of sources.", "pages": [ 47, 48 ] }, { "title": "3.2.1 Compact Binary Systems and Prospects for Detection", "content": "The coalescence of a compact binary produces short-lived and well defined signals of gravitationalwaves and therefore belongs to the most promising category of sources for detection. A compact binary system has two companions, which could be a neutron star (NS) or a black hole (BH), orbiting around the center of mass of the system. The system loses energy and angular momentum by emitting gravitational radiation. This leads to an inspiral of the two bodies toward each other and consequently an increase in rotational frequency of the system. The dynamics of every inspiralling compact binary have three phases which are illustrated in Fig. 3.3 including: emitted gravitational signal in this phase has a characteristic shape with slowly increasing amplitude and frequency. It is often called a chirp waveform. A chirping binary could be considered as an ideal standard candle in the sky [223]; we can measure the luminosity distance by observing gravitational radiation from a chirping binary. The post-Newtonian approximation is valid in this weak-field, slow-motion regime. Three types of compact binaries are possible: such as Advanced-LIGO which would imply an event rate between 0 . 1 and 500 yr -1 for NS-NS coalescences.", "pages": [ 48, 49, 50 ] }, { "title": "3.3 Detection and Data Analysis", "content": "There have been many attempts to detect gravitational-waves beginning with the pioneering work by Joseph Weber in the 1960's. He reported in 1970 coincident excitations of two resonantbar detectors in widely separate laboratories [256, 257]. However, subsequent experiments by other groups (either with the same level of accuracy or better) failed to confirm the reported detections [243]. The first gravitational-wave detectors were metal cylinders and the way that they were supposed to detect gravitational waves was quite simple. If the characteristic frequency of the incident wave is near the resonance frequency of the bar, the response to the wave is magnified and sudden changes in the amplitude of nominally thermal motion of the bar are expected. This effect is similar to an RLC antenna circuit's response to an electromagnetic-wave and we could measure it via piezoelectric transducers. In the late 1990's, before the first generation of gravitational-wave interferometers came online, there were five major bar detectors operating cooperatively in the International Gravitational Event Collaboration (IGEC) [17]. These bars achieved impressive strain amplitude spectral noise densities near 10 -21 / √ Hz, but only in narrow bands of ∼ 1 -30 Hz [16] near their resonant frequencies (ranging from ∼ 700 Hz to ∼ 900 Hz). Today, narrowband resonance bar detectors are almost completely phased out while the broadband interferometer detectors such as LIGO/VIRGO are leading the effort. Almost all of the current operating gravitational-wave detectors and all the proposed ones use interferometry techniques for detection. In the next section we briefly introduce the basics of interferometric detection.", "pages": [ 50 ] }, { "title": "3.3.1 Basics of Interferometer Detectors", "content": "Interferometric gravitational-wave detectors are very similar to the classic 1887 Michelson-Morley interferometer. A simple illustration is shown in Fig. 3.4. The apparatus is composed of two straight, equal-length arms in orthogonal directions. There is a beam splitter at the intersection of the arms which splits the coherent laser beam into two beams directed along each arm. There is a suspended massive mirror at the end of each arm which reflects the beams back to the beam splitter. The returning electromagnetic-wave signals will interfere constructively at the beam splitter, if the lengths of the arms are equal. Studying the interference pattern can show tiny changes in the lengths of the arms due to gravitational waves. The real apparatus is, of course, more sophisticated (see [216]). Gravitational-wave detectors are better thought of as antennae than as telescopes, because their sizes are small compared to the wavelengths they are meant to detect. For example, the LIGO detectors when searching at 4 kHz have L/λ of only about 0.05. This small ratio imply broad antenna lobes. Figure 3.5 shows the antenna lobes for + , × linear polarizations and unpolarized case vs. incident direction for a Michelson interferometer in the long-wavelength limit. As a result, a single interferometer observing a transient event has very poor directionality.", "pages": [ 51 ] }, { "title": "3.3.2 Interferometric Detection on the Earth", "content": "One can think of the ground-based gravitational-waves detectors as having three generations.", "pages": [ 51 ] }, { "title": "3.3.2.1 First Generation", "content": "Prototypes of gravitational-wave detectors since Weber (1960) led eventually to the building of major interferometric detectors on the Earth including: Since the first operation in 1999, LIGO has had three phases so far: Initial-, Enhanced-, and Advanced-LIGO during while significant improvements have been made. According to NSF (2008) LIGO is the largest single enterprise undertaken by NSF, with capital investments of nearly $300 million and operating costs of more than $30 million/year [286]. While not as sensitive as LIGO in the most sensitive band near 150 Hz, VIRGO is more sensitive at low frequencies (below 40 Hz), because of aggressive seismic isolation. This lower reach offers the potential to detect low-frequency spinning neutron-stars that are inaccessible to LIGO. VIRGO's sensitivity range of frequency is from 10 to 10,000 Hz. The VIRGO project is founded by CNRS and INFN on an annual / 10 million budget [106].", "pages": [ 52, 53 ] }, { "title": "3.3.2.2 Second Generation", "content": "The LIGO and VIRGO detectors are now undergoing major upgrades to become Advanced LIGO [132, 171] and Advanced VIRGO [2]. These upgrades are expected to improve their broadband strain sensitivities by an order of magnitude, thereby increasing their effective ranges by the same amount. Since the volume of accessible space grows as the cube of the range, one can expect the advanced detectors to probe roughly 1000 times more volume and therefore have expected transient event rates O(1000) times higher than for the 1st-generation detectors. In parallel, a primarily Japanese collaboration is proceeding to build an underground 3-km interferometer (KAGRA) [168] in a set of new tunnels in the Kamiokande mountain near the famous Super-Kamiokande neutrino detector. Placing the interferometer underground dramatically suppresses noise due to ambient seismic disturbances. In addition, INDIGO [284] -which is a planned LIGO-type observatory in India- has recently received initial approvals by the U.S.A. and Indian governments. The LIGO instrumentation that was initially scheduled to be installed at the 2-km interferometer at Hanford will be transported to India to add to the global network of gravitational-wave detectors, providing better source localization and better sensitivity to the polarization of gravitational-waves. Novel types of interferometers including AGIS [93] and TOBA [10] have been also proposed recently.", "pages": [ 53, 54 ] }, { "title": "3.3.2.3 Third Generation", "content": "With construction of second-generation interferometers well under way, the gravitational wave community has started looking ahead to third-generation underground detectors, for which KAGRA will provide a path finding demonstration. A European consortium is in the conceptual design stages of a 10-km cryogenic underground trio of triangular interferometers called Einstein Telescope [282], which would use a 500-W laser and aggressive squeezing, yielding a design sensitivity an order of magnitude better than the 2nd-generation advanced detectors now under construction. With such capability, the era of precision gravitational wave astronomy and cosmology would open. Large statistics for detections and immense reaches ( ∼ Gpc) would allow new distributional analyses and cosmological probes. LIGO scientists too are starting to consider a 3rd-generation cryogenic detector, with a possible location in the proposed DUSEL underground facility [96, 204]. The sensitivity curves of these detectors with different types of coalescence binary sources are shown in Fig. 3.6. Space-based detectors are needed to detect low-frequency gravitational-waves.", "pages": [ 54 ] }, { "title": "3.3.3 Space-Based Detectors", "content": "Some of the most interesting gravitational wave signals, such as those resulting from the formation and coalescence of black holes in the range 10 3 to 10 6 solar masses, will lie in the region of 10 -4 to 10 -1 Hz. To search for these requires a detector whose strain sensitivity is approximately 10 -23 over relevant timescales. It has been pointed out that the most promising way of looking for such signals is to fly a laser interferometer in space, i.e. to launch a number of drag free spacecraft into orbit and to compare the distances between test masses in these craft using laser interferometry. The sensitivity curve of LISA is shown in Fig. 3.6. An ambitious and long-studied proposed joint NASA-ESA project called LISA (Laser Interferometer Space Antenna) envisioned a triangular configuration (roughly equilateral with sides of 5 × 10 6 km) of three satellites (Fig. 3.7). As discussed above, there are many low-frequency gravitational wave sources expected to be detectable with LISA, and the proposed project has received very favorable review by a number of American and European scientific panels. Nonetheless, primarily for budgetary reasons, the project has been turned down by NASA (2012). Subsequently, NASA and ESA have solicited separate and significantly descoped new proposals. The funding prospects for these new proposals are quite uncertain, with ESA having recently passed over a descoped version of LISA called NGO (New Gravitational-wave Observer) in favor of a mission to Jupiter. Beside LISA-like missions, DECIGO [215] and BBO [77] are other existing possibilities for future spaced-based observatories that have been proposed recently.", "pages": [ 55 ] }, { "title": "3.3.4 Pulsar Timing Arrays", "content": "Detection of stochastic gravitational waves, potentially, can be done by performing precise pulsar timing via radio astronomy. This could be thought of as an entirely different method compared to the interferometry method in LIGO/VIRGO, for instance. Very-low-frequency (VLF) waves ( ∼ several nHz) in the vicinity of the Earth could lead to a quadrupolar pattern in the timing residuals from a large number of pulsars observed at different directions on the sky [92, 136, 217]. Three collaborations have formed in recent years to carry out the precise observations required: (1) The Parkes Pulsar Timing Array (PPTA-Australia) [181], (2) the European Pulsar Timing Array (EPTA-UK, France, Netherlands, Italy) [181], and (3) the North American NanoHertz Observatory for Gravitational Waves (NANOGrav USA and Canada) [156].", "pages": [ 55, 56 ] }, { "title": "3.3.5 Data Analysis", "content": "The most challenging task for gravitational-wave detectors is extracting the signal from noisy data. This issue is less challenging for LISA-like detectors where data is signal-dominated compared to the ground-based detectors such as LIGO which are noise-dominated. Different sources of noise are involved, including seismic noise, thermal noise, photoelectron shot noise. A number of data analysis methods have been derived, which provide useful tools to do this task. The goal of any data analysis method include detection of gravitational waves, inferring the nature of the source from the detailed properties of the wave signal, and testing general relativity. We will discuss this topic in more detail in Chapter 6, focusing on the Matched Filtering method.", "pages": [ 56 ] }, { "title": "PART II Methods", "content": "The framework and methods that will be used in the following parts are introduced in this part, including the methods of the Parametrized Post-Newtonian (PPN) framework, Direct Integration of Relaxed Einstein Equations (DIRE), and Matched Filtering. 'There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance.' -Hippocrates 4", "pages": [ 58, 60 ] }, { "title": "Parametrized Post-Newtonian Theory", "content": "To compare various theories of gravity and also to analyze the significance of various experiments to test the fundamental theory of gravity, two theoretical frameworks have been postulated: the Dicke framework and the Parametrized Post-Newtonian (PPN) framework. The Dicke framework, suggested by Robert Dicke, is particularly powerful for discussing null experiments, for delineating the qualitative nature of gravity, and for devising new covariant theories of gravity. The Dicke formalism has been discussed in more detail in [262]. The PPN framework starts where the Dicke framework leaves off: By analyzing a number of experiments within the Dicke framework one arrives at (among others) two fair-confidence conclusions about the nature of gravity. These are (i) that gravity is associated, at least in part, with a symmetric tensor field, the metric ; and (ii) that the response of matter and fields to gravity is described by ∇· T = 0 , where ∇· is the divergence with respect to the metric, and T is the stress-energy tensor for all matter and non-gravitational fields. These two conclusions in the Dicke framework become the postulates upon which the PPN framework is built. In this chapter, we briefly review the PPN formalism because we will need some part of it in our future calculations and also because it will help us to a better understanding of Part III of this dissertation. This formalism provides a framework which is extremely useful for discussing specific alternative metric theories of gravity including scalar-tensor theories and for analyzing the solar system tests of gravitational effects. We will refer to this chapter when we study the equations of motion for compact binary systems in alternative theories of gravity in Part III. This chapter is mostly based on Will's work in [262, 265]. The main advantage of working in a parametrized post-Newtonian framework is that, in principle, a wide range of metric theories of gravity can be accurately described in this framework only by tuning the values of the PPN parameters for each theory. The PPN formalism provides a useful framework in which comparing the theories and testing gravitational effects are easier to do with very few a priori assumptions about the nature of gravity. The PPN framework is a very practical tool to test alternative theories of gravity in solar system and beyond. Information given by future gravitational wave detection will also provide lots of useful data that can be applied to test alternative theories of gravity, although the PPN framework is less useful for those types of test.", "pages": [ 60, 61 ] }, { "title": "4.1 The Newtonian Limit", "content": "Classic Newtonian mechanics works well on solar system scales. The gravitational field is weak enough and characteristic velocities are such small compared to the speed of light that any general relativistic effect will be extremely small. These two conditions are called weak-field and slow-motion conditions, respectively. Nothing prevents using the Post-Newtonian theory even beyond solar-system scales as long as the weak-field and slow-motion conditions are satisfied. In the solar system, to an accuracy of better than part in 10 5 , light rays travel on straight lines at constant speed, and test bodies move according to where a is the acceleration of moving body, and U is the Newtonian gravitational potential produced by rest-mass density ρ according to Note that we have assumed c = G = 1 . Considering perfect fluids with no viscosity, the Eulerian equations of hydrodynamics are where v is the velocity of an element of the fluid, ρ is the rest-mass density of matter, p is the pressure. Considering a test body momentarily at rest in a static external gravitational field, the body's acceleration a k in a static ( t, x ) coordinate system reduces from ?? to We expect general relativity (or any other alternative theory of gravity) to be the same as Newtonian gravity very far away from the gravitational sources. In another words, we expect the metric in an appropriately chosen coordinate system to reduce to the flat Minkowski metric i.e. To keep everything self-consistent in the Newtonian limit the only choice for the metric components including gravity are to be Given the stress-energy tensor for perfect fluids as this is straightforward to show that the Eulerian equations of motions in Eq. (4.4) are equivalent to where we retain only terms of lowest order in v 2 ∼ U ∼ p/ρ . Beyond the Newtonian limit when we begin to take into account the accuracies greater than a part in 10 5 , we need a more accurate approximation to the spacetime metric that goes beyond or post Newtonian theory (and this is why we called this theory as post-Newtonian theory). For example, for Mercury's additional perihelion shift of ∼ 5 × 10 -7 radians per orbit, the accuracy of the Newtonian gravity is no longer enough, we have to consider the post-Newtonian limits of this problem as well.", "pages": [ 61, 62 ] }, { "title": "4.2 Post-Newtonian Bookkeeping", "content": "For future use, it is very helpful to first develop a bookkeeping system for keeping track of small quantities in our post-Newtonian calculations. Because in the post-Newtonian formalism we often do an expansion in terms of small quantity v/c , it would be useful to compare the order of magnitude of the other quantities with v/c . The Virial theorem in its general form i.e. 2 ×〈 Kinetic Energy 〉 t = 〈 potential energy 〉 t in the effective one-body problem immediately yields µv 2 ∼ µ/r which clearly means The matter making up the Sun and planets is under pressure p , but this pressure is generally smaller than the matter's gravitational energy density ρU i.e. For instance, in the Sun p/ρ ∼ 10 -5 and in the Earth p/ρ ∼ 10 -10 . Other than gravitational energy U , one can also think about other forms of energy such as compressional energy, radiation, and thermal energy. But they are also very small compared to ρ . Defining Π as the specific energy density (ratio of energy density to rest-mass density), Π is ∼ 10 -5 in the Sun and ∼ 10 -10 in the Earth. We can think of the order of magnitude of Π as We assign to these above mentioned small quantities a bookkeeping label that denotes their order of smallness : Later in this dissertation we will neglect the effect of non-gravitational energy density Π in our calculation but we keep it for now to be able to describe all the parameters in the complete PPN formalism. Based on Eq. (4.13), we can conclude that single powers of velocity v are O ( glyph[epsilon1] 1 / 2 ) , U 2 is O ( glyph[epsilon1] 2 ) , Uv is O ( glyph[epsilon1] 3 / 2 ) , and so on. Also since the time evolution of the solar system is governed by the motion of its constituents, we have and thus, Now, we are ready to analyze the post-Newtonian metric using this bookkeeping system. The action for the motion of a point particle in any metric theory of gravity can be written as The integrand in Eq. (4.16) can be considered as a Lagrangian L for a single particle in a metric gravitational field. In the Newtonian limit we can substitute the metric components from Eq. (4.7) to get It is straightforward to confirm that this Lagrangian yields the equations of motion by using the Euler-Lagrange equations. In other words, Newtonian physics can be recovered by using an approximation for the Lagrangian correct to O ( glyph[epsilon1] ) . Therefore L to O ( glyph[epsilon1] 2 ) must give postNewtonian physics. Since half-integer-order terms, such as O ( glyph[epsilon1] 1 / 2 ) and O ( glyph[epsilon1] 3 / 2 ) , contain an odd number of factors of velocity v or of time derivatives ∂/∂t , and these factors are not symmetric under the time reversal operator, half-integer-order terms must be representing energy dissipation or absorption by the system. But what happened to half-integer-order terms, O ( glyph[epsilon1] 1 / 2 ) or O ( glyph[epsilon1] 3 / 2 ) , in the Newtonian Lagrangian? Because of the conservation of rest mass, terms of O ( glyph[epsilon1] 1 / 2 ) don't appear and conservation of energy in the Newtonian limit prevents terms of O ( glyph[epsilon1] 3 / 2 ) . Beyond O ( glyph[epsilon1] 2 ) , different theories may treat things differently. General relativity predicts that the first oddorder terms appear at O ( glyph[epsilon1] 7 / 2 ) , which represents energy lost from the system by gravitational radiation. Terms of O ( glyph[epsilon1] 5 / 2 ) are prohibited by the conservation of post-Newtonian energy in general relativity. Going one step beyond the Newtonian limit i.e. to first post-Newtonian order (1PN), we have to express L to O ( glyph[epsilon1] 2 ) . To do so we have to know the various metric components to an appropriate order as shown in the following, Thus the first post-Newtonian limit of any metric theory of gravity requires a knowledge of For calculation in the second post-Newtonian limit (2PN) we need to know each metric component to an additional power of glyph[epsilon1] higher that what has been shown above for 1PN. Similarly, it can be verified that if one takes the perfect fluid stress-energy tensor which is given by and expand it through the following orders of accuracy: and combine it with the post-Newtonian metric, then the equations of motion T µν ; ν = 0 will yield consistent post-Eulerian equations of hydrodynamics.", "pages": [ 62, 63, 64, 65 ] }, { "title": "4.3 The Most General Post-Newtonian Metric", "content": "The most general post-Newtonian metric can be found by simply writing down metric terms composed of all possible post-Newtonian functions of matter variables, each multiplied by an arbitrary coefficient that may depend on the cosmological matching conditions and on other constants, and adding these terms to the Minkowski metric to obtain the physical metric. Unfortunately, there is an infinite number of such functionals, so that in order to obtain a formalism that is both useful and manageable, we must impose some restrictions on the possible terms to be considered, guided in part by a subjective notation of reasonableness and in part by evidence obtained from known gravitation theories. A list of the restrictions is given in section 4.1d of TEGP, specially: We now can construct a very general form for the post-Newtonian perfect-fluid metric in any metric theory of gravity, expressed in a local, quasi-Cartesian coordinate system moving with respect to the universe rest frame, and in a standard gauge as shown in Eq. (4.22). The only way that that the metric of any one theory can differ from that of any other theory is in the coefficients that multiply each term in the metric. By replacing each coefficient by an arbitrary parameter we obtain a super metric theory of gravity whose special cases (particular values of the parameters) are the post-Newtonian metrics of particular theories of gravity. This super metric is called the parametrized post-Newtonian (PPN) metric, and the parameters are called PPN parameters. The most mature version of the post-Newtonian metric in its most general form is given in [265] as where γ , β , ζ , α 1 , α 2 , α 3 , ζ 1 , ζ 2 , ζ 3 , ζ 4 are 10 PPN parameters and the post-Newtonian potentials are defined to be functions of matter properties as and the preferred-frame potentials are where all above potentials are functions of ( x , t ) while primed functions show the same functions evaluated at ( x ' , t ) . For example, ρ ' and v ' stand for ρ ( x ' , t ) and v ( x ' , t ) , respectively. Notice that w i in 4.24 indicates the coordinate velocity of the PPN coordinate system relative to the mean rest frame of the universe; v i is the coordinate velocity of matter i.e. dx i /dt ; ρ and p are the density and pressure of the matter both measured in a local freely falling frame momentarily comoving with the matter; Π represents internal energy per unit rest mass. It includes all non-rest mass and non gravitational energy, for instance thermal energy and energy of compression. In Eq. (4.22) we are in a nearly globally Lorentz coordinate system in which the coordinates are ( t, x 1 , x 2 , x 3 ) . All coordinate arbitrariness ( gauge freedom ) has been removed by specialization of the coordinates to the standard PPN gauge. For more details about applying Lorentz transformations to the coordinate system and also about the standard PPN gauge see section 4.2 and 4.3 of TEGP.", "pages": [ 65, 66, 67 ] }, { "title": "4.4 The PPN Parameters and Their Significance", "content": "As we explained in Section 4.3, the use of parameters to describes the post-Newtonian limit of metric theories of gravity is called the Parametrized Post-Newtonian (PPN) Formalism . A primitive version of such a formalism was devised and studied by Eddington (1922), Robertson [205], and Schiff [222]. In this formalism, which was developed for solar system tests of general relativity, the Sun is considered to be a non-rotating, spherical, massive object, and planets are modeled as test bodies moving on geodesics of the spacetime metric. The metric in this version of the formalism reads where M is the mass of the Sun, and β and γ are the only PPN parameters in this version. In standard PPN gauge, the parameter β measures the amount of nonlinearity of a theory in g 00 while the parameter γ represents the curvature of spacetime produced by the Sun at radius r . Schiff [220] generalized the metric in Eq. (4.25a) to incorporate rotation (Lense-Thirring effect), and Baierlein [21] developed a primitive perfect-fluid PPN metric. But the pioneering development of the full PPN formalism was initiated by Kenneth Nordtvedt, Jr. [185], who studied the post-Newtonian metric of a system of gravitating point masses. Will [260] generalized the formalism to incorporate matter described by a perfect-fluid. A unified version of the PPN formalism was then presented by Will and Nordtvedt [272] and summarized by Will in [262] (hereafter TTEG). The Whitehead term Φ W was added by Will [261]. Although linear combinations of PPN parameters have been used in Eq. (4.22), it can be seen quite easily that a given set of numerical coefficients for the post-Newtonian terms will yield a unique set of values for the parameters. The linear combinations were chosen in such a way that the parameters α 1 , α 2 , α 3 , ζ 1 , ζ 2 , ζ 3 , and ζ 4 will have special physical significance. Evaluating every PPN parameter in a theory of gravitation is equivalent to measuring some specific properties of the theory.", "pages": [ 67 ] }, { "title": "4.5 Post-Newtonian Limits of Alternative Metric Theories", "content": "The PPN formalism is sufficiently general that a wide range of theories of gravity can be described by this formalism with some specific values for the PPN parameters. The interested reader might refer to TEGP [265] which presents a cookbook for calculating the post-Newtonian limits of many metric theories of gravity. However, in this section we only focus on two major classes of gravitational theories i.e general relativity and scalar-tensor theories of gravity. We show the final post-Newtonian form of the metric tensor in terms of the constants and variables of each theory and read the PPN parameters from that. The field equations in general relativity are given by [see Section 1.2] Considering the stress-energy tensor of matter in the form of a perfect fluid and following the cookbook steps in TEGP, the final form of the metric in general relativity is Keeping all the calculations in the standard PPN gauge, the PPN parameters can be read off immediately Based on table 1.1 and the values of PPN parameters in general relativity one can confirm that this theory is a fully conservative theory of gravity ( α 3 = ζ i = 0 ) and predicts no preferred-frame effects ( α i = 0 ) as we expect. In general scalar-tensor theories of gravity, a dynamical scalar field φ is introduced in addition to the metric tensor g µν . The interaction between φ and g µν is governed by a coupling function ω ( φ ) . If ω = constant the scalar-tensor theory reduces to its specific form of Brans-Dicke theory [58]. The field equations in scalar-tensor theories are derived from the action where the matter action I NG is a function only of matter variables and g µν . It does not depend on the scalar field φ . We choose coordinates (local quasi-Cartesian) in which the metric is asymptotically flat and φ takes the asymptotic value φ 0 . Defining and following the TEGP method we obtain the post-Newtonian metric of general scalar-tensor gravity as where Notice that in going to geometrized units, we have set Comparing Eq. (4.39a) with Eq. (4.22), the PPN parameters in scalar-tensor gravity are [186, 188] Again, α 3 = ζ i = 0 and α i = 0 confirms that scalar-tensor theories are fully conservative theories with no preferred-frame effects. In the limit of ω →∞ , the PPN parameters γ and β reduce to their general relativistic values i.e. unity. Table 4.1 summerizes the PPN parameters of general PPN Parameter relativity and one of the most popular alternative class of theories i.e. general scalar-tensor theories including Brans-Dicke theory.", "pages": [ 68, 69, 70 ] }, { "title": "4.6 Equations of Motion in the PPN Formalism", "content": "We define a conserved density ρ ∗ by where u 0 is the time component of the fluid element's four velocity, and ρ is the locally measured mass density (see Section 5.3 for details). Using the general form of PPN in Eq. (4.22), up to the first post-Newtonian order we find The components of the stress-energy tensor are given to the required order by It is straightforward to calculate the Christoffel symbols from the PPN metric in Eq. (4.22). Having the Christoffel symbols and stress-energy tensor components up to appropriate order, one can substitute them into the equations of motion T µν ; ν = 0 and obtain the PPN equations of hydrodynamics as where where ψ , Φ PF , and Φ PF j are given in Eq. (4.22d) and Eq. (4.24). Note that X = Φ 1 +2Φ 4 -Φ 5 -Φ 6 . 'If we knew what it was we were doing, it would not be called research, would it?' -Albert Einstein 5", "pages": [ 70, 71, 72 ] }, { "title": "DIRE: Direct Integration of Relaxed Einstein Equations", "content": "Direct Integration of the Relaxed Einstein Equations (DIRE) is one of three well-developed approaches to compute analytic, approximate solutions of the nonlinear field equations in general relativity via post-Newtonian methods (the other two methods includes the Blanchet-DamourIyer (BDI) approach [42-46, 86] and the Effective Field Theory (EFT) approach [127]). The DIRE approach has been developed by Will and Pati [190, 191] built upon earlier work by Epstein, Wagoner, Will and Wiseman [108, 251, 274, 278-280]. In this chapter we introduce this approach and show, step by step, how it can be applied to solve the Einstein field equations and obtain the explicit general relativistic equations of motion for non-spinning compact binary systems, including black holes and neutron stars. Here we review what has been done in [190, 191] only up to the lowest post-Newtonian order because of two main reasons: First, showing more details of the calculations and technics that the authors in [190, 191] have used. Second, to provide a well-defined, reference framework in which we can compare our new results with, in the next part of this dissertation. In addition, having the structure of DIRE method in GR will avoid repeating many similar, lengthy steps in some future calculations in this dissertation. In the next part, we will generalize DIRE method from GR to a well-motivated, general class of alternative theories of gravity namely scalar-tensor theories.", "pages": [ 72 ] }, { "title": "5.1.1 The Relaxed Einstein Equations", "content": "The method of DIRE is based on a reformation of the field equations of general relativity into a form known as the relaxed Einstein equations . The main idea is to recast Einstein's field equations from their regular form, to their 'relaxed' form, We choose a particular coordinate system and stick with it hereafter in which This combined with the definition of h µν in Eq. (5.4) is called the De Donder gauge condition in the literature. We also can call this specific coordinate system as harmonic coordinates, simply because Eq. (5.3) requires all the four coordinates to satisfy the curved spacetime scalar wave equation i.e. glyph[square] g x µ = 0 . In Eq. (5.2) the box operator is the flat d'Alambertian, glyph[square] η = η µν ∂ µ ∂ ν , and h µν , referred to as gravitational field , defined as where Equation (5.2) is in the form of a flat spacetime wave equation and therefore its solution can be treated via well-known Green's functions . The equation is called 'relaxed' because it can be solved formally as a functional of source variables without specifying the motion of the source. Here we have to emphasize that h µν plays an important role in gravitational-wave calculations. The spatial components of h µν , evaluated far from the source, describe the gravitational waveform and are directly related to the signal which a gravitational-wave detector measures. The source term in Eq. (5.2), τ µν , is defined to be an effective stress-energy pseudotensor as the sum of a matter part ( T µν ) and a gravitational part ( Λ µν ): where T µν is the stress-energy tensor of matter and all possible non-gravitational fields. Assuming the matter source purely made of perfect fluid we have where p and ρ are the locally measured pressure and energy density, respectively, and u µ is the four-vector of velocity of an element of fluid. The gravitational piece of the effective stress-energy pseudotensor, Λ µν , is given by where t µν LL is the Landau-Lifshitz pseudotensor which is given by To derive the relaxed form of Einstein's equations in Eq. (5.2) from their regular form in Eq. (5.1) the following key identity is useful. This identity is valid in any coordinate system and for any spacetime metric: where G µν and t µν LL are the Einstein tensor and the Landau-Lifshitz pseudotensor, respectively, and The tensor H µανβ has the same symmetry properties as the Riemann tensor, and if we apply ∂ αβ operator to it we immediately obtain which together with identity Eq. (5.10) leads to Eq. (5.2). Before proceeding, we shall discuss some important points about this relaxed form of the field equations compared to its regular form. Up to this point we have not applied any approximation, neither weak-field nor slow-motion approximation. The relaxed Einstein equations in Eq. (5.2) in harmonic coordinates are as exact as the standard Einstein equations in Eq. (5.1). Eqs. ( note that although Eq. (5.2) takes the form of a simple wave equation in harmonic coordinates and doesn't look as difficult as Eq. (5.1), it is actually still very complicated to solve from many aspects. On the right-hand side of the relaxed equation, τ µν is a function of the field, h µν and the derivatives (see Eqs. (5.6, 5.8, 5.9)). In addition, there is a second derivative term, namely h µν ,αβ h αβ , which properly belongs on the left-hand side of the equation where the other second derivative terms in the d'Alembertian operator are. In another words, while we do know the formed Green function solutions for glyph[square] η h µν = η µν ∂ µ ∂ ν we do not for ( η µν -h µν ) ∂ µ ∂ ν , because we are solving for h µν and do not know it before solving the equation. This term causes a deviation from the flat null cones of the background Minkowski spacetime and therefore a modification in the propagation of the field. Fortunately, it has been shown that DIRE recovers the leading manifestations of this effect. Notice that in the regular form of Einstein's equations we have all the geometrical properties of spacetime on the left-hand side and all the matter distribution information (energy-momentum tensor) on the right-hand side. This symmetry does not hold in the relaxed Einstein's equations any more. Generally speaking, by converting to the relaxed form we have not decreased the level of complexity of the equations, we have only changed from a complicated form which we don't know any formalism to solve analytically, to another complicated form for which at least we do have a well-known mechanism for obtaining analytic, if approximate, solutions.. Second, as we mentioned, the right-hand side of Eq. (5.2) depends on h µν , and h µν is the same quantity for which we are trying to solve the equations. Comparing with the classic concept of wave equation, it means what is waving in the left-hand side of the wave equation also plays a role in the source term on the right-hand side of the field (wave) equations. This means that not only the localized matter source generates gravity but also gravity itself generates gravity which is basically everywhere. This is a consequence of non-linearity of the field equations in general relativity. Third, since Λ µν is at least quadratic in h , the relaxed field equations in Eq. (5.2) are very naturally amenable to a perturbative non-linear expansion. If we assume that h µν is suitably small everywhere, then iteration methods can be applied to solve these equations with some hope that the solutions might converge (possibly asymptotically) at the higher orders. Fourth, as an immediate consequence of the harmonic gauge condition, the right-hand side of the relaxed equations Eq. (5.2) is conserved in the sense that τ µν ν = 0 . This can be shown to be equivalent to the covariant equations of motion of matter: where comma and semicolon represent normal partial derivative and covariant derivative operators, respectively.", "pages": [ 72, 73, 74, 75 ] }, { "title": "5.1.2 Source, Near Zone and Radiation Zone", "content": "Consider two non-spinning compact objects, for example two black-holes, two neutron stars, or one black-hole and one neutron star, with masses m 1 and m 2 , orbiting around each other and radiating gravitational waves. We assume that the size S of these compact bodies is very small compared to the separation distance r between them ( S glyph[lessmuch] r ). According to an observer at the center of mass of the system, the companions are located at the positions x 1 and x 2 , and rotate about the common center of mass (denoted by the small red cross in Fig. 5.1) in orbits with the larger mass having the smaller orbit, x 1 , and smaller linear velocity, v 1 . Here we choose the center of mass to be at the origin of our coordinate system, i.e. x µ CM = ( t, 0 , 0 , 0) . For simplicity. Fig. 5.1 shows the orbits to be circular. We are mainly interested in solving the field equations for the field at a point close to the source objects, in order to compute the equations of motion of the system. The vector x shows the position of the field point relative to the origin. We define R to be the distance between the field point and the center of mass of the binary-system. Since we chose the origin to be at the center of mass, R is equal to | x | here. This situation is illustrated in Fig. 5.1. After this point we also assume slow-motion ( v glyph[lessmuch] 1 ) and weak-field ( u glyph[lessmuch] 1 ). We define three spacetime zones around the center of mass of the binary system: (1) The source zone , which includes any point in the world tube T = { x µ | R < S , -∞ < t < ∞} , where S is the radius of a sphere that contains all the matter. Any event that happens inside the source area at anytime belongs to this zone. (2) The near zone , which includes any point inside the world tube D = { x µ | R < R , -∞ < t < ∞} where R ∼ S /v ∼ λ/ 2 π ; λ and v are the wavelength of the radiated gravitational-wave, and the relative velocity of the source bodies, respectively. Note that the near zone includes the source zone. (3) The far zone (radiation zone), which includes all the spacetime outside the near zone, or equivalently F = { x µ | R > R , -∞ < t < ∞} . Fig. 5.2 shows these zones in the spacetime around a binary-system source. For most of the evolution, up to the point where the post-Newtonian approximation breaks down, Rglyph[greatermuch] S . After defining the near zone and far zone we are ready to go back and discuss the standard solutions of the relaxed Einstein equations in Eq.5.2 which are retarded, flat-spacetime Green functions in their integral form: This integral is taken over all spacetime. But the delta function in the integrand reduces the integral to one over the past null cone C emanating from the field point ( t, x ) . That is because the integrand is zero everywhere except when t ' = t - | x -x ' | . This is illustrated in Fig. 5.3. As long as the field point is inside the near zone we can approximately treat the gravitational fields as almost instantaneous functions of the source variables. We can also neglect the retarded solutions or treat them as a small perturbation of instantaneous solutions. However, in the far zone the fully retarded solutions should be evaluated. Anyhow, field point could be either in the near zone or in the far zone. Both of these situations are shown in Fig. 5.4. The intersection of the near-zone world tube D and the hypersurface of the past null cone C is the region denoted by N . We expect that the dominant contribution to the the integral will come from this region, because of the strong effect of the source in this area. We break the integration of Eq. (5.14) over the whole past null cone into two pieces: (1) Integration over the hypersurface N , where the points are close to the matter source and the most important effect comes from, (2) Over the rest of the past null cone i.e. C - N , where gravity alone contributes to the integral, so that We treat these two pieces of the integral a bit differently. Fig. 5.4 shows the situation in two different cases. In the left we see the case in which the field point is inside the near zone. This is relevant to the case that we want to calculate the equations of motion of the compact objects in a binary-system. The right panel of Fig. 5.4 shows the relevant case for evaluating the gravitational waveform and the energy flux in radiation-zone, when the field point is in the far-zone and very far away from the matter source. Depending on if the field point is in the near zone or in the far zone, and if the integral is taken over hypersurface N or hypersurface C - N , we have four possible situations: 1) near-zone field point, near-zone integration 2) near-zone field point, far-zone integration, 3) far-zone field point, near-zone integration, 4) far-zone field point, far-zone integration. All of them have been discussed in [274] in detail. To obtain the equations of motion we focus on near-zone field points . In this case, both x and x ' in Eq. 5.14 are within the near-zone, therefore | x -x ' |≤ 2 R . The value of τ µν varies on a time scale S /v ∼ R . Thereafter we can do a Taylor expansion in powers of the small quantity | x -x ' | . We obtain where M is shown in Fig. 5.5 which represents the intersection of the hypersurface t = constant and the near-zone word tube D . We do not expect the integral in Eq. (5.14) to depend upon the arbitrary boundary R . We integrate over the whole past null cone and the final answer of Eq. (5.14) must be independent of where the radial boundary between the near-zone and farzone is located. But, each piece of this integral either h µν N or h µν C-N individually depends upon R . The only argument that one can make to avoid any inconsistency is that all R -dependent terms must cancel between the inner and outer integrals. This cancellation of R -dependent terms has been shown explicitly in [190]. Thus, to determine the field h µν we don't care about R -dependent terms in h µν N and h µν C-N because they all together will finally cancel out anyway. So, we just keep R -independent terms in each expression, then add them up to obtain the overall h µν . It can be shown that for near-zone field points, the outer integral, i.e. h µν C-N , can be ignored until 3PN order. However, for far-zone field points the outer integrals begin to contribute at 2PN order. Will and Wiseman [274] have calculated the contribution of these terms to the gravitational waveform and energy flux up to 2PN order.", "pages": [ 75, 76, 77, 78, 79, 80 ] }, { "title": "5.1.3 Iteration of the Relaxed Einstein Equations", "content": "Figure 5.6 schematically shows the algorithm for solving the relaxed Einstein equations by iteration. Iteration is a useful tool here because the field itself h µν appears quadratically in the source of the field equation and is assumed to be small. The starting point is h µν 0 = 0 , then construct τ µν 0 ( h 0 ) and find h µν 1 . In another words, starting from N = 1 and knowing h µν 0 based on our knowledge about τ 00 up to Newtonian order (the only survived component of τ µν at this order), in principle we are able to solve the field equation in the next order: glyph[square] h µν 1 = -16 πτ µν ( h µν 0 ) . This gives us h µν 1 . We substitute this recent obtained solution, h µν 1 , to the next-order field equation to get h µν 2 . In principle, this iterative procedure can be continued until the order, needed to achieve a desired accuracy. To derive the equation of motion of the source from the field h µν N , first we have to construct the stress-energy tensor T µν N up to the proper order from the field and then solve T µν N ; µ = 0 at its N -th order. The field h µν N obtained from the N -th iteration is a functional of the matter variables. To compute the gravitational field as a function of spacetime one needs to solve the equations of motion T µν ; µ = 0 to the ( N -1) -th order to obtain the matter variables as functions of spacetime. We have given a rough picture of the iteration procedure required to obtain the equations of motion of the source and to determine the gravitational waveform and energy flux via the DIRE approach. In the remaining sections of this chapter we will present some of the details. We will rederive the equations of motion of non-spinning compact binaries in general relativity up to 1PN order via the DIRE approach (This has been fully done up to 2.5PN order plus the 3.5PN order contributions by Pati and Will [190, 191]). Where needed for future reference, we will quote the complete 2PN expressions. We will refer to them in the next part where we generalize the DIRE approach to calculate the equations of motion of non-spinning compact binaries in scalar-tensor theories of gravity up to 2.5PN order.", "pages": [ 80, 81 ] }, { "title": "5.2 Formal Structure of Near Zone Fields and Expansion to Higher PN Orders", "content": "We introduce a simplified notation for the components of the gravitational field h µν and stressenergy tensor T µν , to make the coming expressions a bit easier to work with: and where we show the leading order dependence on glyph[epsilon1] in the near zone. Recall that glyph[epsilon1] ∼ v 2 ∼ u ∼ ρ/p glyph[lessmuch] 1 . From the definition Eq. (5.4), one can invert the tensor g µν to find g αβ in terms of h µν . Expanding to the required order, we find, where glyph[epsilon1] helps us to keep track of different orders of magnitude for different terms. Note that in Eq. (5.19) we have shown the full metric required for the 2.5PN equations of motion i.e. g 00 to O ( glyph[epsilon1] 7 / 2 ) , g 0 i to O ( glyph[epsilon1] 3 ) , and g ij to O ( glyph[epsilon1] 5 / 2 ) . However, to obtain the equations of motion to 1.5PN order, determining the components of the metric up to one order less than what is shown above for each component would be enough. From above equations in Eq. (5.19), also notice that in order to find the metric g αβ to the desired order. For the 1PN equations of motion we must obtain N and B to O ( glyph[epsilon1] 7 / 2 ) , K i to O ( glyph[epsilon1] 5 / 2 ) , and B ij to O ( glyph[epsilon1] 3 / 2 ) . Note that we treat B ij and its trace B differently simply because B appears in g 00 linearly. The next variable that must be evaluated to solve the relaxed Einstein equations is τ µν which is made of two pieces: T µν and Λ µν (see Eq. (5.6)). We leave the components of T µν in the form introduced in Eq. (5.18) until the final steps of the calculation. To evaluate the components of Λ µν in terms of the field components required for calculating equations of motion up to 2.5PN order, we use Eqs. (5.8,5.9) and obtain: As long as the field point is in the near-zone, we can use the Taylor expansion of the gravitational field introduced in Eq. (5.16) and write the components of h µν as integrals over the time-constant region M (see Fig. 5.5) and their time derivatives. The near-zone expansions of the field components i.e. N , K i , and B ij are then given by where we have define the moments of the system by The index Q is a multi-index, such that x Q denotes x i 1 . . . x i q . The boundary terms N ∂ M , K i ∂ M and B ij ∂ M can be found in Appendix C of [190], but they will play no role in our analysis because they contribute at higher PN orders than we care about. Looking at 5.21, all integrals are well-behaved such that all integrands are constructed from (1) a specific component of the stress-energy pseudo-tensor τ µν , (2) either a power of | x -x ' | (Poisson-like potentials and their generalizations) or a multiple combination of spatial coordinates i.e. x i (multipole moments), and (3) are integrated over a finite domain. Here we re-emphasize that all near-zone integrals are taken over time-constant region of M and we discard any possible R -dependent term because it must cancel with a corresponding term from the far-zone integral. In the near zone, the potentials are either Poisson-like potentials P (the most frequent kind of potential), super-potentials S , or super-duper-potentials SD . For a source f , they are given by the following definitions and satisfy the relevant Poisson equations, We also define potentials based on the densities σ , σ i and σ ij along with the super-potentials and super-duper-potensials Super-duper-potentials begin to show up at 2PN order (only Y at 2PN order and Y i and Y ij at higher orders) while super-potentials begin to contribute at 1PN order. However, Poisson potentials are everywhere; including Newtonian, 1PN, and 2PN terms. A number of potentials occur sufficiently frequently in the PN expansion that it is easier to redefine them specifically, just to make the calculations easier to follow. At Newtonian order there is the Newtonian potential, At 1PN order, frequent potentials are: In Eq. (5.21) we have the implicit integral form of the components of gravitational field h µν in terms of stress-energy pseudo-tensor components. Armed with Eq. (5.20) and starting from Eq. (5.6), we can evaluate the explicit form of the near-zone field components in terms of Poissonlike potentials (see Eq. (5.27), Eq. (5.28)) and multiple-moments (see Eq. (5.22)). To do that we need to evaluate the contribution at each order and be very careful about it. The leading order of magnitude of each field component is shown in Eq. (5.17) but here we need to keep track of the contribution in each PN order separately. So we use the following useful notation. Notice that in this chapter we will do the calculation up to 1.5PN order but here we show the expansion through 2.5PN order, one PN order beyond what we need for the 1.5PN equations of motion: where the subscript on each term indicates the relevant level of PN order in which that particular term leads. For example, N 0 is the leading Newtonian order of the field component N , while N 1 is its leading 1PN contribution, and so on. In other words, in 1PN calculations we do not expect any terms except those with the subscript of 1. Consequently, the subscript of the first term in each line shows the PN order in which the relevant field component begins to contribute. For instance, one can read from Eq. (5.29) that B and B ij show up at 1PN and 2PN for the first time, respectively. From Eq. (5.29) we expect a specific order of magnitude for each subscripted term in these relations, for example N 0 ∼ O ( glyph[epsilon1] ) and N 1 ∼ O ( glyph[epsilon1] 2 ) . In fact, one can check this after evaluating the explicit values of the terms later. Notice that our separate treatment of B and B ij leads to the slightly awkward notational circumstance that, for example, B ii 2 = B 1 . At this point we are ready to deal with the relaxed field equations Eq. (5.2) at the first level of iteration (Newtonian order). At lowest order in the PN expansion (shown as subscript 0 in Eq. (5.29)), we only need to evaluate τ 00 with h µν 0 = 0 , g µν = η µν , so that (see Eq. (5.18)) Other components of τ µν are of higher orders. As a result, at the Newtonian order the tensorial relaxed Einstein equations reduce to a single equation which, with the definition of the Newtonian potential U in Eq. (5.27), has the solution in nearzone This result reproduces Newtonian gravity and confirms the fact that general relativity contains Newtonian gravity at its lowest order when post-Newtonian theory is used. In the next step, using 8.2d, Eq. (5.6), Eq. (5.8), and Eq. (5.32) and keeping only the next generation of higher order terms compared to the first survived terms in the first generation i.e. 5.30, we have Substituting into Eqs. (5.21), and calculating terms through 1.5PN order (e.g. O ( glyph[epsilon1] 5 / 2 ) in N ), we obtain To rederive above equations one needs to use the identities introduced in appendix D of [190], specially the following identity: Using Eq. (5.34) in Eq. (5.19) to the appropriate order, the physical metric to 1.5PN order is obtained as and will be needed in deriving the equations of motion later on.", "pages": [ 81, 82, 83, 84, 85, 86 ] }, { "title": "5.3 Conversion to the Baryon Density ρ ∗ and Equations of Motion in Terms of Potentials", "content": "We treat the source bodies as pressure-free balls of baryons characterized by the 'conserved' baryon mass density ρ ∗ , given by where m is the rest mass per baryon and n is the baryon number density. From the conservation of baryon density, expressed in covariant terms by ( nu µ ) ; µ = 0 = (( √ -g ) -1 ( √ -gnu µ ) ,µ , we see that ρ ∗ obeys the non-covariant, but exact, continuity equation (see Fig. 5.7) where j = ρ ∗ v , v i = u i /u 0 , and spatial gradients and dot products use the Cartesian metric. In terms of ρ ∗ , the stress-energy tensor is given by where v µ = (1 , v i ) . We define the baryon rest mass as such that indicates the baryonic center-of-mass. Therefore, the velocity and acceleration of each body are defined by Using the equations of motion, T µν ; µ = 0 for each fluid element it is not difficult to show that where Γ γ µν are the components of the Christoffel symbols computed from the metric via Our task therefore, is to determine the Christoffel symbols through a PN order sufficient for equations of motion valid through 1.5PN order using the 1.5PN accurate expressions of the metric in Eq. (8.21) (different components of Γ α µν are needed to different accuracy, depending on the number of factors of velocity which multiply them); re-express the Poisson potentials contained in the metric in terms of ρ ∗ , rather than in terms of the 'densities' σ , σ i and σ ij , substitute into Eq. (5.43), and integrate over the A -th body, keeping only terms that do not depend on the bodies' finite size. We must now convert all potentials from integrals over σ , σ i and σ ij to integrals over the conserved baryon density ρ ∗ , defined by Eq. (5.37). From Eqs. (5.18, 5.39), we find where u 0 = ( -g 00 -2 g 0 i v i -g ij v i v j ) -1 / 2 . Substituting the expansions for the metric, Eqs. (5.19), and for the field components Eqs. (5.29) from Eq. (5.32) and Eq. (5.34), we obtain, to the order required for the 1.5PN equations of motion, Substituting these formulae into the definitions for U σ and the other potentials defined in Eqs. (5.28), and iterating successively, we convert all such potentials into new potentials defined using ρ ∗ , plus PN corrections. For example, we find that where henceforth, U , V i , V i 2 , Φ 1 , Φ 2 , and Σ are defined in terms of ρ ∗ (see Appendix A of [190]). At this point everything depends on the conserved baryonic density ρ ∗ , and we are ready to calculate the acceleration of bodyA from Eq. (5.43) and Eq. (5.44) as To do the above integration, first we have to calculate the integrand, which is equal to ρ ∗ times a i . The Christoffel symbols are given in terms of the metric components and their derivatives in Eq. (5.45). Metric components are functions of the field components (see Eq. (5.19)), which we already derived as explicit functions of the potentials defined in Eq. (5.28), up to 1.5PN order in Eqs. (5.32, 5.34) (also see Eq. (5.29)). Applying all these and inserting the iterated forms of all potentials, we obtain the acceleration of a given element of matter through 1.5PN order in the general form of where a i 1 . 5 PN = 0 because the 1.5PN contributions to the metric are all functions of time, which do not survive the gradient used to calculate the Christoffel symbols.", "pages": [ 87, 88, 89 ] }, { "title": "5.4 Two-Body Equations of Motion", "content": "We must now integrate all potentials that appear in the equation of motion, as well as the equation of motion itself given in Eq. (5.51) over the bodies in the binary system. We treat each body as a non-rotating, spherically symmetric fluid ball (as seen in its momentary rest frame), whose characteristic size S is much smaller than the orbital separation ( S glyph[lessmuch] r ). We shall discard all terms in the resulting equations that are proportional to positive powers of S : these correspond to multipolar interactions and their relativistic corrections. We also discard all terms that are proportional to negative powers of S : these correspond to self-energy corrections of PN and higher order. We retain only terms that are proportional to S 0 . Such terms will generally depend only on the mass of each body, but it is conceivable that terms could arise that are proportional to S 0 , but that still depend on the internal structure of each body. It can be shown [191] that such terms cannot appear at 1PN order by a simple symmetry argument. At 2PN order, terms of this kind could appear in certain non-linear potentials, but in fact vanish identically by a subtler symmetry. At 3PN order, such S 0 structure-dependent terms definitely appear, but whether they survive in the final equations of motion is an open question at present. Our assumption that the bodies are non-rotating will imply simply that every element of fluid in the body has the same coordinate velocity, so that v i can be pulled outside any integral. This assumption can be easily modified in order to deal, for example, with rotating bodies. We also assume that each body is suitably spherical. By this we mean that, in a local inertial frame co-moving with the body and centered at its baryonic center of mass, the baryon density distribution is static and spherically symmetric in the coordinates of that frame. We shall evaluate the acceleration consistently for body-1; the corresponding equation for body-2 can be obtained by interchange. At the end, we shall find the centre-of-mass and relative equations of motion. The Newtonian acceleration is straightforward: where ρ ∗ and ρ ∗' are conserved densities at spatial points x and x ' . The denisty ρ ∗ and ρ ∗' vanish anywhere outside the bodies. The first term in the last line of Eq. (5.54) in which both integral points x and x ' are in the same body vanishes by symmetry, irrespective of any relativistic flattening or any other effect (Newton's third law). In the second term in which x is in body-1 and x ' is in body-2, we find that all contributions apart from the leading term are of positive powers in S , and thus are dropped. This is equivalent to fixing x at x 1 and x ' at x 2 . The integral result is as easy as with the second equation obtained from the first by the interchange 1 glyph[harpoonleftright] 2 . These are the wellknown Newtonian equations of motion for body-1 and body-2, re-derived via the post-Newtonian DIRE approach at its lowest order. The 1PN terms are similarly straightforward. A term such as v 2 U ,i is integrated over body-1 by setting v = v 1 and writing U = U 1 + U 2 . With v 2 pulled outside the integral, the integration is equivalent to that of the Newtonian term in Eq. (5.54), with the result v 2 U ,i →-m 2 v 2 1 n i /r 2 . Other 1PN terms involving quadratic powers of velocity ( v i ˙ U , v j V [ i,j ] , Φ ,i 1 and the velocitydependent parts of ˙ V i and X ,i ) are treated similarly. In the non-linear term UU ,i , the term involving U 1 U ,i 1 is of order S i / S 4 , where S i represents a vector, like ( x -x ' ) i that resides entirely within the body. In the two cross terms U 1 U ,i 2 and U 2 U ,i 1 , U 1 and U ,i 1 are of order 1 / S and S i / S 3 respectively. It can be shown (see [191] for details) that the only terms in the product that vary overall as S 0 will have odd numbers of vectors S i , whose integral over body-1 vanishes by spherical symmetry. Only the term from U 2 U ,i 2 contributes. The result is UU ,i →-m 2 2 n i /r 3 . In the terms ˙ V i and X ,i , the acceleration dv i /dt appears. Working to 1PN order, we must insert the Newtonian equation of motion; but working to 2PN order (or higher), we must insert the 1PN (or higher) equations of motion. For ˙ V i , the result using the Newtonian equation of motion is The double integral is integrated over body-1 similarly to the term UU ,i , and the velocitydependent term is integrated similarly to the term v 2 U ,i . The general result of these considerations is that, at 1PN order, only terms are kept in which, in the quantity x -x ' , the two vectors are evaluated at the baryonic center of mass of the two different bodies, respectively, and never within the same body. The resulting 1PN equation of motion is Note that as a natural consequence of the interchange 1 glyph[harpoonleftright] 2 , we have to also convert n i →-n i , because the vector n is a unit vector from body-2 toward the direction of body-1.", "pages": [ 90, 91, 92 ] }, { "title": "5.5 Relative Equations of Motion", "content": "In the previous section we derived the equation of motion up to 1PN order for each star of a compact binary system. In this section we convert the already obtained equations of motion in Section 5.4 to their equivalent equations in the center of mass frame. It is useful to note that the Newtonian equations, given in Eq. (5.55), admit a first integral that corresponds to uniform motion of a 'center of mass' quantity, namely where C i is a constant. Choosing the coordinates so that C i = 0 , we obtain the transformation from individual to relative velocities, to Newtonian order, These expressions can be used in 1PN terms in the equation of motion. Calculating a i 1 -a i 2 , using Eqs. (5.55, 5.57), and substituting Eqs. (5.59), we obtain the final relative equation of motion through 1PN order as where X ≡ x 1 -x 2 , 1 v ≡ v 1 -v 2 , r ≡| X | , n ≡ X /r , m ≡ m 1 + m 2 , η ≡ m 1 m 2 /m 2 , and ˙ r = dr/dt . The coefficients A and B are given by Equation (5.60) with Eq. (5.61) describes the relative motion of the companions in a compact binary system in general relativity with the accuracy of one order of magnitude in glyph[epsilon1] beyond the Newtonian limit, where the components are non-spinning, spherical, very small compared to the separation distance, slowly moving compared to the speed of light, and far away enough from each other such that the tidal gravitational field of each body at the other body can be neglected. We showed, in this chapter, how the DIRE method works to order 1PN in GR. To learn how DIRE is applied at 2PN order in general relativity see [190, 191] and at 2PN order in scalar-tensor theories of gravity see part III of this dissertation. In the following we quote the 2PN and 2.5PN coefficients in the relative equations of motion for an inspiralling compact binary system in general relativity [191]. The reader might compare Eqs. (5.61, 5.62) in general relativity with the corresponding expressions for scalar-tensor theories given by Eqs. (10.13). 'The scientist is not a person who gives the right answers, he's one who asks the right questions.' -Claude Lévi-Strauss 6", "pages": [ 92, 93, 94 ] }, { "title": "Parameter Estimation", "content": "We begin this chapter with a general discussion of data analysis methods in gravitational-wave astronomy. We then focus on the matched filtering technique and introduce the basics of this method. We end this chapter with an example to show how matched filtering method can be applied to do parameter estimation for a compact binary source of gravitational-waves. We will use these same methods in Part IV where we apply Fisher matrix analyses to bound the graviton mass and to constrain the deviation from Lorentz symmetry in quantum-mechanical inspired, Lorentz-violating theories of gravity.", "pages": [ 94 ] }, { "title": "6.1 Gravitational-Wave Data Analysis", "content": "As we discussed earlier in Chapter 3, the observation of gravitational waves requires a very precise data analysis strategy, which is different from conventional astronomical data analysis in many ways. There are several reasons why this is so. Sathyaprakash and Schutz [214] have listed some of them as: In this chapter we consider the problem of detection of gravitational-wave signals embedded in a background of noise of a detector, and the question of estimation of their parameters. This led data analysts to develop a useful set of tools to search for gravitational-wave signals. A very powerful method to detect a signal in noise that is optimal by several criteria consists of correlating the data with a template that is matched to the expected signal. This matchedfiltering technique is a special case of the maximum likelihood detection method. In this chapter we review the theoretical foundation of the method and we show how it can be applied to the case of a very general deterministic gravitational-wave signal buried in a stationary and Gaussian noise. Among all the potential candidates of gravitational-wave sources, inspiralling compact binaries are amongst the most promising. This is a result of the ability to model the phase and amplitude of the signals quite accurately and consequently to achieve maximum signal-to-noise ratio (SNR) by using matched filtering techniques. Even though gravitational-wave signals have not been detected yet, we can already investigate the performance of the detectors from a parameter estimation point of view. The relevant information is the distribution of the measured values (e.g., component masses, time of coalescence) and the error bounds on their variances. The Fisher information matrix is a convenient tool to obtain these error bounds. More details on the Fisher matrix analysis and the matched filtering technique will be given in Section 6.2. Indeed, in the cases that will interest us, the Fisher information matrix can easily be computed because inspiralling compact binaries can be modeled analytically. The covariance matrix was derived in [79, 114, 154] using Newtonian waveforms, extended to second post-Newtonian order (2PN) [167, 197], and revisited up to 3.5 PN order [11, 12]. The main advantage of the covariance matrix is that once analytical expressions are available, expected error bounds can be calculated quickly for any type of component masses. Moreover, the errors are expected to fall off as the inverse of the SNR. However, the analytical expressions are valid in the strong-signal approximation case only. Since the first detection of gravitational-wave signals is expected to be in a low-SNR regime (below 20), the Fisher information matrix may not be the best tool to estimate error bounds in practice. There are other methods for estimating errors bounds that are based on simulations, and they should be able to correctly estimate error bounds even at low SNRs. However, these methods are computationally much more intensive compared to the Fisher information matrix formalism. For instance, in [208], the authors use a Bayesian analysis framework (for binary neutron star signals) so as to estimate the signal's parameters and their errors. The posterior integration is carried out using Markov Chain Monte Carlo (MCMC) methods. In [19, 22], the authors compared the error bounds given by the Fisher information matrix with those from Monte Carlo simulations. They found that in the case of black-hole neutron-star binaries ( (1 . 4 , 10) M glyph[circledot] ), the covariance matrix underestimates the error bounds by a factor of 2 at a SNR of 10 (chirp mass errors). This discrepancy vanishes when the SNR is approximately 15 for a Newtonian waveform and 25 for a 1PN waveform. It was also stated that the inclusion of higher order terms would be computationally quite intensive [71]. A very important development was the work by Cutler et al. [78] where it was realized that for the case of coalescing binaries matched filtering was sensitive to very small post-Newtonian effects of the waveform. Thus these effects can be detected. This leads to a much better verification of Einstein's theory of relativity and provides a wealth of astrophysical information that would make a laser interferometric gravitational-wave detector a true astronomical observatory complementary to those utilizing the electromagnetic spectrum. Figure 6.1 shows a schematic outline of the way in which LIGO and Virgo searches can be broken down. As one moves from left to right on the diagram, waveforms increase in duration, while as one moves from top to bottom, a priori waveform definition decreases. Populating the upper left corner is the extreme of an inspiraling compact binary system of two neutron stars in the regime where corrections to Newtonian orbits can be calculated with great confidence. Populating the upper right corner are isolated, known, non-glitching spinning neutron stars with smooth rotational spindown and measured orientation parameters. Populating the lower left corner of the diagram are supernovae, rapid bursts of gravitational radiation for which phase evolution cannot be confidently predicted, and for which it is challenging to make even coarse spectral predictions. At the bottom right one finds a stochastic, cosmological background of radiation for which phase evolution is random, but with a spectrum stationary in time. Between these extremes can live sources on the left such as the merger phases of a BH-BH coalescence. On the right one finds, for example, an accreting neutron star in a low-mass X-ray binary system where fuctuations in the accretion process lead to unpredictable wandering phase. The matched filtering technique can be applied as long as the waveform is known (gray area in Fig. 6.1). Solving the field equations and obtaining the gravitational waveform as a known expression, one can use it as a template to do matched filtering and hence measure the properties of gravitational source. For instance, in Section 6.3 we show how we can use the matched filtering method to estimate the parameters of a compact binary system, such as the masses and spins of the companions. In Part IV we use the same method to bound the graviton's mass [179, 197] and to constrain the deviation from Lorentz symmetry [179] in alternative theories of gravity.", "pages": [ 94, 95, 96 ] }, { "title": "6.2 Matched Filtering: Theory", "content": "Various work by various authors, including Finn [113] and Cutler and Flanagan [79] have put the theory and measurement of gravitational-wave signals on a firm statistical foundation, rather similar to that underlying the theory of radar detection [138, 253]. Here in this section we introduce the theory of matched filtering and parameter estimation for our future purposes in Part IV. To extract the gravitational-wave signal h ( t ; θ ) from noisy detector data, we need to be armed with some standard mechanism. When a signal of the form h ( t ; θ ) has passed through the detectors (a network of detectors), this data analysis mechanism should allow us to determine the value of the source parameters θ and the measurement error ∆ θ = θ -˜ θ , where ˜ θ denotes the true value. It is useful to define p ( θ | s ) as the probability that the gravitational-wave signal is characterized by the parameters θ , where the detector output is s ( t ) and a signal h ( t ; θ ) -for any value of the parameters θ - is present. Finn in [113] has derived an expression for p ( θ | s ) . The detector output signal is composed of gravitational-wave signal h ( t ; θ ) and the stationary random (Gaussian) function of detector noise n ( t ) such that Note that being a stationary and Gaussian random process for the detector noise, n ( t ) , is a crucial assumption. Finn shows that where p (0) ( θ ) is the a priori probability that the signal is characterized by θ (this represents our prior information regarding the possible value of the parameters) and where the constant of proportionality is independent of θ . In a given measurement, characterized by the particular detector output s ( t ) , the true values of the source parameters can be estimated by maximizing the value of probability distribution function and locating the parameter θ at this maximized p ( θ | s ) which in this case θ = ˆ θ . This is the so-called maximum-likelihood estimator [253]. The inner product operator ( · | · ) is defined such that [79] The inner product in Eq. (6.3) is defined so that the probability for the noise n ( t ) to have a particular realization n 0 ( t ) is given by p ( n = n 0 ) ∝ exp[ -( n 0 | n 0 ) / 2] . The noise spectral density S n ( f ) in Eq. (6.3) is twice the Fourier transform of the autocorrelation function of the noise detector which is defined for f > 0 only, and C n ( τ ) is the autocorrelation function of the noise detector where 〈·〉 denotes a time average (It is assumed that the noise has zero mean). All of the statistical properties of the detector noise can be summarized by its autocorrelation function. Notice that in Eq. (6.3) ' ∗ ' denotes complex conjugation and ' ˜ ' shows the Fourier transformation e.g. We define ρ , the signal-to-noise ratio (SNR) associated with the measurement, to be the norm of the signal h ( t ; θ ) , evaluated at θ = ˆ θ , where p ( θ | s ) is maximum and therefore θ = ˆ θ is the estimated value of the source parameters. In the limit of large values of SNR, to which we henceforth specialize, p ( θ | s ) will be strongly peaked about this value. We now derive a simplified expression for p ( θ | s ) appropriate for this limiting case of high SNR values. First of all, we assume that p (0) ( θ ) is nearly uniform near θ = ˆ θ . This indicates that the prior information is practically irrelevant to the determination of the source parameters; we shall relax this assumption below. Then, denoting we have that ξ is minimum at θ = ˆ θ . It follows that this can be expanded as where ∆ θ a = θ a -ˆ θ a , comma represents partial derivative with respect to the parameter θ (for example ξ ,a = ∂ξ/∂θ a ), and summation over repeated indices is understood. We assume that ρ is sufficiently large that the higher-order terms can be neglected. Calculation yields and we again assume that ρ is large enough that the first term can be neglected (see Cutler and Flanagan [79] for details). Therefore, in the limit of high SNR values, Eq. (6.2) can be well approximated by a Gaussian form distribution as where evaluated at θ = ˆ θ , is the Fisher information matrix [138] that is the most crucial quantity that has to be evaluated in the matched filtering technique. From Eq. (6.11) it can be established that the variance-covariance matrix Σ ab is given by Here, 〈·〉 denotes an average over the probability distribution function Eq. (6.11), and Γ -1 represents the inverse of the Fisher matrix. We define the measurement error in the parameter θ a to be (no summation over repeated indices), and -based on the above defined σ a and σ b - the correlation coefficient between parameters θ a and θ b as by definition each c ab must lie in the range ( -1 , 1) . In the next section, we clarify how to use the method of matched filtering by giving an example. For a specific SNR value, with knowing ( a ) the anticipated noise spectral density of a gravitational-wave detector and ( b ) the waveform template accurate to the appropriate post-Newtonian order, in Section 6.3 we describe how one can use the Fisher matrix approach to calculate σ a and c ab .", "pages": [ 97, 98, 99, 100 ] }, { "title": "6.3 Matched Filtering to Parameter Estimation: An Example", "content": "In this section we apply the matched filtering Fisher matrix analysis to a specific example. This example has been studied by Poisson and Will [197] and we review it here. The techniques and methods that we show in this example are same as those that we will apply in Part IV of this dissertation. The detailed expression for the post-Newtonian waveform is complicated: the dependence on the various angles (position of the source in the sky, orientation of the detector, orientation of the polarization axes) is not simple, and the waves have several frequency components given by the harmonics of the orbital frequency (assuming that the orbit is circular [172, 193]). A Fourier domain waveform (the so-called as TaylorF2 template), which is the most often employed PN approximant, is given by where the amplitude A ∝ M 5 / 6 Q ( angles ) /r , ( r is the distance to the source, Q is a function of the various angles mentioned above) and the phase is Here we introduce all the variables in Eq. (6.17): where φ c and t c are (formally) the values of Φ and t at F = ∞ . Of course, the signal can not be allowed to reach arbitrarily high frequencies; it must be cut off at a frequency F = F i corresponding to the end of the inspiral. We put πMF i = ( M/r i ) 3 / 2 = 6 -3 / 2 ; r i = 6 M is the Schwarzschild radius of the innermost circular orbit for a test mass moving in the gravitational field of a mass M 1 . where χ i = S i /m i 2 ; S 1 , S 2 are the spin angular momentum of each companion, and ˆ L is the unit vector in the direction of total orbital angular momentum. Defining dimensionless, symmetric, mass-ratio parameter η as we can rewrite the last equation in Eq. (6.22) as M = η 3 / 5 M . The main purpose of this section is to estimate the anticipated accuracy with which the various parameters such as M , η, β , and σ can be determined during a gravitational-wave measurement. At this point we have to specify the anticipated noise spectral density of the detector. In this example we just follow Poisson and Will [197] and use the following analytic expression for LIGO-VIRO-type detectors. where S 0 is a normalization constant irrelevant for our purposes, and f 0 the frequency at which S n ( f ) is minimum; we set f 0 = 70 Hz, which is appropriate for advanced LIGO sensitivity [285]. To mimic seismic noise we assume that Eq. (6.24) is valid for f > 10Hz only, and that S n ( f ) = ∞ for f < 10Hz . Although Eq. (6.24) is not the most updated analytic expression, it is ideal for our purposes to show the application of the Fisher matrix analysis method. We will use the most updated version of the noise spectral density for different detectors in Part IV. We now substitute Eq. (6.16) into Eq. (6.7) and calculate the signal-to-noise ratio. We readily obtain where the integrals I ( q ) represent various moments of the noise spectral density: where x = f/f 0 and the minimum and maximum of x in this case is put equal to 1 / 7 (corresponding to f m in = 10 and f 0 = 70 for LIGO) and (6 3 / 2 πMf 0 ) -1 (corresponding to f max = f I SCO ), respectively. As the next step toward the computation of the Fisher matrix, we calculate the derivatives of ˜ h ( f ) with respect to the following seven parameters By taking derivatives of the Fourier domain waveform in Eqs. (6.17, 6.16) with respect to all parameters in Eq. (6.27) we obtain where v ≡ ( πMf ) 1 / 3 and the index numbers in the left hand sides of the above equations correspond to different components of θ in Eq. (6.27), respectively. Notice that ˜ h , 1 which corresponds to ln A is the only one among the above expressions which does not have an imaginary part 2 . In Eq. (6.28), we also have defined Finally, the components of Γ can be obtained by evaluating the inner products ( h ,a | h ,b ) using Eq. (6.3) as where different components of ˜ h ,a are given by Eqs. (6.28, 6.29, 6.30) and S n ( f ) in this specific example is given by Eq. (6.24). The Γ ab 's can all be expressed in terms of the parameters θ , the signal-to-noise ratio ρ , and the integrals I ( q ) . The resulting expressions are too numerous and lengthy to be displayed here. To double check this example and for our future use, we developed a computer code 3 . Starting with the same initial conditions, we re-produced exactly the results shown in tables II&III of [197]. The variance-covariance matrix Σ ab can now be obtained from Eqs. (6.13), and the measurement errors and correlation coefficients computed from Eqs. (6.14, 6.15). Before doing so, however, we must first state our assumptions regarding the prior information available on the source parameters. We assume the SNR value to be ρ = 10 everywhere and that the companions are spin-less so that β = σ = 0 . and", "pages": [ 100, 101, 102, 103 ] }, { "title": "Motion and Gravitational Radiation in Scalar-Tensor Gravity", "content": "This part is based on a published paper in Physical Review D. [178] in which we adapt the Newtonian method of DIRE to scalar-tensor theory, coupled with an approach pioneered by Eardley for incorporating the internal gravity of compact, self-gravitating bodies. Explicit equations of motion for non-spinning binary systems (including neutron stars and black holes) are derived to 2 . 5 post-Newtonian order or O ( v/c ) 5 beyond Newtonian gravity. 'A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it.' -Max Planck 7", "pages": [ 104, 106 ] }, { "title": "7.1 Introduction", "content": "The anticipated detection of gravitational waves by a network of ground-based laser-interferometric observatories promises a new way of 'listening' to the universe in the high-frequency band. A future space-borne interferometer would open the low-frequency band and pulsar timing arrays may soon begin exploring the nano-Hertz region of the gravitational-wave spectrum. In addition to providing a wealth of astrophysical information, these observations also hold the promise of providing tests of Einstein's theory of general relativity in the strong-field, dynamical regime. The 'inspiralling compact binary'- a binary system of neutron stars or black holes (or one of each) in the late stages of inspiral and coalescence - is a leading potential source for detection. Given the expected sensitivity of the ground-based interferometers, stellar-mass compact binaries could be detected out to hundreds of megaparsecs, while for a space interferometer, inspirals involving supermassive black holes could be heard to cosmological distances. In order to maximize the detection capability and the science return of these observatories, extremely accurate, theoretically generated 'templates' for the gravitational waveform emitted during the inspiral phase must be available. This means that correction terms in the equations of motion and gravitational-wave signal must be calculated to high orders in the post-Newtonian (PN) approximation to general relativity, which, roughly speaking, is an expansion in powers of v/c ∼ ( Gm/rc 2 ) 1 / 2 (for a review and references see [214]). Contributions to the waveform from the merger phase of the two objects and from the 'ringdown' phase of the final vibrating black hole also play an important role. The detected gravitational-wave signals can also be used to test Einstein's theory in the radiative regime, particularly for waves emitted by sources characterized by strong-field gravity, such as inspiraling compact binaries. One way to study the potential for this is to check the consistency of a hypothetical observed waveform with the predicted higher-order terms in the general relativistic PN sequence, which depend on very few parameters (only the two masses, for non-spinning, quasi-circular inspirals). Another is to examine the constraints that could be placed on specific alternative theories using gravitational-wave observations [14, 33, 34, 218, 232, 237, 266, 267, 275]. Most of these analyses have incorporated only the dominant effect that distinguishes the chosen theory from general relativity, such as dipole radiation or the wavelength-dependent propagation of a massive graviton (see, however [293]). Some authors have taken a different approach by proposing parametrized versions of the gravitational waveform model [179, 180, 294], inspired by the parametrized post-Newtonian (PPN) formalism used for solar-system experiments, and analysing the bounds that could be placed on those theory-dependent parameters by various gravitational-wave observations. Yet the authors of these frameworks were limited by the fact that for many alternative theories of gravity, only the leading terms in the waveform model have been derived. In addition, the existing parametrizations of the gravitational waveform make the implicit assumption that the gravitational wave signal during the inspiral depends only on the masses of the orbiting compact bodies (in the spinless case), and not on their internal structure. This is true in general relativity, which satisfies the Strong Equivalence Principle, but is known to be violated by almost every alternative theory that has ever been studied. In scalar-tensor theory, for example, the internal gravitational binding energy of neutron stars has a definite effect on the motion and gravitational-wave emission, and since the binding energy can amount to as much as 20 percent of the total mass-energy of the body, the effects can be significant. In order to determine the full nature of the gravitational-wave signal in an alternative theory of gravity, the strong internal gravity of each body must be accounted for somehow, even in a PN expansion. To make the situation even more interesting, binary black holes play a special role within the scalar-tensor class of alternative theories. Based on evidence from a 1972 theorem by Hawking [134], together with known results from first-post-Newtonian theory, it is likely that in a broad class of scalar-tensor theories, binary black hole motion and gravitational radiation emission are observationally indistinguishable from their GR counterparts . This conjecture will be discussed in more detail later in Chapter 11. Scalar-tensor gravity is the most popular and well-motivated class of alternative theories to general relativity. Apart from the long history of such theories, dating back more than 50 years to Jordan, Fierz, Brans and Dicke [59], scalar-tensor gravity has been postulated as a possible low-energy limit of string theory. In addition, a wide class of so-called f ( R ) theories, designed to provide an alternative explanation for the acceleration of the universe to the conventional darkenergy model, can be recast into the form of a scalar-tensor theory (for reviews, see [88, 118]). Measurements in the solar system and in binary pulsar systems already place strong constraints on key parameters of such theories, notably the coupling parameter ω 0 . Yet these tests probe only the lowest-order, first post-Newtonian limit of these theories, some aspects of their strong-field regimes (related to the strong internal gravity of the neutron stars in binary pulsars) and the lowest-order, dipolar aspects of gravitational radiation damping. These considerations have motivated us to develop the full equations of motion and gravitational waveform for compact bodies in a class of scalar-tensor theories to a high order in the PN sequence. It should be acknowledged that we do not expect any big surprises. Damour and EspositoFarèse [85] have shown on general grounds that the available constraints on the scalar-tensor coupling constant ω 0 derived from solar-system experiments imply that scalar-tensor differences from GR will be small to essentially all PN orders, except for certain regions of scalar-tensor theory space where non-linear effects inside neutron stars, called 'spontaneous scalarization', can occur. It is therefore unlikely that we will be able to point to a qualitatively new test of scalar-tensor gravity to be performed with gravitational waves. Nevertheless we expect to provide a complete and consistent waveform model to an order in the PN approximation comparable to the best models from GR. With this model it will be possible to carry out parameter estimation analyses for gravitational waves from binary inspiral, and to compare the bounds with those from earlier work that either confined attention to the leading dipole term, such as [33], or assumed extreme mass ratios, such as [293].", "pages": [ 106, 107, 108 ] }, { "title": "7.2 An Overview", "content": "We will use the approach known as Direct Integration of the Relaxed Einstein Equations (DIRE) that we described in Chapter 5. DIRE is based on a framework originally developed by Epstein and Wagoner [108, 251, 252], extended by Will, Wiseman and Pati [190, 191, 274, 278], and applied to numerous problems in post-Newtonian gravity [159, 160, 182, 255, 268, 299]. As we discussed earlier in Chapter 5, DIRE is a self-contained approach in which the Einstein equations are cast into their 'relaxed' form of a flat-spacetime wave equation together with a harmonic gauge condition, and are solved formally as a retarded integral over the past null cone of the field point. The 'inner', or near-zone part of this integral within a sphere of radius λ , a gravitational wavelength, is approximated in a slow-motion expansion using standard techniques; the 'outer' part, extending over the radiation zone, is evaluated using a null integration variable. DIRE is rather easily adapted to scalar-tensor theories, so that the same methods that have been worked out for GR can be applied here. It is possible that many other theories that generalize the standard action of general relativity in four spacetime dimensions by adding various fields could be cast in a similar form, permitting a systematic study of their predictions for compact binary inspiral beyond the lowest order in the PN approximation. Indeed another motivation for this work is to lay out a template for possible extensions to other theories of gravity, such as the Einstein-Aether theory [152] or TeVeS [27]. Specifically, the theories we address here are described by the action given by Eq. (1.10) that we recall here as where R is the Ricci scalar of the spacetime metric g αβ , φ is the scalar field, of which ω is a function. Throughout, we use the so-called 'metric' or 'Jordan' representation, in which the matter action S m involves the matter fields m and the metric only; φ does not couple directly to the matter (see [84] for example, for a representation of this class of theories in the so-called 'Einstein' representation). We exclude the possibility of a potential or mass for the scalar field. In order to incorporate the internal gravity of compact, self-gravitating bodies, we adopt an approach pioneered by Eardley [100], based in part on general arguments dating back to Robert Dicke, in which one treats the matter energy-momentum tensor as a sum of delta functions located at the position of each body, but assumes that the mass of each body is a function M A ( φ ) of the scalar field. This reflects the fact that the gravitational binding energy of the body is controlled by the value of the gravitational constant, which is directly related to the value of the background scalar field in which the body finds itself. Consequently, the matter action will have an effective dependence on φ , and as a result the field equations will depend on the 'sensitivity' of the mass of each body to variations in the scalar field, holding the total number of baryons fixed. The sensitivity of body A is defined by For neutron stars, the sensitivity depends on the mass and equation of state of the star and is typically of order 0 . 2 ; in the weak-field limit, s A is proportional to the Newtonian self-gravitational energy per unit mass of the body. From the theorem of Hawking, for stationary black holes, it is known that s BH = 1 / 2 . This part of the dissertation (Chapters 7-11) reports the results of a calculation of the explicit equations of motion for binary systems of non-spinning compact bodies, through 2 . 5 PN order, that is, to order ( v/c ) 5 beyond Newtonian theory. The post-Newtonian corrections at 1PN and 2PN orders are conservative; we obtain from them expressions for the conserved total energy and linear momentum, and obtain the 2-body Lagrangian from which they can be derived. There are also terms in the equations of motion at 1 . 5 PN and 2 . 5 PN orders. These are gravitational-radiation reaction terms. Terms at 1 . 5 PN order do not occur in general relativity (see Section 5.4), but in scalar-tensor theories with compact bodies, they are the result of the emission of dipole gravitational radiation. At 2 . 5 PN order, one finds the analogue of the general relativistic quadrupole radiation, together with PN correction effects related to monopole and dipole radiation. Not surprisingly the expressions for these quantities are complicated, much more so than their counterparts in general relativity. On the other hand, they depend on a relatively small number of parameters, related to the value of ω ( φ ) far from the system, where φ = φ 0 , along with its derivatives with respect to ϕ ≡ φ/φ 0 , and the sensitivities s 1 and s 2 of the two bodies, and their derivatives with respect to φ . The parameters and their definitions are shown in Table 7.1. At Newtonian order, the 'bare' gravitational coupling constant G is related to the asymptotic value of the scalar field, but for two-body systems of compact objects, the coupling is given by the combination Gα , where where ω 0 = ω ( φ 0 ) . At 1 PN order there are two body-dependent parameters, ¯ γ and ¯ β A , A = 1 , 2 (see Table 7.1 for definitions of the parameters). For non-compact objects, where s A glyph[lessmuch] 1 , ¯ γ = γ -1 and ¯ β A = β -1 , where γ and β are precisely the PPN parameters for scalar-tensor theory, as listed in 4.42. At 2 PN order, there are two additional parameters δ A and χ A . Most of the parameters in Table 7.1 can be related directly to parameters defined in [84, 85]. Here we will quote the bottom-line result: the two-body equation of motion, expressed in relative coordinates, X ≡ x 1 -x 2 , through 2 PN order. This equation is ready-to-use, for example in calculating time derivatives of radiative multipole moments in determining the gravitationalwave signal. The equation has the form where r ≡| X | , n ≡ X /r , m ≡ m 1 + m 2 , η ≡ m 1 m 2 /m 2 , v ≡ v 1 -v 2 , and ˙ r = dr/dt . We use units in which c = 1 but for reasons to be discussed later, we will not set G = 1 , in contrast to the notation used earlier in this dissertation. The leading term is Newtonian gravity. The next group of terms are the conservative terms, of integer PN order, while the final group are dissipative radiation-reaction terms, of half-odd-integer PN order. The coefficients A and B are given explicitly in Eqs. (10.13). Several things are worth noting about these equations (and indeed about all the two-body equations shown later in the next chapters). In the general relativistic limit ω 0 → ∞ , or ζ → 0 , the equations (including the 2 . 5 PN terms) reduce to those of general relativity, as", "pages": [ 108, 109, 110 ] }, { "title": "2nd post-Newtonian", "content": "determined by many authors [50, 83, 131, 148, 164, 191, 240]. Considering scalar-tensor theories, one might compare the values of coefficients A and B in Eq. (7.4) given in Eqs. (10.13) with their corresponding values in general relativity given in Eqs. (5.61, 5.62). At 1 PN order, the equations agree with the standard scalar-tensor equations, both for weakly self-gravitating bodies in the general class of theories [186] (shown within the PPN framework in Sec. 6.2 and 7.3 of [262]), and for arbitrarily compact bodies in pure Brans-Dicke theory (as displayed in Sec. 11.2 of [262]). Although a number of authors have obtained partial results in scalar-tensor theory at 2 PN order, notably the metric sufficient to study light deflection at 2 PN order [91, 289], and the generic structure of the 2 PN Lagrangian for N compact bodies [85], our explicit formulae for the 2 PN and 2 . 5 PN contributions to the two-compact-body equations of motion are new. The energy loss that results from the 1 . 5 PN and 2 . 5 PN terms in the equations of motion is in complete agreement with the energy flux calculated to the corresponding order by Damour and Esposito-Farèse [84]. The other interesting limit is that in which both bodies are black holes. Assuming that Hawking's result that s BH = 1 / 2 applies equally for binary black holes as for isolated black holes, we find that the parameters ¯ γ , ¯ β A , ¯ δ A and ¯ χ A all vanish, and α = 1 -ζ = (3 + 2 ω 0 ) / (4 + 2 ω 0 ) . In this case the equations reduce identically to those of general relativity through 2 . 5 PN order, with Gαm A replacing of Gm A for each body. In other words, if each mass is rescaled by (4 + 2 ω 0 ) / (3 + 2 ω 0 ) , the scalar-tensor equations of motion for binary black holes, including the 2.5PN terms, become identical to those in general relativity. Again this applies to all the equations of motion and related quantites (total energy, Lagrangian), whether for the individual bodies or for the relative motion. Since the masses of bodies in binary systems are measured purely via the Keplerian dynamics of the system, the rescaling is unmeasurable, and therefore, the dynamics of binary black holes in this class of theories is observationally indistinguishable from the dynamics in general relativity. Assuming, as we believe will be the case, that this is also true for the gravitational wave emission, the conclusion is that gravitational-wave observations of binary black hole systems will be unable to distinguish between these two theories. If only one member of the binary system is a black hole, then α = 1 -ζ , and ¯ γ = ¯ β A = 0 , so that even at 1 PN order, the equations of motion are identical to those of general relativity, after rescaling each mass. Only at 1 . 5 PN order and above do differences between the two theories occur for the mixed binary system, because of the non-vanishing of S -in the dipole radiation reaction term, and the non-vanishing of ¯ δ 1 (if body 1 is the neutron star) in the 2 PN terms. However, in this case all the deviations from general relativity depend on a single parameter Q , given by where s 1 is the sensitivity of the neutron star. In particular, all reference to the parameters λ 1 and λ 2 disappears, and the motion through 2 . 5 PN order is identical to that predicted by pure BransDicke theory. If this conclusion holds true for the gravitational-wave emission, then gravitationalwave observations of mixed black-hole neutron-star binaries will be unable to distinguish between Brans-Dicke theory and its generalizations. The only caveat is that, for a given neutron star, generalized scalar-tensor theories can predict very different values of its un-rescaled mass and its sensitivity from those predicted by pure Brans-Dicke. Now we turn to the detailed calculations.", "pages": [ 111, 112 ] }, { "title": "7.3.1 Field equations and equations of motion", "content": "We begin by recasting the field equations of scalar-tensor theory into a form that parallels as closely as possible the 'relaxed Einstein equations' used to develop post-Minkowskian and postNewtonian theory in general relativity. Referring back to Section 5.1.1 will be useful. The original field equations of scalar-tensor theory as derived from the action of Eq. (7.1) take the form of Eqs. (1.11) that we recall here as where T µν is the stress-energy tensor of matter and non-gravitational fields, G µν is the Einstein tensor constructed from the physical metric g µν , φ is the scalar field, ω ( φ ) is a coupling function, glyph[square] g denotes the scalar d'Alembertian with respect to the metric, and commas and semicolons denote ordinary and covariant derivatives, respectively. We work throughout in the metric or 'Jordan' representation of the theory, in contrast to the 'Einstein' representation used, for example in Section 1.3.1 and [84]. Normally, such as for a perfect-fluid source, the matter stress-energy tensor depends only on the matter field variables and the physical metric g µν , not on the scalar field, and accordingly the term ∂T/∂φ does not appear in the field equations. But in dealing with a system of selfgravitating bodies, we will adopt an approach pioneered by Eardley [100]. Because φ controls the local value of the gravitational constant in and near each body in this class of theories, the total mass of each body, including its self-gravitational binding energy, may depend on the scalar field. Thus, as long as each body can be regarded as being in stationary equilibrium during its motion, Eardley proposed letting each mass be a function of φ , namely M A ( φ ) . With this assumption, T µν takes the form where τ is proper time measured along the world line of body A and u µ A is its four-velocity. The indirect coupling of φ to matter via the binding energy is responsible for the term ∂T/∂φ in the field equations. From the Bianchi identity applied to Eq. (7.6a), the equation of motion is with the right-hand-side vanishing in the perfect-fluid case. From the compact body form of T µν in Eq. (7.7), it can then be shown that the equation of motion for each compact body takes the modified geodesic form or in terms of coordinate time and ordinary velocities v α , These equations of motion could also be derived directly from the effective matter action, S m = ∑ A ∫ A M A ( φ ) dτ . Equation (7.7) can equally well be taken to describe a pressureless perfect fluid (dust), simply by letting the mass of each particle be a constant, independent of φ .", "pages": [ 113, 114 ] }, { "title": "7.3.2 Relaxed field equations in scalar-tensor gravity", "content": "To recast Eq. (7.6a) into the form of a 'relaxed' Einstein equation, we recall the discussion of Section 5.1.1. Defining the quantities we show that the following is an identity, valid for any spacetime, where t µν LL is the Landau-Lifshitz pseudotensor [see Eqs. (5.9) for an explicit formula]. To incorporate scalar-tensor theory into this framework, we assume that, far from any isolated source, the metric takes its Minkowski form η µν , and that the scalar field φ tends to a constant value φ 0 . We define the rescaled scalar field ϕ ≡ φ/φ 0 . We next define the conformally transformed metric ˜ g µν by and the gravitational field ˜ h µν by the equation From Eq. (7.13) it can be shown that this is equivalent to We now impose the 'Lorentz' gauge condition which is equivalent to Substituting Eqs. (7.6a), (7.6b), (7.14) and (7.16) into (7.12), we can recast the field equation (7.6a) into the form where glyph[square] η is the flat spacetime d'Alembertian with respect to η µν , and where where where the notation [( -g ) t µν LL ](˜ g µν ) denotes that the Landau-Lifshitz piece should be calculated using only ˜ g , in other words, exactly as in general relativity, except using the conformal metric, rather than the physical metric. The scalar field equation can also be rewritten in terms of a flat-spacetime wave equation, of the form where In principle, Eqs. (7.11a) and (7.14) can be combined to give g µν in terms of ϕ and ˜ h µν , although in practice, we will express it as a PN expansion. The final result will be the relaxed field equations (7.18) - (7.22) expressed entirely in terms of ˜ h µν , ϕ , and the matter variables. The next task will be to solve these equations iteratively in a post-Newtonian expansion in the near-zone. Formally the solutions of these wave equations can be expressed using the standard retarded Green function, in the form where the integration is over the past flat spacetime null cone of the field point ( t, x ) . We will expand these integrals in the near-zone, and incorporate a slow-motion, weak-field expansion in terms of a small parameter glyph[epsilon1] ∼ v 2 ∼ m/r ; the strong-field internal gravity effects will be encoded in the functions M A ( φ ) . 'Science never solves a problem without creating ten more.' 8", "pages": [ 114, 115, 116, 118 ] }, { "title": "8.1 Formal Structure of The Near-Zone Fields", "content": "Following Eq. (5.17), we reintroduce a simplified notation for the field ˜ h µν and the scalar field ϕ : where we show the leading order dependence on glyph[epsilon1] in the near zone. To obtain the equations of motion to 2.5PN order, we need to determine the components of the physical metric and ϕ to the following orders: g 00 to O ( glyph[epsilon1] 7 / 2 ) , g 0 i to O ( glyph[epsilon1] 3 ) , g ij to O ( glyph[epsilon1] 5 / 2 ) , and ϕ to O ( glyph[epsilon1] 7 / 2 ) . From the Eqs. (7.11a, 7.14), one can invert to find g µν in terms of ˜ h µν and ϕ to the appropriate order in glyph[epsilon1] , as in PWI, Eq. (4.2). Expanding to the required order, we find (compare to Eq. (5.19)) In Eqs. Eqs. ((8.2) we do not distinguish between covariant and contravariant components of quantities such as K i or B ij , since their indices are assumed to be raised or lowered using the Minkowski metric, whose spatial components are δ ij . We now define a set of provisional 'densities' following the convention of Blanchet and Damour [45] (given in Eqs. (5.18)), but adding a separate density for the scalar field equation: The second contribution to σ s will be non-zero only in the case where our system consists of gravitationally bound bodies, whose internal structure could depend on the environmental value of ϕ . Because of the way we have formulated the relaxed scalar-tensor equations, the quantity Λ µν has exactly the same form as in Eqs. (5.20) to the 2PN order needed for our work. The additional scalar stress-energy pseudotensor is new and given by where ω ' 0 ≡ ( dω/dϕ ) 0 . The near-zone expansions of the fields N , K i , B ij and Ψ are then given by Eq. (5.21) and where the scalar moments I Q s and M s are defined by Again, the index Q is a multi-index, such that x Q denotes x i 1 . . . x i q . The integrals are taken over a constant time hypersurface M at time t out to a radius R , which represents the boundary between the near zone and the far zone. The structure of the expansions for N N , K i N and B ij N is identical to the structure in Chapter 5 because the source τ µν satisfies the conservation law τ µν ,ν = 0 , a consequence of the Lorentz gauge condition. However, no such explicit conservation law applies to τ s ; nevertheless, in a post-Newtonian expansion, we will be able to show, for example, that the term glyph[epsilon1] 3 / 2 ˙ M s actually vanishes to lowest PN order, and thus contributes only beginning at glyph[epsilon1] 5 / 2 order; the other terms involving time derivatives of M s will also be boosted to one higher PN order. The time derivatives of the dipole moments I j s do not vanish in general; this is related to the well-known phenomenon of dipole gravitational radiation that can occur in scalar-tensor theories. The boundary terms N ∂ M , K i ∂ M and B ij ∂ M can be found in Appendix C of PWI, but they will play no role in our analysis. As in Chapter 5, we will discard all terms that depend on the radius R of the near-zone; these necessarily cancel against terms that arise from integrating over the remainder of the past null cone; those 'outer' integrals can be shown to make no contribution to the near zone metric to the PN order at which we are working. In the near zone, the potentials are Poisson-like potentials and their generalizations. Most were defined in [190], but we will need to define additional potentials associated with the scalar field. For a source f , we use the definition of the Poisson potential P ( f ) in Eq. (5.23a). We also use the definition of potentials based on the 'densities' σ , σ i and σ ij and σ s constructed from T αβ and from T -2 ϕ∂T/∂ϕ in Eqs. (5.24) plus a new potential along with the super- and superduper-potentials defined in Eq. (5.25, 5.26) and their obvious counterparts X i , X s , and so on. A number of potentials occur sufficiently frequently in the PN expansion that is it useful to define them specifically. There are the the 'Newtonian' potentials, The potentials needed for the post-Newtonian limit are (compare to Eq. (5.28)): Useful 2PN potentials include:", "pages": [ 118, 119, 120, 121 ] }, { "title": "8.2 Expansion of Near-Zone Fields to 2.5PN Order", "content": "In evaluating the contributions at each order, we shall use the notation defined in 5.29 plus a similar notation for the scalar sector as where the subscript on each term indicates the level (1PN, 2PN, 2.5PN, etc.) of its leading contribution to the equations of motion.", "pages": [ 121 ] }, { "title": "8.2.1 Newtonian, 1PN and 1.5PN solutions", "content": "At lowest order in the PN expansion, we only need to evaluate τ 00 = ( -g ) T 00 /φ 0 + O ( ρglyph[epsilon1] ) = σ/φ 0 + O ( ρglyph[epsilon1] ) (recall that σ ii ∼ glyph[epsilon1]σ ), and τ s = σ s / [ φ 0 (3 + 2 ω 0 )] , where ω 0 ≡ ω ( φ 0 ) . Since both densities have compact support, the outer integrals vanish, and we find Consider the case where we are dealing with pure perfect fluids, with no compact bodies having sensitivity factors s A . Then to Newtonian order, σ = σ s , U = U s , and the metric to Newtonian order is given by the leading term in Eq. (8.2b), We therefore identify the coefficient of U in g 00 as the effective Newtonian gravitational coupling constant, G , given by in agreement with our earlier definition of G today in Eq. (4.41). However, we will not set G = 1 as is conventional in general relativity, in order to highlight the fact that it is an effective gravitational constant linked to the asymptotic value of φ , which could, for example, vary with time as the universe evolves. For future use, we also recall ζ and λ 1 from Eq. (4.38) and define a new parameter λ 2 as A consequence of these definitions is that It is worth pointing out that ω 0 enters at Newtonian order, via the modified coupling constant G of Eq. (8.16). It is then clear, by virtue of the expansion ω ( φ ) = ω 0 + ( dω/dϕ ) 0 Ψ + ( d 2 ω/dϕ 2 ) 0 Ψ 2 / 2 + . . . , that the parameter λ 1 will first contribute at 1 PN order, λ 2 will first contribute at 2 PN order, and so on. To this order, ( -g ) = 1 + 4 GU (1 -ζ ) -8 GU s ζ + O ( glyph[epsilon1] 2 ) . Then, through PN order, the required forms for τ µν and τ s are given by Substituting into Eqs. (5.21), and calculating terms through 1.5PN order (e.g. O ( glyph[epsilon1] 5 / 2 ) in N ), we obtain In Eq. (8.20g), we have used the fact (to be verified later) that, because of the conservation of baryon number, and assuming that our compact bodies have stationary internal structure, M s ( t ) is constant to the lowest PN order. Thus, rather than contributing to Ψ 0 . 5 as shown in Eq. (8.5), the term -2 ˙ M s contributes to Ψ 1 . 5 ; similarly the term in Ψ 1 . 5 involving three time derivatives of M s actually contributes to Ψ 2 . 5 . The physical metric to 1.5PN order is then given by", "pages": [ 122, 123, 124 ] }, { "title": "8.2.2 2PN and 2.5PN solutions", "content": "At 2PN and 2.5PN order, we obtain, from Eqs. (7.19), (5.8) and (8.4), Outer integrals and boundary terms contribute nothing, so we obtain All solutions obtained so far must be substituted into Eqs. (7.19), (7.22), (5.8) and (8.4) to obtain τ 00 , τ ii and τ s to the required order, Substituting into Eqs. (5.21a), (5.21c) and (8.5) and evaluating terms through O ( glyph[epsilon1] 7 / 2 ) , and verifying that the outer integrals and surface terms make no R -independent contributions, we obtain, 'In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.' -Galileo Galilei 9", "pages": [ 124, 125, 126, 128 ] }, { "title": "9.1 Energy-Momentum Tensor and The Conserved Density", "content": "We now must expand the effective energy-momentum tensor, Eq. (7.7) in a PN expansion to the required order, including the φ dependence of the masses M A . We first expand M A ( φ ) about the asymptotic value φ 0 : We then define the dimensionless 'sensitivities' Note that the definition of s ' A used in [265] and [5] has the opposite sign from our definition. Recalling that φ = φ 0 (1 + Ψ) we can write where we define the constant mass for each body m A ≡ M A 0 and the definition of S ( s A ; Ψ) is clearly given above in terms of the sensitivities. In general relativity, neglecting pressure, the stress energy tensor can be written as (see Eqs. (5.39)) where ρ ∗ is identified as the 'baryonic', or 'conserved' mass density, ρ ∗ = mn √ -g u 0 , where n is the number density of baryons, and m is the rest mass per baryon. It satisfies an exact continuity equation ∂ρ ∗ /∂t + ∇· ( ρ ∗ v ) = 0 , and implies that the baryonic mass of any isolated body is constant. Here we identify the 'baryons' as our compact point masses with constant mass m A , so that Thus, we can rewrite Eq. (7.7) in the form where ρ ∗ is given by Eq. (5.37), and where we have substituted u µ = u 0 v µ , with v µ = dx µ /dt = (1 , v ) being the ordinary velocity. We have dropped the subscript from the variable s in S because it will be assigned a label A wherever the delta function that is implicit in ρ ∗ corresponds to body A . Thus, we arrive at a conversion from the σ -densities of Eq. (5.18) to ρ ∗ , given by To convert σ s , recall that and that ϕ = 1 + Ψ , ∂/∂ϕ = ∂/∂ Ψ . Consequently Defining we can write Substituting the expansion for the metric, Eq. (8.2), and for the metric potentials, Eq. (5.29), we obtain to the 2.5PN order required for the equations of motion, where U σ , U sσ and V i σ are defined in terms of the σ -densities. Substituting these formulas into the definitions of U σ , U sσ and the other potentials defined in terms of σ , we can convert all potentials into new versions defined in terms of ρ ∗ , plus PN corrections. For example, we find that the 'Newtonian' potentials U σ and U sσ become while the relevant PN potentials become where all potentials are now defined in terms of the density ρ ∗ , and including, where needed, the sensitivity factors s , a s and b s . In manipulating these expressions, we have made use of the identities, valid for any function f , Σ( sf ) = [Σ( f ) -Σ s ( f )] / 2 and Σ( x i f ) = x i Σ( f ) -X ,i ( f ) . The potentials U and U s will henceforth be given by In some cases we will use the same notation as before, to avoid a proliferation of hats, tildes or subscripts. We redefine the Σ , X and Y potentials by and their obvious counterparts X i , X ij , X s , Y i , Y ij , Y s , and so on. With this new convention, all the potentials defined in Eqs. (8.10) can be redefined appropriately.", "pages": [ 128, 129, 130, 131, 132, 133 ] }, { "title": "9.2 Equations of Motion in Terms of Potentials", "content": "Pulling together all the potentials expressed in terms of ρ ∗ , inserting into the metric, Eq. (8.2), calculating the Christoffel symbols, we obtain from Eq. (7.10) the equation of motion where s We next turn to the problem of expressing these equations explicitly in terms of positions and velocities of each body in a two-body system. 'Science is a way of thinking much more than it is a body of knowledge.' -Carl Sagan", "pages": [ 133, 134, 136, 138 ] }, { "title": "Equations of Motion for Two Compact Objects", "content": "We now wish to calculate the equation of motion for a member of a compact binary system. To do this, we integrate ρ ∗ dv i /dt over body 1, and substitute Eq. (9.25) and then Eqs. (9.26) (9.30). We follow closely the methods already detailed in Chapter 5 based on [191] (hereafter referred to as PWII) for evaluating the integrals of the various potentials, and so we will not repeat those details here. Readers should consult Sec. III and Appendices B, C, and D of PWII for details. In structural terms almost all of the potentials that appear in the 2PN terms in scalar-tensor theory also appear in general relativity, apart from the differences in the types of densities that generate the potentials, for example U s vs. U , X s vs. X , Φ s 2 s vs. Φ 2 , and so on. The only 2PN term that does not appear in GR involves the potential P ( ˙ U s U ,i s ) , but this can be evaluated using the methods described in Chapter 5. Similarly, at 2.5PN order most of the moments that appear here also appear in GR, only a few, notably the scalar monopole and dipole moments M s and I i s are new. Particularly new is the appearance of a 1 . 5 PN order term generated by the scalar dipole moment; this, of course, is the radiation-reaction counterpart of the well-known dipole gravitational radiation prediction of scalar-tensor theories.", "pages": [ 138 ] }, { "title": "10.1 Conservative 1 PN and 2 PN Terms", "content": "We begin with the conservative Newtonian, 1PN and 2PN terms. The results are, at Newtonian and 1PN orders. where r ≡| x 1 -x 2 | , n ≡ ( x 1 -x 2 ) /r , and where the parameters α , ¯ γ , and ¯ β A are defined in Table 7.1. Note that under the interchange (1 glyph[harpoonleftright] 2) , n →-n . At 2PN order, we find 2 (2 PN ) { →-} where ¯ δ A and ¯ χ A are defined in Table 7.1. It is straightforward to show that these equations of motion can be derived from a two-body Lagrangian, given by As in general relativity, the Lagrangian contains acceleration-dependent terms at 2 PN order, and thus the Euler-Lagrange equations are ( d 2 /dt 2 )( δL/δa i ) -( d/dt )( δL/δv i ) + δL/δx i = 0 . The equations of motion (absent radiation-reaction terms) admit the usual conserved quantities. The energy is given to 2PN order by while the total momentum is given by", "pages": [ 138, 139, 140, 141 ] }, { "title": "10.2 Radiation-Reaction Terms", "content": "At 1 . 5 PN order, the leading dipole radiation reaction term is given by Because we will be working to 2 . 5 PN order, the scalar dipole moment I i s must be evaluated to post-Newtonian order, and when time derivatives of that moment generate an acceleration, the post-Newtonian equations of motion must be inserted. Explicit two-body expressions for I i s and the other moments needed for the radiation-reaction terms are provided in an Appendix. In addition to evaluating the direct 2 . 5 PN terms from Eq. (9.30) for two bodies, we must include the 1 . 5 PN contributions to the accelerations that occur in the 1 PN terms ˙ V i , X ,i and X ,i s that appear in Eq. (9.27). At 2 . 5 PN order, the final two-body expressions take the form We shall defer calculating the moments and their time derivatives explicitly until the next subsection, where we obtain the relative equation of motion.", "pages": [ 141, 142 ] }, { "title": "10.3 Relative Equation of Motion", "content": "We now wish to find the equation of motion for the relative separation x = x 1 -x 2 , through 2 . 5 PN order. We take the PN contributions to the equation of motion for body 1 and body 2 and calculate d 2 x /dt 2 = a 1 -a 2 . We must then express the individual velocities v 1 and v 2 that appear in post-Newtonian terms in terms of v ≡ v 1 -v 2 . Since velocity-dependent terms show up at 1 PN order, we need to find the transformation from v 1 and v 2 to v to 1 . 5 PN order so as to keep all corrections through 2 . 5 PN order. To do this we make use of the momentum conservation law which the momentum is given in Eq. (10.5). But because of the contributions of dipole radiation reaction at 1 . 5 PN order, the momentum is not strictly conserved because of the recoil of the system in response to the radiation of linear momentum at dipole order. By combining Eqs. (10.5) and (10.6), it is straightforward to show that the following quantity is constant through 1 . 5 PN order: Setting C i = 0 and combining this with the definition of v , we find that where where m and η are the total mass and reduced mass ratio, ψ = δm/m = ( m 1 -m 2 ) /m , and We also need to evaluate the multipole moments that appear in the radiation-reaction terms to the appropriate order, and then calculate their time derivatives, inserting the equations of motion to the appropriate order as required. Explicit formulae for the moments are displayed in Appendix A.2. Combining all the various PN contributions consistently, we arrive finally at the relative equation of motion through 2 . 5 PN order, as given in Eq. (10.12) i.e. where again r ≡| X | , n ≡ X /r , m ≡ m 1 + m 2 , η ≡ m 1 m 2 /m 2 , v ≡ v 1 -v 2 , and ˙ r = dr/dt . Here we display the coefficients A and B as: where in the two last equations Here the subscripts ' + ' and ' -' on various parameters denote sums and differences, so that, for a chosen parameter τ i we define where τ can be either ¯ β , ¯ δ , or ¯ χ . However, note that S + , S -are already defined in Eqs. (10.11) explicitly. Comparing relative equations of motion in scalar-tensor theories i.e. Eqs. (10.13) with their correspondin expressions in general relativity i.e. Eqs. (5.61, 5.62), it clearly shows that scalartensor geavity gives general relativistic expressions plus some extra terms.", "pages": [ 142, 143, 144 ] }, { "title": "10.4 Energy Loss Rate", "content": "We now wish to evaluate the rate of energy loss that is induced by the radiation-reaction terms in the equations of motion. Because those equations of motion contain both 1 . 5 PN as well as 2 . 5 PN contributions, we will have not only the normal 'quadrupole' order contributions to the energy loss rate analogous to those that appear in general relativity, but also dipole contributions that are in principle larger by a factor of 1 /v 2 . Since the conventional 'counter' for keeping track of contributions to the waveform and energy flux in the wave-zone denotes the GR quadrupole terms as 'Newtonian' or 0 PN order, the dipole terms will, by this reckoning, be of -1 PN order. To evaluate the energy loss correctly through 'Newtonian' order, we first express the conserved energy in relative coordinates to 1 PN order. Using the transformations (10.9) and (10.10) to 1 PN order, we obtain We then calculate dE/dt , inserting the 1 . 5 PN and 2 . 5 PN acceleration terms into the leading term v · a , and inserting only the 1 . 5 PN terms wherever accelerations occur in the time derivative of the 1 PN terms. Beginning with the leading term, and expressing the 1 . 5 PN acceleration in the form a 1 . 5 PN = ( D/r 3 )(3 ˙ r n -v ) , where D = 4 ηζ ( Gαm ) 2 S 2 -/ 3 , we find for the -1 PN term ( dE/dt ) -1 PN = µ ( D/r 3 )(3 ˙ r 2 -v 2 ) . This can be simplified by exploiting the identity Thus ( v 2 -3˙ r 2 ) /r 3 can be written as the total time derivative of a quantity that can be absorbed as a 1 . 5 PN correction to the definition of E , leaving ( dE/dt ) -1 PN = µ ( D/r 3 )( x · a ) . Inserting the Newtonian acceleration for a , we obtain This is in agreement with earlier calculations of the energy flux due to dipole gravitational radiation [100, 265]. However, since we are working to Newtonian order in the energy loss, we also need to include the 1 PN contributions to the acceleration that appears in Eq. (10.17), yielding a contribution given by µD ( Gαm/r 4 )( A 1 PN + ˙ r 2 B 1 PN ) , where A 1 PN and B 1 PN are given by Eqs. (10.13b). We then combine this with the other Newtonian order terms generated from dE/dt , leading to an expression of the general form We now use an identity derived from the Newtonian equations of motion, This is applicable at this PN order provided that the integers s and p are non-negative, q ≥ 2 and 2 s + p +2 q = 7 . Using the three possible cases ( s, p, q ) = (1 , 1 , 2) , (0 , 3 , 2) , (0 , 1 , 3) , we can freely manipulate the values of three of the six coefficients p i in Eq. (10.19). The idea is to combine terms on the right-hand-side of Eq. (10.19) into a total time derivative, to move that to the left-hand-side and then to absorb it into a meaningless redefinition of E (see for example, [149, 150] for discussion). Thus one can easily arrange for p 4 , p 5 and p 6 to vanish. It then turns out that the coefficient p 3 of the term proportional to v 2 ˙ r 2 is proportional to the combination of the 2 . 5 PN equation-of-motion coefficients 5 a 1 +3 a 3 -15 b 1 -5 b 3 . An inspection of Eqs. (10.14) reveals that this combination miraculously vanishes. Pulling everything together, we obtain the final expression for the energy loss rate, where These results are in complete agreement with the total energy flux to -1 PN and 0 PN orders, as calculated by Damour and Esposito-Farèse [84]. 1 'Science, my lad, is made up of mistakes, but they are mistakes which it is useful to make, because they lead little by little to the truth.' -Jules Verne, Journey to the Center of the Earth 11", "pages": [ 144, 145, 146, 148 ] }, { "title": "Discussion", "content": "We have used the DIRE approach based on post-Minkowskian theory to derive the explicit equations of motion in a general class of massless scalar-tensor theories of gravity for compact binary systems through 2 . 5 PN order. Here we discuss the results, and compare our work with related work on scalar-tensor gravity and equations of motion.", "pages": [ 148 ] }, { "title": "11.1 General Remarks and Comparison with Other Results", "content": "We begin by noting that, not surprisingly, the expressions are considerably more complicated than the corresponding general relativistic expressions (compare Eqs. (10.12, 10.13) with Eqs. (5.60, ?? ) and Eqs. (1.2, 1.3, 5.4) in PWII). Given that the results depend on the function ω ( φ ) and its first and second derivatives, on the masses of each body, and on the sensitivities of each body and their derivatives, it is somewhat remarkable that the final equations of motion depend on a rather small number of parameters, as shown in the right-hand column of Table 7.1. The parameter α combines with G to yield an effective two-body Newtonian coupling constant. It is not a universal constant, as it depends symmetrically on the sensitivities of each body. The parameter ¯ γ and the body-dependent parameter ¯ β A govern the post-Newtonian corrections, while the body-dependent parameters ¯ δ A and ¯ χ A govern the 2 PN corrections. In the radiation-reaction terms, the sensitivities s A occur explicitly along with ¯ γ and ¯ β A . The relative simplicity of the parameters at 1 PN and 2 PN orders has been noted before. Damour and Esposito-Farèse [84, 85] (DEF hereafter) studied a class of multi-scalar-tensor theories, but worked in the Einstein representation, where the gravitational action was pure general relativity, augmented by a free action for the scalar fields. This is a non-metric representation of the theory, since the scalar field(s) couple to normal matter via a function A ( ϕ ) (here we will - - - focus on a single scalar field). For a compact body with mass ˜ m ( ϕ ) (using the Eardley ansatz), the effective matter action depends on the product A ( ϕ ) ˜ m ( ϕ ) . The scalar field φ of our Jordan representation is given by φ = A ( ϕ ) -2 , and 3 + 2 ω ( φ ) = ( d ln A/dϕ ) -2 . Using a diagrammatic approach, DEF showed that the important quantities involved derivatives of A ( ϕ ) ˜ m ( ϕ ) with respect to ϕ , and consequently (in our language) ω and s A and their derivatives always combined in specific ways, leading to relatively few parameters. Table 11.1 gives a dictionary that translates from our parameters to those of DEF for the case of two bodies. Interestingly, our parameters ¯ δ A do not appear in DEF's list, so far as we could tell. In the 1 PN limit, Will [265] wrote down a general N -body Lagrangian for compact selfgravitating bodies that could span a wide class of metric theories of gravity that embody postGalilean invariance (so-called 'semi-conservative' theories of gravity), and that have no 'Whitehead' potential in the post-Newtonian limit. Comparing our Lagrangian of scalar-tensor theory with the 2-body limit of Eq. (11.62) of [265], we can translate between our parameters and the coefficients G ab , B ab , and D abc of [265], as shown in Table 11.1. The factor 1 -2 s A appears throughout these equations. This quantity is often called the 'scalar charge' of the object. From the point of view of the Einstein representation of scalartensor theory, it is easy to see how this factor arises. The scalar field appears in the gravitational part of the action only in a kinetic term g µν ϕ ,µ ϕ ,ν (we assume that there is no potential V ( ϕ ) ). It does not couple to gravity other than via the metric in the kinetic term. The effective matter action for a compact body depends on the product A ( ϕ ) M ( ϕ ) . Varying this product with respect to ϕ yields the quantity where we used the fact that ln φ = -2 ln A ( ϕ ) . Thus the factor 1 -2 s and its derivatives naturally control the source of the scalar field, as can be seen clearly in Eq. (9.12e). Defining a scalar charge for body A in a two-body system by we see that the quantities S ± are given by The scalar charge, or sensitivity of a given body depends on its internal structure. For weakly gravitating bodies, s ≈ -Ω /M glyph[lessmuch] 1 , where Ω ≡ -(1 / 2) G ∫ ∫ ρ ∗ ρ '∗ | x -x ' | -1 d 3 xd 3 x ' is the Newtonian self-gravitational binding energy . For neutron stars, values of the sensitivities range from 0 . 1 to 0 . 3 , depending on the mass and equation of state of the body [276, 297] and can vary dramatically, depending on the specific form of ω ( φ ) [84].", "pages": [ 148, 149, 150 ] }, { "title": "11.2 Weakly Self-Gravitating Systems", "content": "In the post-Newtonian limit with weakly self-gravitating systems, the sensitivities s i are themselves of order glyph[epsilon1] . If one is working purely at 1 PN order, then the effects of sensitivities in the 1 PN terms of Eq. (10.1) will be of 2 PN order. So the only effect of the bodies' sensitivities in this case will come from the coefficient α in the Newtonian term. Consider a specific example: body 1 with sensitivity s 1 resides in the field of body 2 , with sensitivity zero. The acceleration of body 1 is then given by and thus the body's Newtonian acceleration will depend on its internal structure, a violation of the Strong Equivalence Principle, commonly known as the Nordtvedt effect. In the PPN framework [265], the Nordtvedt effect is normally expressed in terms of Ω . Alternatively, since M ≈ m 0 + Ω , we have that Ω /M = d ln M/d ln G . Taking into account Eq. (8.16), we can connect the sensitivity s to Ω by where is the parameter defined in TEGP (see Eqs. (5.36) and (5.38)) such that the PPN parameter β = 1+Λ in scalar-tensor theory (note the relationship between φ 0 and G , which is set equal to unity in TEGP). We also have that γ = 1 -2 ζ . We can then express the acceleration of body 1 as The coefficient in front of Ω 1 /m 1 is precisely 4 β -γ -3 , as in the standard PPN framework. In the 1 PN terms in Eq. (10.1), for weakly self-gravitating systems, it is easy to see from Table 7.1 that in the limit s i → 0 , α → 1 , the parameters ¯ γ and ¯ β i tend to the PPN parameters γ -1 and β -1 , respectively, as shown in Table 11.1, and thus our equations of motion at 1 PN order agree with the standard ones for 'point' masses in scalar-tensor theory. The radiation-reaction results can also be compared with existing work. The -1 PN energy loss due to dipole gravitational radiation reaction, Eq. (10.18) is in complete agreement with calculations of the dipole energy flux [100, 264, 265]. In comparing Eq. (10.18) with Eqs. (10.84) and (10.136) of [265], the additional factor of [1 + 4Λ(2 + ω 0 )] 2 arises from the relation (11.5) between s and Ω /M . For weakly self-gravitating bodies, the Newtonian-order energy loss simplifies by virtue of setting all sensitivities equal to zero. In this case, with α = 1 , ¯ γ = -2 ζ , ¯ β + = β -1 = Λ , ¯ β -= 0 , S -= 0 , and S + = 1 , we obtain These agree completely with Eq. (10.136) of [265].", "pages": [ 150, 151 ] }, { "title": "11.3 Binary Black Holes", "content": "Roger Penrose was probably the first to conjecture, in a talk at the 1970 Fifth Texas Symposium, that black holes in Brans-Dicke theory are identical to their GR counterparts [242]. Motivated by this remark, Thorne and Dykla showed that during gravitational collapse to form a black hole, the Brans-Dicke scalar field is radiated away, in accord with Price's theorem, leaving only its constant asymptotic value, and a GR black hole [242]. Hawking [134] proved on general grounds that stationary, asymptotically flat black holes in vacuum in BD are the black holes of GR. The basic idea is that black holes in vacuum with non-singular event horizons cannot support scalar 'hair'. Hawking's theorem was extended to the class of f ( R ) theories that can be transformed into generalized scalar-tensor theories by Sotiriou and Faraoni [235]. For a stationary single body, it is clear from Eq. (9.12e) that, if s = 1 / 2 and all its derivatives vanish, the only solution for the scalar field is φ ≡ φ 0 , and hence the equations reduce to those of general relativity. In the Einstein representation, this corresponds to A ( ϕ ) M ( ϕ ) = constant, so that the scalar field decouples from any source, and thus must be either constant or singular. Consequently, stationary black holes are characterized by s = 1 / 2 . Another way to see this is to note that, because all information about the matter that formed the black hole has vanished behind the event horizon, the only scale on which the mass of the hole can depend is the Planck scale, and thus M ∝ M Planck ∝ G -1 / 2 ∝ φ 1 / 2 . Hence s = 1 / 2 . If s A = 1 / 2 for each black hole in a binary system, then, as we discussed in the introduction, all the parameters ¯ γ , ¯ β A , ¯ δ A , ¯ χ A , and S ± vanish identically, and α = 1 -ζ . But since α appears only in the combination with Gαm A , a simple rescaling of each mass puts all equations into complete agreement with those of general relativity, through 2 . 5 PN order. But is s A = 1 / 2 really true for binary black holes? If the orbital timescale is long compared to the dynamical (quasinormal mode) timescale of each black hole, then it is plausible to assume that Hawking's theorem holds for each black hole, at least up to some PN order. On the other hand, one could imagine a situation where each hole is distorted by the tidal forces from the companion hole, or where gravitational radiation flowing across the event horizons disrupts the stationarity needed for Hawking's theorem. In PN language, these kinds of effects are known to be of an order higher than the 2 . 5 PN order achieved in this paper, so perhaps some non-GR effects might emerge at sufficiently high PN order. Can a perturbation of the scalar field be supported sufficiently by strong gravity or by time varying fields to make any difference? Or, without matter to support it, does any scalar perturbation get radiated away on a quasinormalmode timescale, which is short compared to the orbital timescale, except during the merger of the two black holes? Preliminary evidence from numerical relativity supports the latter scenario: Healy et al. [135] introduced a very large Brans-Dicke type scalar field into the initial data of a binary black hole merger and found that, while the field affected the inspiral while it lasted, it was radiated away rather quickly, although it was not possible from the numerical data to fully quantify this. It should be pointed out that there are ways to induce scalar hair on a black hole. One is to introduce a potential V ( φ ) , which, depending on its form, can help to support a non-trivial scalar field outside a black hole. Another is to introduce matter. A companion neutron star is an obvious choice, and such a binary system in scalar-tensor theory is clearly different from its general relativistic counterpart (see the next subsection). Another possibility is a distribution of cosmological matter that can support a time-varying scalar field at infinity. This possibility has been called 'Jacobson's miracle hair-growth formula' for black holes, based on work by Jacobson [145, 151]. Whether it is possible to incorporate such ideas into our approach is a subject for future work. These considerations motivate us to formulate a conjecture along the following lines: Consider a scalar-tensor theory of gravity with no potential for the scalar field, and consider two black holes with non-singular event horizons in a vacuum (no normal matter), asymptotically flat spacetime with φ at spatial infinity constant in time. Following an initial transient period short compared to the orbital period, the orbital evolution and gravitational radiation from the binary system are identical to those predicted by GR, after a mass rescaling, independent of the initial scalar field configuration. Aspects of this conjecture could be addressed by numerical simulations that extend the work of [135]. It may also be possible to address it partially by generalizing Hawking's theorem to a situation that is not strictly stationary, but yet still retains some symmetry, such as a helical Killing vector. This will be the subject of future work.", "pages": [ 151, 152, 153 ] }, { "title": "11.4 Black-Hole Neutron-Star Binary Systems", "content": "Finally, we note the unusual circumstance that, if only one of the members of the binary system, say body 2, is a black hole, with s 2 = 1 / 2 , then α = 1 -ζ , ¯ γ = ¯ β A = 0 , and hence, through 1 PN order, the motion is again identical to that in general relativity. This result is actually implicit in the post-Newtonian equations of motion for compact binaries in Brans-Dicke theory displayed in Eq. (11.91) of [265], but was never stated explicitly there. At 1 . 5 PN order, dipole radiation reaction kicks in, since s 1 < 1 / 2 . In this case, S -= S + = α -1 / 2 (1 -2 s 1 ) / 2 , and thus the 1 . 5 PN coefficients in the relative equation of motion (10.12) take the form where At 2 PN order, ¯ χ A = ¯ δ 2 = 0 , but ¯ δ 1 = Q = 0 . In this case, the 2 PN coefficients in (10.12) take the form glyph[negationslash] Finally, the 2 . 5 PN coefficients in Eq. (10.13h) have the form while the coefficients in the energy loss rate simplify to We find, somewhat surprisingly, that the motion of a mixed compact binary system through 2 . 5 PN order differs from its general relativistic counterpart only by terms that depend on a single parameter Q , as defined by Eq. (11.10). Furthermore, all reference to the parameters λ 1 and λ 2 , related to derivatives of the coupling function ω ( φ ) , has disappeared, in other words, the motion of mixed compact binary systems in general scalar-tensor theories through 2 . 5 PN order is formally identical to that in standard Brans-Dicke theory. The only way that a generalized scalar-tensor theory affects the motion differently than pure Brans-Dicke theory is through the value of the un-rescaled mass m 1 and the sensitivity s 1 for a neutron star of a given central density and total number of baryons. The general conclusions reached in this work about binary black holes and mixed binaries in scalar-tensor gravity were obtained from the near-zone gravitational fields. If these conclusions continue to hold for the gravitational-wave signal, then gravitational-wave observations of binary black holes will be unable to distinguish between general relativity and scalar-tensor theories, and observations of mixed black-hole neutron-star binaries will be essentially unable to distinguish between general scalar-tensor theories and Brans-Dicke theory (Fig. 11.1 illustrates this fact). The radiative part of this problem, which will involve a derivation of the gravitational waveform to 2 PN order, together with the energy flux, will be the subject of future work.", "pages": [ 153, 154 ] }, { "title": "Constraining Lorentz-Violating, Modified Dispersion Relations with Gravitational Waves", "content": "This part is based on a published paper in Physical Review D. [179] in which we construct a parametrized dispersion relation that can reproduce a range of known Lorentz-violating predictions and investigate their impact on the propagation of gravitational waves. We show how such corrections map to the waveform observable and to the parametrized post-Einsteinian framework, proposed to model a range of deviations from General Relativity. Given a gravitational-wave detection, the lack of evidence for such corrections could then be used to place a constraint on Lorentz violation. The constraints we obtain are tightest for dispersion relations that scale with small power of the graviton's momentum and deteriorate for a steeper scaling. 'Science may be described as the art of systematic oversimplification.' - Karl R. Popper 12", "pages": [ 156, 158 ] }, { "title": "Introduction and Foundations", "content": "In this chapter we first start with a brief introduction to declare the possibility of testing alternative theories of gravity by studying gravitational-wave signals emitted from inspiralling compact binary sources. Second, we propose a general, parametrized dispersion relation for Lorentzviolating theories which will be useful to do parameter estimation of the source and bounding the parameters of this modified dispersion relation, specially the parameter that presents the deviation from Lorentz symmetry. We also give an overview on the next following chapters of this part. The obtained bounds on the mass of graviton and on the deviation from Lorentz symmetry are also summarized.", "pages": [ 158 ] }, { "title": "12.1 Introduction", "content": "After a century of experimental success, Einstein's fundamental theories, ie. the special theory of relativity and the General theory of Relativity (GR), are beginning to be questioned. As an example, consider the observation of ultra-high-energy cosmic rays. In relativity, there is a threshold of ∼ 5 × 10 19 eV (GZK limit) for the amount of energy that charged particles can carry, while cosmic rays have been detected with higher energies [40]. On the theoretical front, theories of quantum gravity also generically predict a deviation from Einstein's theory at sufficiently large energies or small scales. In particular, Lorentz violation seems ubiquitous in such theories. These considerations motivate us to study the effects of Lorentz violation on gravitational wave observables. Einstein's theory will soon be put to the test through a new type of observation: gravitationalwaves. Such waves are (far-field) oscillations of spacetime that encode invaluable and detailed information about the source that produced them. For example, the inspiral, merger and ringdown of compact objects (black holes or neutron stars) are expected to produce detectable waves that will access horizon-scale curvatures and energies. Gravitational waves may thus provide new hints as to whether Einstein's theory remains valid in this previously untested regime. For more details about gravitational-waves see Chapter 3. Gravitational-wave detectors are today a reality. As we mentioned earlier in Chapter 3, ground-based interferometers, such as the Advanced Laser Interferometer Gravitational Observatory (Ad. LIGO) [1, 132, 285] and Advanced Virgo [288], are currently being updated, and are scheduled to begin data acquisition by 2015. Second generation detectors, such as the Einstein Telescope (ET) [200, 282] and the Laser Interferometer Space Antenna (LISA) [199, 281], are also being planned for the next decade. Recent budgetary constraints in the United States have cast doubt on the status of LISA, but the European Space Agency is still considering a descoped, LISA-like mission (an NGO, or New Gravitational Observatory). The detection of gravitational waves is, of course, not a certainty, as the astrophysical event rate is highly uncertain. However, there is consensus that advanced ground detectors should observe a few gravitational-wave events by the end of this decade. r t Some alternative gravity theories endow the graviton with a mass [265]. Massive gravitons would travel slower than the speed of light, but most importantly, their speed would depend on their energy or wavelength. Since gravitational waves emitted by compact binary inspirals chirp in frequency, gravitons emitted in the early inspiral will travel more slowly than those emitted close to merger, leading to a frequency-dependent gravitational-wave dephasing, compared to the phasing of a massless general relativistic graviton. This fact is shown schematically in Fig. 12.1. If such a dephasing is not observed, then one could place a constraint on the graviton mass [267]. A Lorentz-violating graviton dispersion relation leaves an additional imprint on the propagation of gravitational waves, irrespective of the generation mechanism. Thus a bound on the dephasing effect could also bound the degree of Lorentz violation. Note that our use of the term 'graviton' is not meant to imply that geavitational-wave detectors will observe individual gravitons. The detected waves are perfectly classical, i.e. they contain enourmous numbers of gravitons. In this work, we construct a framework to study the impact of a Lorentz-violating dispersion relation on the propagation of gravitational waves. We begin by proposing a generic, but quantum-gravitational inspired, modified dispersion relation, given by where m g is the mass of the graviton and A and α are two Lorentz-violating parameters that characterize the GR deviation ( α is dimensionless while A has dimensions of [energy] 2 -α ). We will assume that A / ( cp ) 2 -α glyph[lessmuch] 1 . When either A = 0 or α = 0 , the modification reduces to that of a massive graviton. When α = (3 , 4) , one recovers predictions of certain quantum-gravitation inspired models. This modified dispersion relation introduces Lorentz-violating deviations in a continuous way, such that when the parameter A is taken to zero, the dispersion relation reduces to that of a simple massive graviton. The dispersion relation of Eq. (12.1) modifies the gravitational waveform observed at a detector by correcting the phase with certain frequency-dependent terms. In the stationaryphase approximation (SPA), the Fourier transform of the waveform is corrected by a term of the form ζ ( A ) u α -1 , where u = π M f is a dimensionless measure of the gravitational-wave frequency with M being the 'chirp mass'. We show that such a modification can be easily mapped to the recently proposed parametrized post-Einsteinian framework (ppE) [75, 294] for an appropriate choice of ppE parameters. In deriving the gravitational-wave Fourier transform we must assume a functional form for the waveform as emitted at the source so as to relate the time of arrival at the detector to the gravitational-wave frequency. In principle, this would require a prediction for the equations of motion and gravitational-wave emission for each Lorentz violating theory under study. However few such theories have reached a sufficient state of development to produce such predictions. On the other hand, it is reasonable to assume that the predictions will be not too different from those of general relativity. For example, Will argued [267] that for a theory with a massive graviton, the differences would be of order ( λ/λ g ) 2 , where λ is the gravitational wavelength, and λ g is the graviton Compton wavelength, and λ g glyph[greatermuch] λ for sources of interest. Similar behavior might be expected in Lorentz violating theories. The important phenomenon is the accumulation of dephasing over the enormous propagation distances from source to detector, not the small differences in the source behavior. As a result, we will use the standard general relativistic wave generation framework for the source waveform.", "pages": [ 158, 159, 160 ] }, { "title": "12.2 An Overview", "content": "With this new waveform model described in the previous section, we then carry out a simplified (angle-averaged) Fisher-matrix analysis to estimate the accuracy to which the parameter ζ ( A ) could be constrained as a function of α , given a gravitational-wave detection consistent with general relativity. We perform this study with a waveform model that represents a non-spinning, quasi-circular, compact binary inspiral, but that deviates from general relativity only through the effect of the modified dispersion relation on the propagation speed of the waves, via Eq. (12.1). To illustrate our results, we show in Table 12.1 the accuracy to which Lorentz-violation in the α = 3 case could be constrained, as a function of system masses and detectors for fixed signal-to-noise ratio (SNR). The case α = 3 is a prediction of 'doubly special relativity'. The bounds on the graviton mass are consistent with previous studies [14, 33, 158, 267, 275, 291] (for a recent summary of current and proposed bounds on m g see [36]). The table here means that given a gravitational-wave detection consistent with GR, m g and A would have to be smaller than the numbers on the third and fourth columns respectively. × Let us now compare these bounds with current constraints. The mass of the graviton has been constrained dynamically to m g ≤ 7 . 6 × 10 -20 eV through binary pulsar observations of the orbital period decay and statically to 4 . 4 × 10 -22 eV with Solar System constraints (see e.g. [36]). We see then that even with the inclusion of an additional A parameter, the projected gravitational wave bounds on m g are still interesting. The quantity A has not been constrained in the gravitational sector. In the electromagnetic sector, the dispersion relation of the photon has been constrained: for example, for α = 3 , A glyph[lessorsimilar] 10 -25 eV -1 using TeV γ -ray observations [39]. One should note, however, that such bounds on the photon dispersion relation are independent of those we study here, as in principle the photon and the graviton dispersion relations need not be tied together. We must stress that, in this work, we only deal with Lorentz-violating corrections to the gravitational wave dispersion relation, and thus, we deal only with propagation effects and not with generation effects . Generation effects will in principle be very important, possible leading to the excitation of additional polarizations, as well as modifications to the quadrupole expressions. Such is the case in several modified gravity theories, such as Einstein-Aether theory and HořavaLifshitz theory [27, 28, 53, 90, 115, 141, 153, 184, 189, 201, 210, 211, 290]. Generically studying the generation problem, however, is difficult, as there does not exist a general Lagrangian density that can capture all Lorentz-violating effects. Instead, one would have the gargantuan task of solving the generation problem within each specific theory. The goal of this piece of work, instead, is to consider generic Lorentz-violating effects in the dispersion relation and focus only on the propagation of gravitational waves. This will then allow us to find the corresponding ppE parameters that represent Lorentz-violating propagation. Thus, if future gravitational wave observations peak at these ppE parameters, then one could suspect that some sort of Lorentz-violation could be responsible for such deviations from General Relativistic. Future work will concentrate on the generation problem. The remainder of this part deals with the details of the calculations and is organized as follows. In Chapter 13, we introduce and motivate the modified dispersion relation, given by Eq. (12.1), and derive from it the gravitational-wave speed as a function of energy and the new Lorentz-violating parameters. In the same chapter, Section 13.2, we study the propagation of gravitons in a cosmological background as determined by the modified dispersion relation and graviton speed. We find the relation between emission and arrival times of the gravitational waves, which then allows us in Section 13.3 to construct a restricted post-Newtonian gravitational waveform to 3 . 5 PN order in the phase [ O ( v/c ) 7 ] . We also discuss the connection to the ppE framework. In Chapter 14, we calculate the Fisher information matrix for Ad. LIGO, ET and a LISA-like mission and determine the accuracy to which the compact binary's parameters can be measured, including a bound on the graviton and Lorentz-violating Compton wavelengths. In secion 14.4 we present some conclusions and discuss possible avenues for future research. 'We know very little, and yet it is astonishing that we know so much, and still more astonishing that so little knowledge can give us so much power.' -Bertrand Russell 13", "pages": [ 161, 162, 164 ] }, { "title": "Gravitational Waves in Lorentz-Violating Gravity", "content": "In this chapter we study how some specific properties of gravitational-waves change in Lorentz violating theories of gravity. We are specifically interested in how modifications in the speed of propagation affect the observed waveforms. Knowing about these modifications is required to do parameter estimation analyses for Lorentz violating theories in the next chapter. In this chapter we also show how one can map the calculations to the parametrized post-Einsteinian formalism.", "pages": [ 164 ] }, { "title": "13.1.1 Massive Graviton Theories", "content": "In general relativity, gravitational waves travel at the speed of light c because the gauge boson associated with gravity, the graviton, is massless. Modified gravity theories, however, predict modifications to the gravitational-wave dispersion relation, which would in turn force the waves to travel at speeds different than c . The most intuitive, yet purely phenomenological modification one might expect is to introduce a mass for the graviton, following the special relativistic relation From this dispersion relation, together with the definition v/c ≡ p/p 0 , or v ≡ c 2 p/E , one finds the graviton speed [267] where m g , v g and E are the graviton's rest mass, velocity and energy.", "pages": [ 164 ] }, { "title": "13.1.2 Lorentz-Violating Theories", "content": "Different alternative gravity theories may predict different dispersion relations from Eq. (13.1). A few examples of such relations include the following: For more details about each of the alternative theories listed above, see Chapter 2. Of course, the list above is just representative of a few models, but there are many other examples where the graviton dispersion relation is modified [29, 30]. In general, a modification of the dispersion relation will be accompanied by a change in either the Lorentz group or its action in real or momentum space. Lorentz-violating effects of this type are commonly found in quantum gravitational theories, including loop quantum gravity [55] and string theory [67, 239]. Modifications to the standard dispersion relation are usually suppressed by the Planck scale, so one might wonder why one should study them. Recently, Collins, et al. [73, 74] suggested that Lorentz violations in perturbative quantum field theories could be dramatically enhanced when one regularizes and renormalizes them. This is because terms that would vanish upon renormalization due to Lorentz invariance do not vanish in Lorentz-violating theories, leading to an enhancement after renormalization [120]. Although this is an appealing argument, we prefer here to adopt a more agnostic viewpoint and simply ask the following question: What type of modifications would enter gravitationalwave observables because of a modified dispersion relation and to what extent can these deviations be observed or constrained by current and future gravitational-wave detectors? In view of this, we postulate the parametrized dispersion relation of Eq. (12.1). One can see that this model-independent dispersion relation can be easily mapped to all the ones described above, in the limit where E and p are large compared to m g , but small compared to the Planck energy E p . More precisely, we have Of course, for different values of ( A , α ) we can parameterize other Lorentz-violating corrections to the dispersion relation. One might be naively tempted to think that a p 3 or p 4 correction to the above dispersion relation will induce a 1 . 5 or 2 PN correction to the phase relative to the massive graviton term. This, however, would be clearly wrong, as p is the graviton's momentum, not the momentum of the members of a binary system. With this modified dispersion relation the modified graviton speed takes the form To first order in A , this can be written as and in the limit E glyph[greatermuch] m g it takes the form Notice that if A > 0 or if m 2 g c 4 /E 2 > | A | E α -2 , then the graviton travels slower than light speed. On the other hand, if A < 0 and m 2 g c 4 /E 2 < | A | E α -2 , then the graviton would propagate faster than light speed.", "pages": [ 165, 166 ] }, { "title": "13.2 Propagation of Gravitational Waves", "content": "We now consider the propagation of gravitational waves that satisfy the modified dispersion relation of Eq. (12.1). Since we may consider sources at very great distances, we must consider the propagation in a cosmologycal background spacetime. Consider the Friedman-RobertsonWalker background where a ( t ) is the scale factor with units of length, and Σ( χ ) is equal to χ , sin χ or sinh χ if the universe is spatially flat, closed or open, respectively. Here and henceforth, we use units with G = c = 1 , where a useful conversion factor is 1 M glyph[circledot] = 4 . 925 × 10 -6 s = 1 . 4675 km. In a cosmological background, we will assume that the modified dispersion relation takes the form where | p |≡ ( g ij p i p j ) 1 / 2 . Consider a graviton emitted radially at χ = χ e and received at χ = 0 . By virtue of the χ independence of the t -χ part of the metric, the component p χ of its 4momentum is constant along its worldline. Using E = p 0 , together with Eq. (13.7) and the relations we obtain where p 2 χ = a 2 ( t e )( E 2 e -m 2 g -A | p e | α ) . The overall minus sign in the above equation is included because the graviton travels from the source to the observer. Expanding to first order in ( m g /E e ) glyph[lessmuch] 1 , and A /p 2 -α glyph[lessmuch] 1 and integrating from emission time ( χ = χ e ) to arrival time ( χ = 0 ), we find Consider gravitons emitted at two different times t e and t ' e , with energies E e and E ' e , and received at corresponding arrival times ( χ e is the same for both). Assuming ∆ t e ≡ t e -t ' e glyph[lessmuch] a/ ˙ a , then where Z ≡ a 0 /a ( t e ) -1 is the cosmological redshift, and where we have defined and where m g /E e = ( λ g f e ) -1 , with f e the emitted gravitational-wave frequency, E e = hf e and λ g = h/m g the graviton Compton wavelength. Notice that when α = 2 , then the A correction vanishes. Notice also that λ A always has units of length, irrespective of the value of α . The distance measure D α is defined by where a 0 = a ( t a ) is the present value of the scale factor. For a dark energy-matter dominated universe D α and the luminosity distance D L have the form where H 0 ≈ 72 km s -1 Mpc -1 is the value of the Hubble parameter today and Ω M = 0 . 3 and Ω Λ = 0 . 7 are the matter and dark energy density parameters, respectively. Before proceeding, let us comment on the time shift found above in Eq. (13.11). First, notice that this equation agrees with the results of [267] in the limit A → 0 . Moreover, in the limit α → 0 , our results map to those of [267] with the relation λ -2 g → λ -2 g + λ -2 A . Second, notice that in the limit α → 2 , the ( a ( t e ) E e ) 2 -α in Eq. (13.10) goes to unity and the A correction becomes frequency independent. This makes sense, since in that case the Lorentz-violating correction we have introduced acts as a renormalization factor for the speed of light.", "pages": [ 166, 167, 168 ] }, { "title": "13.3 Modified Waveform in the Stationary Phase Approximation", "content": "We consider the gravitational-wave signal generated by a non-spinning, quasi-circular inspiral in the post-Newtonian approximation. In this scheme, one assumes that orbital velocities are small compared to the speed of light ( v glyph[lessmuch] 1 ) and gravity is weak ( m/r glyph[lessmuch] 1 ). Neglecting any amplitude corrections (in the so-called restricted PN approximation), the plus- and cross-polarizations of the metric perturbation can be represented as where A ( t ) is an amplitude that depends on the gravitational-wave polarization (see e.g. Eq. (3 . 2) in [267]), while f ( t ) is the observed gravitational-wave frequency, and Φ c and t c are a fiducial phase and fiducial time, respectively, sometimes called the coalescence phase and time. The Fourier transform of Eq. (13.16) can be obtained analytically in the stationary-phase approximation, where we assume that the phase is changing much more rapidly than the amplitude [95, 292]. We then find where f is the gravitational-wave frequency at the detector and In these equations, M e = η 3 / 5 m is the chirp mass of the source, where η = m 1 m 2 / ( m 1 + m 2 ) is the symmetric mass ratio. We can now substitute Eq. (13.11) into Eq. (13.20) to relate the time at the detector to that at the emitter. Assuming that α = 1 , we find glyph[negationslash] glyph[negationslash] while for α = 1 , we find The quantities ( ¯ t c , ¯ ¯ t c ) and ( ¯ φ c , ¯ ¯ φ c ) are new coalescence times and phases, into which constants of integration have been absorbed. We can relate t e -t ec to f e by integrating the frequency chirp equation for non-spinning, quasi-circular inspirals from general relativity [267]: where we have kept terms up to 1 PN order. In the calculations that follow, we actually account for corrections up to 3 . 5 PN order, although we don't show these higher-order terms here (they can be found e.g. in [61]). After absorbing further constants of integration into ( ¯ t c , ¯ Φ c , ¯ ¯ t c , ¯ ¯ Φ c ) , dropping the bars, and re-expressing everything in terms of the measured frequency f at the detector [note that ˙ f 1 / 2 = ( df e /dt e ) 1 / 2 / (1 + Z ) ], we obtain with the definitions where the numerical coefficient glyph[epsilon1] = 1 for LIGO and ET, but glyph[epsilon1] = √ 3 / 2 for a LISA-like mission (because when one angle-averages, the resulting geometric factors depend slightly on the geometry of the detector). The coefficients ( c n , glyph[lscript] n ) can be read up to n = 7 in Appendix A.3. In these equations, u ≡ π M f is a dimensionless frequency, while M is the measured chirp mass, related to the source chirp mass by M = (1 + Z ) M e . The frequency f max represents an upper cut-off frequency where the PN approximation fails. The dephasing caused by the propagation effects takes a slightly different form depending on whether α = 1 or α = 1 . In the general α = 1 case, we find glyph[negationslash] glyph[negationslash] glyph[negationslash] where the parameters β and ζ are given by glyph[negationslash] In the special α = 1 case, we find where β remains the same, while and we have re-absorbed a factor into the phase of coalescence. As before, notice that in the limit A → 0 , Eq. (13.27) reduces to the results of [267] for a massive graviton. Also note that, as before, in the limit α → 0 , we can map our results to those of [267] with λ -2 g → λ -2 g + λ -2 A , i.e. in this limit, the mass of the graviton and the Lorentzviolating A term become 100% degenerate. In the limit α → 2 , Eq. (13.11) becomes frequencyindependent, which then implies that its integral, Eq. (13.20), becomes linear in frequency, which is consistent with the α → 2 limit of Eq. (13.27). Such a linear term in the gravitationalwave phase can be reabsorbed through a redefinition of the time of coalescence, and thus is not observable. This is consistent with the observation that the dispersion relation with α = 2 is equivalent to the standard massive graviton one with a renormalization of the speed of light. When α = 1 , Eq. (13.11) leads to a 1 /f term, whose integral in Eq. (13.20) leads to a ln( f ) term, as shown in Eq. (13.22). Finally, notice that, in comparision with the phasing terms that arise in the PN approximation to standard general relativity, these corrections are effectively of (1+3 α/ 2) PN order, which implies that the α = 0 term leads to a 1PN correction as in [267], the α = 1 case leads to a 2 . 5 PN correction, the α = 3 case leads to a 5 . 5 PN correction and α = 4 leads to a 7 PN correction. This suggests that the accuracy to constrain λ A will deteriorate very rapidly as α increases.", "pages": [ 168, 169, 170, 171 ] }, { "title": "13.4 Connection with the PPE Framework", "content": "Recently, there has been an effort to develop a framework suitable for testing for deviations from general relativity in gravitational-wave data. In analogy with the parametrized post-Newtonian (PPN) framework [187, 260, 261, 265, 269, 272], the parametrized post-Einsteinian (ppE) framework [75, 247, 294] suggests that we deform the gravitational-wave observable away from our GR expectations in a well-motivated, parametrized fashion. In terms of the Fourier transform of the waveform observable in the SPA, the simplest ppE meta-waveform is where ( α ppE , a ppE , β ppE , b ppE ) are ppE, theory parameters. Notice that in the limit α ppE → 0 or β ppE → 0 , the ppE waveform reduces exactly to the SPA GR waveform. The proposal is then to match-filter with template families of this type and allow the data to select the best-fit ppE parameters to determine whether they are consistent with GR. We can now map the ppE parameters to those obtained from a generalized, Lorentz-violating dispersion relation: Quantum-gravity inspired Lorentz-violating theories suggest modified dispersion exponents α = 3 or 4 , to leading order in E/m g , which then implies ppE parameters b ppE = 2 and 3 . Therefore, if after a gravitational wave has been detected, a Bayesian analysis with ppE templates is performed that leads to values of b ppE that peak around 2 or 3 , this would indicate the possible presence of Lorentz violation [75]. Notice however that the α = 1 case cannot be recovered by the ppE formalism without generalizing it to include ln u terms. Such effects are analogous to memory corrections in PN theory. At this point, we must spell out an important caveat. The values of α that represent Lorentz violation for quantum-inspired theories ( α = 3 , 4 ) correspond to very high PN order effects, i.e. a relative 5 . 5 or 7 PN correction respectively. Any gravitational-wave test of Lorentz violation that wishes to constrain such steep momentum dependence would require a very accurate (high PN order) modeling of the general relativistic waveform itself. In the next chapter, we will employ 3 . 5 PN accurate waveforms, which are the highest-order known, and then ask how well ζ and β can be constrained. Since we are neglecting higher than 3 . 5 PN order terms in the template waveforms, we are neglecting also any possible correlations or degeneracies between these terms and the Lorentz-violating terms. Therefore, any estimates made in the next section are at best optimistic bounds on how well gravitational-wave measurements could constrain Lorentz violation. 'We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress.' -Richard Feynman 14", "pages": [ 171, 172, 174 ] }, { "title": "Parameter Estimation in Lorentz-Violating Gravity", "content": "In this chapter, we perform a simplified Fisher analysis, following the method outlined for compact binary inspiral in [79, 114, 197], to get a sense of the bounds one could place on λ g and λ A given a gravitational-wave detection that is consistent with general relativity. We begin by summarizing some of the basic ideas behind a Fisher analysis, introducing some notation. We then apply this analysis to an Adv. LIGO detector, an ET detector and a LISA-like mission.", "pages": [ 174 ] }, { "title": "14.1 Fisher-Matrix Parameter Estimation Method", "content": "Based on the Fisher matrix method that we reviewed in Chapter 6, we will work with an angleaveraged response function, so that the templates depend only on the following parameters: where each component of the vector θ is dimensionless. We recall that A is an overall amplitude that contains information about the gravitational-wave polarization and the beam-pattern function angles. The quantities Φ c and t c are the phase and time of coalescence, where f 0 is a frequency characteristic of the detector, typically a 'knee' frequency, or a frequency at which S n ( f ) is a minimum. The parameters M and η are the chirp mass and symmetric mass ratio (see the definitions in Eq. (6.22)), which characterize the compact binary system under consideration. The parameters β and ζ describe the massive graviton and Lorentz-violating terms respectively. Recalling Eq. (6.7), the SNR value for the templates in Eq. (13.24) is simply where we have redefined the integrals I ( q ) from Eqs. (6.26) (written specifically for Ad. LIGO) to a more general case as with x ≡ f/f 0 . The quantity g ( x ) is the rescaled power spectral density, defined via g ( x ) ≡ S h ( f ) /S 0 for the detector in question, and S 0 is an overall constant. When computing the Fisher matrix, we will replace the amplitude A in favor of the SNR, using Eq. (14.2). This will then lead to bounds on β and ζ that depend on the SNR and on a rescaled version of the moments J ( q ) ≡ I ( q ) /I (7) . In the next sections, we will carry out the integrals in Eq. (14.3), but we will approximate the limits of integration by certain x min and x max [33]. The maximum frequency will be chosen to be the smaller of a certain instrumental maximum threshold frequency and that associated with a gravitational wave emitted by a particle in an innermost-stable circular orbit (ISCO) around a Schwarzschild black hole (BH): f max = 6 -3 / 2 π -1 η 3 / 5 M -1 . The maximum instrumental frequency will be chosen to be (10 5 , 10 3 , 1) Hz for Ad. LIGO, ET and LISA-like, respectively. The minimum frequency will be chosen to be the larger of a certain instrumental minimum threshold frequency and, in the case of a space mission, the frequency associated with a gravitational wave emitted by a test-particle one year prior to reaching the ISCO. The minimum instrumental frequency will be chosen to be (10 , 1 , 10 -5 ) Hz for Ad. LIGO, ET and a LISA-like mission, respectively. Once the Fisher matrix has been calculated, we will invert it using a Cholesky decomposition to find the variance-covariance matrix, the diagonal components of which give us a measure of the accuracy to which parameters could be constrained. Let us then define the upper bound we could place on β and ζ as ∆ β ≡ ∆ 1 / 2 /ρ and ∆ ζ ≡ ¯ ∆ 1 / 2 /ρ , where ∆ and ¯ ∆ are numbers. Combining these definitions with Eqs. (13.28) and (13.29), we find, for α = 1 , the bounds: glyph[negationslash] Notice that the direction of the bound on λ A itself depends on whether α > 2 or α < 2 ; but because A = ( λ A /h ) α -2 , all cases yield an upper bound on A . For the case α = 1 , we find In the remaining sections, we set β = 0 and ζ = 0 in all partial derivatives when computing the Fisher matrix, since we derive the error in estimating β and ζ about the nominal or a priori general relativity values, ( β, ζ ) = (0 , 0) .", "pages": [ 174, 175 ] }, { "title": "14.2 Detector Spectral Noise Densities", "content": "We model the Ad. LIGO spectral noise density via [180] Here, f 0 = 215 Hz, S 0 = 10 -49 Hz -1 , and f s = 10 Hz is a low-frequency cutoff below which S h ( f ) can be considered infinite for all practical purposes The initial ET design postulated the spectral noise density [180] where f 0 = 100 Hz, S 0 = 10 -50 Hz -1 , f s = 1 Hz , and The classic LISA design had an approximate spectral noise density curve that could be modeled via (see eg. [23, 33]): where S NSA h ( f ) = [ 9 . 18 × 10 f 1 Hz - 52 ( - 4 ) and with ∆ f = T -1 mission the bin size of the discretely Fourier transformed data for a classic LISA mission lasting a time T mission and κ glyph[similarequal] 4 . 5 the average number of frequency bins that are lost when each galactic binary is fitted out. Recently, the designs of LISA and ET have changed somewhat. The new spectral noise density curves can be computed numerically [32, 139, 213] and are plotted in Fig. 14.1, and Fig. 14.2. Notice that the bucket of the NGO noise curve has shifted to higher frequency, while the new ET noise curve is more optimistic than the classic one at lower frequencies. The spikes in the latter are due to physical resonances, but these will not affect the analysis. In the remainder of this chapter, we will use the new ET and NGO noise curves to estimate parameters.", "pages": [ 176, 177 ] }, { "title": "14.3 Results", "content": "We plot the bounds that can be placed on ζ by using different detectors in Fig. 14.3, Fig. 14.4, and Fig. 14.5 as a function of the α parameter. Fig. 14.3 corresponds to the bounds placed with Ad. LIGO and ρ = 10 ( D L ∼ 160 Mpc , Z ∼ 0 . 036 for a double neutron-star inspiral), Fig. 14.4 corresponds to ET and ρ = 50 ( D L ∼ 2000 Mpc , Z ∼ 0 . 39 for a double 10 M glyph[circledot] BH inspiral) and Fig. 14.5 corresponds to NGO and ρ = 100 ( D L ∼ 20 , 000 Mpc , Z ∼ 2 . 5 for a double 10 5 M glyph[circledot] BH inspiral). When α = 0 or α = 2 , ζ cannot be measured at all, as it becomes 100% correlated with either standard massive graviton parameters. Thus we have drawn vertical lines in those cases. As the figures clearly show, the accuracy to which ζ can be measured deteriorates rapidly as α becomes larger. In fact, once α > 4 , we find that ζ cannot be confidently constrained anymore because the Fisher matrix becomes non-invertible (its condition number exceeds 10 16 ). Attempting to constrain values of α > 5 / 3 becomes problematic not just from a data analysis point of view, but also from a fundamental one. The PN templates that we have constructed contain general relativity phase terms up to 3 . 5 PN order. Such terms scale as u 2 / 3 , which corresponds to α = 5 / 3 . Therefore, trying to measure values of α ≥ 5 / 3 without including the corresponding 4PN and higher-PN order terms is not well-justified. We have done so here, neglecting any correlations between these higher order PN terms and the Lorentz-violating terms, in order to get a rough sense of how well Lorentz-violating modifications could be constrained. The bounds on β and ζ are converted into a lower bound on λ g and and upper bound on λ A in Table 14.1 for α = 3 and binary systems with different component masses. Given a gravitationalwave detection consistent with general relativity, this table says that λ g and λ A would have to be larger and smaller than the numbers in the seventh and eight columns of the table respectively. _ In addition, this table also shows the accuracy to which standard binary parameters could be measured, such as the time of coalescence, the chirp mass and the symmetric mass ratio, as well as the correlation coefficients between parameters. Different clusters of numbers correspond to constraints with Ad. LIGO (top), New ET (middle) and NGO (bottom) (see caption for further details; specifically, notice the different units for the numbers in each section of the table) Although our results, presented in Fig. 14.3, Fig. 14.4, and Fig. 14.5, suggest bounds on ζ of O (10 3 -10 5 ) for the α = 3 case, the dimensional bounds in Table 14.1 suggest a strong constraint on λ A . This is because in converting from ζ to λ A one must divide by the D 3 distance measure. This distance is comparable to (but smaller than) the luminosity distance, and thus, the longer the graviton propagates the more sensitive the constraints are to possible Lorentz violations. Second, notice that the accuracy to which many parameters can be determined, e.g. t c , ∆ M , and ∆ η , degrades with total mass because the number of observed gravitational-wave cycles decreases. Third, notice that the bound on the graviton Compton wavelength is not greatly affected by the inclusion of an additional parameter in the α = 3 case, and is comparable to the one obtained in [267] for LIGO. In fact, we have checked that in the absence of λ A we recover Table II in [267]. We now consider how these bounds behave as a function of the mass ratio. Figure 14.6 plots the bound on the graviton Compton wavelength and Fig. 14.7 plots the Lorentz-violating _ 162 Chapter 14. Parameter Estimation in in Lorentz-Violating Gravity Ro 14.1: able T 4.03% 100 for ts, efficien co in is g λ , 10 = ρ of units in is g λ 10 of units in is 0.259% 0.00124% 4 10 NGO 0.00434% 4 10 0.0574% 5 10 1 m Detector 0.0374% 1.4 LIGO d. A 0.267% 1.4 10 10 10 ET 5 10 6 10 Compton wavelength λ A as a function of η both for Ad. LIGO and α = 3 , with systems of different total mass. Notice that, in general, both bounds improve for comparable mass systems, even though the SNR is kept fixed. With all of this information at hand, it seems likely that gravitational-wave detection would provide useful information about Lorentz-violating graviton propagation. For example, if a Bayesian analysis were carried out, once a gravitational wave is detected, and the ppE parameters peaked around b ppE = 2 or 3 , this could possibly indicate the presence of some degree of Lorentz violation. Complementarily, if no deviation from general relativity is observed, then one could constrain the magnitude of A to interesting levels, considering that no bounds exist to date.", "pages": [ 177, 178, 179, 180, 181, 182 ] }, { "title": "14.4 Conclusions and Discussion", "content": "We studied whether Lorentz symmetry-breaking in the propagation of gravitational waves could be measured with gravitational waves from non-spinning, compact binary inspirals. We considered modifications to a massive graviton dispersion relation that scale as A p α , where p is the graviton's momentum while A and α are phenomenological parameters. We found that such a modification introduces new terms in the gravitational-wave phase due to a delay in the propagation: waves emitted at low frequency, early in the inspiral, travel slightly slower than those emitted at high frequency later. This results in an offset in the relative arrival times at a detector, and thus, a frequency-dependent phase correction. We mapped these new gravitational-wave phase terms to the recently proposed ppE scheme, with ppE phase parameters b ppE = α -1 . We then carried out a simple Fisher analysis to get a sense of the accuracy to which such dispersion relation deviations could be measured with different gravitational-wave detectors. We found that indeed, both the mass of the graviton and additional dispersion relation deviations could be constrained. For values of α > 4 , there is not enough information in the waveform to produce an invertible Fisher matrix. Certain values of α , like α = 0 and 2 , also cannot be measured, as they become 100% correlated with other system parameters. In deriving these bounds, we have made several approximations that force us to consider them only as rough indicators that gravitational waves can be used to constrain generic Lorentzviolation in gravitational-wave propagation. For example, we have not accounted for precession or eccentricity in the orbits, the merger phase of the inspiral, the spins of the compact objects or carried out a Bayesian analysis. We expect the inclusion of these effects to modify and possibly worsen the bounds presented above by roughly an order of magnitude, based on previous results for bounds on the mass of the graviton [14, 34, 158, 237, 267, 275, 291]. However, the detection of N gravitational waves would lead to a √ N improvement in the bounds [36], while the modeling of only the Lorentz-violating term, without including the mass of the graviton, would also increase the accuracy to which λ A could me measured [75]. Future work could concentrate on carrying out a more detailed data analysis study, using Bayesian techniques. In particular, it would be interesting to compute the evidence for a general relativity model and a modified dispersion relation model, given a signal consistent with general relativity, to see the betting-odds of the signal favoring GR over the non-GR model. A similar study was already carried out in [75], but there a single ppE parameter was considered. Another interesting avenue for future research would be to consider whether there are any theories (quantum-inspired or not) that predict fractional α powers or values of α different from 3 or 4 .", "pages": [ 182, 183, 184 ] }, { "title": "A.1 Basic Facts", "content": "A", "pages": [ 188 ] }, { "title": "Evaluations", "content": "(b) The location of field points relative to the origin and relative to each other and, where Using the following straghtforward relations help following the procedure to obtain above results.", "pages": [ 188, 191, 192 ] }, { "title": "A.2 Multipole Moments for Two-Body Systems", "content": "Here we evaluate the multipole moments that appear in the radiation reaction expressions (10.6) and (10.7) to the order required to obtain 2 . 5 PN-accurate contributions. The scalar dipole moment I i s in Eq. (10.6) must be evaluated to 1 PN order. Substituting τ s from Eq. (8.19e) and σ s from Eq. (9.12e) to 1 PN order into Eq. (8.6a), we obtain Most of the multipole moments that appear in the 2 . 5 PN expressions (10.7) can be evaluated to the lowest PN order, so that we may write and and The exception to this rule is the scalar monopole moment M s = ∫ M τ s d 3 x ; formally it contributes at 0 . 5 PN order, as can be seen in Eq. (8.5), but its leading contribution is constant in time, and hence it is the 1 PN correction that matters. Inserting τ s and σ s from Eqs. (8.19e) and (9.12e) to 1 PN order, we obtain Since the first term is constant, it can be dropped.", "pages": [ 192, 193 ] }, { "title": "A.3 Phase of the Gravitational Waveform to 3.5PN order", "content": "The phasing expression of Eq. (6.17) was valid to 2PN order. Here we quote the full expression, which has been calculated through 3.5PN order, as in Eq. (3.18) in [61] where v = ( πMf ) 1 / 3 , and γ = 0 . 577216 · · · is the Euler constant.", "pages": [ 193 ] }, { "title": "B.1.1 Integration of 1PN potentials", "content": "At 1PN order, the integration is straightforward. We consider a particular example Since we are interested in integration over body 1, the coordinate x only needs to be integrated over body 1 instead of the entire space beacuase ρ ∗ is zero everywhere else. There is, however, no such restriction on ρ ∗ ' , and the x ' coordinate is to be evaluated over bodies 1 and 2, as these are the only bodies in the problem. Therefore, the integral splits into two integrals, one over each body. We consider these two pieces indivisually. First, the integral with both the x and x ' coordinates evaluated over body 1 is a self-integral over body 1. With v = v 1 + ¯ v ' , we have B", "pages": [ 194 ] }, { "title": "Calculations", "content": "where The integral that involves v 2 1 is zero because by symmetry the integral is automatically zero. The second integral of Eq. (B.2) is integrated with x ranging over body 1 and x ' over body 2. Let v ' = v 2 + ¯ v ' , so we have where we have approximated | x -x ' | as r , the distanse of separation between the two bodies, and n i is a unit vector in the direction of x 1 -x 2 . Note that the second term, involving v 2 · ¯ v ' integrates to zero because ∫ 2 ρ ∗ ' ¯ v ' d 3 x = 0 by the definition of center of mass. Combining Eq. (B.3) and Eq. (B.6), we have If the integral involves an additional coordinate x '' and an additional conserved density ρ ∗'' , there will be 4 integrals from all combinations of permutating ρ ∗' and ρ ∗'' between bodies 1 and 2.", "pages": [ 195 ] }, { "title": "B.1.2 Integration of 2PN potentials - part I", "content": "At 2PN order, there are many more terms to integrate in the equation of motion than at 1PN order (compare the number of terms between Eq. (10.1) and Eq. (10.2)), and it is much work to include all possible combinations of permutating the various ρ ∗ between bodies 1 and 2. Therefore, we neglect terms that involve any self-integrals, and only consider terms that involve the masses and velocities of the bodies, and the distance of separation between them. First we express the integrals in terms of the conserved densities, velocities, and the coordinates. Then, if (a) for each coordinate, a corresponding conserved density also appears, and (b) there are no 'triangle' terms in any of the denominators, then there is no problem of divergence, and the integration is straightforward. We simply associate x with body 1, then assign the coordinates x ' and x '' to either body 1 or body 2 in such a way that all distances that appear in denominators of the integral are a difference between a coordinate associated with body 1 and a coordinate associated with body 2. There should be no distance in any denominators of the integral that is a difference between two coordinates associated with the same body. Such terms would be singular in the point mass limit, and our procedure is to discard such terms. For example, where in the first example both x ' and x '' are associated with body 2, but in the second example, x ' is associated with body 2 while x '' is associated with body 1.", "pages": [ 195, 196 ] }, { "title": "B.1.3 Integration of 2PN potentials - part II", "content": "Not all terms in Eq. (10.2) can be integrated by the method described in the last section. Potentials such as P ij 2 and related potentials such as H , G 1 , G 2 have to be integrated with the use of the integral Using this integral, integration of P ij 2 ,k becomes where a = x -x '' , b = x -x ''' , c = x '' -x ''' , A = + | a | + | b || c | , and the hat notation denotes a unit vector. We then integrate Eq. (B.11) over all possibilities by associating the coordinates x '' and x ''' with bodies 1 and 2 in turn, and keep only finite terms. The coordinate x is associated with body 1, so when the coordinates x '' and x ''' are both assigned to body 1, the result is a self integral, which we discard. Now, if we assign the coordinate x '' to body 1 and x ''' to body 2, then a → 0 , b = r , and c = r , where r = x 1 -x 2 . Special care is needed in taking the limit of | a |→ 0 as ther emay be finite terms associated with the limit. For example, let | a | = ε glyph[lessmuch]| r | and r = | r | and consider the term | a | -1 A -2 ( δ ik -ˆ a i ˆ a k )( ˆ b j +ˆ c j ) , where we have discarded terms that diverge as ε -1 as well as terms that tend to zero in the limit of ε → 0 . Note that although, to the leading order, the term | a | -1 A -2 ( δ ik -ˆ a i ˆ a k )( ˆ b j +ˆ c j ) diverges as ε -1 as ε → 0 , there is a finit contribution to the integral that we could not have obtained had we naively, and incorrectly, discarded the entire term. Repeating the process of associating the coordinates x '' and x ''' to bodies 1 and 2 , the final result of ∫ P ij 2 ,k becomes where the first line of Eq. (B.13) is obtained by associating x '' with body 1 and x ''' with body 2 , so that a → 0 , b = r , and c = r . The second line of the equation is obtained by associating x '' with body 2 and x ''' with body 1 , so that a = r , b → 0 , and c = -r . Finally, the third line of the equation is obtained by associating both x '' and x ''' with body 2, so that a = r , b = r , and c → 0 . For potentials P ij 2 ,k , the procedure of integration is similar. One difference is that, once Eq. (B.10) is used to simplify the integral, we may need to take derivatives with respect to different coordinates than the ones taken in P ij 2 ,k . Another difference is that many integrals have quantities such as v '' , and so we assign them to the appropriate bodies consistent with the assignment of the associated coordinates, for example, ( x '' for the case with v '' ).", "pages": [ 196, 197, 198 ] }, { "title": "B.1.4 Integration of 2PN potentials - part III", "content": "Of all the potentials at 2PN order, the potential H ≡ P ( U ij P ij 2 ) most difficult to integrate it is 'doubly triangular'. Although the principle of integration is same as that in Appendix A.1, there is no closed form expression such as Eq. (B.10) that simplifies the 'triangular' potentials, and the integration must be carried out on mathematical software. We first simplify H by partial integration where we have used the formula The first term of Eq. (B.14) vanishes because it can be converted to surface integral at infinity. So the 2PN potential that we need to integrate becomes The second term of Eq. (B.16) involving U ,k Φ 2 ,k can be integrated using the methods of Appendix A.1, but all other terms in the equation have to be integrated using a mathematical software. As an example we consider the third term of Eq. (B.16) where y 1 = x ' -x 1 and y 2 = x ' -x 2 , and we use a point mass expression for U . Since the coordinate x ' is not assigned to either body, the potentials U and U ,k , which are functions of x ' , are then the sum of the potential form each body. Of the six terms in Eq. (B.17), the first term involves solely y 1 and is a self-integral, which we discard. The second term becomes where we first note that since x is assigned to body 1 , ( x -x ' ) j / | x -x ' | 3 is simply -y j 1 / | y 1 | 3 . We next perform a partial integration and discard the surface term. The partial integration is not always necessary and is done in this case to avoid having Maple crash. We then make the substitution z = | y 2 | and express | y 1 | in terms of z and the distance of separation between the two bodies r = | x 1 -x 2 | . We also choose the ( z, θ, φ ) spherical coordinate system such that the θ = 0 axis is parallel to x 1 -x 2 , and integrate over the azimuthal angle φ . Since the integral in Eq. (B.18) must in the end be proportional to n (the only vector in the problem), we need only to evaluate the projection of the integral onto the z -axis. This integral can be done either by hand or using a mathematics software package, and we used Maple 14 for our calculations. We first integrate over u from -1 to 1 ; because the result contains terms proportional to ( z 2 -r 2 ) -k , we then integrate z from ε to r and from r to infinity. While one can show that the integral over z ∼ r is non-singular, splitting the integral avoids having Maple crash. We then expand the result in powers of ε , and discard all terms proportional to ε -2 , ε -1 , and ε k , and keep only the terms independent of ε . For the integral in Eq. (B.18), we obtain the following result, Therefore, the contribution of the integral in Eq. (B.18) to Eq. (B.17) is zero. As another example, we consider the third term of Eq. (B.17). where we make similar substitutions as the previous example, except this time we let z = | y 1 | , and note that y 1 · y 2 = z · ( z + r ) = z 2 = rz cos θ because r is projected onto the z axis. Note that integration by parts is not necessary for this example, and that the final result is exact. Since we only keep terms that are independent of ε , the contribution of this terms to Eq. (B.17) is -(3 / 5) m 2 1 m 2 n j /r 4 . Repeating this procedure for all terms in Eq. (B.17), we obtain the final result", "pages": [ 198, 199, 200 ] }, { "title": "B.2 Results of Integration of 2PN Potentials", "content": "In this section we give the results of integration of 2PN potentials that appear in Chapter 8 and Chapter 9. The integrated results of individual terms of Eq. (10.2) are: -1 m 1 ∫ 1 ρ ∗ [1 -ζ -(2 λ 1 + ζ )(1 -2 s )] v 2 Φ s 2 s,j d 3 x = [1 -ζ -(2 λ 1 + ζ )(1 -2 s 1)](1 -2 s 1 )(1 -2 s 2 ) m 1 m 2 v 2 1 n j r 3 , (B.39) 4 m - 1 ∫ 1 - - 1 r v 3 ∗ i s i 2 s,j 3 ( v · 2 ) j ρ v UV d x = 4(1 2 s 1 )(1 2 s 2 ) m 1 m 2 n , (B.73) 2 m 1 ∫ 1 - - - - v r 2 2 3 ρ ∗ (1 2 s ) U Φ s 1 ,j 3 d x = 2(1 2 s 1 )(1 2 s 2 ) m 2 2 j n , (B.93) 4 1 m 1 ∫ 1 ρ ∗ [1 -ζ +(2 λ 1 + ζ )(1 -2 s )] [Σ s (Σ( a s U s ))] ,j d 3 x = -4[1 -ζ +(2 λ 1 + ζ )(1 -2 s 1 )] a s 1 (1 -2 s 2 ) 2 m 1 m 2 2 n j r 4 , (B.132) -4 m 1 ∫ 1 ρ ∗ H s,j d 3 x = -4 { 1 5 (1 -2 s 1 )( s 1 -s 2 ) m 2 1 m 2 n j r 4 +2(1 -2 s 1 )(1 -2 s 2 ) m 1 m 2 2 n j r 4 + 1 4 (1 -2 s 2 ) 2 m 3 2 n j r 4 } , (B.147) 4 m 1 ∫ 1 ρ ∗ [2 λ 1 (1 -2 s )+ ζ (2 a s -s -2 s 2 )] UU s U s,j d 3 x = -4[2 λ 1 (1 -2 s 1 )+ ζ (2 a s 1 -s 1 -2 s 2 1 )] (1 -2 s 2 ) 2 m 3 2 n j r 4 , (B.157) + ζ 2 (2 a s +12 s a s -4 b s -s -2 s 2 -8 s 3 )] U 2 s U s,j d 3 x = [(8 λ 2 1 -2 ζλ 1 -2 λ 2 )(1 -2 s 1 ) +6 λ 1 ζ (2 a s 1 -s 1 -2 s 2 1 ) + ζ 2 (2 a s 1 +12 s 1 a s 1 -4 b s 1 -s 1 -2 s 2 1 -8 s 3 1 )] (1 -2 s 2 ) 3 m 3 2 n j r 4 , (B.158) (B.166)", "pages": [ 200, 201, 204, 206, 210, 211, 212, 213 ] } ]

[ { "title": "Department of Physics", "content": "Dissertation Examination Committee: Clifford M. Will, Chair Francesc Ferrer, Co-Chair Mark G. Alford Gregory Comer Ram Cowsik Renato Feres", "pages": [ 1 ] }, { "title": "Star Clusters and Dark Matter as Probes of the Spacetime Geometry of Massive Black Holes", "content": "by Laleh Sadeghian Doctor of Philosophy in Physics Washington University in St. Louis, 2013 Professor Clifford Will (Chair) and Professor Francesc Ferrer (Co-Chair) This thesis includes two main projects. In the first part, we assess the feasibility of a recently suggested strong-field general relativity test, in which future observations of a hypothetical class of stars orbiting very close to the supermassive black hole at the center of our galaxy, known as Sgr A glyph[star] , could provide tests of the so-called no-hair theorem of general relativity through the measurement of precessions of their orbital planes. By considering how a distribution of stars and stellar mass black holes in the central cluster would perturb the orbits of those hypothetical stars, we show that for stars within about 0.2 milliparsecs (about 6 light-hours) of the black hole, the relativistic precessions dominate, leaving a potential window for tests of no-hair theorems. Our results are in agreement with N-body simulation results. In the second part, we develop a fully general relativistic phase-space formulation to consider the effects of the Galactic center supermassive black hole Sgr A glyph[star] on the dark-matter density profile and its applications in the indirect detection of dark matter. We find significant differences from the non-relativistic result of Gondolo and Silk (1999), including a higher density for the spike and a larger degree of central concentration. Having the dark matter profile density in the presence of the massive black hole, we calculate its perturbing effect on the orbital motions of stars in the Galactic center, and find that for the stars of interest, relativistic effects related to the hair on the black hole will dominate the effects of dark matter.", "pages": [ 14 ] }, { "title": "Acknowledgements", "content": "In my PhD program I have had the great opportunity to work with two advisors which I will always treasure the lessons that I learned form both of them. Foremost, I would like to express my sincere gratitude to my advisor Prof. Clifford M. Will for the continuous support of my PhD study and research, his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I should also mention that his sense of humor was always appreciated. Overally I could not have imagined having a better advisor and mentor for my PhD study and I simply do not have the words to thank him enough. Also, my sincere gratitude and heartfelt thanks goes to my other advisor Professor Francesc Ferrer for his continuous support, guidance and encouragement. his unflagging enthusiasm and energy impressed me all the time and I have always felt very lucky and fortunate for having him as my advisor. I have benefitted a lot from his knowledge and experience and I am very grateful to him because of his generosity with his time. I wish to thank the members of my dissertation committee, Prof. Mark Alford, Prof. Ram Cowsik, Prof. Gregory Comer and Prof. Renato Feres for their time, guidance and helpful comments and suggestions. I am also very grateful to all the faculty, staff and graduate students in the Department of Physics at Washington University for providing a very calm and friendly atmosphere and my special thanks goes to Sai Iyer who has been very kind and patient and always willing to lend his service whenever I approached him. I acknowledge and appreciate him for all of his helps. I would also like to thank Claud Bernard, Luc Blanchet, Joe Silk, David Merritt, Scott Hughes, K. G. Arun, Ryan Lang and Daniel Hunter for their helpful comments and discussions during this work. I would never have achieved what I have achieved without the unconditional love and support I have received from my parents, Soraya and Mohammad, and my siblings, Nadia, Nahid, and Shahin. Finally, I am infinitely grateful for the love and support I have gotten from my husband Saeed who has been also a great officemate and colleague for me. The research presented in this thesis was supported in part by the National Science Foundation, Grant Nos. PHY 06-52448, 09-65133, 12-60995 & 0855580, the U.S. DOE under contract No. DE-FG02-91ER40628, the National Aeronautics and Space Administration, Grant No. NNG06GI60G, and the Centre National de la Recherche Scientifique, Programme Internationale de la Coopération Scientifique (CNRS-PICS), Grant No. 4396. I also gratefully acknowledge the Institut d'Astrophysique de Paris and University of Florida for their hospitality during the completion part of my research.", "pages": [ 12 ] }, { "title": "1.1 Black Holes", "content": "The simplest description of black holes says a black hole is a region of spacetime from which gravity prevents anything, including light, from escaping. It is an object created when a massive star collapses to a size smaller than twice its geometrized mass, thereby creating such strong spacetime bending that its interior can no longer communicate with the external universe. Black holes were first predicted using solutions of the equations of General Relativity (GR); these equations predict specific properties for their external geometry. If the black hole is non-rotating, then its exterior metric is be that of Schwarzschild, which is the exact, unique, static and spherically symmetric solution of Einstein's equation in vacuum. In Schwarzschild coordinates, the line element for the Schwarzschild metric has the form where G is Newton's constant and we use units in which c = 1 . The surface of the black hole, i.e., the horizon, is located at r = 2 Gm . Only the region on and outside the black hole's surface, r ≥ 2 Gm , is relevant to external observers. Events inside the horizon can never influence the exterior. In that region of spacetime, r glyph[greatermuch] 2 Gm , where the geometry is nearly flat, Newton's theory, d v / d t = ∇ Φ( r ) , where Φ( r ) is the Newtonian gravitational potential, can be obtained from the approximate line element For Schwarzschild metric, in the limit r glyph[greatermuch] 2 Gm , Φ( r ) = -Gm/r . Consequently, m is the mass that governs the Keplerian motions of test masses in the distant, Newtonian gravitational field and we can call m in Eq. (1.1) Keplerian mass of the black hole. If the black hole is rotating with angular momentum J , its exterior geometry is given by the Kerr metric. The Kerr metric is given in Boyer-Lindquist coordinates, which are a generalization of Schwarzschild coordinates, by where a is the Kerr parameter, related to the angular momentum J by a ≡ J/m ; Σ 2 = r 2 + a 2 cos 2 θ , and ∆ = r 2 + a 2 -2 Gmr . We will assume throughout that a is positive. Just as the electromagnetic potentials Φ and A i of a charge and current distribution can be expanded in a sequence of multipole moments (dipole, quadrupole, magnetic dipole, etc), so too can part of the exterior metric of the Kerr black hole. In a coordinate system that is a variant of the Boyer Lindquist coordinates, the 00 and 0 φ components of the Kerr metric describing the exterior of a rotating black hole can be expanded as where Φ = (1 + g 00 ) / 2 , and A φ = -g 0 φ / 2 sin 2 θ . The quantities Q glyph[lscript] and J glyph[lscript] are mass and current multipole moments respectively and P glyph[lscript] (cos θ ) and ˜ P glyph[lscript] (cos θ ) are suitable angular functions. The zero degree mass moment is equal to the mass of the black hole, Q 0 = m , and the degree one current moment is its angular momentum, J 1 = J .", "pages": [ 16, 17 ] }, { "title": "1.1.1 The Black Hole No-Hair Theorem", "content": "One important property of black holes predicted by GR is commonly known as the no-hair theorem . The no-hair theorem states that, once a black hole achieves a stable condition after formation, it has only three independent physical properties: mass m , angular momentum J , and charge Q . The exterior geometry of a black hole is completely governed by these three parameters. In fact, any two black holes that share the same values for these parameters are indistinguishable. It is widely agreed that processes involving the matter in which they are embedded will rapidly neutralize astrophysical black holes, and so from now on, we only consider neutral black holes, Q = 0 . The no-hair theorem establishes the claim that black holes are uniquely characterized by their mass m and spin J , i.e., by only the first two multipole moments of their exterior spacetimes [2-6]. As a consequence of the no-hair theorem, all higher-order moments are already fully determined and turn out to obey the simple relation [7, 8] where a ≡ J/m is the spin parameter, and the multipole moments are written as a set of mass multipole moments Q glyph[lscript] which are nonzero for even values of glyph[lscript] and as a set of current multipole moments J glyph[lscript] which are nonzero for odd values of glyph[lscript] . The specific relation that we are going to use in testing the no-hair theorem is, for glyph[lscript] = 2 :", "pages": [ 17, 18 ] }, { "title": "1.2 The Massive Black Hole at the Galactic Center", "content": "Observation indicates that most galaxies contain a massive compact dark object in their centers whose mass lies in the range 10 6 M glyph[circledot] < m < few × 10 9 M glyph[circledot] [9, 10]. It is widely believed that these dark objects are Massive Black Holes (MBHs), and that they exist in the centers of most, if not all galaxies. Their number density and mass scale are broadly consistent with the hypothesis that they are now-dead quasars, which were visible for a relatively short time in their past as extremely luminous Active Galactic Nuclei (AGN), powered by the gravitational energy released by the accretion of gas and stars [11]. It is also possible that low-mass MBHs like the one in the Galactic center (GC) have acquired most of their mass by mergers with other black holes. Some present-day galaxies have AGN, although none as bright as quasars. However, most present-day galactic nuclei are inactive, which implies that accretion has either almost ceased or switched to a non-luminous mode. Their inactivity is not due to the lack of gas supply; most galaxies have more than enough to continue powering an AGN. The 'dimness problem' is one of the key issues of accretion theory, which deals with the physics of flows into compact objects. The MBH in the center of Milky Way is the nearest example of a central galactic MBH. It was first detected as an unusual non-thermal radio source, Sagittarius A glyph[star] (Sgr A glyph[star] ). Over the following decades, observations across the electromagnetic spectrum, together with theoretical arguments, established with ever-growing confidence that Sgr A glyph[star] is at the dynamical center of the Galaxy and that it is associated with a very massive and compact dark mass concentration. This has ultimately led to the nearly inescapable conclusion that the dark mass is a black hole. The Galactic MBH is quite normal. Like most MBHs, it is inactive. With m ∼ (3 -4) × 10 6 M glyph[circledot] , it is one of the least massive MBHs discovered. What makes it special is its proximity. At ∼ 8 kpc ( 1 ps = 3 . 26 light years ) from the Sun, the Galactic black hole is ∼ 100 times closer than the MBH in Andromeda, the nearest large galaxy, and ∼ 2000 times closer than galaxies in Virgo, the nearest cluster of galaxies. For this reason it is possible to observe today the stars and gas in the immediate vicinity of the Galactic MBH at a level of details that will not be possible for any other galaxy in the foreseeable future. In spite of its relative proximity, observations of the GC are challenging due to strong, spatially variable extinction by interstellar dust, which is opaque to optical-UV wavelengths. As a result, observation of the GC must be conducted in the infrared. Using the highest angular resolution obtained at near-infrared wavelength at mid 1990s, a large population of faint stars orbiting the center of the Galaxy was discovered [12-14]. The orbital periods of these stars are on the scale of tens of years and since the initial discovery, one of these stars has been observed to make a complete orbit around the center. The detection of stars orbiting the dynamical center of the Galaxy has given us quantitative information about the mass, size and position of the dark mass at the center and has confirmed the idea that we have a MBH at the Galactic center. Inside ∼ 0 . 04pc , there are no bright giants, and only faint blue stars are observed with orbital periods on the scale of tens of years. This population is known as the 'S-stars' or 'S-cluster', after their identifying labels. Deep near-IR photometric and spectroscopic observations of that region were all consistent with the identification of these stars as massive main sequence stars. There is no indication of anything unusual about the S-stars, apart from their location very near the MBH. Because of the huge mass ratio between a star and the MBH, stars orbiting near it, are effectively test particles. This is to be contrasted with the gas in that region, which can be subjected to non-gravitational forces due to thermal, magnetic or radiation pressure. These can complicate the interpretation of dynamical data and limit its usefulness. The term 'near' is taken here to mean close enough to the MBH so that the gravitational potential is completely dominated by it, but far enough so that the stars can survive, i.e. beyond the MBH event horizon, or beyond the radius where stars are torn apart by the black hole's tidal gravitational field. In this range, stars directly probe the gravitational field of the MBH. The event horizon of the MBH in the GC is much smaller than the orbital radius for the stars that have been observed to date, and so effects due to GR lead to deviations from Newtonian motion that are unmeasurable at present. To first order, the stellar orbits can be treated as Keplerian, which substantially simplifies the analysis. However, with accurate enough astrometric observations it may be possible to detect post-Newtonian effects in the orbits and to probe GR. We will discuss this more specifically in the next section in the context of testing the no-hair theorem, and with more details in Chapter 2.", "pages": [ 18, 19, 20 ] }, { "title": "1.3 Testing the Black Hole No-Hair Theorem at the Galactic Center", "content": "There seems to be every expectation that, with improved observing capabilities, a population of stars closer to the MBH than the S-stars, will eventually be discovered, making orbital relativistic effects detectable. This makes it possible to consider doing more than merely detect relativistic effects, but rather to provide the first test of the black hole no-hair theorem, which demands that Q 2 = -J 2 /m , to see if the central dark mass at the GC is truly a GR black hole. If the black hole were non-rotating ( J = 0 ), then its exterior would be that of Schwarzschild, and the most important relativistic effect would be the advance of the pericenter. If it is rotating, then two new phenomena occur, the dragging of inertial frames and the effects of the hole's quadrupole moment, leading not only to an additional pericenter precession, but also to a precession of the orbital plane of the star. These precessions are smaller than the Schwarzschild effect in magnitude because they depend on the dimensionless angular momentum parameter χ = a/ ( Gm ) = J/ ( Gm 2 ) , which is always less than one, and because they fall off faster with distance from the black hole. However, accumulating evidence suggests that the MBH should be rather rapidly rotating, with χ larger than 0 . 5 and possibly as large as 0 . 9 , so these effects could be significant. It has been suggested that if a class of stars were to be found with orbital periods of fractions of a year, and with sufficiently large orbital eccentricities, then the frame-dragging and quadrupoleinduced precessions could be as large as 10 µ arcsecond per year [15]. The precession of the orbital plane is the most important effect in testing the no-hair theorem, because it depends only on J and Q 2 ; the Schwarzschild part of the metric affects only the pericenter advance because its contributions are spherically symmetric, and thus cannot alter the orbital plane. In order to test the no-hair theorem, one must determine five parameters: the mass of the black hole, the magnitude and two angles of its spin, and the value of the quadrupole moment. The Kepler-measured mass is determined from the orbital periods of stars, but may require data from a number of stars to fix it separately from any extended distribution of mass. Then to measure J and Q 2 , it is necessary and sufficient to measure precessions in the orbital planes for two stars in non-degenerate orbits. Detecting such stars so close to the black hole, and carrying out infrared astrometry to 10 µ arcsec per year accuracy will be a challenge. However, if this challenge can be met with future improved adaptive optics systems currently under study, such as GRAVITY [16], it could lead to a powerful test of the black hole paradigm.", "pages": [ 20, 21 ] }, { "title": "1.4 Complications in Testing the Black Hole No-Hair Theorem", "content": "As we discussed, observations of the precessing orbits of a hypothetical class of stars very near the MBH in the GC could provide measurements of the spin and quadrupole moment of the hole and thereby test the no-hair theorem of GR. However, in assessing the feasibility of such strong-field GR tests, one must inevitably address potential complications, notably the perturbing effect of the other stars that may also reside in a cluster close to the black hole and a possible distribution of dark matter (DM) particles in the GC. These perturbing effects will be the focus of this thesis, and will be detailed in Chapters 2 and 3.", "pages": [ 21 ] }, { "title": "1.4.1 Perturbing Effects of Stars in the Surrounding Cluster", "content": "N -body simulations, have shown that for a range of possible stellar and stellar-mass black hole distributions within the central few milliparsecs (mpc) of the black hole, there could exist stars in eccentric orbits with semi-major axes less than 0 . 2 milliparsecs for which the orbital-plane precessions induced by the stars and black holes would not exceed the relativistic precessions [17]. These conclusions were gleaned from thousands of simulations of clusters ranging from seven to 180 stars and stellar mass black holes orbiting a 4 × 10 6 M glyph[circledot] maximally rotating black hole, taking into account the long-term evolution of the system as influenced by close stellar encounters, dynamical relaxation effects, and capture of stars by the black hole. In Chapter 2, we study the extent to which the conclusions of these complex N -body simulations can be understood, at least within an order of magnitude, using analytic orbit perturbation theory. After a brief review of orbit perturbation theory, we calculate the average change in the orientation of the orbital plane of a given 'target' star orbiting the massive black hole, as determined by its inclination and ascending node angles i and Ω , induced by the Newtonian gravitational attraction of a distant third star (which could be either inside or outside the target star's orbit). The perturbing accelerations are expanded in terms of multipoles through glyph[lscript] = 3 . We then calculate the root-mean-square variation of each orbit element, averaged over all possible orientations of the perturbing star's orbit, and averaged over a distribution of orbits in semi-major axis and eccentricity, arguing that this will give an estimate of the 'noise' induced by the graininess of the otherwise spherically symmetric perturbing environment. Our analytic estimates of this 'noise' will turn out to be consistent with the results from the N-body simulations, and will demonstrate that, for a range of possible distributions of stars in the central region, a test of the no-hair theorem will still be possible.", "pages": [ 21, 22 ] }, { "title": "1.4.2 Perturbing Effects of Dark Matter", "content": "Another perturbing factor which can cause precessions in stellar motions is DM. To study the effect of DM on stellar motions in the GC, we need to have the DM density in that region. In order to derive an accurate density profile of DM particles in the GC, the effect of the MBH on the DMparticles distribution, should be taken into account. Calculations by Gondolo and Silk ([18], GS hereafter) have shown that for a pre-existing cusped DM halo, adiabatic (i.e. slow) growth of the MBH pulls the DM particles into a dense 'spike'. The calculation in GS was based on a Newtonian analysis, with some relativistic effects introduced in an ad hoc fashion, but because of the strong gravitational field near the MBH, a more reliable and realistic prediction for the DM density profile demands a fully general relativistic calculations. In Chapter 3, we report the first, fully relativistic calculation of the density profile of DM particles near a Schwarzschild black hole in the adiabatic growth model. We find significant differences with the conclusions of GS very close to the hole, but we are in complete agreement with them at large distances. We use these relativistically correct density distributions to calculate the perturbing effect of the DM distribution on stellar motion in the GC for the hypothetical target stars to test the no-hair theorem and also the for S2 star in the S-stars cluster. The perturbing effect of the DM distribution depends on whether or not the dark-matter particles self-annihilate. The DM density distribution and therefore its perturbing effect also depends on whether the DM particle can self-annihilate or not. We will show that the perturbing effects of the DM mass distribution are too small to affect the possibility of testing the no-hair theorems using stars very close to the black hole.", "pages": [ 22 ] }, { "title": "1.5 Dark Matter Evidence and Distribution", "content": "We observe some 'anomalies' in astrophysical systems, with sizes ranging from sub-galactic to cosmological scales, that can be explained by assuming the existence of a large amount of unseen, DM. Therefore, DM can be studied in different scales. In the following, we review the evidence for DM at these different scales although we will be primarily interested in the sub-galactic domain.", "pages": [ 23 ] }, { "title": "1.5.1 Galaxy Cluster and Galactic Scales", "content": "Agalaxy cluster gave the first evidence of DM. In 1933, F. Zwicky [19] calculated the gravitational mass of the galaxies within the Coma cluster using the observed velocities of outlying galaxies and obtained a value more than 400 times greater than expected from their luminosity, which his interpretation was that most of the matter controlling the motion of the galaxies must be dark. Today, using the modern value of the Hubble constant and taking into account that there is baryonic gas in the galaxy cluster, bring down the amount of DM to 25 times the baryonic matter which still makes it clear that the great majority of matter appears to be dark. The most convincing and direct evidence for the DM existance on galactic scales, comes from the observations of the rotation curves of galaxies, namely the graph of circular velocities of stars and gas as a function of their distance from the galactic center. Observed rotation curves usually exhibit a characteristic flat behavior at large distances, i.e. out towards, and even far beyond, the edge of the visible disk. Fig. 1.1 is a typical example [1]. In Newtonian dynamics the circular velocity is expected to be where as usual, m ( r ) = 4 π ∫ ρ ( r ) r 2 d r , and ρ ( r ) is the mass density profile. If ρ vanishes outside the visible disk, then m ( r ) is constant beyond the visible disk, and v ( r ) should be falling as 1 / √ r . The fact that the observed v ( r ) is approximately constant implies the existence of a halo with m ( r ) ∝ r and a mass density profile closely resembling that of an isothermal sphere, i.e., ρ ∝ 1 /r 2 at distances of few kiloparsec. Although there is a consensus about the shape of DM halos at intermediate distances, DM distribution is unclear in the innermost regions of galaxies. The observed rotation velocity associated with DM in the inner parts of disk galaxies is found to rise approximately linearly with radius which leads to mass ∝ r 3 and therefore constant density. This solid-body behavior can be interpreted as indicating the presence of a central core in the DM distribution, spanning a significant fraction of the visible disk [20]. Observations of dwarf spheroidal galaxies also seem to favor a constant density of DM in the inner parts [21]. On the other hand, N-body simulations indicate a steep power-law-like behavior for the DM distribution at the center. The results of N-body simulations are based on the ( Λ )CDM paradigm, where the most of the mass-energy of our universe consists of collisionless cold dark matter (CDM) in combination with a cosmological constant Λ . This Λ CDM paradigm provides a comprehensive description of the universe at large scales. However, despite its great successes, it should be kept in mind that the cusp and the central DM distribution are not predicted from first principles by Λ CDM. Rather these properties are derived from analytical fits made to darkmatter-only numerical simulations. While the quality and quantity of these simulations has improved by orders of magnitude over the years, there is as yet no 'cosmological theory' that explains and predicts the distribution of DM in galaxies from first principles. In the early 1990s, the first results of numerical N-body simulations of DM halos based on the collisionless cold dark matter (CDM) prescription became available. 'Cold' dark matter is dark matter composed of constituents with a free-streaming length much smaller than the ancestor of a galaxy-scale perturbation. These did not show the observed core-like behavior in their inner parts, but were better described by a steep power-law mass density distribution, the so-called cusp . The presence of a cusp in the center of a CDM halo is one of the earliest results derived from cosmological N-body simulations. The first simulations indicated an inner distribution ρ ∼ r α with α = -1 [22]. They did not rule out the existence of central cores, but noted that these would have to be smaller than the resolution of their simulations ( ∼ 1 . 4 kpc ). Subsequent simulations, at higher and higher resolutions, made the presence of cores in simulated CDM halos increasingly unlikely. In addition to finite resolution, the other limitation of N-body simulations is that the role of the baryons at small radius is ignored in their calculations. A systematic study by Navaro et al. [23, 24] of simulated CDM halos, derived assuming many different sets of cosmological parameters, found that the innermost DM density distribution could be well described by a characteristic α = -1 slope for all simulated halos, independent of mass and size. A similar general result was found for the outer mass profile, with a steeper slope of α = -3 : where ρ 0 is related to the density of the universe at the time of halo collapse and a is the characteristic radius of the halo. This kind of profile is also known as the 'NFW profile'. In Chapter 3, we will consider the constant and Hernquist distribution functions as examples of cored and cuspy models, respectively. The advantage of considering the Hernquist density profile which, like the NFW profile, is ∝ 1 /r for small r , is that for the Hernquist model we have a closed analytical distribution function which allows us to study the effect of adiabatic growth of the MBH on the DM distribution using adiabatic invariants.", "pages": [ 23, 24, 25 ] }, { "title": "1.5.2 Cosmological Scales", "content": "As we have seen, on distance scales of the size of galaxies and clusters of galaxies, the evidence of DM appears to be compelling. Despite this, the observations discussed do not allow us to determine the total amount of DM in the Universe. The theory of Big Bang nucleosynthesis gives a good estimate of the amount of ordinary (baryonic) matter at around 4 - 5 percent of the critical density (the density required to have a universe with a flat spatial section); while evidence from large-scale structure and other observations indicates that the total matter density is substantially higher than this [25]. The Cosmic Microwave Background (CMB) fluctuations imply that at present the total energy density is equal to the critical density. This means that the largest fraction of the energy density of the universe is dark and nonbaryonic. It is not yet clear what constitutes this dark component. Combining the data on CMB, large scale structure, gravitational lensing and high-redshift supernovae, it appears that the dark component is a mixture of two types of constituents. More precisely, it is composed of dark matter and dark energy. The cold dark matter has zero pressure and can cluster, contributing to gravitational instability, but it does not emit light, which means that it does not have electromagnetic interactions. Various (supersymmetric) particle theories provide us with natural candidates for the cold dark matter, among which Weakly Interacting Massive Particles (WIMPs) are the most favored at present. The nonbaryonic cold dark matter contributes only about 25 percent of the critical density. The remaining 70 percent of the missing density comes in the form of nonclustered dark energy with negative pressure. It may be either a cosmological constant ( pressure = -energy density ) or a scalar field (quintessence) with pressure = ω × energy density , where ω is less than -1 / 3 today [26].", "pages": [ 25, 26 ] }, { "title": "1.6 Dark Matter Candidates", "content": "The evidence for non-baryonic DM is compelling at all observed astrophysical scales. Candidates for nonbaryonic DM are hypothetical particles such as axions, or supersymmetric particles. The most widely discussed models for nonbaryonic DM are based on the cold dark matter hypothesis, and the corresponding particle is most commonly assumed to be for instance a WIMP. WIMPs interact through a weak-scale force and gravity, and possibly through other interactions no stronger than the weak force. Because of their lack of electromagnetic interaction with normal matter, WIMPs would be dark and invisible through normal electromagnetic observations and because of their large mass, they would be relatively slow moving and therefore cold. Their relatively low velocities would be insufficient to overcome their mutual gravitational attraction, and as a result WIMPs would tend to clump together. Although WIMPs are a more popular DM candidate, there are also experiments searching for other particle candidates such as axions. The axion is a hypothetical elementary particle postulated to resolve the strong CP problem in quantum chromodynamics. Observational studies to detect DM axions through the products of their decay are underway, but they are not yet sufficiently sensitive to probe the mass regions where axions would be expected to be found if they are the solution to the DM problem.", "pages": [ 26 ] }, { "title": "1.7 Indirect Detection of Dark Matter", "content": "Indirect dark matter searches measure the annihilation and/or decay products of DM from astrophysical systems. Schematically, they measure the rate for DM DM → SM SM or DM → SM SM , depending on whether dark matter particles annihilate or decay where DM represents the dark matter particle and SM represents any standard model particle. In many instances, the particle represented by SM is unstable, and decays into other particles (for example, photons or neutrinos) that are observable in detectors. In order to best interpret the results from indirect searches, we must have a good idea as to both how the dark matter is distributed in halos, and what standard model particles the dark matter preferentially annihilates or decays into. One of the main possibilities for indirect detection of DM particles is to search for high-energy gamma rays, positrons, antiprotons, or neutrinos produced by WIMP pair annihilations in the Galactic halo. In particular, the flux of gamma rays in a given direction is proportional to the square of the DM particle density and since the DM density is expected to be largest towards the Galactic center, the flux of such exotic gamma rays should be highest in that direction. In other words, the innermost region of our galaxy is one of the most promising targets for the indirect detection of DM and it is important that we know the DM density profile in the vicinity of the Galactic center MBH. In Chapter 3, to study the effect of the MBH, we developed a fully general relativistic phase-space formulation, allowed the central black hole to grow adiabatically, holding the general relativistic adiabatic orbital invariants fixed, and incorporated a relativistically correct condition for particle capture by the black hole. The result showed significant differences with the semi-relativistic result of Gondolo and Silk [18], including a bigger spike in the halo density close to the black hole. Finally having the dark matter profile density in presence of the MBH, we also calculated its perturbing effect on the orbital motions of stars in the Galactic center. 2", "pages": [ 27, 28 ] }, { "title": "Testing the Black Hole No-Hair Theorem at the Galactic Center", "content": "In this chapter we start with the well-known Kepler problem to introduce the notation and review the necessary equations which need to be generalized to the non-spherical cases in order to study Keplerian orbits in space. Then we introduce the basic equations of orbit perturbation theory and derive the general relativistic effects of the central massive black hole on the orbits of stars as one of the applications of this theory. This provides the test of the no-hair theorem in the innermost region of the galactic center. Then we study the perturbing effect of a distribution of stars on the orbit of a target star.", "pages": [ 28 ] }, { "title": "2.1.1 The Kepler Problem", "content": "The simplest Newtonian problem is that of two 'point' masses in orbit about each other, frequently called the 'Kepler problem'. In Kepler's problem, we have a body of mass m 1 , position r 1 , velocity v 1 = d r 1 / d t , and acceleration a 1 = d v 1 / d t , and a second body of mass m 2 , position r 2 , velocity v 2 = d r 2 / d t , and acceleration a 2 = d v 2 / d t . We place the origin of the coordinate system at the center of mass, so that m 1 r 1 + m 2 r 2 = 0 . The position of each body is then given by in which m ≡ m 1 + m 2 is the total mass and r ≡ r 1 -r 2 the separation between bodies. Similar relations hold between v 1 , v 2 , and the relative velocity v ≡ v 1 -v 2 = d r / d t . For the relative acceleration a ≡ v 1 -v 2 = d v / d t we have where r ≡ | r | is the distance between the bodies, and ˆ n ≡ r /r , is a unit vector that points from body 2 to body 1. The total energy and the angular momentum of the system are given by is the reduced mass of the system. It is simple to verify explicitly using Eq. (2.2) that d E/ d t = 0 and d L / d t = 0 . The constancy of E and L are a result of the fact that the potential Gm/r that governs the effective one-body problem of Eq. (2.2) is static and spherically symmetric. The constancy of L implies that all the motion lies in a plane perpendicular to L and it is fixed. So, we are free to choose our coordinates so that the z -axis is parallel to L , and the motion occurs in the xy -plane. Converting from Cartesian to polar coordinates in the orbital plane using x = r cos φ and y = r sin φ , we see that where h , called the angular momentum per unit reduced mass, is constant. Writing r = r ˆ n , where ˆ n = cos φ ˆ e x +sin φ ˆ e y , we see that where is a vector in the orbital plane orthogonal to ˆ n . From this we see that where We now take the component of Eq. (2.2) in the radial direction, and note that where ˙ r ≡ ˆ n · v , and we have used the fact that ˆ n · dˆ n / d t = 0 , and that h 2 = | r × v | 2 = r 2 ( v 2 -˙ r 2 ) . The result is a differential equation for the radial motion, Multiplying by ˙ r and integrating once, we find the 'first integral' of the equation, where from Eq. (2.3), we can see that ˜ E is the energy per unit reduced mass. It is useful to rewrite Eq. (2.12) in the form where we define the effective radial potential This must be combined with the equation for the angular motion, Now we try to find a parametric solution to the equations, which is a solution of the form r = r ( λ ) , φ = φ ( λ ) , where λ is a parameter which will depend on t . Consider Eq. (2.11), and insert the fact that d / d t = ˙ φ d / d φ = ( h/r 2 )d / d φ , to obtain Using 1 /r as the variable, we can recast this equation into the form The homogenous solution can be written as A cos( φ -B ) , where A and B are arbitrary constants. Combining this with the inhomogeneous solution m/h 2 , and redefining the constants, we obtain the solution for 1 /r in terms of the parameter φ , given by where e and ω fill in for the two arbitrary constants A and B , and Notice that a solution with e < 0 is equivalent to one with e > 0 , but with ω → ω + π ; henceforth we will adopt the convention that e is positive. The angle f ≡ φ -ω is called the true anomaly . The curve described by Eq. (2.18) can be shown to be a conic section , an ellipse if the quantity e < 1 , a hyperbola if e > 1 , and a parabola if e = 1 , with the origin r = 0 at one of the foci of the curve. The parameter e is called the eccentricity of the orbit. Notice that r is a minimum when φ = ω ; this is the point of closest approach in the orbit, called the pericenter , and ω is called the angle of pericenter and simply fixes the orientation of the orbit in the xy -plane. For the e < 1 case, the point where φ = ω + π is the point of greatest separation, called the apocenter . The pericenter and apocenter distances are thus given by The sum of these is the major axis of the ellipse, so we define the semi-major axis a to be As a result, we can also write the solution for 1 /r in the form The quantity p = a (1 -e 2 ) is called the semi-latus rectum . From Eqs. (2.15) and (2.18), it is straightforward to derive the following useful formulae, valid for arbitrary values of e : So far we have determined the orbit as a function of φ , with three arbitrary constants, a , e , and ω , called orbit elements . To complete the parametric solution we need to determine φ as a function of time or as a function of some parameter related to time. From Eq. (2.15), we obtain where T , called the time of pericenter passage , is the fourth orbit element required to complete our solution in the orbital plane. For e < 1 , we can integrate over a complete orbit, and obtain the orbital period It is common to define the mean angular frequency or mean motion n ≡ 2 π/P , so that n 2 a 3 = Gm . Now carrying out the integral in Eq. (2.27) explicitly, we can find that where the variable u is called the eccentric anomaly , and is related to f by In terms of the eccentric anomaly, the radius of the orbit is given by This set of equations, called Kepler's solution for the two body problem is a convenient parametric solution for orbit determinations, since for given values of the orbit elements a , e , ω and T , one chooses t , solves Eq. (2.29) for u , then substitutes that into Eqs. (2.30) and (2.31) to obtain f ( t ) and r ( t ) , and thence x ( t ) and y ( t ) . Similar parametric solutions can be obtained for hyperbolic orbits, in terms of hyperbolic functions. There is one curious feature of our solution for the Kepler problem, and that is that the orientation of the orbit is fixed in the orbital plane, i.e. the angle of pericenter ω is a constant. It is not related to the spherical symmetry of the potential or to its time independence; these led only to the conservation of L and E and to the integrability of the equations. The constancy of ω is the result of a deeper symmetry embedded in the Kepler problem, associated with the 1 /r nature of the potential. One can define another vector associated with the orbital motion, often called the Runge-Lenz vector, given by where h = r × v . Substituting r = r n , with r given by Eq. (2.18), along with ˆ n = ˆ e x cos φ + ˆ e y sin φ and Eqs. (2.23) and (2.24), it can be shown that which is a vector of magnitude e pointing toward the pericenter. However, using the equation of motion Eq. (2.2), it is easy to show that so that R is another constant of the motion. Since e is constant by virtue of Eq. (2.26), this implies that ω is constant. But in this case, the 1 /r nature of the potential is crucial; had one substituted an equation of motion derived from a potential 1 /r 1+ glyph[epsilon1] , or 1 /r + α/r 2 , R would no longer be constant, even though E and L would stay constant and the problem would remain completely integrable.", "pages": [ 28, 29, 30, 31, 32, 33 ] }, { "title": "2.1.2 Keplerian Orbits in Space", "content": "In order to consider more realistic problems, we are interested in perturbations in our two-body problem which may be caused by gravitational forces exerted by external bodies, by the effects of multipole moments resulting from tidal or rotational perturbations, or by general relativistic contributions. Such effects will not be spherically symmetric in general, and so the orientation of the orbit will be important. So in this section we will review the full Keplerian orbit in space. The conventional description of the full Keplerian orbit in space goes as follows: we first establish a reference XY plane and a reference Z direction. For planetary orbits, the reference plane is the plane of the Earth's orbit, called the ecliptic plane , and the Z direction is perpendicular to the ecliptic plane is in the same sense as the Earth's north pole (ignoring the 23 · tilt). For Earth orbiting satellites, it is the equatorial plane. For binary star systems, it is the plane of the sky. Within each reference plane, the X -direction must be chosen in some conventional manner. We now define the inclination i of the orbital plane to be the angle between the positive Z direction and a normal to the plane (where the direction of the normal is defined by the direction of the angular momentum of the orbiting body). This tilted plane then intersects the reference XY plane along a line. We define the angle of the ascending node or nodal angle Ω to be the angle between the X axis and the intersection line where the body 'ascends' from below the reference plane (the negative Z side) to above it. The pericenter angle ω is the angle measured in the orbital plane from the line of nodes to the pericenter. These three angles then fix the orientation of the orbit in space. Within the orbital plane, the orbit is determined by the three remaining orbit elements a , e , and T . The true anomaly f is measured in the orbital plane from the pericenter to the location of the body. The orbit elements which uniquely identify a specific orbit are illustrated in Fig. 2.1. Given a unit vector ˆ n pointing from the center of mass to the body, it is straightforward to express ˆ n in terms of the XYZ basis: We can relate all the six orbit elements a , e , ω , Ω , i and T directly to the position r and velocity v of a body in a Keplerian orbit at a given time t . The first step is to use r and v to form the vectors where R is the Runge-Lenz vector. Given h 2 = Gmp = Gma (1 -e 2 ) , we can identify the orbit elements in terms of quantities constructed from r and v in the XYZ coordinates: Given these elements, and the Keplerian solution Eq. (2.18), the final orbit element T , the time of pericenter passage is given by the integral where f = φ -ω . The actual orbit is then given by r ( t ) = r ˆ n , with r given by either Eq. (2.22) or Eq. (2.31), and with the appropriate relation between the true anomaly f or the eccentric anomaly u and time t .", "pages": [ 33, 34, 35, 36 ] }, { "title": "2.1.3 Osculating Orbit Elements and the Perturbed Kepler Problem", "content": "Suppose the equation of motion for our effective two-body problem is no longer given by Eq. (2.2), but by something else: where A is a perturbing acceleration, which may depend on r , v and time. The solution of this equation is no longer a conic section of the Kepler problem. However, whatever the solution is, at any given time t 0 , for r ( t 0 ) , v ( t 0 ) , there exists a Keplerian orbit with orbit elements e 0 , a 0 , ω 0 , Ω 0 , i 0 and T 0 that corresponds to those values, as we constructed in the previous section. In other words there is a Keplerian orbit that is tangent to the orbit in question at the time t 0 , commonly called the osculating orbit . However, because of the perturbing acceleration, at a later time, the orbit will not be the same Keplerian orbit, but will be tangent to a new osculating orbit, with new elements e ' , a ' and so on. The idea then is to study a general orbit with the perturbing acceleration A by finding the sequence of osculating orbits parametrized by e ( t ) , a ( t ) , and so on. If the perturbing acceleration is small in a suitable sense, then since the orbit elements of the original Kepler motion are constants, we might hope that the osculating orbit elements will vary slowly with time and by small amounts. Mathematically, this approach is identical to the method of variation of parameters in solving differential equations, such as the harmonic oscillator with a slowly varying frequency. In this case, we replace our Keplerian solution for the motion with the following definitions : where the unit vectors ˆ n , ˆ λ , and ˆ h are given by Note that ˆ n × ˆ λ = ˆ h . In the pure Kepler problem, we saw that the orbit elements (apart from T ) were obtained from the constant vectors h and R ; now we calculate their time derivatives, using the perturbed equation of motion Eq. (2.43), with the result We now decompose the perturbing acceleration into components along the orthogonal directions ˆ n , ˆ λ , and ˆ h by where R , S , and W are sometimes referred to as the radial or 'cross-track', tangential or 'intrack', and out-of-plane components of the acceleration, respectively. With these definitions we obtain Note that, because h · ˙ h = h ˙ h , we immediately conclude that We can now systematically develop equations for the variations with time of the osculating orbit elements. For example, since h · ˆ e Z = h cos i , then ˙ h · ˆ e Z = ˙ h cos i -h sin i (d i/ d t ) = r S cos i -r W cos ( ω + f ) sin i , with the result that d i/ d t = ( r W /h ) cos ( ω + f ) . Similarly, since h · ˆ e Y = -h sin i cos Ω , then taking the derivative of both sides and subtracting our previous result for ˙ h , ˙ h and d i/ d t , we obtain sin i ˙ Ω = ( r W /h ) sin ( ω + f ) . To obtain ˙ e , we note that e ˙ e = R · ˙ R , and use the fact that R = ˆ n cos f -ˆ λ sin f . For ˙ a , we use the definition h 2 = Gma (1 -e 2 ) , from which ˙ a/a = 2 ˙ h/h +2 e ˙ e/ (1 -e 2 ) . For ˙ ω , we use the fact that R · ˆ e Z = e sin i sin ω , combined with previous results for ˙ e and d i/ d t . The final equations for the osculating orbit elements are Notice that the orbit elements a and e are affected only by components of A in the plane of the orbit, while the elements Ω and i are affected only by the component out of the plane. The pericenter change has both, but this is because of the combination of intrinsic, in-plane perturbations (the first two terms) with the perturbation of the line of nodes from which ω is measured (the third term). In fact it is customary to define an angle of pericenter which represent a kind of angle measured from the reference X-direction, rather than from the nodal line. The variation of this angle is given by Although we have discussed this from the point of view of perturbations, Eqs. (2.57)-(2.61) are exact ; they are merely a reformulation of the three second-order differential equations for r ( t ) , Eq. (2.43), as a set of six first-order differential equations for the osculating elements (we have not displayed the sixth equation, related to the time orbit element T ). Given a set of functional forms for A in terms of the orbit elements, an exact solution of these equations is an exact solution of the original equations. What makes this formulation so useful is that, when A = 0 , the solutions for the orbit elements are constants. If the perturbation represented by A is small in a suitable sense, one expects the changes in the elements to be small. Therefore we can find a first-order perturbation solution by inserting the constant zeroth order values of the elements into the right-hand side, and simply integrating the equations with respect to t . In principle, we could go to higher order by inserting this first-order solution back into the right-hand side and integrating again, and so on. It is sometimes more convenient to integrate the equations with respect to the true anomaly f rather than t . To relate the two when dealing with an osculating orbit, we recall that f = φ -ω , and that φ is measured from the line of nodes, thus φ can change both because of the orbital motion, but also by an amount -cos i ∆Ω if Ω is changing. Hence, since from Eqs. (2.44) - (2.46) we can write r 2 d φ/ d t ≡ | r × v | = h , we have Of course, if we are integrating the equations only to first order, we can drop the terms involving d ω/ d t and dΩ / d t and use d f/ d t = h/r 2 .", "pages": [ 36, 37, 38, 39 ] }, { "title": "2.2 Testing the No-Hair Theorem Using the Galactic Center Black Hole", "content": "If a class of stars orbits the central black hole in short period ( ∼ 0 . 1 year), high eccentricity ( ∼ 0 . 9 ) orbits, they will experience precessions of their orbital planes induced by both relativistic frame dragging and the quadrupolar gravity of the black hole. Here we are going to apply the orbit perturbation theory that we introduced in the previous sections to study this phenomenon for the galactic center massive black hole. We will see that observation of the precessions of the orbital planes will lead to determination of the spin J and the quadrupole moment Q 2 of the black hole. By having J and Q 2 we can test the specific relation which the black hole no-hair theorem requires between these parameters and the mass of the black hole i.e. Q 2 = -J 2 /m .", "pages": [ 39, 40 ] }, { "title": "2.2.1 Orbit Perturbations in Field of a Rotating Black Hole", "content": "For the purpose of testing the no-hair theorem it suffices to work in the post-Newtonian limit. The post-newtonian limit is the weak-field and slow-motion limit of general relativity in which a quantity of interest is expressed as an expansion in powers of a post-Newtonian parameter glyph[epsilon1] ∼ v 2 ∼ U where U is the gravitational potential. The leading term in the expansion is the Newtonian term and it is labeled as 0 PN term. The term of order glyph[epsilon1] is the first-post-Newtonian correction, and it is labeled as 1 PN term and so on. Consider a two-body system where a body of negligible mass is in the field of a body with mass m , angular momentum J and quadrupole moment Q 2 . The equation of motion of the test body in the first-post-Newtonian limit is given by where r and v are the position and velocity of the body, ˆ n = r /r , ˙ r = ˆ n · v , h = r × v , ˆ h = h /h , and ˆ J = J / | J | (see, e.g. [27]). The first line of Eq. (2.65) corresponds to the Schwarzschild part of the metric (at post-Newtonian order), the second line is the frame-dragging effect, and the third line is the the effect of the quadrupole moment (formally a Newtonian-order effect). For an axisymmetric black hole, the symmetry axis of its quadrupole moment coincides with its rotation axis, given by the unit vector ˆ J . As illustrated in Fig. 2.2, the star's orbital plane is defined by the unit vector ˆ e p along the line of nodes and the unit vector in the orbital plane ˆ e q orthogonal to ˆ e p and ˆ h i.e. ˆ e q = ˆ h × ˆ e p . With these definitions, then The polar angels α and β define the direction of the black hole's angular momentum J in the ˆ e p , ˆ e q , ˆ h coordinate system, so that All the terms in Eq. (2.65) except the first term, which is the Newtonian acceleration, are perturbing terms, and by using Eqs. (2.66) and (2.67) we can find the radial, tangential, and out-of-plane components of the perturbing terms as following By substituting R , S , and W from Eqs. (2.68)-(2.70) in Eqs. (2.57)-(2.61), we get the rate of change of the each orbit element. To derive the total change of an orbit element over one orbit, we need to integrate over one orbit i.e. integrating over f from 0 to 2 π : where x could be any of the orbit elements. We recall the relations r = p/ (1 + e cos f ) , ˙ r = he sin f/p , v 2 = ( he sin f/p ) 2 +( h (1 + e cos f ) /p ) 2 , and h = √ Gmp (see Eqs. (2.44)-(2.48)). Now to study the precessions of the orbit, we derive the total changes in i , Ω , and glyph[pi1] , which are the three orbit angles defining the orientation of the orbit in space. To first order we get where χ ≡ J/ ( Gm 2 ) is the dimensionless spin parameter of the black hole which is always less than one for Kerr black hole and Q 2 = -J 2 /m . To get an idea of the astrometric size of these precessions, we define an angular precession rate amplitude ˙ Θ i = ( a/D ) A i /P , where D is the distance to the galactic center and P = 2 π ( a 3 /Gm ) 1 / 2 is the orbital period. Using where m = 4 × 10 6 M glyph[circledot] , D = 8 kpc , we obtain the rates, in µ arcseconds per year where we have assumed Q 2 = -G 2 m 3 χ 2 . The observable precessions will be reduced somewhat from these raw rates because the orbit must be projected onto the plane of the sky. For example, the contributions to ∆ i and sin i ∆Ω are reduced by a factor of sin i ; for an orbit in the plane of the sky, the plane precessions are unmeasurable. For the quadrupole precessions to be observable, it is clear that the black hole must have a decent angular momentum ( χ > 0 . 5 ) and that the star must be in a short period high-eccentricity orbit.", "pages": [ 40, 41, 42, 43 ] }, { "title": "2.2.2 Testing the No-Hair Theorem", "content": "Although the pericenter advance is the largest relativistic orbital effect, it is not the most suitable effect for testing the no-hair theorem. The pericenter advance is affected by a number of complicating phenomena including any distribution of mass (such as dark matter or gas) within the orbit. Even if it is spherically symmetric, such a distribution of matter will generally contribute to the pericenter advance because it might induce derivations from the pure Keplerian 1 /r potential. By contrast, the precessions of the node and inclination are relatively immune from such effects. Any spherically symmetric distribution of mass has no effect on these orbit elements [15]. As a consequence of Eqs. (2.72) and (2.73) we have the purely geometric relationship, From the measured orbit elements and their drifts for a given star, Eq. (2.81) gives the angle β , independently of any assumption about the no-hair theorem. This measurement then fixes the spin axis of the black hole to lie on a plane perpendicular to the star's orbital plane that makes an angle β relative to the line of nodes. The equivalent determination for another stellar orbit fixes another plane; as long as the two planes are not degenerate, their intersection determines the direction of the spin axis, modulo a reflection through the origin. This information is then sufficient to determine the angles α and β for each star. Then, from the magnitude determined for each star, together with the orbit elements, one can solve for J and Q 2 to see if the Q 2 = -J 2 /m relation demanded by the no-hair theorem holds. So, in principle we see that observations of the precessing orbits of stars very near the massive black hole in the galactic center could provide measurements of the spin and quadrupole moment of the hole and thereby test the no-hair theorems of general relativity. But since the galactic center is likely to be populated by a distribution of stars and small black holes, their gravitational interactions will also perturb the orbit of any given star. In the next sections, we will estimate the effects of such perturbations using analytic orbital perturbation theory to see if the relativistic spin and quadrupole effects of the central massive black hole dominates the effects of stellar cluster perturbation. These estimates will allow us to assess whether the proposed test of the black hole no-hair theorem is going to be feasible.", "pages": [ 43, 44 ] }, { "title": "2.3.1 Perturbation by a third body", "content": "In Newtonian theory, the acceleration a 1 of a target star with mass m 1 and the acceleration of the Galactic center black hole with mass m 2 in the presence of a perturbing star with mass m 3 are given by where r ab = r a -r b and r ab = | r ab | . The equation of motion for the effective two-body problem is Since m 1 glyph[lessmuch] m 2 , Eq. (2.85) is basically the acceleration of the target star. For a perturbing star inside the orbit of the target star ('intenal' star), with r 32 glyph[lessmuch] r 12 , we have where the capitalized superscripts denote multi-indices , so that r L 32 ≡ r i 32 r j 32 . . . r k glyph[lscript] 32 , and similarly for the partial derivatives; 〈 . . . 〉 denotes a symmetric trace-free product (STF). A STF product is symmetric on all indices; furthermore, contracting any pair of indices gives zero. For example applying a gradient successively to (1 /r ) gives STF products: where ∂ k ≡ ∂/∂x k . In Eqs. (2.87) the combination of unit vectors in each case is symmetric on all indices, because the partial derivatives commute, and also contracting on any pair of indices automatically gives zero, because, for example for Eq. (2.87c), δ ij ∂ ijk r -1 = ∇ 2 ∂ k r -1 = ∂ k ∇ 2 r -1 = 0 for r = 0 . glyph[negationslash] Using Eq. (2.86) the i -component of ∇ (1 /r 13 ) is Substituting Eq. (2.88) in Eq. (2.85), the equation of motion can be expanded as where r ≡ r 12 and R ≡ r 23 . For a perturbing star outside the orbit of the target star ('external' star), with r 12 glyph[lessmuch] r 23 , the expansion takes the form Because m 1 glyph[lessmuch] m 2 and m 3 glyph[lessmuch] m 2 , and because in what follows we are only concerned with orbital plane effects, we can replace both m 1 + m 2 and m 1 + m 2 + m 3 with a single m , effectively the mass of the massive black hole. Establishing a reference XY plane and a reference Z direction, we have the standard 'osculating' orbital elements including i , Ω , ω , a , e , f , and glyph[pi1] here. The unit vector ˆ n pointing from the MBH to the target star, and the orthogonal unit vectors ˆ λ and ˆ h are given by Eqs. (2.49)-(2.51) where ˆ h is normal to the orbital plane. We also have the osculating orbit definitions r ≡ p/ (1 + e cos f ) , h ≡ | r × v | ≡ ( GMp ) 1 / 2 , d φ/ d t ≡ h/r 2 , and p ≡ a (1 -e 2 ) for the target star, and R ≡ p ' / (1 + e ' cos F ) , h ' ≡ | R × V | ≡ ( GMp ' ) 1 / 2 , d φ ' / d t ≡ h ' /R 2 , and p ' ≡ a ' (1 -e ' 2 ) for the perturbing star, along with its orbital elements i ' , Ω ' and ω ' . Here the perturbing acceleration A is everything in Eqs. (2.89) and (2.90) except the leading acceleration -GM r /r 3 . In the internal perturbing star case the first three terms of the expansion are where ˆ n ≡ r /r and ˆ N ≡ R /R . Similarly, for the external perturbing star we have We use Eqs. (2.57)-(2.61) to calculate the variations with time of the target star's orbit elements, which means we need to derive the components of the perturbing terms along ˆ n , ˆ λ , and ˆ h denoted as R , S , and W in subsection 2.1.3 respectively. We will work in first-order perturbation theory, whereby we express R , S and W in terms of osculating orbit variables, set the orbit elements equal to their constant initial values in the right-hand side of Eqs. (2.57)-(2.61), and then integrate with respect to time.", "pages": [ 44, 45, 46, 47 ] }, { "title": "2.3.2 Time Averaged Variations in Orbit Elements", "content": "We want to use Eqs. (2.57)-(2.61) to calculate the time averaged rates of change of the orbit elements of the target star, given by d x/ d t ≡ T -1 ∫ T 0 (d x/ d t )d t , where T is the longest relevant timescale, and x is the element in question. For an internal perturbing star, T would be the period of the target star P . where f is the true anomaly. Assuming that the shorter period P ' is much shorter than the longer period P , we can split the longer period P to small pieces, each equal to P ' . Then d x/ d f in Eq. (2.97) will be the rate of change of x with f while the perturbing star completes one orbit ( ∆ f = 2 π ) and we can write where d t ' and F are the time element and the true anomaly of the perturbing star, respectively and d t ' = ( r ' 2 /h ' )d F , valid to first order in perturbation theory. Using the osculating orbit definitions, Eq. (2.98) can be written as For an external perturbing star, T would be the orbital period of the perturbing star P ' and by a similar argument, it is straightforward to show that Eq. (2.99) gives the time-averaged rates of change of the orbital elements of the target star in this case too. By way of illustration, we show here the time-averaged changes of orbital elements for the glyph[lscript] = 1 term induced by an external star (Eq. (2.91)), for the special case i ' = 0 and Ω ' = 0 : where B ext = (2 π/P )( m 3 /m )( p/p ' ) 3 . For arbitrary orientations i ' and Ω ' the expressions are much more complicated. We have also found the analogous expressions for the glyph[lscript] = 2 and glyph[lscript] = 3 terms. These are smaller than the glyph[lscript] = 1 results by factors of p/p ' and ( p/p ' ) 2 , respectively. We used a trick described in Appendix A, which allows us to get analytical forms of the integrations over f and F easily by Maple or Mathematica. For an internal star, the glyph[lscript] = 1 term (Eq. (2.94)) contributes no time-averaged variation of any of the elements. The glyph[lscript] = 2 contributions scale as B int = (2 π/P )( m 3 /m )( p ' /p ) 2 , while the glyph[lscript] = 3 contributions are smaller by a factor of p ' /p . Again, the general expressions are long, so we will not display them here. Since the orbital energy of the target star is proportional to 1 /a , Eq. (2.100) simply reflects the absence of a secular energy exchange mechanism between the target and perturbing stars at first order in the perturbations. As a side remark, Eqs. (2.101) and (2.102) together imply that (1 -e 2 ) 1 / 2 cos i is a constant, so that a decreasing inclination produces an increasing eccentricity; in planetary dynamics this is known as the Kozai mechanism [28].", "pages": [ 47, 48, 49 ] }, { "title": "2.3.3 Average Over Orientations of Perturbing Stellar Orbits", "content": "With the time-averaged changes in the orbital elements due to one perturbing star in hand, we now turn to the changes caused by a distribution of perturbing stars. We will assume a cluster of stars whose orbital orientations ( i ' , Ω ' , ω ' ) are randomly distributed. We will discuss the distributions in a ' and e ' later. The 'orientation-average' of a function F ( i ' , Ω ' , ω ' ) will be defined by We then find that 〈 d x/ d t 〉 = 0 for all four orbit elements e , i , Ω and glyph[pi1] , for both internal and external stars. The reason is easy to understand: the averaging process is equivalent to smearing the perturbing stars' mass over a concentric set of spherically symmetric shells. The target star will thus be moving in what amounts to a spherically symmetric, 1 /r potential and its orbit elements will therefore be constant, just as in the pure Kepler problem. But for a finite number of stars, the potential will not be perfectly spherically symmetric, even if the orientations are randomly distributed. It is the effect of this discreteness that we wish to estimate. We do this by calculating the root-mean-square (r.m.s.) angular average [ 〈 (d x/ d t ) 2 〉 ] 1 / 2 . This will give an estimate of the 'noise' induced in the orbital motion of the target star by the surrounding matter. We will then compare this noise with the relativistic effects that we wish to measure. Here we list the r.m.s. orientation averages for d i/ d t and dΩ / d t for internal and external stars, and for all glyph[lscript] ≤ 3 . It turns out that cross terms between different glyph[lscript] values vanish. We can also see that the contribution of the ( Gm 3 /R 2 ) ˆ N term in Eq. (2.91) is zero. where We will focus on the r.m.s change in the direction of ˆ h , the normal to the orbital plane which can be expressed in terms of r.m.s changes in i and Ω . Squaring both sides of Eq. (2.54) gives Substituting Eq. (2.56), ˙ h = r S , in Eq. (2.117), we have Then we note that adding the squares of Eqs. (2.60) and (2.61) gives us exactly the right hand side of Eq. (2.118) and therefore we can write which leads to The leading contributions, corresponding to the glyph[lscript] = 2 contribution from internal stars, and to the glyph[lscript] = 1 contribution from external stars are given by Note although the perturbing terms are due to randomly distributed stars, the r.m.s changes of the individual elements i and Ω depend on ω and i (see Eqs. (2.106)-(2.115)). The reason is that i and Ω depend on the choice of reference plane, and ω is measured from the line of nodes and i is the inclination angle between the reference plane and the orbital plane. So they also depend on the reference plane and if we chose a different reference plane, ω and i would be different, and so we might expect 〈 (dΩ / d dt ) 2 〉 and 〈 (d i/ d t ) 2 〉 to depend on the orientation of the orbital ellipse relative to the nodal line. On the other hand, from Eqs. (2.121) and (2.122) we can see that 〈 (d h/ d t ) 2 〉 is independent of ω . It is because ˆ h is a vector in space, it knows nothing about the arbitrary choice of reference plane, and hence its variation can't depend on ω . For future use, we define the angular r.m.s. rate of change of the orbital orientation by d θ/ d t ≡ 〈 (d h/ d t ) 2 〉 1 / 2 .", "pages": [ 49, 51, 52 ] }, { "title": "2.3.4 Average Over Size and Shape of Perturbing Stellar Orbits", "content": "We now integrate over the semi-major axes a ' and eccentricities e ' of the perturbing stars. We will use a distribution function of the form N g ( a ' ) h ( e ' 2 )d a ' d e ' 2 , where N is a normalization factor, set by the condition N = N/ I , where N is the total number of stars in the distribution, and where the limits of integration will be determined by the limiting orbital elements for those stars. Since at the end we are going to compare our results with N-body simulations by Merritt et al ([17], hereafter referred to as MAMW), we will consider the same range of parametrized models for the dependences g ( a ' ) and h ( e ' 2 ) as was used in their simulations, and will consider clusters that contain both stars and stellar-mass black holes. The variables a ' and e ' will be constrained by a number of considerations. The minimum pericenter distance r min for any body will be given by the tidal-disruption radius for a star, and the capture radius for a black hole. This will therefore give the bound For r min we will use the estimates These are derived in Appendix B. However our analytic formulae for the r.m.s. orientation-averaged variations are valid only in the limits p ' /p glyph[lessmuch] 1 or p/p ' glyph[lessmuch] 1 for internal and external stars, respectively. But since our target star is embedded inside the cluster of stars, there may well be perturbing stars that do not satisfy either constraint. On the other hand, an encounter between the target star and another star that is too close could perturb the orbit so strongly that it will be unsuitable for any kind of relativity test. Because we are looking only for an estimate of the statistical noise induced by the cloud of stars, we will try three approaches in order to capture the range of perturbations induced by the cluster. Integration I. Because Eqs. (2.121) and (2.122) are valid only in the extreme limits where the perturbing star is always far from the target star (so that the higher-order terms are suitably small), we cut out of the stellar distribution any stars that violate this constraint. This yields the following conditions on the allowed orbital elements of the perturbing stars: (i) for an internal star, we demand that r ' max = a ' (1 + e ' ) of the perturbing star be less than r min = a (1 -e ) of the target star; (ii) for an external star, we demand that r ' min = a ' (1 -e ' ) of the perturbing star be greater than r max = a (1 + e ) of the target star. For an internal star, we thus have the two conditions, The maximum values of e ' and a ' allowed under these conditions are For an external star, we have the two conditions where a max is the outer boundary of the cluster, chosen to be large enough that the effects of stars beyond this boundary are assumed to be negligible. Following MAMW, we choose a max = 4 mpc. The maximum e ' and minimum a ' allowed are thus Thus the average of a function F ( a ' , e ' ) over this distribution will be given by where However, instead of substituting N = N/ I , we substitute where This amounts to assuming that all N stars in the cluster happen to have orbit elements that satisfy our constraint. Thus the average of the function F ( a ' , e ' ) will be given by Note that if F = 1 , we get 〈F〉 = N . In our simple model, we are treating the stars and black holes as independent distributions, so the mean value of F can be written as a sum over the two normalized distributions, where the only differences between the integrals for the distributions are the perturbing object mass m 3 , the value of r min , which affects only the integrals J 1 and I 1 , and the number of particles, N S for stars, and N B for black holes, with N = N B + N S ; for later use, we define N B /N S ≡ R . Hence we obtain For the r.m.s. variations in d h/ d t , we include all the higher-order terms shown in Eqs. (2.106)(2.115). Integration II. Taking the ratio of the higher glyph[lscript] contributions to the orbit element variations to the leading glyph[lscript] contribution (see Eqs. (2.106)-(2.115)) reveals that the parameter controlling the relative size of the higher-order terms is the ratio a ' / [ a (1 -e 2 )] for internal stars, and a/ [ a ' (1 -e ' 2 )] for external stars. Requiring each of these ratios in turn to be less than one, we repeat the integrals, but with new limits of integration given by This condition permits closer encounters than the condition imposed in Integration I . Here as well, we include all higher-order contributions to the r.m.s. variations. Integration III . In an attempt to include even closer encounters between the target star and cluster stars, we adopt a fitting formula for the r.m.s. perturbations of the orbital plane that interpolates between the two limits of very distant internal and very distant external stars. A simple formula that achieves this is given by where we use only the lowest-order contributions to the r.m.s. variations, given by Eqs. (2.121) and (2.122). In this case the average over the distributions becomes where the integrals now take the form with I = J (1) , thereby including the full distribution of stars.", "pages": [ 52, 53, 54, 55 ] }, { "title": "2.3.5 Numerical Results", "content": "In order to compare our analytic estimates with the results of the N-body simulations of MAMW, we will adopt as far as possible the same model assumptions. We parametrize the distribution functions g ( a ' ) and h ( e ' 2 ) according to g ( a ' ) = a ' 2 -γ , and h ( e ' 2 ) = (1 -e ' 2 ) -β , where γ ranges from 0 to 2, and β ranges from -1 to 0.5. The values ( γ, β ) = (2 , 0) correspond to a mass segregated distribution with isotropic velocity dispersion. We will choose a ' max = 4 mpc, arguing that the perturbing effect of the cluster outside this radius is negligible by virtue of the increasing distance from the target star and the more effective 'spherical symmetry' of the mass distribution. We will assume that the cluster contains stars each of mass 1 M glyph[circledot] and black holes each of mass 10 M glyph[circledot] , and will consider values of the ratio of the number of black holes to the number of stars to be R = 0 and R = 1 (MAMW also consider the ratio R = 0 . 1 ). The main difference between stars and black holes in our integrals is the factor m 2 3 , so there will simply be a relative factor of 100 between the black hole contribution and the stellar contribution, apart from the small effect of the difference in r min between stars and black holes. Of the 22 stellar distribution models listed in Table I of MAMW, we consider only the 15 models with either R = 0 or R = 1 ; these are listed in Table 2.1. While N denotes the total number of objects within 4 mpc, the parameter M glyph[star] , chosen to parallel the notation of MAMW, denotes the approximate total mass within one mpc of the black hole, and gives an idea of the perturbing environment around a close-in target star. Figure 2.3 shows the results for the three stellar distribution models 9, 11 and 12 in Table 2.1; In these models γ = 2 and β = 0 and they have an equal number of 1 M glyph[circledot] stars and 10 M glyph[circledot] black holes. The three cases correspond to a total number of perturbing bodies within a radius of four mpc of 7, 21 and 72, respectively. The target star has eccentricity e = 0 . 95 , and its semi-major axis a ranges from 0 . 1 to 2 mpc. Plotted is the rate of precession of the vector perpendicular to the orbital plane, d θ/ d t ≡ 〈 (d h/ d t ) 2 〉 1 / 2 , observed at the source, in arcminutes per year, calculated using three ways of carrying out the integrals over the stellar distribution. The dashed line denotes Integration I , in which all perturbing stars are assumed to be sufficiently far from the target star at all times that their pericenters are outside its apocenter or that their apocenters are inside its pericenter. The solid line denotes Integration II , in which closer encounters are permitted, limited by demanding that all perturbing stars be on orbits such that the higher glyph[lscript] contributions to d θ/ d t be at worst comparable to the contribution at lowest order in glyph[lscript] . The dot-dashed line denotes Integration III , which uses a fitting formula that interpolates between the extreme limits of a perturbing star well outside the target star, and a perturbing star well inside the target star; in this case the integration is over the entire stellar distribution. The orange band in each panel denotes the value of d θ/ d t corresponding to an astrometric precession rate dΘ / d t of 10 µ arcsecond per year as seen from Earth, given by where ˜ a is the semi-major axis in units of mpc; we use 8 kiloparsecs as the distance to the galactic center. Also plotted are the rates of precessions due to the frame-dragging and quadrupolar effects of a Kerr black hole, given by [15] where P = 2 π ( a 3 /Gm ) 1 / 2 is the orbital period, A J and A Q 2 are the amplitude of precessions given in Eqs. (2.76) and (2.77), and where χ = J/Gm 2 is the dimensionless Kerr spin parameter, set equal to its maximum value of unity in Fig. 2.3. Because Integration I keeps the stars far from the target star, the precessions are small. By contrast, the fitting formula of Integration III is large for very close encounters, so not surprisingly, the precessions from that method are large. Integration II gives results intermediate between the two. Interestingly, the spread between these methods is roughly consistent with the spread between individual precessions obtained in the N -body simulations of MAMW. This can be seen in the top panel of MAMW, Fig. 7, which corresponds to the middle panel of Fig. 2.3 (to properly compare the two figures, one must translate between d θ/ d t and dΘ / d t ). It can also been in the bottom panel of MAMW Fig. 5, where the points labelled by × indicate the mean precessions in the absence of black hole spin, for the same three stellar distributions as are shown in Fig. 2.3. Thus we regard our three integration methods as giving a reasonable estimate of the range of stellar perturbations. Comparing the three stellar distributions shown in Fig. 2.3, we see that the effects vary roughly as N 1 / 2 ∝ M 1 / 2 glyph[star] , as expected, from the nature of our r.m.s. calculation. We consider eight different stellar distribution models, and for seven of them, consider models with equal numbers of stars and black holes, and models with only stars, totaling 15 models. In all but one case, the precessions are generally smaller than the ones shown in Fig. 2.3, and that case is a centrally condensed model with a non-isotropic velocity dispersion leading to a preponderance of highly eccentric orbits. We conclude that, for a target star in a very eccentric orbit with a < 0 . 2 mpc, there is a reasonable possibility of seeing relativistic frame-dragging and quadrupole effects above the level of 10 µ arcsec/yr without undue interference from stellar perturbations. We also show in Appendix C that the effects of tidal deformations on the orbital planes of stellar orbits are negligible. To illustrate the differences between different models of the stellar distribution, Fig. 2.4 shows the predicted precessions for a target star at 0 . 1 mpc with e = 0 . 95 , for all 15 model distributions. The crosses and the error bars indicate the range of results from the three integration models. Models with γ = 0 or 1 generally give smaller precessions than those with γ = 2 . The latter models are more centrally condensed, and lead to larger perturbations of a close-in target star. For the same value of ( γ, β, M glyph[star] ) , models with equal numbers of stars and black holes ( R = 1) lead to larger perturbations than those with pure stars ( R = 0 ); the former models are more 'grainy' (smaller N ), and so the effects are larger by roughly N 1 / 2 R =0 /N 1 / 2 R =1 . Models 14 and 15 ( β = 0 . 5) have an excess of stars in highly eccentric orbits, thus leading to larger precessions.", "pages": [ 55, 56, 57, 58, 59 ] }, { "title": "2.3.6 Conclusions", "content": "We have used analytic orbital perturbation theory to investigate the rate of precession of the orbital plane of a target star orbiting the galactic center black hole Sgr A glyph[star] induced by perturbations due to other stars in the central cluster. We found that, although the results have a wide spread, they compare well with the distribution of precessions obtained using N -body simulations. One feature not included in our analysis is the fact that orbital planes in a real cluster are not randomly distributed, but become somewhat correlated over the long-term evolution of the cluster. Whether these correlations are large enough to have a significant effect on our estimates is an open question. Within our assumptions, however, we find a range of possible models for the cluster of objects within the central 4 mpc of the black hole in which it may still be possible to detect relativistic precessions of the orbital planes at the 10 µ arcsec/yr level. 3", "pages": [ 60, 62 ] }, { "title": "Dark Matter Distributions Around Massive Black Holes: A General Relativistic Analysis", "content": "In this chapter we start with reviewing the non-relativistic phase-space formulation to study the effects of the adiabatic i.e. slow growth of the massive black hole on the dark-matter density profile. Then we develop a fully general relativistic phase-space formulation to consider these effects and we find the dark matter distribution in vicinity of the Galactic center supermassive black hole Sgr A glyph[star] which has significant differences with the non-relativistic results. Having the dark matter profile density in the presence of the massive black hole, we calculate its perturbing effect on the orbital motions of stars in the Galactic center, and find that for the stars of interest, relativistic effects related to the hair on the black hole will dominate the effects of dark matter.", "pages": [ 62 ] }, { "title": "3.1 Growing a Black Hole in a Dark Matter Cluster: Newtonian Analysis", "content": "In this section we begin with a purely Newtonian analysis of the process of growing a black hole slowly within a pre-existing DM halo. This is an example of a process in which a system responds adiabatically to a slowly varying potential. In such a situation, the use of action-angle variables enables us to predict how a distribution of particles will respond to changes in the gravitational field that confine it. As discussed below, when the process of growing a black hole within a pre-existing DM halo is adiabatic, the gravitational potential changes slowly enough so that the constants of the motion of the DM particles vary smoothly while keeping the action variables invariant. A brief consideration of the physical conditions close to the GC will convince us that the requirements for adiabatic evolution are likely to be met. The central MBH will dynamically dominate a region of radius r h = Gm BH /σ 2 , where m BH is the mass of the black hole and σ is the velocity dispersion of the DM particles outside the radius of influence. The dynamical timescale inside r h can be estimated as t dyn = r h /σ , which for the Milky Way turns out to be about 10 4 yr, taking m BH ∼ 4 × 10 6 M glyph[circledot] and estimating from the velocity dispersion of the stars σ ≈ 66 km/s. On the other hand we can estimate the shortest timescale for growth of the black hole as the Salpeter timescale t S = m BH / ˙ m Edd ≈ 5 × 10 7 yr , where ˙ m Edd is the usual Eddington accretion timescale. Hence, the dynamical timescale inside r h is much shorter than the typical timescale for black hole growth. In addition, since the DM is assumed to be collisionless, the relaxation timescale will always be longer than the evolutionary timescale (This is not necessarily the case for the stellar population close to the central cusp) [29, 30]. We generally follow the approach used by Binney and Tremaine [30] and Quinlan et al. [31]. In addition to reproducing the non-relativist results in [31], which extended the study of the isothermal sphere carried out in [32], this will set the stage for our fully general relativistic analysis. We will use c = 1 throughout this chapter.", "pages": [ 62, 63 ] }, { "title": "3.1.1 Basic Equations", "content": "Given a distribution function f ( E,L ) , which is normalized to give the total mass M of the halo upon integration over phase-space, the physical mass density is given by: where the energy and angular momentum per unit mass E and L ≡ | L | are functions of velocity and position, defined by where Φ( r ) is the Newtonian gravitational potential. We now change integration variables from v to E , L , and the z-component of angular-momentum L z , using the relation where the Jacobian is given by the determinant of the matrix where ˙ r = v r = r · v /r . For the ∂L/∂v i components, we used the relation L 2 = r 2 ( v 2 -˙ r 2 ) ; e.g. for v x we have To express the Jacobian in terms of the components of v we use also we have therefore So, Eq. (3.4) can be written as To perform the integrations in Eq. (3.1) using Eq. (3.3), we need to write the Jacobian in terms of L and L z : where we used r · L = 0 . An alternative derivation of the Jacobian in terms of L and L z uses the metric components in spherical coordinates, g rr = 1 , g θθ = r 2 , and g φφ = r 2 sin 2 θ , in Eq. (3.5) which leads to where we used L z ≡ v φ = g φφ v φ . Solving Eq. (3.12) for v θ gives Therefore, we can write Combining Eqs. (3.14) and (3.4) we again find Eq. (3.10). Including a factor of 4 to take into account the ± signs of v θ and v r available for each value of E and L , we obtain and thus the physical density The limits on L z are derived by demanding that v θ should be real in Eq. (3.14). We will also assume throughout that the distribution function is independent of L z ; as a result we can integrate over L z between the limits ± L sin θ , to obtain Eq. (1) in [31]: The limits of integration are set in part by the fact that | v r | must be real. Solving the energy per unit mass of each particle, E = Φ( r ) + (1 / 2)( v 2 r + L 2 /r 2 ) , for v r we have and thus L ranges from 0 to [2 r 2 ( E -Φ( r ))] 1 / 2 , while E ranges from Φ( r ) to E max , the maximum energy that a bound particle could have. We thus have Hence, given a distribution function, f ( E,L ) , we can use Eq. (3.20) to find the density, ρ ( r ) , which acts as the source of the gravitational potential Φ( r ) . We will also encounter the situation where ρ ( r ) is known, e.g. from fits to numerical simulations, and we would like to find the distribution function. In the next subsection we review Eddington's method, which allows us to construct the distribution function from the density density.", "pages": [ 63, 64, 65, 66, 67 ] }, { "title": "3.1.2 Eddington's Method", "content": "Following the terminology in Binney and Tremaine ([30], BT hereafter), we define a new gravitational potential and a new energy. If Φ 0 is some constant, then let the relative potential Ψ( r ) and the relative energy E of a particle be defined by where H is the Hamiltonian of the system. In practice, Φ 0 is chosen to be such that f > 0 for E > 0 and f = 0 for E ≤ 0 . If an isolated system extends to infinity, Φ 0 = 0 and the relative energy is equal to the binding energy. The relative potential of an isolated system satisfies Poisson's equation in the form subject to the boundary condition Ψ( r ) → Φ 0 as | x | → ∞ . Suppose we observe a spherical system that is confined by a known spherical potential Φ( r ) . Then it is possible to derive for the system a unique distribution function that depends on the phase-space coordinates only through the Hamiltonian H ( r , v ) . Here we express this distribution function as a function of the relative energy f ( E ) . Using Eq. (3.1), since f depends on the magnitude v of v and not its direction, we can immediately integrate over angular coordinates in velocity space. We then have where we have used Eq. (3.21) and assumed that the constant Φ 0 in the definition of E has been chosen such that f = 0 for E ≤ 0 . It can be shown that Ψ is a monotonic function of r in any spherical system, therefore we can regard f as a function of Ψ instead of r . Thus Differentiating both sides of Eq. (3.24) with respect to Ψ , we obtain Equation (3.25) is an Abel integral equation having the solution An equivalent formula is This result is due to Eddington [33], and it is called Eddington's formula . It implies that, given a spherical density distribution, we can recover a distribution function depending only on the Hamiltonian that generates a model with the given density. In general, there might be multiple distribution functions that generate a given density, and Eddington's formula gives us the one which is isotropic in the velocity space. However, there is no guarantee that the solution f ( E ) to Eqs. (3.26) will satisfy the physical requirement that it be nowhere negative. Indeed, we may conclude from Eq. (3.26a) that a spherical density distribution f ( r ) in the potential Φ( r ) can arise from a distribution function depending only on the Hamiltonian if and only if is an increasing function of E .", "pages": [ 67, 68 ] }, { "title": "3.1.3 Adiabatic Invariants", "content": "We next imagine a point mass growing slowly at the center of a pre-existing distribution of particles. Systems like this where potential variations are slow compared to a typical orbital frequency are called adiabatic . It can be shown using the action-angle formalism ([30], Section 3.6.) that the actions of particles, ∮ pdq , for each independent coordinate and conjugate momentum are constant during such adiabatic changes of potential. For this reason such action integrals are often called adiabatic invariants . So, as the gravitational potential near the point mass changes because of the growth of the point mass, each particle responds to the change by altering its energy E and angular momentum L and L z , holding the adiabatic invariants I r , I θ , and I φ fixed, where The constancy of I θ and I φ implies that L and L z remain constants, no surprise considering the assumed spherical symmetry. But when the potential evolves from the initial potential Φ ' to a new potential Φ that includes the point mass, E ' evolves to E such that In [32], it has been shown that for an adiabatic growth of a point mass inside a cluster, the conservation of the adiabatic invariants of each particle leads to the invariance of the distribution function f ( E,L ) = f ' ( E ' , L ' ) . In Appendix D we review this argument of [32] and also generalize it to the relativistic analysis. So, by equating radial actions in Eq. (3.28) and solving to obtain the relation E ' = E ' ( E, L ) , the new distribution function is then assumed to be given by the original distribution function f ' , where E ' is expressed in terms of E and L . Note that, in a Newtonian analysis for a potential dominated by a point mass, Φ( r ) = -Gm/r , and Considering what we reviewed here, the density in the presence of the point mass may then be expressed as", "pages": [ 68, 69 ] }, { "title": "3.2 Growing a Black Hole in a Dark Matter Cluster: Relativistic Analysis", "content": "Given a system of particles characterized by a distribution function f (4) ( p ) , there is a standard prescription for writing down the mass current four-vector [34]: where µ is the particle's rest mass, p and p µ represent the four-momentum, g is the determinant of the metric, and d 4 p is the four-momentum volume element; the distribution function is again normalized so that the total mass of the halo is M . As in the Newtonian case, we wish to change variables from p µ to variables that are related to suitable constants of the motion. In the absence of a black hole, and for a spherically symmetric cluster, the constants would be the relativistic energy E , the angular momentum and its z -component ( L, L z ) , together with the conserved rest-mass µ = ( -p µ p µ ) 1 / 2 . A black hole that forms at the center will generically be a Kerr black hole, whose constants of motion are E , L z , µ , plus the so-called Carter constant C . In the limit of spherical symmetry, such as for the case of no black hole or for a central Schwarzschild black hole, C → L 2 . We will therefore begin by changing coordinates in the phase-space integral from p µ to E , C , L z , and µ assuming that the background geometry is the Kerr spacetime. We will find that the loss of spherical symmetry and the dragging of inertial frames that go together with the Kerr geometry make the problem considerably more complex. Further study of this case will be deferred to future work. Taking the limit of a Schwarzschild black hole simplifies the analysis, and allows us to formulate the adiabatic growth of a non-rotating black hole in a fully relativistic manner.", "pages": [ 70 ] }, { "title": "3.2.1 Kerr Black Hole Background", "content": "The Kerr metric is given in Boyer-Lindquist coordinates by where G is Newton's constant, m is the mass, a is the Kerr parameter, related to the angular momentum J by a ≡ J/m ; Σ 2 = r 2 + a 2 cos 2 θ , and ∆ = r 2 + a 2 -2 Gmr . We will assume throughout that a is positive, and use units in which c = 1 . Timelike geodesics in this geometry admit four conserved quantities: energy of the particle per unit mass, E , angular momentum per unit mass, L z , Carter constant per unit (mass) 2 , C , and the norm of the four momentum, The version of the Carter constant used here has the property that, in the Schwarzschild limit ( a → 0 ), C → L 2 , where L is the total conserved angular momentum per unit mass. We want to convert from the phase space volume element d 4 p to the volume element d E d C d L z d µ , using the relation where the Jacobian is given by the determinant of the matrix where we used the fact, which follows from the Kerr metric, that g 2 0 φ -g 00 g φφ = ∆sin 2 θ . Again including a factor of 4 to take into account the ± signs of p θ and p r in contrast to the quadratic nature of C and the norm of p µ , and using the fact that √ -g = Σ 2 sin θ , we obtain If the particles described by the distribution have the same rest mass, and if we again assume that the three-dimensional distribution function is normalized as before, then f (4) ( p ) ≡ µ -3 f ( E , C ) δ ( µ -µ 0 ) , and thus we can integrate over µ , to obtain We again assume that f is independent of L z . This may be compared with Eq.(3.17); J 0 is related to the density ρ , the relativistic energy E replaces E , C plays the role of L 2 , Σ 2 ∆ replaces r 4 , and four-velocities u r and u θ replace ordinary velocities v r and v θ . By definition, J µ ≡ ρu µ , where ρ is the mass density as measured in a local freely falling frame, and u µ is the four-velocity of an element of the matter, which can be expressed in the form u µ ≡ γ (1 , v j ) , where v j ≡ u j /u 0 = J j /J 0 , and using u µ u µ = -1 leads to γ = ( -g 00 -2 g 0 j v j -g ij v i v j ) -1 / 2 . Thus, once the components of J µ are known, then the v j components and therefore, u 0 = γ can be determined, and from that ρ = J 0 /u 0 can be found. Alternatively, because the norm of u µ is -1 , ρ = ( -J µ J µ ) 1 / 2 . In particular, if J µ has no spatial components, then u 0 = ( -g 00 ) -1 / 2 and ρ = √ -g 00 J 0 . The four-velocity components u r and u θ can be expressed in terms of the constants of the motion by suitably manipulating Eqs. (3.34c) and (3.34d), leading to where From Eq. (3.39), it is clear that, since u r and u θ are equally likely to be positive as negative for a given set of values for E , C and L z , the components J r and J θ of the current must vanish. Furthermore, since u 0 = -E and u φ = L z , we have that Even if we assume that f is independent of L z , the presence of the term in V ( r ) [Eq. (3.41)] that is linear in L z implies that J φ will not vanish in general, and thus the distribution of matter will have a flux in the azimuthal direction. This, of course, is the dragging of inertial frames induced by the rotation of the black hole, an effect that will be proportional to the Kerr parameter a . In this case the density may be obtained from where Ω ≡ J φ /J 0 . If a = 0 , then J φ = 0 , and ρ = -J 0 ( g φφ / ∆) 1 / 2 = -J 0 ( -g 00 ) 1 / 2 = √ -g 00 J 0 . The three-dimensional region of integration over E , C and L z is complicated. The energy E is bounded above by unity if unbound particles are to be excluded from consideration. The variables are bounded by the two-dimensional surfaces defined by u θ = 0 and u r = 0 , the latter depending on the value of r . A final bound is provided by the condition that if a given particle has an orbit taking it close enough to the black hole to be captured, it will disappear from the distribution. For a given E and L z there is a critical value of C , below which a particle will be captured. No analytic form for this condition has been found to date, although for nonrelativistic particles for which E = 1 is a good approximation, Will [35] found an approximate analytic expression for the critical value of C .", "pages": [ 70, 71, 72, 73 ] }, { "title": "3.2.2 Schwarzschild Black Hole Background", "content": "We now restrict our attention to the Schwarzschild limit, a = 0 , in which Σ 2 = r 2 , C = L 2 , u θ = ( L 2 -L 2 z sin -2 θ ) 1 / 2 and The metric components are g 00 = -g -1 rr = -1 + 2 Gm/r , and g 0 φ = 0 . Substituting these relations, along with the fact that u 0 = -g 00 E , we write J 0 in the form and we observe that J φ = 0 . We then integrate over L z between the limits ± L sin θ explicitly to obtain We again assume that E is bounded above by unity; E and L are also bounded by the vanishing of V ( r ) and by the black hole capture condition. Unlike the Kerr case, the capture condition in Schwarzschild can be derived analytically. We wish to find the critical value of L such that an orbit of a given energy E , and L will not be 'reflected' back to large distances, but instead will continue immediately to smaller values of r and be captured by the black hole. The turning points of the orbit are given by the values of r where V ( r ) = 0 . The critical values of E , L are those for which the potential has an extremum at that same point, that is where d V ( r ) / d r = 0 . The chosen sign for V ( r ) also dictates that this point should be a minimum of V ( r ) , that is that d 2 V ( r ) / d r 2 > 0 , corresponding to an unstable extremum. We obtain from the condition d V ( r ) / d r = 0 the standard solution for the radius of the unstable circular orbit in Schwarzschild r = 6 Gm/ { 1 + [1 -12( Gm/L ) 2 ] 1 / 2 } . Substituting this into the condition V ( r ) = 0 and solving for L , we obtain the critical value Notice that, for E = 1 , L c = 4 Gm , corresponding to the unstable marginally bound orbit in Schwarzschild at r = 4 Gm , while for E = √ 8 / 9 , L c = 2 √ 3 Gm , corresponding to the innermost stable circular orbit at r = 6 Gm . The range of integration of the variables is therefore as follows: L is integrated from L min = L c to the value given by V ( r ) = 0 , namely In fact, using Eq. (3.40), L max is the value such that for L ≤ L max , u r is real. The energy E is then integrated between its minimum value and unity. That minimum value is found by solving V ( r ) = 0 with L = L c , and is given by The regions of integration for various values of r are shown in Fig. 3.1. For a given r , the region is a triangle bounded by the critical capture angular momentum on the left, the maximum energy E = 1 at the top, and the condition V ( r ) = 0 on the triangle's lower edge. For r = 6 Gm , the lower edge of the region is the long dashed line shown (red in color version). As r increases above 6 Gm the lower edge of the triangle moves upward and the right-hand vertex moves rightward, as shown by the dotted and dot-dashed lines in Fig. 3.1 (blue and green in color version). For values of r decreasing below 6 Gm , the lower edge of the triangle moves upward and leftward as shown by the short dashed line in Fig. 3.1 (violet in color version). At r = 4 Gm , E min = E max = 1 2 ∝) (E/ and L min = L max = 4 , and the volume of phase space vanishes. This implies that, irrespective of the nature of the distribution function, the density of particles must vanish at r = 4 Gm ; this makes physical sense, since any bound particle that is capable of reaching r = 4 Gm is necessarily captured by the black hole and leaves the distribution. This is a rather different conclusion from the one reached by Gondolo and Silk ( [18], GS hereafter), who argued that the density would generically vanish at r = 8 Gm . The specific shape of this phase space region for small r will play a central role in determining the density distribution near the black hole. In the Schwarzschild limit, the four-velocity components are given by u φ = L z , u θ = ( L 2 -L 2 z sin -2 θ ) 1 / 2 , and u r = [ E 2 -(1 -2 Gm/r )(1 + L 2 /r 2 )] 1 / 2 , so that the adiabatic invariants are", "pages": [ 73, 74, 75 ] }, { "title": "3.2.3 Example: Constant Distribution Function", "content": "To illustrate the application of these results, we consider the special, albeit unrealistic case of a constant distribution function f ( E , L ) = f 0 . Then f is still constant after applying the adiabatic condition. Since f is independent of L , we can do the L integration explicitly to obtain from which we obtain the density Substituting Eq. (3.48) and integrating over E numerically between the limits shown in Eq. (3.50), we obtain the number density plotted in Fig. 3.2. GS [18] attempted to incorporate the relativistic effects of the black hole within a Newtonian context as follows. First they approximated the energy E by E = 1 + E , so that, to Newtonian order, the denominator in Eq. (3.47) is ≈ [2( E + Gm/r ) -L 2 /r 2 ] 1 / 2 , and E d E ≈ d E . For the critical capture angular momentum they adopted the approximation L c = 4 Gm , the value corresponding to E = 1 , while for the minimum energy, they adopted the value of E for which the denominator vanishes for that critical angular momentum. For the constant distribution function the integrals can be done analytically, with the result [GS, Eq. (6)] In Fig. 3.2 we plot Eq. (3.54) for comparison with the relativistic result. The two distributions agree completely at large distances, as expected. The GS distribution vanishes at r = 8 Gm , and is a factor of three smaller at its peak than the fully relativistic distribution.", "pages": [ 76, 77 ] }, { "title": "3.3 Application: the Hernquist Model", "content": "The luminosity density of many elliptical galaxies can be approximated as a power law in radius at both the largest and smallest observable radii, with a smooth transition between these power laws at intermediate radii [36]. Numerical simulations of the clustering of dark matter (DM) particles suggest that the mass density within a dark halo has a similar structure [30]. For these reasons much attention has been devoted to models in which the density is given by where ρ 0 and a are the two parameters of the system. With β = 4 these models have particularly simple analytic properties and are known as Dehnen models [36-38]. The model with α = 1 and β = 4 is called a Hernquist model [39], while that with α = 2 and β = 4 is called Jaffe model [40]. Another dark halo model is given by Eq. (3.55) with α = 1 and β = 3 ; this is called the NFW model after Navarro, Frenk, and White [41]. Note that the Hernquist and NFW models have the same behavior for small r . However, the Hernquist model has the advantage that we can find its distribution function as a closed analytical function using Eddington's formula [30]. Therefore, in this section we choose the Hernquist model as the initial distribution of DM particles before the growth of the black hole; then, we derive how the growth of a Schwarzschild black hole will redistribute the DM distribution.", "pages": [ 77 ] }, { "title": "3.3.1 Newtonian Analysis", "content": "The Hernquist model is a spherically symmetric matter distribution whose density is given by where ρ 0 and a are the two scale factors. The corresponding Newtonian gravitational potential of this model is where M is the total mass of the cluster with M = 2 πρ 0 a 3 . The distribution function that is consistent with this potential is given by the (properly normalized) Hernquist form where where we adopt the following dimensionless quantities: where m is the mass of the black hole. With these definitions, the density Eq. (3.20) becomes: where ˜ L 2 max = 2 x 2 ( ˜ ψ -˜ glyph[epsilon1] ) and ˜ f H (˜ glyph[epsilon1] ) is given by Eq. (3.59). Normally we would have ˜ L min = 0 , and ˜ glyph[epsilon1] max ( x ) = ˜ ψ ( x ) . But we will allow the more general limits in order to include for comparison the GS ansatz for incorporating black-hole capture effects, namely ˜ L min = 4 ˜ m ( GM/a ) 1 / 2 and ˜ glyph[epsilon1] max ( x ) = ˜ ψ ( x )(1 -8 ˜ mM/xa ) . When we now grow a point mass adiabatially within the Hernquist model, the argument ˜ glyph[epsilon1] ' of the initial distribution (3.59) becomes a function of ˜ glyph[epsilon1] and L by equating the radial actions: and using the fact that ˜ L ' = ˜ L from the angular action. Hence the density around the point mass in a Hernquist profile takes the form: where ˜ L 2 max = 2 x 2 ( ˜ m/x -˜ glyph[epsilon1] ) . From Eq. (3.30), the radial adiabatic invariant for a point mass potential in dimensionless variables is We see that it diverges for glyph[epsilon1] → 0 , corresponding to the least bound particle. We will have to be careful when matching the radial actions in this limit. For the Hernquist potential, with ˜ ψ = 1 / (1 + x ) an analytic formula cannot be found for the radial invariant and thus it will have to be evaluated numerically. To this end, it is convenient to transform the integration in the following way. First, combine the three terms inside the square root to get We solve for the three roots of the numerator, of which the two positive roots give the turning points x + and x -, while the third root x neg is always negative. We then rewrite the function in the square root as: which is positive in the region x -≤ x ≤ x + . We now make a change of variables x = t ( x + -x -) + x -, which brings the integral into the domain [0 , 1] : This makes it much easier to control the integration numerically, since we can make sure that the roots have the right signs and ordering, and no numerical round-off errors will change that within the domain. For ˜ L 2 = 0 , the radial invariant can be integrated analytically, with the turning points x -= 0 and x + = 1 /glyph[epsilon1] -1 , and we use this fact in the code. The radial invariant is again divergent for glyph[epsilon1] → 0 . Since we are only interested in finding a solution in the domain (0 , 1] , we simply define the value there to be a very large number, and use a bracketing algorithm. For numerical work, it is also convenient to remap the integral (3.63) for ρ ( r ) into a square domain. This is a particular case of a set of transformations discovered by Duffy [42]. We make a change of variables, (˜ glyph[epsilon1], ˜ L 2 ) → ( u, z ) , that maps the domain of integration in Eq. (3.61) onto the square [0 , 1] × [0 , 1] : where we emphasize that ˜ L 2 max depends on u . The jacobian is: With this change, the integral in Eq. (3.61) reads: where the arguments of the distribution function are given in Eq. (3.70). This will have the effect of making our codes faster and more stable. One of the advantages is that the integrable singularity that was originally in a corner ( ˜ glyph[epsilon1] = ˜ ψ , ˜ L 2 = 0 ) of the integration domain has now been transferred to a line, depending only on the variable z . Using the GS conditions for ˜ L min and ˜ glyph[epsilon1] max and carrying out the numerical integrations, we obtain the curve labeled 'Non-relativistic' in Fig. 3.3.", "pages": [ 78, 79, 80, 81 ] }, { "title": "3.3.2 Relativistic Analysis", "content": "We now apply these considerations to the relativistic formalism. Here we define ˜ glyph[epsilon1] in terms of the relativistic energy E per unit particle mass using the other definitions in Eqs. (3.60) will be the same. Using these definitions, and the relation ρ = -J 0 ( -g 00 ) 1 / 2 along with Eq. (3.47), we find where ˜ f (˜ glyph[epsilon1] ) is again given by Eq. (3.59), and where we used E = 1 for the maximum energy of the bound particles which leads to ˜ glyph[epsilon1] min = 0 . Compare the last equation of (3.74) to Eq. (3.61). To consider the growth of the central black hole and its capture effects, we use Eqs. (3.48)-(3.50) as the limits of the integrals of Eq. (3.74), which in terms of the dimensionless parameters have the form As in the non-relativistic case, in order to grow a point mass adiabatically within the Hernquist model, the argument ˜ glyph[epsilon1] ' of the initial distribution function becomes a function of ˜ glyph[epsilon1] and ˜ L by equating the radial actions and using the fact that ˜ L ' = ˜ L from the angular action. Hence, the density around a relativistic point mass in a Hernquist profile takes the form: The difference here is that in equating the radial actions in Eq. (3.62), we use the relativistic expression for the point-like mass radial action i.e. Eq. (3.51a) which in terms of dimensionless variables can be written as where x + and x -are the two turning points. The integration in Eq. (3.77) will have to be evaluated numerically. Now we take the same steps as we used to get Eq. (3.68): first we combine the terms inside the square root to get We solve for the three roots of the numerator, of which the two positive roots give the turning points x + and x -, while the third x neg is always negative. We then rewrite the function in the square root as: which is positive in the region x -≤ x ≤ x + . We now make a change of variables x = t ( x + -x -) + x -, which brings the integral into the domain [0 , 1] : As before, this leads to easier numerical control. For ˜ L 2 = 0 , the radial invariant can be integrated analytically, with the turning points x -= 0 and x + = ˜ m/ (˜ glyph[epsilon1] (1 -˜ glyph[epsilon1] GM/ 2 a )) : and we use this fact in the code. The radial invariant is again divergent for glyph[epsilon1] → 0 but we are only interested in finding a solution in the domain (0 , 1] . For the Hernquist potential we use the same equations as the non-relativistic calculations. Again we remap the integral in Eq. (3.76) into a square domain using the Duffy transformations. The only difference here is that ˜ L 2 min also depends on u . With these changes, the integral in Eq. (3.76) reads: where the arguments of the distribution function are given in Eq. (3.70). The numerical integrations yield the curve labeled 'Relativistic' in Fig. 3.3.", "pages": [ 81, 82, 83 ] }, { "title": "3.3.3 Profile Modification due to Self-annihilation", "content": "Our calculations so far give the DM distribution as it reacts to the gravitational field of the growing black hole. In addition, the DM density will decrease if the particles self-annihilate. In fact, if we take into account the annihilation of DM particles, the density cannot grow to arbitrary high values, the maximal density being fixed by the value is [43]: where σv is the annihilation flux (cross-section times velocity), m χ is the mass of the DM particle, and t bh is the time over which the annihilation process has been acting, which we take it to be ≈ 10 10 yr [18]. The probability for DM self-annihilation is proportional to the square of the density, This expression can be derived by noting that the annihilation rate per particle is Γ = nσv , therefore ˙ n = -n Γ = -n 2 σv and ρ = nm χ . If we call the output of our code neglecting annihilations ρ ' ( r ) and the final profile reprocessed by this process ρ sp ( r ) , we can integrate Eq. (3.84) as follows: which gives: Our calculations do not include the effect of the gravitational field of the halo in the final configuration. This is a good approximation close to the black hole, but far away from the center the effect of the black hole is negligible and the DM density will be described by the halo only. We take care of this fact by simply adding the initial Hernquist profile, given in Eq. (3.56) to the calculated spike. We expect this approximation to be good, except possibly in the transition region. The result is the curve labeled 'DM annihilation' in Fig. 3.3. We show in Fig. 3.3 the results of our numerical calculations. In the non-relativistic limit, they are a good match to the calculation in GS.", "pages": [ 83, 84 ] }, { "title": "3.3.4 Periastron Precession with a Dark Matter Spike", "content": "As we mentioned in Chapter 2, the presence of the DM density at the GC can perturb the orbits of stars in that region. For related articles see [44, 45]. A spherically symmetric distribution of dark matter will cause pericenter precessions in orbital motions, but will not change the orientation of the orbital planes. But to get an upper bound on the possible effect of a nonspherical distribution of dark matter on the orbits of potential no-hair-theorem target stars, it is useful to determine the pericenter precession. For this we need the dark matter mass including the spike inside a given radius r , which we obtain by integrating our density profile, m(r)= 4 π ∫ r 2 ρ ( r )d r . The result for both the self-annihilating and non-self-annihilating cases, is shown in Fig. 3.4. As can be seen from Fig. 3.4, we can approximate the total mass of the DM in the region between 10 and 10 4 Schwarzschild radii by a power-law function: which leads to the following additional acceleration term in the equation of motion of a star orbiting the black hole: where ˆ n ≡ r /r . Since the perturbing term in Eq. (3.88) has only the radial component R , using Eq. (2.59) for the rate of change with angle of the pericenter of an orbit, d ω/ d f , we have where we used Eq. (2.64), which for calculations of the first order perturbation, reduces to d f/ d t = h/r 2 . Substituting Eq. (3.88) in Eq. (3.89) and using r = p/ (1 + e cos f ) and h 2 = Gmp , we get To get the changes of ω over one orbit, we integrate Eq. (3.90) over the true anomaly f from 0 to 2 π to obtain where, using the change of variable of integration described in Appendix A, for various values of q , we get the forms for f q ( r ) shown in Table 3.1. Now from Fig. 3.4, we can see that the power q in Eq. (3.87) can be chosen to be 1 or 3 depending on whether the DM particles self-annihilate or not, respectively. Using r 0 = r Sch × 10 4 = (2 Gm ) × 10 4 ≈ 4 . 6 mpc , assuming a black hole mass m = 4 × 10 6 M glyph[circledot] , we can read off the values × Table 3.2: Astrometric precession rates as seen from the Earth in units of µ arcsec/yr; ˙ Θ J and ˙ Θ Q 2 denote orbital plane precessions, while the others denote pericenter precessions. of m 0 : To get an estimation of the pericenter precession effect of stars at the GC as seen from Earth caused by the DM distribution including the spike, we use our previous definition for the angular precession rate amplitude as seen from the Earth in Chapter 2, which is ˙ Θ DM = ( a/D )∆ ω/P , where D is the distance to the GC and P = 2 π ( a 3 /m ) 1 / 2 is the orbital period. Using m = 4 × 10 6 M glyph[circledot] and D = 8 kpc , we obtain the rates for the non-self-annihilating ( q = 1 ) and self-annihilating ( q = 3 ) DM particles distributions in microarcseconds per year: where we used Eq. (3.91) and the numbers in Eq. (3.92). To compare the rate of precession of periastron of a star rotating the MBH induced by the DM particles distributions with the relativistic effects of the MBH at the center, in Table 3.2, we provide numerical results for the S2 star and for a hypothetical target star which is closer to the center and could be used for the test of the no-hair theorem. Shown are the periastron precessions rates as seen from Earth from the Schwarzschild part of the metric and from the two dark matter distributions ( ˙ Θ S , ˙ Θ DM , ann . , and ˙ Θ DM , non -ann . , respectively) and the orbital plane precessions from the frame dragging and quadrupole effects ( ˙ Θ J and ˙ Θ Q 2 , respectively). In Fig. 3.5, using Eqs. (2.75)-(2.77) and Eq. (3.91), we plot the periastron precessions at the source given in the following equations, for a maximum rotating MBH ( χ = 1 ) and a higheccentricity target star with e = 0 . 95 : As can be seen from Table 3.2 and Fig. 3.5, for hypothetical target stars in eccentric orbits with semi-major axes less than 0 . 2 milliparsec, which could be used to test the no-hair theorem, the periastron precessions induced by the DM distribution at the center do not exceed the relativistic precessions. Because the pericenter advance due the dark matter distribution is so small , we argue that it is reasonable to consider this as a good estimate for the upper limit on the precession of orbital planes that might be induced by a non-spherical component of the DM distribution that would be generated by a rotating central black hole. That non-spherical part is likely to be a small perturbation of the basic DM distribution because the effects of frame dragging and the quadrupole moment are relativistic effects that fall off faster with distance than the basic Newtonian gravity of the hole. As a result, we can conclude that a dark matter distribution near the black hole will not significantly interfere with a test of the black hole nohair theorem. Furthermore, if the dark matter particles are self-annihilating, their effects will be utterly negligible. On the other hand, for S2-type stars, if future capabilities of observational precision reach the level of 10 µ arcsec per year, the perturbing effect of the DM distribution on stellar motion at the GC could be marginally detectable if the DM particles are not self-annihilating, as would be the case if they were axions, for example. If they are self-annihilating, the effects of a DM distribution on the outer cluster of stars will be unobservable.", "pages": [ 84, 85, 86, 87, 88, 89 ] }, { "title": "APPENDICES", "content": "A", "pages": [ 92 ] }, { "title": "A Useful Change of Variables", "content": "In calculating the time averaged rates of change of the orbit elements of the target star given by Eq. (2.97), we encounter integrals such as which can not be done analytically by Maple or Mathematica. To find the analytical result for these kind of integrals we rewrite Eq. (A.1) as where the second term comes from letting f → π -f . Depending on the value of n , this gives a sum of integrals of the form which can be evaluated analytically easily by Maple. For example where Writing every P n,m integral as a sum of Q n,m integrals simplify the calculations, and minimally, it allows us to give analytical expressions for many steps. B", "pages": [ 92, 93, 94 ] }, { "title": "Minimum Distance for a Stellar or Black Hole Orbit", "content": "A star that approaches too close to the black hole will be tidally disrupted and be removed from the stellar distribution. An estimate of this distance is given by the 'Roche radius', r Roche ≈ R (2 M/m ) 1 / 3 , where R is the radius of the star, and M and m are the black-hole and stellar masses, respectively. For a solar-type star, the radius R may be estimated using the empirical formula R ≈ R glyph[circledot] ( m star /m glyph[circledot] ) 0 . 8 . Thus we obtain r star min ≈ R glyph[circledot] ( m star /m glyph[circledot] ) 0 . 47 (2 m/m glyph[circledot] ) 1 / 3 . Putting in numbers gives the first of Eqs. (2.125). A stellar-mass black hole will not be tidally disrupted, but can be captured directly if its energy and angular momentum are such that there will be no turning point in its radial motion. For equatorial orbits in the Kerr geometry (in Boyer-Lindquist coordinates), the equation of radial motion has the form (d r/ d τ ) 2 = ˜ E 2 -V ( r ) , where τ is proper time, ˜ E is the relativistic energy per unit m bh of the orbiting black hole where m bh is the mass of the orbiting stellar mass black hole , and where ˜ m = Gm , a = J/m , β = ˜ L 2 z -a 2 ˜ E 2 , and α = ˜ L z -a ˜ E , where J is the angular momentum of the central black hole and ˜ L z is the angular momentum per unit m bh of the orbiting black hole. The critical angular momentum for capture is given by that value such that the turning point occurs at the unstable peak of V ( r ) . Since the orbiting stars and black holes are in nonrelativistic orbits, we can set ˜ E ≈ 1 . Under these conditions, it is straightforward to show that where the upper (lower) sign corresponds to prograde (retrograde) orbits. For a/ ˜ m = 1 , the critical angular momenta are 2 ˜ m and -2(1 + √ 2) ˜ m . Converting to the language of orbital elements, where L 2 z = m 2 bh Gma (1 -e 2 ) , we find in the large e limit, L 2 z ≈ 2 m 2 bh Gmr p where r p is the pericenter distance of the stellar mass black hole orbit. The result is that This ranges from 2 Gm to 11 . 6 Gm for a/ ˜ m = 1 and is 8 Gm for a = 0 (Schwarzschild). We adopt the latter value as a suitable estimate; inserting numbers gives the second of Eqs. (2.125). C", "pages": [ 94, 95, 96 ] }, { "title": "Effects of Tidal Deformations", "content": "Even if stars survive tidal disruption on passing very close to the MBH at pericenter, they will be tidally distorted, and these distortions can affect their orbits. However, we argue that, for the stellar orbits of interest, these effects are negligible. For example, the rate of pericenter advance due to tidal distortions is given by (Eq. (12.31) of [27]) where k 2 is the so-called 'apsidal constant' of the star, a dimensionless measure of how centrally condensed it is. Inserting R = R glyph[circledot] ( m/m glyph[circledot] ) 0 . 8 , we obtain The variations in ı and Ω scale in exactly the same way, but are further suppressed by the sine of the angle by which the tidal bulge points out of the orbital plane, resulting from the rotation of the star coupled with molecular viscosity, leading to a lag between the radial direction and the tidal bulge. This angle is expected to be very small. Thus we can conclude that, as far as perturbations of the orbital planes are concerned, tidal distortions will not be important. D", "pages": [ 96, 98 ] }, { "title": "Distribution Function Invariance in Adiabatic Growth of a Point Mass", "content": "Young has shown in [32] that for the adiabatic growth of a black hole in the center of a star cluster, the conservation of the two adiabatic invariants, namely the angular momentum L and the radial action I r of each star, leads to the invariance of the distribution function. In this appendix we first review his argument in our notation for the adiabatic growth of the central black hole in the distribution of dark matter particles and then we show that the result holds in the general relativistic domain too. As the black hole grows, the gravitational potential evolves from the initial potential Φ ' to a new potential Φ that includes the point mass and a dark matter particle, initially with conserved quantities ( E ' , L ) in E -L space, moves to ( E,L ) such that I r ( E,L ) = I ' r ( E ' , L ) , therefore: where N ( E,L ) is the density of particles in E -L space. The number of particles in phase space for a spherically symmetric system is where we used the same change of variables that we have in Chapter 3 to get Eq. (3.18) assuming the distribution function is independent of L z . The corresponding number of dark matter particles in E -L space with energy E in [ E,E + d E ] and angular momentum L in [ L, L +d L ] in the d E d L volume element is N ( E,L )d E d L and to equate this with Eq. (D.2), we need to integrate Eq. (D.2) over all values of r . Assuming the distribution function is independent of position we have: where r ± are the turning points of the dark matter particles equation of motion and P ( E,L ) is the orbital period of the dark matter particle. Equation (D.4) agrees with Eq. (26a) of Young's paper [32]. According to the definition of the radial action I r ( E,L ) in Eq. (3.27) we have: and using I r ( E,L ) = I ' r ( E ' , L ) leads to where P ' ( E ' , L ) = ∮ d r/ √ 2 E ' -2Φ ' ( r ) -L 2 /r 2 . Substituting Eq. (D.4) for N ( E,L ) in Eq. (D.1) gives: Now by using Eqs. (App.D-6) and (App.D-7), we get the invariance of the distribution function (Eq. (29) of Young's paper): where we used d E ' = ( ∂E ' /∂E | L ) d E . So by equating the radial actions and deriving the E ' = E ' ( E,L ) relation, we will have the final distribution function. Now we generalize the derivation of Eq. (D.8) to the relativistic formalism for the growth of a Schwarzschild black hole. Here we need to use the relativistic radial action given in Eq. (3.51a). Similar to the non-relativistic case, the conservation of the number of particles in phase space gives: To get a similar equation to Eq. (D.2), we need to use the relativistic Jacobi to change the variables. In spherical symmetry limit, the Jacobi is similar to what we have in Eq. (3.47): where v r = √ E 2 -(1 -2 Gm/r )(1 + L 2 /r 2 ) . Therefore, if f ( r, E , L ) = f ( E , L ) , by integrating Eq. (D.10) over r , for the number of particles in d E d L volume element we get Note that the differences of Eq. (D.12) with the non-relativistic case (Eq. (D.4)), are an extra factor of E and the definition of v r . Also the P ( E , L ) in Eq. (D.12) is not the orbital period of the dark matter particle's orbit measured by an observer sitting at infinity. In fact, since v r = d r/ d τ , P ( E , L ) is the orbital period measured by the clock moving with the particle. Using the definition of I r ( E , L ) in Eq. (3.51a) we have Assuming I r ( E , L ) = I ' r ( E ' , L ) , Eq. (D.13) results in Substituting Eq. (D.12) in Eq. (D.9) gives again since d E ' = ( ∂ E ' /∂ E ) | L d E , using Eqs. (D.14) and (D.15) leads to the invariance of the distribution function in the relativistic formalism:", "pages": [ 98, 99, 100, 101 ] } ]

[ { "title": "Squeezing of optomechanical modes in detuned Fabry-Perot interferometer", "content": "Andrey A. Rakhubovsky ∗ 1 and Sergey P. Vyatchanin 1 1 Physics Department, Moscow State University, Moscow 119992 Russia (Dated: June 6, 2021) We carry out analysis of optomechanical system formed by movable mirror of Fabry-Perot cavity pumped by detuned laser. Optical spring arising from detuned pump creates in the system several eigen modes which could be treated as high-Q oscillators. Modulation of laser power results in parametric modulation of oscillators spring constants thus allowing to squeeze noise in quadratures of the modes. Evidence of the squeezing could be found in the light reflected from the cavity.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The purpose of gravitational waves detection leads to construction of large-scale antennas like LIGO [1, 2], VIRGO [3] and GEO [4]. Very high sensitivity of these devices is limited by a number of noises. In the low frequency range ( below ∼ 50 Hz) the prevailing sources of noise are seismic ones, at middle frequencies ( ∼ 50 -200 Hz) thermal noises dominate and in high frequency range (over 200 Hz) photon shot noise prevails. However the technical improvement of antennas by compensation and suppression of these and other noises will allow to achieve sensitivity level defined only by quantum noise which for continuous position measurement has lowest boundary defined by Standard Quantum Limit (SQL) [5-8]. SQL is the optimal combination of two noises of quantum nature: fluctuations of mirror motion caused by random photon number falling onto its surface and photon counting noise. One of the ways to overcome the SQL is the implementation of so-called optical rigidity (optical spring) effect [8-11] which is based on a fact that in a detuned Fabry-Perot interferometer the circulating power and consequently the radiation pressure is dependent on the distance between the mirrors. It has been shown in a number of papers [12-18] that interferometers using optical springs exhibit sensitivity below the SQL. In a system utilizing optical spring there are two degrees of freedom (in case of one pump): a mechanical one and an optical one. Interaction of these coordinates gives birth to several eigen modes (the number of which is equal to the system degrees of freedom number) each of which is characterized by its own resonance frequency and damping. Free evolution of the system can be represented as a sum of eigen modes each of which is an oscillator with its eigen frequency and has corresponding damping. In principle one can make transfer from the conventional coordinates to eigen ones and consider the evolution of the system as evolution of these new oscillators. Using the curve of susceptibility of the system one can estimate the average energy stored in each equivalent eigen mode oscillator. It is well known [5, 8, 19] that an oscillator with average energy E a exhibits quantummechanical behavior if the condition is met. Here ω and Q are eigen frequency and quality factor of oscillator. Estimations [20] show that for system such as interferometer Advanced LIGO these conditions are well fulfilled. As a consequence one can expect the corresponding eigen modes to be observed in quantum state despite the fact that the (effective) mass of mechanical oscillator in considered interferometer is equal to 10 kg. Given this motivation it is interesting to look for any experimental scenario giving an ability to bring one of the eigen modes to non-classical state. Currently the development of techniques aimed on preparation of mechanical resonator in quantum state by means of optomechanical interaction is well underway including investigations of micromembranes [21], microtoroids [22], optomechanical crystals [23], pulse-pumped optomechanical cavities [24] and even large-scale gravitational-wave detectors [25]. One of the most important problems inherent to optomechanical devices is relatively large losses of mechanical microoscillator. Replacement of material spring by optical one may decrease the losses in mechanical system and thus provide an experimental possibility of realisation of quantum regimes. For the initial consideration it is also worth to show the very possibility of manipulation with optomechanical modes. To prove this possibility we demonstrate the mechanism to perform quadrature squeezing of shot noise caused fluctuations in these modes using parametric modulation of spring constant which has been previously considered as a tool to squeeze fluctuations in cavity mirror motion [26].", "pages": [ 1 ] }, { "title": "II. DESCRIPTION OF MODEL", "content": "We consider a gravitational-wave detector Advanced LIGO with a signal recycling mirror (SRM) having an amplitude reflectivity ρ and power recycling mirror (PRM); a scheme of antenna is presented in fig. 1. Antenna consists of a Michelson interferometer with addi- ional mirrors forming Fabry-Perot (FP) cavities with mean distance L between mirrors in arms. Input mirrors have amplitude transmittivities T and masses m and output mirrors have the same masses m and are completely reflective. We assume that all mirrors are lossless. The interferometer is pumped by laser having frequency resonant for FP cavities in arms. Dynamics of this interferometer could be split into two modes: namely differential one and common one. The optical fields in the modes represent difference and sum of the fields in arms respectively and carry information about difference and sum between lengthenings of arm cavities. Each mode is characterized by optical detuning and decay rate introduced by displacement of corresponding recycling mirror: SRM for differential mode and PRM for common one. It has been demonstrated [15] that each of this modes could be equivalently described using a model of FP cavity (see fig. 1, bottom). We focus our attention to differential mode. The end mirror of equivalent cavity has mass equal to reduced mass of four mirrors of Michelson interferometer µ = m 4 and is completely reflective. The displacement of this mirror is equal to difference of arms lengthening of initial interferometer: y = ( x e -x i ) -( y e -y i ). Distance separating the mirrors is equal to one in initial interferometer, namely L . The pump in equivalent scheme should be characterized by the same relaxation rate Γ and detuning ∆ as in initial scheme and mean circulating power P two times bigger than one in initial interferometer. Adetuned pump creates in FP cavity an optical spring, that is the radiation-pressure force which depends on the distance separating mirrors of the cavity. A system with only one optical spring is always unstable because a single pump introduces either negative damping or negative rigidity [9-12]. A few ways to avoid instabilities have been proposed amongst which there are implementation of feedback [15] or utilization of additional pump [27, 28]. The latter way has been investigated in details and proven experimentally with mirror of gram-scale [29]. However in laboratory-scale experiment it should be easier to utilize a proper feedback. Hence we consider a scheme of FP cavity with one pump and a feedback instead of two pumps. Input optical fluctuations are described by annihilation operator a i , operator a o describes output fluctuations. To avoid instability caused by optical spring we measure phase quadrature a o 2 of output fluctuations with homodyne detector and send its derivative over time as a feedback force to the movable mirror. The dynamics of interferometer could be described in terms of two degrees of freedom. One of those is annihilation operator b of optical fluctuations inside the interferometer (actually, its amplitude quadrature b 1 ) and another is mechanical displacement y . Equations of motion for this system could be written as follows (in capacity of mechanical coordinate we use dimensionless displacement z ): Derivation of these equations from the hamiltonian of the system and exact definition of coefficients are presented in Appendix A. We use dimensionless parameters defined the same way as in [20, 28]. Parameter g stands for optical relaxation rate, coupling coefficient A is proportional to pump power, α is the coefficient of feedback. The right parts of equations represent fluctuational forces acting on corresponding degree of freedom. As of optical one the force ν 1 represents input vacuum fluctuations and for mechanical one ν 2 describes vacuum fluctuations reflected by the cavity and transmitted by feedback. We neglect all other fluctuational forces including thermal and seismic ones. Depending on device in consideration this first step assumption could be either realistic or not. Some calculations of thermal noise influence is presented in Appendix B. In particular they show that in configuration of Advanced LIGO interferometer the impact of thermal noises is only slightly below the one of quantum fluctuations. The equations (1) look similar to ones for oscillator coupled to a free mass (or two coupled oscillators one of which has partial frequency equal to zero). However we would note the difference in signs in front of the coupling terms (proportional to A ): in the case of conventional coupled oscillators these signs coincide; in our case the signs differ. This is the reason of instability which appears in absence of feedback and this is also the reason of possibility for eigen frequencies of such system to coincide (so called double resonance case [13, 16, 17] - let us remind that in case of two coupled oscillators coincidence of eigen frequencies is impossible).", "pages": [ 1, 2, 3 ] }, { "title": "III. PARAMETRIC SQUEEZING OF EIGEN MODES AMPLITUDES", "content": "As in the case of conventional coupled oscillators one can treat the system in terms of eigen modes. The evolution of oscillator system could be written as a sum of eigen oscillations: Index in the sum runs from 1 to the number of system degrees of freedom (2 for set (1)). This equation could be treated as transformation from conventional coordinates ( b 1 ; z ) to the eigen ones. As in eqn. (2) one can present evolution of each of eigen modes as combination of oscillations at corresponding eigen frequencies ω i and slow (compared to these oscillations) changing of amplitude. In case of free evolution (zero right parts in eqns. (1)) quantities g i ∼ e -γ i t i.e. they freely decay with relaxation rates γ i into the equilibrium values. Vectors glyph[vector]v i represent distribution of amplitudes in corresponding eigen modes. Suppose that we have a possibility to modulate the power of pumping laser. This will result in modulation of coupling coefficient and hence it will cause shifting of eigen frequencies of system. It is well known [30] that modulation of eigen frequency of oscillator with frequency twice bigger than its own one results in squeezing of noise in its quadratures. In this paper we show that modulation of the pumping power performs squeezing of noise in quadratures of eigen modes amplitudes. Consider we harmonically modulate the pumping power at frequency 2 p so coupling coefficient in (1) depends on time: A → A (1 + 2 | m | cos(2 pt + φ )). Assuming the quantities g i to be slow (i.e. not to significantly change on times compared to mechanical periods 2 π/ω i ) one can plug the expression (2) into system (1) and to get rid of rapidly oscillating terms by averaging over a period 2 π/p . This procedure results in shortened equations for amplitudes g i . For one with number j ( j = 1 , 2) the equation takes form: Here we have introduced a set of vectors glyph[vector] Π j built to be orthogonal to all v i except one: ( glyph[vector] Π j glyph[vector]v i ) = δ ji , where δ ji is a Kronecker delta (in the case of two conventional coupled oscillators vectors glyph[vector]v i are itself orthogonal and system of glyph[vector] Π i is unnecessary). Vectors glyph[vector] w i are defined as follows: Underlining used for ν j is to emphasise that this quantity is obtained in right part of equation by procedure of keeping only slow (in comparison to terms oscillating with frequency p ) quantities. To make ν j e iω j t fulfilling this criterion one should keep in ν j only spectral components close to ω j . From now on we will focus on the amplitude of one of modes taking into account that all the results obtained for this mode are similar for other ones. For clarity let us set j = 1 thus considering the first mode. Also we suppose the modulation to happen at frequency twice bigger than the one of first mode: p = ω 1 . First let us consider the simplest case when difference between eigen frequencies ω 1 -ω 2 is large in respect to decay rates γ i . In this case the exponential multipliers in eqn. (3) should be considered as fast oscillating ones and the equations for amplitudes g i decouple. In this case the equation for g 1 takes the following form in spectral domain Here x stands for normalized spectral frequency (see appendix A), γ 1 is the decay rate of first mode; glyph[epsilon1] 11 is a quantity proportional to modulation strength, the general expression for it could be easily deduced from eqn. (3): We also write the equation for g † 1 ( -x ) by hermitian conjugation of eqn. 4 and replacement x →-x . Taking proper combinations of these equations yields the expressions for corresponding quadratures of g 1 . For simplicity let glyph[epsilon1] 11 = glyph[epsilon1] ∗ 11 = | glyph[epsilon1] 11 | which is achievable by proper choosing φ , in this case the mentioned combinations reduce to sum or difference: One can conclude from these equations that sum quadrature is squeezed due to parametric modulation and the difference quadrature is antisqueezed. The measure of squeezing is the spectral density which for arbitrary quantity d ( ω ) is given by expression where the angle brackets mean averaging. Obvious calculations prove that spectral densities of right parts of both equations (5a) and (5b) coincide and the equations differ by additional damping | glyph[epsilon1] 11 | (positive or negative for different quadratures) introduced by modulation. In absence of modulation glyph[epsilon1] 11 = 0 and there is no discrepancy between quadratures. If modulation is applied then glyph[epsilon1] 11 has nonzero value and quadratures spectral densities differ. One can also estimate the critical level of modulation characterized by value m c , this is a level when negative damping added to the differential quadrature g 1 -g † 1 becomes equal to its own damping: γ 1 -| glyph[epsilon1] 11 | = 0 thus making quadrature instable. Given this value of modulation coefficient damping in the sum quadrature is twice bigger than its own damping γ 1 . This means that quadrature noise squeezing is limited by factor of two. In general case of arbitrary eigen frequencies ω 1 , 2 the equations (3) could not be solved in such an obvious way, but in this case one can rewrite these equations to switch from spectral components g j ( x ) to the same components but shifted by frequency, namely g j ( x -( ω j -ω 1 )). The equations take form This set represents a system of linear algebraic equations and could be solved by applying Kramer's rule. Using the solution we obtain the expression for plus or minus quadratures of j -th mode G ( ± ) j defined as follows and calculate spectral densities of these quadratures. In coincidence with expectations in absence of parametric modulation both plus and minus quadratures of first mode have equal spectral densities. If modulation is applied the plus quadrature G (+) 1 appears to be squeezed and the minus quadrature G ( -) 1 to be antisqueezed. The bigger is modulation coefficient the more significant the (anti-) squeezing effect is. It could be easily shown that squeezing is also limited by a factor of two in this case.", "pages": [ 3, 4 ] }, { "title": "IV. READOUT", "content": "The important question is whether it is possible to see this squeezing in output light. The answer to this question is positive. Output fluctuations a o are defined by input fluctuations a i and light inside cavity b , in particular for phase quadrature of a o the following expression is valid: Plugging the solution obtained in previous section for g 1 , 2 into the equation (2) we can thus express the output fluctuations through quantities g i . It is easy to show (see appendix C) that quantity is proportional to the quadrature G ( ± ) j ( x ). In time domain measurement of this quantity is equivalent to multiplication of measured phase quadrature with cosine of frequency ω j with proper phase: Estimations of A ( ± ) j spectral density demonstrate that it is possible to see squeezing in output light. The spectral densities of these quantities are plotted in fig. 2 being normalized to the shot noise level. The limitation of squeezing by factor of two shows up in output light too.", "pages": [ 4 ] }, { "title": "V. CONCLUSION", "content": "In this paper we have shown the possibility of quadrature noise squeezing of eigen modes amplitudes in an optomechanical system. As a model of latter we have used a Fabry-Perot cavity with movable mirror and a detuned pump. The eigen modes in this system are defined by interaction between optical and mechanical degrees of freedom. Detuned pump transforms the mirror which is initially a free-mass into a harmonic oscillator with spring coefficient provided by means of optical rigidity. In comparison with material rigidity peculiar to usual optomechanical devices such as microtoroids or membranes etc [21-24] optical rigidity is characterized by very low noises level. The another advantage of using optical springs is the very ability of spring constant manipulation by means of pump power modulation. Comparison of impact exerted by quantum and thermal noises upon quadratures (see Appendix B) reveals that both fluctuations have their affects of same order. This gives us an ability to talk about quantum noise squeezing of optomechanic quadratures in gravitational wave detectors (Advanced LIGO [2], Einstein Telescope [37]) and in prototypes (Glasgow University prototype [31], AEI Hannover prototype [32] and Gingin facility [33]) Despite the fact that we consider a feedback in our model, the feedback itself is unnecessary to achieve the squeezing as it serves only to stabilize the eigen frequencies of the system. The stabilization could be performed by means of an auxiliary pump with power significantly lower than one in main pump. However implementation of feedback is more feasible in small-scale experimental setups like gravitational-wave antenna prototypes [3133]. Realization of proposed scheme in experiment could demonstrate non-classical behavior of optomechanical mode which could be useful for tests of quantum mechanics applied to macroscopic mechanical objects. We would like to underline it is not pure optical or pure mechanical, but an optomechanical degree of freedom that may exhibit quantum behavior. Also a quantum state if created in an eigen mode could be transmitted into another one (for example, by modulation of pump with difference frequency ( ω 2 -ω 1 )). This gives us a potential playground for creation and transmission of quantum states between optomechanical modes which could be useful for problems of quantum information.", "pages": [ 4, 5 ] }, { "title": "Acknowledgments", "content": "The authors would thank Stefan Hild for stimulating discussions. We also are grateful to Yanbei Chen, Farid Khalili and Stefan Danilishin. Authors are supported by the Russian Foundation for Basic Research Grant No. 08-02-00580-a and NSF grant PHY-0967049.", "pages": [ 5 ] }, { "title": "Appendix A: Derivation of initial equations", "content": "The hamiltonian of considered system (without feedback) could be written as follows [34]: Here B ( B † ) is annihilation (creation) operator of optical mode, y ( p ) is position (momentum) operator of movable mirror. Also ω c = ω 0 -∆is the frequency of cavity fundamental mode closest to the pump frequency ω 0 . The difference between these frequencies is denoted as ∆, please note the sign of this detuning: for the pump tuned onto the right slope of optical resonance curve (blue detuned pump) ∆ is positive . K = ω c /L is the coupling coefficient between light mode and mirror motion. H bath is the hamiltonian describing interaction of the system with bath and the bath itself. H pump is the same for pump. If we then suppose the optical operators to consist of strong classical part ¯ B (which is real positive number characterizing mean optical power P inside cavity: ¯ B = √ P/ glyph[planckover2pi1] ω 0 ) and quantum fluctuations b so that B = ¯ B + b and switch to the frame rotating with frequency ω 0 , then the linearized hamiltonian should take the form One then should write down Heisenberg equations for position and momentum of mirror and amplitude ( b 1 ) and phase ( b 2 ) operators of light defined in following way: To obtain the equations of motion (1) one should eliminate p and b 2 and derive the forces acting from the pump and bath. In consideration above we have not given concrete expressions to the bath and pump hamiltonians so in order to perform inclusion of damping, feedback and optical fluctuational forces in a correct way we use semiclassical approach writing down the equations of motion in spectral form [35]: Here Ω is spectral frequency, τ is the time it takes light to travel between mirrors (one way), α fb is a coefficient of feedback, i.e. the force fed to the mirror is equal to f fb = -µα fb ˙ a o 2 . The dimensionless parameters are defined as follows. Switching to dimensionless parameters from dimensional ones using the definitions listed above yields the system Finally by transferring from frequency domain to time one following rule -i Ω → ∂ t one can write down the system (1) For numerical calculations through this paper we have used the following set of parameters", "pages": [ 5, 6 ] }, { "title": "Appendix B: Thermal fluctuations", "content": "In order to account thermal fluctuations in the model one should include corresponding fluctuational force in the right part of the equation of motion for mechanical degree of freedom. This results in formal replacement Here -x 2 z th is thermal fluctuational force. If we consider coating Brownian noise as the source of this force (in Advanced LIGO coating fluctuations are dominant thermal noises), then its spectral density is given by expression [36]: Here f is spectral frequency (measured in Hz), w is a radius of a beam spot on mirror's surface, Y, σ are Young modulus and Poisson ratio of substrate respectively, Y 1 , 2 , σ 1 , 2 are the same quantities for alternating layers, d 1 , 2 = Nλ/ 4 n 1 , 2 are total thicknesses of quater wavelength layers, N is the number of layers' pairs, λ -optical wavelength, n 1 , 2 are refraction indices of layers. T is the temperature of mirrors coating, k B is Bolzmann's constant. The numerical parameters that we use for estimations are listed in table I. As the thermal noise is not correlated to vacuum noises, inclusion of the last term in equation (B1) reveals in estimations of spectral densities of quadratures only by additional term in expression for spectral density of ν 2 : The convenient factor of thermal noise influence on eigen mode amplitude spectral density is the following ratio: Here S th is the spectral density of output field quadrature A (+) 1 in case of only thermal noises present and S q is the spectral density of the same quadrature provided by quantum noises. Both spectral densities are calculated for the case of absent modulation. Obviously there is a possibility to speculate on quantum noise squeezing if factor ξ is lesser than unity. Calculations done for parameters of Advanced LIGO [2] yield the value ξ aLIGO = 0 . 82 which means that despite the fact that limitations imposed on the antenna sensitivity by coating Brownian noises are smaller then quantum noise imposed ones, their influence on the eigen mode amplitude are still comparable with the influence of latter. However the estimations for future antenna Einstein Telescope [37] provide much more optimistic value ξ ET = 0 . 15. Estimations for some experimental prototypes show that thermal noises are dominant in these devices. In particular for Glasgow University prototype [31] ξ GP = 2 . 7, for AEI Hannover prototype [32] ξ AEI = 1 . 7 and for Gingin facility [33] ξ Gingin = 3 . 8. The parameters used for these estimations are listed in table II. Calculations for Einstein Telescope differ from another devices by used wavelength λ = 1 . 55 µ m and temperature T = 10 K.", "pages": [ 6 ] }, { "title": "Appendix C: Readout", "content": "In output field we measure phase quadrature a o 2 which is proportional to b 1 that carries information about squeezing. Further we discuss which combination of spectral components of b 1 contains squeezing evidence taking into account that corresponding combination of a o 2 components represents the same combination with some input fluctuations a i 2 added. For spectral components of b 1 shifted in frequency domain by amount of ω j one can write the following expression using eqn. (2): Here we use freedom in definition of glyph[vector]v i and let V i = ( glyph[vector]v i ) 1 = ( glyph[vector]v i ) ∗ 1 . An equation similar to one above could be written for b 1 ( x -ω j ): We now take sum or difference of these quantities and obtain expression containing sum of quadratures G ( ± ) j that are of our interest glyph[negationslash] Remind that we assumed g i to be slow amplitudes which means that spectral components of these quantities are situated close to zero frequency. Note that due to resonant multiplyers (expressions in square brackets) in eqn. (6) one can estimate the width of band containing g i spectra with order of corresponding mode decay rate γ i which is much smaller than difference between eigen frequencies ω i -ω j (for i = j ). This consideration is also valid for quadratures G i hence in definition of B ( ± ) j all summands differ from zero in different frequency bands (not overlapping because of narrowness), or in other words in each frequency band there is not more than one non-zero summand in this definition. In particular at frequencies close to zero one can use quite exact expression As we actually measure a o 2 we need to consider a combination of spectral components of this quantity containing B ( ± ) j . It is obvious that corresponding combination is given by the following expression which represents a sum of desired quadrature and input fluctuations.", "pages": [ 7 ] } ]

[ { "title": "Mu-In Park ∗", "content": "The Institute of Basic Sciences, Kunsan National University, Kunsan, 573-701, Korea", "pages": [ 1 ] }, { "title": "Abstract", "content": "Recently Hoˇrava proposed a renormalizable quantum gravity, without the ghost problem, by abandoning Einstein's equal-footing treatment of space and time through the anisotropic scaling dimensions. Since then various interesting aspects, including the exact black hole solutions have been studied but no rotating black hole solutions have been found yet, except some limiting cases. In order to fill the gap, I consider a simpler three-dimensional set-up with z = 2 and obtain the exact rotating black hole solution. This solution has a ring curvature singularity inside the outer horizon, like the four-dimensional Kerr black hole in Einstein gravity, as well as a curvature singularity at the origin. The usual mass bound works also here but in a modified form. Moreover, it is shown that the conventional first law of thermodynamics with the usual Hawking temperature and chemical potential does not work, which seems to be the genuine effect of Lorentz-violating gravity due to lack of the absolute horizon. PACS numbers: 04.20.Jb, 04.20.Dw, 04.60.Kz, 04.60.-m, 04.70.Dy", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Recently Hoˇrava proposed a renormalizable gravity theory, without the ghost (i.e., unitarity) problem, which reduces to Einstein gravity in IR but with improved UV behaviors, by abandoning Einstein's equal-footing treatment of space and time through the anisotropic scaling dimensions, [ t ] = -1 , [ x ] = -z with the dynamical critical exponents ( z > 1) [1]. Since then various aspects have been studied, in particular several exact black hole solutions have been found [2-10]. But no rotating black hole solutions have been found yet, except some limiting cases [11] and so there have been some gap in Hoˇrava gravity for describing our real black holes in the sky, which can be even nearly extremal, for example, c /lscript J/GM 2 > 0 . 98 in GRS 1915+105 [12] for the speed of light c /lscript . In order to fill the gap, in this paper I consider the three-dimensional set-up with z = 2, instead of studying the more challenging four-dimensional Hoˇrava gravity with z = 3. By solving the three coupled non-linear equations for the three-dimensional z = 2 Hoˇrava gravity with the general axisymmetric metric ansatz, I obtain the exact rotating black hole solution and study its physical properties. This solution has a ring curvature singularity inside the outer horizon, like the four-dimensional Kerr black hole in Einstein gravity, as well as a curvature singularity at the origin. The usual mass bound works also here but in a modified form. Moreover, it is shown that the conventional first law of thermodynamics with the usual Hawking temperature and chemical potential does not work, which seems to be the genuine effect of Lorentz-violating gravity due to lack of the absolute horizon.", "pages": [ 2 ] }, { "title": "II. THE ROTATING BLACK HOLE IN THREE-DIMENSIONAL HO ˇ RAVA GRAVITY", "content": "Using the ADM decomposition of the metric the three-dimensional renormalizable action with z = 2 [13, 14], up to surface terms, is given by 1 where κ = 16 πG 3 , is the extrinsic curvature, R is the Ricci scalar of the Euclidean two-geometry, λ, ξ are the IR Lorentz-violating parameters, and Λ is the cosmological constant. Note that in two-spatial dimensions all curvature invariants can be expressed by the Ricci scalar due to the identities, R ijkl = ( g ik g jl -g il g jk ) R/ 2 , R ij = g ij R/ 2. Here, I do not consider the terms which depend on a i ≡ ∂ i N/N and ∇ j a i , which can change the IR as well as UV behaviors a lot from that of (2). Moreover, I do not consider the term of ∇ 2 R [13] either since the qualitative structure of the solutions I will get is expected to be similar, as in the four dimensions [8]. Let me consider now an axially symmetric solution with the metric ansatz (I adopt the convention of c /lscript ≡ 1, hereafter) Note that there is no angle ( φ ) dependance in the metric due to the circular symmetry in the two-dimensional space even with the rotation. By substituting the metric ansatz into the action (2), the resulting reduced Lagrangian, after angular integration, is given by where the prime ( ' ) denotes the derivative with respect to r . Note that there is only the ξ dependance but no λ dependance in the Lagrangian. The equations of motions are by varying the functions N , N φ , and f , respectively. For arbitrary α , Λ and ξ , I obtain the general solution with Here, I have set W ( ∞ ) ≡ 1 , N φ ( ∞ ) ≡ 0 by choosing the appropriate coordinate system, without loss of generality, but they can be conventionally kept as independent parameters for the analysis of the mass and angular momentum of the solution. Note that the parameters a, c are restricted to zero or positive values, i.e., a, c ≥ 0, or equivalently, and α ≥ 0, for the real-valued metric functions f, N , and N φ . For large r and small α , the solution expands as It is easy to check that, in the limit of α → 0, the solution reduces to the BTZ black hole solution (with ξ = 1) [19] The non-vanishing curvature invariants are and (15) shows a ring curvature singularity when W = 0, i.e., at for the rotating solution, as well as a curvature singularity at r = 0 in both R and K ij K . ij Note that the existence of a ring singularity is analogous to the four-dimensional Kerr black hole case but the singularity at r = 0 is not. For the BTZ black hole in Einstein gravity ( α = 0 , ξ = 1), the curvature singularity at r = 0 in R is canceled by K ij K ij -K 2 and the remainders, which become the boundary terms in the action, in the (covariant) three curvature scalar R (3) , resulting the finite value: R (3) = R + K ij K ij -K 2 -f ' /r -f '' = 6Λ. This means that the curvature singularity at r = 0 in R is the artifact of the time-foliation and not the physical singularity in Einstein gravity. But for the general solutions (9), the curvature singularities at r = 0 in (14) and (15) are covariant in the foliation preserving diffeomorphism such that they are physical singularities. For asymptotically AdS, i.e., Λ < 0 2 , the solution (9) has two horizons generally where f and N vanish simultaneously, i.e., the apparent and Killing horizons coincide, and the Hawking temperature for the outer horizon r + is given by /negationslash from the regularity of the horizon in the Euclidean space-time, as usual. There is another Killing horizon when W = 0, i.e., N = W √ f = 0 with f = 0, at r = r ring but this is not the event horizon since one can escape from (or reach to) the horizon in a finite time. 3 In Fig.1 (left), the temperature T + vs. the outer horizon radius r + is plotted. For non-vanishing c , i.e., α, J /negationslash = 0, there are two instances of the vanishing temperature: (a) The first case is the usual extremal black hole limit, where the inner horizon r + meets the outer horizon r + at and the integration constant gets the minimum (Fig.1 (right)). This is the ground state in the usual black hole system and the outer horizon can not be smaller than r ∗ + ; T + < 0 for r + < r ∗ + and this reflects a pathology of the region ( for some related discussions, see Ref. [22]). (b) The second case is the instance when W | r + = 0, i.e., when the outer horizon r + meets the ring curvature singularity r ring of (16). If α is small enough so that a > 1 /e 2 , i.e., -8 α Λ /ξ 2 < 1 -1 /e 2 , then this instance does not really occur since the outer horizon is always larger than the radius of the ring curvature singularity r ring , i.e., r ring < r ∗ + ≤ r + . In this case the zero temperature is arrived when r + meets r -at r ∗ + before meets r ring . This shows that the ring curvature singularity is safely protected by the outer horizon for the small α . However, if α is not so small so that a ≤ 1 /e 2 , i.e., -8 α Λ /ξ 2 ≥ 1 -1 /e 2 , then there is the chance when r + meets r ring from outside, i.e., r + ≥ r ring . But even in this case the ring singularity would not be naked since the zero temperature, i.e., the ground state is arrived by merging r + → r ring , and r + can not be smaller than r ring : There is the ring singularity 'on' the horizon, but this does not affect the outer region ( > r + ) by the definition of the event horizon. In this case the second instance of the zero temperature is arrived before reaching the extremal black hole, except the case a = 1 /e 2 , where the extremal and ring singularity radius are degenerate, r ∗ + = r ring . The ergo-region is defined by g tt = -N 2 + r 2 ( N φ ) 2 ≥ 0 with its boundary at ∣ and this region is outside of the outer horizon r + , i.e., r + ≤ r erg since f ( r erg ) = g rr ( r erg ) ≥ 0 is required by (20). Here, the properties W ( r erg ) > 0 from r erg ≥ r + > r ring and N φ > 0 are used. Another peculiar property of the general solution is that there is the counter-rotating region inside the outer horizon r + (Fig.2). It is interesting to note that the turning point of N φ , i.e., N φ ' = 0 is at the location of the ring singularity r = r ring from N φ ' = W J /r 3 = 0 and the counter-rotating region starts at r count = ( ηc/a ) 1 / 4 ( < r ring < r + ) with η ≈ 0 . 0308 which solves N φ = 0 ( J /negationslash = 0).", "pages": [ 2, 3, 4, 5, 6 ] }, { "title": "III. THE UNUSUAL THERMODYNAMICS", "content": "The thermodynamics of Lorentz-violating black holes has not been well established yet 4 . In order to study this subject, I start by computing the conserved mass and angular momentum of the rotating black solution (9). To this ends, let me consider the variation of the total action I total = I + B with boundary terms B at space-like infinity such that the boundary variation ( δI )( ∞ ) is canceled by δB and there remain only the bulk terms in δI total which vanish when the equations of motions hold. Then for the class of fields that approach our solution (9) at infinity, one finds which defines the canonical mass and angular momentum /negationslash as the conjugates to the asymptotic displacements N ( ∞ ) and N φ ( ∞ ), respectively, when kept as independent parameters. In order that the curvature singularities are not naked, i.e., satisfying the cosmic censorship, one needs the mass bound condition with the monotonic function which can vary in [0 , 1] as x 2 ≡ ξ 2 / ( -8Λ α ) varies in [1 , ∞ ]. In the BTZ limit, x 2 = ∞ , ξ = 1, one has the usual mass bound ( χ = 1) but even for the other more general classes of 1 ≤ χ ( x ) < ∞ so that 0 ≤ χ < 1, the mass bound still works for each theory parameterized by x , but in a modified form. Now in order to study the first law of black hole thermodynamics, let me consider the variation of the mass M as a function of J and r + , with Then, in order to see whether the first law of thermodynamics in the conventional form works with the usual Hawking temperature T + of (17) and the chemical potential Ω + = -N φ | + , let me define the black hole entropy function S with as a function of r + and J . Then, from (26), (27), and (28), one can find /negationslash but ∂ J ∂ r + S -∂ r + ∂ J S = 0, for arbitrary non-vanishing α , J , and finite r + . The lengthy result for the non-integrability is not so impressive to be shown here but, in order to grasp how the Lorentz violation and the angular momentum affect the non-integrability, I show its leading term /negationslash and the full results in the numerical plots (Fig. 3). These results show that the entropy is not integrable by the non-relativistic higher curvature corrections ( α = 0) for the rotating and finite black holes. The infinite barriers at α and J are due to α ≤ -ξ 2 / 8Λ (11) and M ≤ χ ( x ) J √ -Λ / | ξ | (23). This proves that the entropy can not be defined in the conventional form of the first law of thermodynamics with the usual Hawking temperature and chemical potential.", "pages": [ 6, 7, 8 ] }, { "title": "IV. DISCUSSION", "content": "In conclusion, I have obtained the rotating black hole solution in the three-dimensional Hoˇrava gravity where the Lorentz symmetry is broken by the higher-spatial derivatives in UV. Here, it is remarkable that the existence of the rotating black hole does not depend much on the existence nor the momentum dependance of speed, i.e., no absolute speed limit, of gravitons. Actually, in our case there would be no graviton mode at the linear perturbation 5 from the similar analysis in four-dimensional Hoˇrava gravity since the calculation is not sensitive to the dimensionality of space [5, 24]. The status of its full, non-linear analysis is still unclear and needs more elaborative works with some ingenious separation of the genuine constraints, which being left as a further work. And I have also shown that the mass bound condition still works in the new solution, analogous to the mass bound GM 2 ≥ c /lscript J for Kerr black hole in Einstein gravity. However, I have shown that the first law of thermodynamics can not be written in the conventional form with the usual Hawking temperature and the chemical potential such that the entropy function can not be defined for the generically rotating and Lorentz-violating black holes. The existence of Hawking temperature implies the Hawking radiation and this can be proved quite generally without knowing much details of the solutions (see for example Ref. [25]). So, we have the black holes which generate the Hawking radiation but without the black hole entropy. Actually the notion of 'Hawking radiation without black hole entropy' has been studied in the context of analogue black holes [26], previously. In our case, this seems to be a genuine effect of the Lorentz-violating gravity due to lack of the absolute horizon which can leak the information depending on the matter's momentum scale. This may be compared with other Lorentz-violating black holes, called Lifshitz black holes, where the first law of thermodynamics does not hold for a generic member of a class of black holes [27]. The study of rotating black holes in four-dimensional Hoˇrava gravity and their black hole thermodynamics would be quite a challenging problem. As a possible resolution for the failure of the usual black hole thermodynamics, one might try to consider the first law of thermodynamics in the form of dM = ˜ TdS + Ω + dJ with an unusual 'temperature' function ˜ T = ˜ T ( r + , J ) and its associated entropy function S = S ( r + , J ), instead of the standard one (28). However, the usual interpretations of ˜ T and S as the thermal temperature of Hawking radiation and the black hole entropy, respectively, need to be justified. For non-rotating black holes, or more generally oneparameter family of black hole solutions, one can always consider the standard first law of thermodynamics dM = T + dS with the appropriate entropy function S = S ( r + ) [3]. In our three dimensional case, one can easily find T + = ¯ hbr + (1 -√ a ) / 4 π , S = 2 πr + ξ/ 4 G ¯ h with the black hole horizon r + = (2 M /b (1 -√ a )) 1 / 2 and this becomes the usual black hole entropy in three dimensions [19] with ξ = 1, i.e., no Lorentz violation in IR. Here, it is interesting to note that the UV Lorentz violation parameter α does not affect the usual entropy formula but affect only the value of r + through the parameters a and b from (10), in contrast to four-dimensional black holes [3, 23], where UV Lorentz violation terms produce logarithmic corrections to the entropy formula. On the other hand, the modification of the entropy from the usual area ( perimeter , in our case) law comes from IR Lorentz violation parameter ξ but a thermodynamic interpretation of the modified entropy is not obvious.", "pages": [ 8, 9 ] }, { "title": "Acknowledgments", "content": "I would like to thank Gungwon Kang for giving some inspiration. This work was supported by the Korea Research Foundation Grant funded by Korea Government(MOEHRD) (KRF-2010-359-C00009).", "pages": [ 9, 10 ] } ]

[ { "title": "High Scale SUSY Breaking From Topological Inflation", "content": "Keisuke Harigaya a , Masahiro Kawasaki b,a and Tsutomu T. Yanagida a a Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan b Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "The recently observed mass ∼ 125 GeV for the Higgs boson suggests a highenergy scale SUSY breaking, above O (10) TeV. It is, however, very puzzling why nature chooses such a high energy scale for the SUSY breaking, if the SUSY is a solution to the hierarchy problem. We show that the pure gravity mediation provides us with a possible solution to this puzzle if the topological inflation is the last inflation in the early universe. We briefly discuss a chaotic inflation model in which a similar solution can be obtained.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The ATLAS and CMS collaborations recently discovered a standard-model like Higgs boson of mass about 125 GeV [1]. This observed Higgs mass, together with non-discovery of superpartners at LHC, suggests that the supersymmetry (SUSY) breaking scale is much higher than we expected, say above O (10) TeV [2]. However, if the SUSY is a solution to the hierarchy problem and hence its breaking is biased toward low energy scales, a crucial question naturally arises ; why does nature choose such a high energy scale for the SUSY breaking [3]? We show, in this letter, that the pure gravity mediation model recently proposed to explain the 125 GeV Higgs mass [4] (for a similar model, see also [5]) provides us with a possible explanation for the high scale SUSY breaking if the topological inflation is the last inflation in the early universe. It is believed that our universe experienced the quasi-exponential expansion called inflation [6] at its very early stage. Inflation makes our universe homogeneous and flat, which solves conceptual problems of big-bang cosmology, and also dilutes harmful relics like monopoles. Furthermore, quantum fluctuations of the inflaton (= a scalar field that drives inflation) become classical by the cosmic expansion during inflation and result in density perturbations of the universe [7]. Inflation predicts nearly scale-invariant, adiabatic and gaussian density perturbations, which are consistent with the recent observations of the cosmic microwave radiation (CMB) [8]. Thus, the inflationary universe successfully describes our universe. However, most inflation models have so called initial value problem [9], that is, they require tuning for the initial conditions of the inflaton and other relevant fields. Among many models, chaotic inflation [10] and topological inflation [11] are free from the initial value problem. It is well known that chaotic inflation occurs naturally from large field fluctuations at the Planck time. In topological inflation models, some discrete symmetry is spontaneously broken and topological defects (domain walls) are formed in the early universe. If the scalar field forming the defects has the vacuum expectation value larger than the Planck scale ( M pl glyph[similarequal] 2 . 4 × 10 18 GeV), the region inside the domain wall undergoes inflation. Since the defect formation is inevitable, inflation takes place naturally as long as the universe lives until the start of inflation. The longevity of the universe is not a problem in the open universe. Furthermore, open universes are likely created through tunneling in quantum cosmology [12]. We show, in this letter, that there is an upper bound of the reheating temperature, T R < ∼ 10 10 GeV, if the topological inflation takes place in the early universe. In the pure gravity mediation model, the wino is the LSP and it is the unique candidate of dark matter (DM) in the universe. The number density of the wino is almost proportional to the reheating temperature T R as long as its mass is lower than 1 TeV [4]. We thus obtain a lower bound of the wino mass to explain the observed DM density as m wino > ∼ 200 GeV. The lower bound of the wino mass is translated to the lower bound on the gravitino mass, m 3 / 2 > ∼ O (10) TeV in the pure gravity mediation model, implying scalar masses > ∼ O (10) TeV. Thus, there is a cosmological reason why the SUSY breaking scale is higher than O (10) TeV in the pure gravity mediation model if the topological inflation is the last inflation in our universe. We briefly note, in the last section, that a similar conclusion can be obtained in a chaotic inflation model.", "pages": [ 2, 3 ] }, { "title": "2 Topological Inflation Model", "content": "It was pointed out long time ago [13, 14] that the topological inflation takes place for the following simple super potential and Kahler potential with U (1) R × Z 2 symmetry in the supergravity: where g 2 n ( n = 1 , 2 , .. ) , k 1 and k 2 are coupling constant. We take a unit of the Planck scale M pl = 1, here and hereafter. We have assumed that the expansions of super and Kahler potentials in the fields, X and φ , are well defined as long as the absolute values of the fields are within the Planck scale. Thus, it is reasonable to consider all constants g 2 n , k 1 and k 2 are at most O (1). For our analysis we neglect the higher order terms with n ≥ 2 in the super potential, for simplicity. This may be valid as long as the expansion in φ converges sufficiently fast. In the followings, we assume this property about the expansion in the inflaton φ . We have neglected possible higher order terms already in the Kahler potential in Eq. (2). We assume this model throughout this letter and consider that the main conclusion of this letter does not change even if we adopt another model for the topological inflation. g ≡ g 2 is chosen to be real and positive by a phase rotation of φ . X and φ have U (1) R charges 2 and 0, respectively. We also assume that X is even and φ is odd under the Z 2 , which is essential for the topological inflation to take place. The potential has a vacuum (see Eq. (1) with n = 1), As shown in Ref. [14], the topological inflation takes place if 〈 φ 〉 > ∼ 1 / √ 2 ( g glyph[lessorsimilar] 4). Since 〈 φ 〉 < ∼ O (1) for g = O (1), it is consistent with our assumption of neglecting higher order terms in the super and the Kahler potential. The scalar potential derived from Eqs. (1) with ( n = 1) and (2) is, for | X | and | φ | glyph[lessmuch] 1, The X field quickly settles down to the origin if k 2 glyph[lessorsimilar] -1, so hereafter we take X = 0. We can identify the inflaton field as the real part of φ . Using ϕ = √ 2Re( φ ), the potential is rewritten for ϕ glyph[lessmuch] 1 as From this potential we obtain the e -folds N as where ϕ f is the field value of ϕ at the end of inflation. The slow roll parameters are given by For the inflation to produce the observed curvature perturbation, the inflaton potential satisfies V 3 / 2 ( ϕ N ) /V ' ( ϕ N ) glyph[similarequal] 5 × 10 -4 for N = 50 -60, which leads to for κ = 0 . 01, where we have used ϕ f glyph[similarequal] 1. The inflaton mass is m φ glyph[similarequal] v 2 √ 2 g glyph[similarequal] 10 13 GeV for g glyph[similarequal] 1. The spectral index n s and tensor to scalar ratio r are given by Thus, r is written as a function of N s , s The prediction of the topological inflation is shown in Fig. 1 together with WMAP 7year constraint [8]. For the spectral index n s glyph[similarequal] 0 . 94 -0 . 98 which is consistent with WMAP 7 year data, r glyph[similarequal] (3 -4) × 10 -4 . Therefore, the tensor mode that the topological inflation produces will be not detected even in the future satellite experiments such as CMBPol [15] and LiteBIRD [16]. We find that the value of the coupling constant g is further restricted. If g is too small, the vacuum 〈 φ 〉 = √ 2 /g is far above one. On the other hands, a factor of exp( | φ | 2 ) in the inflaton potential lifts up the potential above φ glyph[similarequal] 1. Therefore, the potential becomes a old-inflation type and the topological inflation fails. Now let us estimate the lower bound for g . The potential of the inflaton field is, for X = 0, In the second line we have imposed k 1 + g glyph[similarequal] 1 which is required by the observation of the spectral index. The potential becomes a old-inflation type if there exists a point such that ∂V/∂ϕ = 0 for 0 < ϕ < 2 / √ g . By a simple algebraic calculation, we find that the condition #1 is required. On the other hand, if g is too large, k 1 glyph[similarequal] 1 -g is negatively large. From Eq. (13), one can see that the inflaton potential becomes singular before ϕ reaches the vacuum. In order to avoid such behavior, the condition is required. Eqs. (14) and (15) are consistent with our assumption that g is of order one. Now, let us estimate the reheating temperature after the inflation. There are four types of R and Z 2 invariant interactions which contribute dominantly to the inflaton decay. One originates from the Kahler potential of the form where Ψ is any field which is lighter than the half of the inflaton mass and c ''' nm is a coupling constant of order one. However, with this type of interactions, the matrix element of the inflaton decay is proportional to square of the mass of Ψ for any n and m . Since the mass of Ψ should be smaller than the half of the inflaton mass, decay width is suppressed in comparison with the mode described below at least by the factor of 16. Therefore, we ignore the contribution from this type of interactions. The other dominant interaction originates from the Kahler potential of the form where H u and H d are the up and down type Higgs field and c '' is the coupling constant of order one. This term is allowed since H u H d has a vanishing U (1) R charge in the pure gravity mediation model in order to achieve an appropriate value for the Bµ and µ term [4]. This Kahler potential leads to the inflaton interaction with the Higgs scalar as The decay rate of the inflaton to Higgs bosons is given by where we have used the inflaton mass m φ = v 2 √ 2 g . The decay rate into Higgsinos is the same as this value. There also exist interactions originate from the super potential. The small value of the v 2 required for the topological inflation in Eq. (9) is considered as a result of some new symmetry breaking, otherwise it should be O (1). For instance, consider a parity under which both of X and v 2 transform as odd, then v 2 X is a completely neutral under all symmetries except for U (1) R (the R charge of X is two). The small value of v is regarded as a small breaking of the parity. Thus, a term is allowed in the super potential. This super potential leads to the inflaton interaction with the Higgs bosons as The decay rate due to this operator is given by The last dominant interactions originate from the gauge kinetic functions: where a, b, · · · are the indices for gauge group and c is a coupling constant of order one. This kinetic function leads to inflaton interactions with the gauge fields in the standard model as where F a µν is the field strength of the gauge boson A a µ , and Re and Im denote the real part and the imaginary part, respectively. Then the decay rate of the inflaton to the gauge fields is given by N A = 1 + 3 + 8 = 12 is the number of the gauge fields. The contribution from the decay into gauginos is exactly the same one. Adding above contributions together, the decay rate of the inflaton is given by From this decay rate we obtain the reheating temperature T R as Using Eqs. (9) and (10), the reheating temperature is written as a function of the spectral index n s , which is shown in Fig. 2. We can see that T R glyph[similarequal] (2 -7) × 10 9 GeV in the present model. This relatively low reheating temperature is a consequence of the Z 2 and the U (1) R symmetry, which is essential for the topological inflation. Taking into account O (1) ambiguity in the coupling constants c 's, we safely conclude T R < ∼ 10 10 GeV.", "pages": [ 3, 4, 5, 6, 7, 8 ] }, { "title": "3 Lower Bound on the Gravitino Mass in the Pure Gravity Mediation Model", "content": "In the previous section, we have shown that the maximal reheating temperature is about T R < ∼ 10 10 GeV in a reasonable parameter range if the topological inflation is the last inflation in our universe. Now we are at the point to show that there is indeed a lower bound on the SUSY breaking energy scale, m 3 / 2 > ∼ O (10) TeV. First of all, the wino is the LSP and the unique candidate for the DM in the pure gravity mediation model [4]. The wino has a large annihilation cross section and hence the thermal wino can not provide a sufficient density for the observed DM as long as the mass of the wino m wino < ∼ 1 TeV. Thus, we must invoke a non thermal wino production in the early universe. The most promising process is the gravitino decay, since the gravitino is much heavier than the wino in the pure gravity mediation model. Let us discuss the gravitino production in the high-energy thermal particle scattering in the early universe. Since the production of the gravitino occurs by Planck suppressed interactions, the production is more effective for higher temperatures. Therefore, the abundance of the gravitino is determined by the reheating temperature and given by [17] Here, n 3 / 2 is the number density of the gravitino and s is the entropy density of the universe. The abundance of the wino from the decay of the gravitino is the same as Eq. (29), n wino = n 3 / 2 . After all, the energy fraction of the wino dark matter in the present universe Ω wino , 0 is given by #2 where s 0 glyph[similarequal] 2 . 2 × 10 -11 eV 3 is the entropy density of the present universe, ρ cr , 0 glyph[similarequal] 8 . 1 h -2 × 10 -11 eV 4 is the critical density of the present universe and h is the scale factor for Hubble constant defined by H 0 = 100 h km sec -1 Mpc -1 . The anthropic bound of the dark matter density is known as Ω DM h 2 > ∼ 0 . 1 [18]. And as we have seen in the previous section, the reheating temperature T R is bounded from above as T R < ∼ 10 10 GeV if the topological inflation is the last inflation in the early universe. Therefore, we obtain the lower bound on the wino mass as In the pure gravity mediation model, the wino mass is determined by the contribution from the anomaly mediation and the threshold correction by the Higgsino loop. Both contributions are of the same order and the wino mass has the upper bound as [4] Therefore, we obtain the lower bound for the SUSY breaking scale m SUSY ;", "pages": [ 8, 9, 10 ] }, { "title": "4 Conclusions and Discussion", "content": "We have discussed the topological inflation in the previous section. However, as pointed out in the introduction, the chaotic inflation [10] is also interesting since there is no initial value problem. The chaotic inflation is easily constructed by using a shift symmetry in supergravity [19]. The Kahler potential is a function of φ + φ † and the super potential is given by where φ is the inflaton. The inflaton φ can have an interaction in the Kahler potential, where d is a coupling constant of O (1). It is easy to see that the upper bound of the reheating temperature T R < ∼ 10 10 GeV can be obtained for d < ∼ O (1). In this letter, we consider relatively high energy scales for the SUSY breaking. However, there is a very natural parameter region where the observed DM density is explained by a mixed wino-bino thermal relic DM of mass O (1) GeV and sfermion masses are O (100) GeV, even in the pure gravity mediation model [20]. However, this parameter region is excluded by too much non thermal DM production due to the overproduction of the gravitino in the inflaton decay [21]. We have assumed the R parity conservation throughout this letter. However, if the R parity is broken, the present argument is not applicable. The model which connects the SUSY breaking and the Peccei-Quinn symmetry breaking dynamics [20] may be interesting to understand the high scale SUSY breaking, if it is the case.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "This work is supported by Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 14102004 (M.K.), No. 21111006 (M.K.), No. 22244021 (T.T.Y.), and also by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The work of K.H. is supported in part by a JSPS Research Fellowships for Young Scientists.", "pages": [ 10 ] } ]

[ { "title": "A black ring on the Taub-bolt instanton in five dimensions", "content": "Cristian Stelea, 1 Marian C. Ghilea 2 1 Faculty of Physics, 'Alexandru Ioan Cuza' University 11 Bd. Carol I, Iasi, 700506, Romania", "pages": [ 1 ] }, { "title": "Abstract", "content": "Using a solution generating technique, we derive a new exact solution describing a charged static black ring on the Taub-bolt gravitational instanton in five dimensions. Unlike the black ring constructed on the self-dual Taub-NUT instanton, it turns out that it is possible to find values of the parameters for which the static black ring is in equilibrium and the conical singularities disappear. We compute its conserved charges and discuss some of its thermodynamic properties.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "While the physics of four dimensional black holes is pretty much understood, in recent years one witnessed remarkable developments in the study of higher dimensional black holes (for a review see [1, 2]). There are many motivations to embark in such a study. For instance, higher than four dimensions are required by string/M-theory, which is candidate for a unified field theory, aiming to provide us with a consistent theory of quantum gravity. Another motivation for studying gravity in higher dimensions is intrinsic, as a way to better understand the nature of the theory itself. In particular, one should be able to answer the question: which properties are characteristic to four dimensions and which are common to gravity in higher than four dimensions. One important advance in this direction was the Emparan and Reall's discovery of an exact solution in five dimensions describing an asymptotically flat rotating black ring [3, 4]. The black ring provided the first nontrivial example that known properties of the four dimensional black holes, such as the uniqueness theorem, do not hold in higher than four dimensions. Indeed, unlike their four dimensional counterparts, which necessarily posses a spherical horizon [6] as a consequence of the topological censorship theorem [7, 8], black holes in higher dimensions can have other horizon topologies as long they admit nonnegative scalar curvature [9, 10]. The black ring is a prime example of such an exotic black hole, as it has a ring topology S 2 × S 1 of the horizon, with rotation in the S 1 direction. Moreover, in certain conditions it can carry the same mass and angular momenta as the spherical Myers-Perry black hole [11, 12, 13]. As a consequence, the uniqueness theorems for black holes in four dimensions cannot be extended to the five dimensional case without further assumptions of additional symmetry and specification of the rod structure [14]. In the black ring solution the rotation along the S 1 direction creates a centrifugal force, which opposes the collapse of the black ring under its own gravity. If these forces are not in balance there are conical singularities in the space-time to keep the black ring in equilibrium. For a certain value of the angular momentum these competing forces will balance and the conical singularities disappear, rendering the solution regular on and outside the horizon. A more general solution describing a balanced rotation black ring, with rotation not only along S 1 but also along the azimuthal direction of S 2 has been presented in [15]. This solution generalized the Emparan and Reall's black ring but also the black rings found in [16, 17], which had rotation only along the azimuthal direction of S 2 . The general unbalanced black ring solution was subsequently found in [18, 19] and, demanding the absence of the conical singularities, it reduces to the balanced rotating black ring. Following the discovery of the rotating black ring, its generalization to black Saturn [20] and multi-black rings have been found in five dimensions [21, 22, 23, 24]. Black rings carrying electric/dipole charges have also been studied. For instance, a static black ring with an electric charge has been considered in [25]. In that work is was found that the static black ring cannot be stabilized against its own gravitational collapse even in presence of an electric charge. However, the static charged black ring could be stabilized and the conical singularities eliminated if one immerses the system in a background electric field. The drawback of this construction was that the black ring is no longer asymptotically flat. Configurations involving charged black rings in various theories, including string theory, have been considered in [26, 27, 28, 29, 30, 31]. Concentric supersymmetric black rings in five dimensions were first constructed in [32, 33]. Black rings in higher than five dimensions have also been found, albeit in numerical form [34, 35]. Recently, Chen and Teo introduced in [36] a new class of five-dimensional black hole solutions, the so-called black holes on gravitational instantons. For these solutions, in absence of black holes, the background geometry is a direct product of a trivial time direction with a four dimensional Ricci flat gravitational instanton having U (1) × U (1) symmetry [37]. For the asymptotically flat black rings the spatial part of the background corresponds to four dimensional flat space. Another interesting case is provided by the so-called squashed black holes [38, 39, 40, 41], for which the corresponding spatial part of the background is given by the self-dual Taub-NUT gravitational instanton. The asymptotic geometry is the same with that encountered in the case of the Kaluza-Klein magnetic monopole [42, 43]. Given the properties of the self dual Taub-NUT instanton, black holes in such backgrounds look fivedimensional in the near-horizon region, while asymptotically they resemble four-dimensional objects with a compactified fifth dimension. A static black ring in this background has been constructed in [44] and subsequently generalized to a rotating black ring in [45]. The most general double-rotating black ring on the Taub-NUT instanton was recently found in [46]. If one considers black holes/rings carrying Maxwell charges, there is a plethora of solutions (see for instance [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]). The squashed black holes have also been generalized to include Yang-Mills fields in [60]. Similarly to the case of asymptotically flat black holes/rings, one can prove uniqueness theorems for KK black holes by assuming additional symmetry and specification of other invariants [61]. There are also known solutions describing extremal black holes on the Eguchi-Hanson [62, 63, 64, 65, 66] and also the Atiyah-Hitchin instanton [67, 68]. Now, instead of the self-dual Taub-NUT instanton one could use the so-called Taub-bolt solution [69], which is a non-self-dual asymptotically locally flat gravitational instanton (with the same asymptotics as the Taub-NUT instanton) while having a space-like direction with finite norm at infinity. Both the Taub-NUT and the Taub-bolt solutions can be obtained from the Lorentzian Taub-NUT solution by a Wick rotation of the time coordinate and also an analytical continuation of the nut parameter. The solution describing a static black hole on Taub-bolt instanton was recently discovered by Chen and Teo [36]. Its thermodynamics, including also an electric charge, has been investigated in [70]. This single black hole solution has been recently generalized to a system of two non-extremal black holes sitting at the two turning points of the Taub-bolt geometry [71]. Surprisingly, the conical singularities in this double black hole solution can be completely eliminated by careful choices of the various parameters and the double black hole system remains in equilibrium. In this paper we are interested in constructing a five dimensional static black ring solution on the Taub-bolt instanton. The solution generating method that we shall use has been previously applied to construct multi-black hole objects in spaces with Kaluza-Klein asymptotics [58, 59, 71]. The main idea of this method is to map a general static electrically charged axisymmetric solution of Einstein-Maxwell theory in four dimensions to a five-dimensional static electrically charged axisymmetric solution of the Einstein-Maxwell system. The final five dimensional solution is determined up to the choice of a harmonic function. By carefully choosing the form of this function, one can construct the appropriate rod structures to describe the desired configuration of a black holes/rings on the Taub-bolt instanton. For example, as it was shown in [71], starting from the four dimensional Reissner-Nordstrom solution one is able to recover the solution describing a five dimensional black hole sitting at one of the turning points of the Taub-bolt instanton. Another simple choice of the harmonic functionl leads us to the desired solution describing a five dimensional static and charged black ring on the Taub-bolt instanton. The main result of this paper is showing that in the Taub-bolt background one can keep this static black ring in equilibrium, that is, it is possible to completely eliminate the conical singularities in this system. To our knowledge, this is the first example of a regular static black ring solution in spaces with Kaluza-Klein asymptotics. The structure of this paper is as follows: in the next section we derive the solution describing a static charged black ring on the Taub-bolt background. In section 3 we discuss the rod structure of this solution and compute its conserved charges. Finally, we end with a summary of our work and avenues for further research.", "pages": [ 2, 3, 4 ] }, { "title": "2 The black ring on the Taub-bolt instanton", "content": "Start with the static electrically charged black hole solution written here in Weyl form: where we define in general while here a 2 = -σ , and a 3 = σ . Note that σ = √ m 2 -q 2 , where m denotes the mass and q the electric charge of the black hole, while Ψ is the electric potential in four dimensions. Then the corresponding solution of the Einstein-Maxwell system in five dimensions with Lagrangian where F (2) = dA (1) can be written as: Here A and C are constants, while h is an harmonic function, which is so far arbitrary. By carefully choosing the form of h , one can construct the appropriate rod structure to describe the wanted configuration of a black ring on the Taub-bolt instanton. In this case, the appropriate choice turns out to be: where -a 1 = R 1 > σ and a 4 = R > σ . The second factor corresponds to a 'correction' that has to be taken into account for the black hole horizon in the four-dimensional seed solution, while the first and the third factors simply correspond to finite rods along the χ -direction, the first one which starts at a 1 and ends at a 2 , while second one starts at a 3 and ends at a 4 . Once the form of h has been specified for this particular solution, the remaining function γ can be obtained by quadratures using the equations: By integrating (6) one simply obtains: where K 0 is a constant to be fixed later. Also, the function H is the so-called 'dual' of h 1 and it is easily evaluated to be: up to a additive constant factor. Then the final solution in five dimensions can be written in the form: where we defined the following functions: In the following section we shall prove that for certain values of the parameters A and C this solution describes a regular static charged black ring on the Taub-bolt instanton.", "pages": [ 4, 5 ] }, { "title": "3 Properties of the black ring solution", "content": "So far the constants A and C were kept arbitrary. However, in order to have the right asymptotics at infinity, it turns out that one has to impose the following condition A 2 = C 2 +1. This follows from asking that F Σ → 1 in the asymptotic region. Let us consider the rod structure of the above solution. Following the procedure outlined in [37, 5], note that there are four turning points that divide the z -axis into five rods as follows: 2 is the Euclidian surface gravity on this rod. is the surface gravity of the black ring. is the Euclidian surface gravity on this rod. To have a proper black ring solution, it is necessary that the rod directions l 2 and l 3 be parallel as in Figure 1. This can be easily achieved if one takes R 1 = R . Finally, let us pick the value of the constant K 0 = 4. Then, if one defines the nut charge n = 2 AC ( R -σ ), to obtain the rod structure that corresponds to the Taub-bolt instanton [37], one has to impose the following condition: Similarly to the case of a single black hole on the Taub-bolt instanton, if one parameterizes the constants A 2 = 1 1 -α 2 and C 2 = α 2 1 -α 2 such that A 2 = C 2 +1 is trivially satisfied for α < 1, then the nut parameter can be written as n = 2 α ( R -σ ) 1 -α 2 , while the parameter α has to satisfy the simple condition: Then the final rod structure becomes: where the black ring surface gravity is: Note that there is the constraint R > 2 σ 3 to keep α < 1. Finally, to ensure regularity of the background geometry, the following identifications of the coordinates ( χ, ϕ ) have to be made [36]: In conclusion, we have shown that the static black ring solution on the Taub-bolt instanton can become nonsingular, as long as one constrains the parameters as above.", "pages": [ 6, 7 ] }, { "title": "3.1 Particular limits", "content": "In the uncharged case, one simply takes q = 0, that is σ = m . One should note that the uncharged black ring is still free of conical singularities as long as condition (17) is satisfied. Another limit of interest is when the black ring is removed from the Taub-bolt instanton. This limit corresponds to taking m = σ = 0. Then A 2 = 4 3 , while C 2 = 1 3 and the nut charge becomes n = 4 R 3 . It can be easily shown that the five-dimensional geometry can be cast in a Taub-NUT-like form with a trivial time direction: after performing the coordinate transformations: as long as one takes R = √ M 2 -n 2 . According to the previous discussion, the fivedimensional background is regular if the nut charge is n = 4 R 3 . This constrains the value of the Taub-NUT mass to be M = 5 | n | 4 , which is the regularity condition of the Taub-bolt instanton. In conclusion, the four-dimensional Taub-NUT-like part of the geometry corresponds indeed to that of the Taub-bolt instanton. Finally, in the extremal limit, one should take q = m , that is σ = 0 and the black ring horizon degenerates to a single point. The four-dimensional part of the metric can be cast into a Taub-NUT-like form with R = √ M 2 -n 2 . The five dimensional extremal solution becomes: where we defined: while f ( r ) is the same as in (20). For M = 5 | n | 4 the four dimensional geometry in the right bracket is precisely that of the Taub-bolt instanton. A preliminary analysis shows that the horizon of the extremal black ring degenerates as the S 1 direction of the black ring collapses to zero in this limit, such that there is a curvature singularity on the horizon. This resembles the situation of an extremal black ring in a asymptotically flat background.", "pages": [ 8 ] }, { "title": "3.2 Conserved charges and thermodynamics", "content": "The asymptotic region is found after performing the coordinate transformations: and taking the limit r → ∞ . Since the asymptotic region is the same with that of the Kaluza-Klein magnetic monopole, to compute the conserved charges the easiest way is to use the counter-terms method, as described for instance in [72] or [73]. One obtains the conserved mass M and the gravitational tension T in the form: The electric charge is easily evaluated using Gauss formula, with the result Q = √ 3 Lq 4 G , while the electric potential on the horizon is found to be: where L = 8 πn is the length of the χ circle at infinity, while G is the gravitational constant in five dimensions. Note that Q Φ H = 3( m -σ ) L 4 G , upon using the relation q 2 = m 2 -σ 2 . Finally, the area of the black ring horizon is computed to be: and the entropy of the black ring is S = A H 4 G . The Hawking temperature is easily evaluated using the surface gravity of the horizon k E . In this particular case one obtains: It is now easy to verify that the following Smarr relation is satisfied:", "pages": [ 9 ] }, { "title": "4 Summary", "content": "In this work we derived an exact solution describing a static and charged black ring on the Taub-bolt instanton. The solution generating technique that we used has been previously employed in [58, 59, 71] to obtain exact solutions describing general configurations of charged black holes in five dimensional spaces whose asymptotic geometry resembles that of the KK magnetic monopole. Generically, for multi-black hole systems, the presence of conical singularities is unavoidable, as they are needed to keep the system static. This is not an impediment in discussing the thermodynamic properties of such spaces, since it turns out that the gravitational action is still well-defined [74, 75, 76, 77, 59]. In section 2 we derived the solution describing a static charged black ring on the Taub-bolt instanton. In section 3 we showed that one can pick the parameters such that the rod structure of the background is the same as that of the Taub-bolt instanton. Further imposing appropriate identifications on the coordinates the conical singularities are avoided and the solution becomes completely regular. To our knowledge, this is the first regular solution describing a static black ring in spaces with KK asymptotics, without using Kaluza-Klein bubbles to stabilize the ring. We studied particular limits of the obtained black ring solution. Finally, by using a counter-terms method we computed the conserved charges and showed that a Smarr relation is satisfied in this case. As avenues for further work, it would be interesting to consider more general configurations, containing black holes and black rings. For example, one should be able to construct a solution describing a static black Saturn on the Taub-bolt instanton and study its properties. More generally, one should be able to derive a rotating black ring on the Taub-bolt instanton and also find rotating generalizations of the black Saturn and bi/di-rings systems. Another interesting solution would correspond to a black ring in the same Taub-bolt background, for which the S 1 direction corresponds to the compactified dimensions χ . However, to find such a solution one should recourse to a seed solution that is not static. Work on these subjects is currently in progress and it will be reported elsewhere.", "pages": [ 9, 10 ] }, { "title": "Acknowledgements", "content": "The work of C. S. was financially supported by POSDRU through the POSDRU/89/1.5/S/49944 project. Marian C. Ghilea is supported by the U.S. Department of Energy under Grant No. DE-SC0004036.", "pages": [ 10 ] } ]

[ { "title": "Spontaneous Spacetime Reduction and Unitary Weak Boson Scattering at the LHC", "content": "Hong-Jian He a,b,c and Zhong-Zhi Xianyu a a Institute of Modern Physics and Center for High Energy Physics, Tsinghua University, Beijing 100084, China b Center for High Energy Physics, Peking University, Beijing 100871, China c Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China", "pages": [ 1 ] }, { "title": "Abstract", "content": "Theories of quantum gravity predict spacetime dimensions to become reduced at high energies, a striking phenomenon known as spontaneous dimensional reduction (SDR). We construct an effective electroweak theory based on the standard model (SM) and incorporate the TeV-scale SDR, which exhibits good high energy behavior and ensures the unitarity of weak gauge boson scattering. This also provides a natural solution to the hierarchy problem in the presence of scalar Higgs boson. We demonstrate that this model predicts unitary longitudinal weak boson scattering, and can be discriminated from the conventional 4d SM by the WW scattering experiments at the CERN LHC. PACS numbers: 04.60.-m, 11.80.-m, 12.60.-i Phys. Lett. B (in Press)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Spontaneous dimensional reduction (SDR) [1] is a striking phenomenon, showing that the spacetime dimensions effectively equal 3+1 at low energies, but get reduced toward 1+1 at high energies. This is predicted by a number of quantum gravity approaches [1], including the causal dynamical triangulation, the exact renormalization group method, the loop quantum gravity, the high-temperature string theory, and Hoˇrava-Lifshitz gravity, etc. The SDR is expected to greatly improve the ultraviolet (UV) behavior of the standard model (SM) of particle physics. Recently, this property of the quantum gravity has called phenomenological interests, and has found applications to astrophysics and collider phenomenologies in a different context [2]. In addition, some hints of a TeV scale SDR have been noted [2] from the observations of alignment of high energy cosmic rays [3]. The SU (2) L ⊗ U (1) Y gauge structure of the SM is well established for electroweak forces. In the conventional SM, it is linearly realized and spontaneously broken by the Higgs mechanism [4], leading to massive weak gauge bosons ( W ± , Z 0 ). A Higgs boson is predicted and is crucial for the renormalizability and unitarity of the SM. Recently, ATLAS and CMS collaborations have found signals for a new particle with mass around 125GeV at the LHC, which are somewhat different from the SM Higgs boson (especially in the diphoton mode), although still consistent with the SM expectations within about 2 σ statistics [5]. Hence, the true mechanism of electroweak symmetry breaking is awaiting further explorations at the LHC, and the possible new physics beyond the SM Higgs boson is highly anticipated. Given the fact that no other new particles have been detected so far, it is tantalizing to explore alternative new physics sources at the TeV scale, beyond the conventional proposals such as extra dimensions, supersymmetry and strong dynamics at the TeV scale. In this Letter, we will explore the quantum gravity effect of SDR at TeV scale, and study its applications to the electroweak sector of the SM. We conjecture that the TeV Scale SDR can play a key role to unitarize weak gauge boson scattering in the theory without or with a light Higgs boson. As a first example, we will show that the perturbative unitarity is maintained by the TeV scale SDR in scenarios without a Higgs boson, where the recently observed 125 GeV boson [5] can be something else, such as a dilaton-like particle [6]. Without a Higgs boson, the SM electroweak gauge symmetry SU (2) L ⊗ U (1) Y becomes nonlinearly realized [7] and the three Goldstone bosons are converted to the longitudinal polarizations of ( W ± , Z 0 ) after spontaneous symmetry breaking. Such a minimal Higgsless SM loses traditional renormalizability [7] and violates unitarity of weak boson scattering at the TeV scale [8], hence it is incomplete. We show that the TeV-scale SDR can provide a new way to unitarize the WW scattering, and will be discriminated from the SM at the LHC. Then, we study the SM with a non-standard Higgs boson of mass around 125 GeV under the TeV-scale SDR (called the Higgsful SM). We will show that the corresponding WW scattering cross sections become unitary at TeV scale under the SDR, but exhibit different behaviors from the conventional 4d SM. We note that different ways of unitarizing the longitudinal WW scattering around TeV scale reflect the underlying mechanisms of electroweak symmetry breaking (EWSB), and will be discriminated by the WW scattering experiments as a key task of the LHC [9].", "pages": [ 1, 2 ] }, { "title": "2. The TeV Scale SDR", "content": "Despite lacking a full theory of quantum gravity that could precisely describe the SDR, we are modest and approach this problem by using the effective theory formulation [10]. In particular, to mimic the result from the causal dynamical triangulation [11], we parameterize the spacetime dimension n = n ( µ ) as a smooth function of the energy scale µ (which we call the dimensional flow by following Calcagni[12]), such that n ( µ ) → 4 under µ → 0 in the infrared region as supported by all low energy experiments, and n ( µ ) → 2 at a certain UV scale Λ UV . We can make a simple choice for the dimensional flow, where the index γ > 1 is a model-dependent parameter, determined by the nonperturbative dynamics of quantum gravity. As simple realizations, we may set, γ = 2 or 1.5 . Before finding a unique full theory of the quantum gravity, other variations of (1) are possible [11, 12], but this will not affect the main physics features of the present analysis. An easy choice for Λ UV would be the Planck scale. But it is a very interesting and intriguing possibility that the nonperturbative dynamics of quantum gravity drives Λ UV down to O (TeV) [2]. If this happens, a number of difficulties associated with the EWSB and W/Z mass-generations in the SM can be resolved without introducing additional ad hoc hypothetical dynamics. When the quantum gravity effects show up at the TeV scale, they will induce effective operators causing sizable anomalous Higgs couplings to WW ( ZZ ) and fermions in the low energy effective theory. This will violate perturbative unitarity at TeV energy scale in the conventional 4d setup [13][14]. However, we show that under the TeV-scale SDR, the weak boson scattering amplitudes will still be unitarized through the reduction of spacetime dimensions. Furthermore, the presence of TeV scale SDR also provides a natural solution to the hierarchy problem since the 4d quadratically divergent radiative corrections to Higgs boson mass is rendered to be logarithmic in n = 2 spacetime and thus harmless.", "pages": [ 2 ] }, { "title": "3. The Standard Model with SDR", "content": "As an effective theory description of the SDR, we encode the information of dimensional flow n = n ( µ ) into the measure of spacetime integral d ρ , and replace all integral measure d 4 x in the action functional by d ρ . A rigorous mathematical construction of d ρ is given by Ref. [12], but the detail is not needed here. All we need to know is that the mass-dimension of this measure is [d ρ ] = -n , where n = n ( µ ) is the dimensional flow in Eq. (1). It is enough to define the measure d ρ formally by d n x , with n a scaledependent quantity. Thus, we can write down the action of the theory, S = ∫ d n x L = ∫ d n x ( L G + L F ) , where L G and L F are the gauge and fermion parts of the SM Lagrangian. We will focus on the gauge sector for the current study. We first consider the gauge Lagrangian with Higgs boson removed, where the gauge field strength W a µν = ∂ µ W a ν -∂ ν W a µ + g/epsilon1 abc W b µ W c ν and B µν = ∂ µ B ν -∂ ν B µ . In the above, θ w = arctan( g ' /g ) represents the weak mixing angle and connects the gauge-eigenbasis ( W 3 µ , B µ ) to the mass-eigenbasis ( Z 0 µ , A µ ) . Eq. (2) contains ( W,Z ) mass terms in unitary gauge and can be made gauge-invariant in the nonlinear realization of SU (2) L ⊗ U (1) Y gauge symmetry, where D µ Σ = ∂ µ Σ + i 2 gW a µ τ a Σ -i 2 g ' B µ Σ τ 3 , and Σ = exp[i τ a π a /v ] with { π a } the Goldstone bosons. Eq.(3) gives, M W = 1 2 gv , where the parameter v will be fixed by the low energy Fermi constant G F = ( √ 2 v 2 ) -1 . The Lagrangian (2) derives directly from the SM in unitary gauge after removing the Higgs boson; while Eq. (3) is just the lowest order electroweak chiral Lagrangian of the SM [7]. We can further embed the Higgs boson as a singlet scalar h 0 in this formulation by extending the Lagrangian (3) as follows [13][14], /negationslash /negationslash where ∆ κ ≡ κ -1 and ∆ κ ' ≡ κ ' -1 denote the anomalous gauge couplings of the Higgs boson h 0 with WW and ZZ . The conventional 4d SM is just a special case, ∆ κ = ∆ κ ' = 0, and λ 3 = λ 4 = λ 0 = 3 M 2 h /v 2 , in the general effective Lagrangian (4). For non-zero anomalous couplings ∆ κ, ∆ κ ' = 0 and/or λ 3 , λ 4 = λ 0 , as induced by the effects of TeV scale SDR, the scalar field h 0 becomes a non-standard Higgs boson. We will study how to discriminate such as non-SM Higgs particle from the conventional 4d SM in Sec. 5. We also note that our Lagrangian L is manifestly Lorentz invariant and thus all particles' dispersion relations remain unchanged, as in [12]. This is because our formulation is based on the framework of [12], where it is shown that the action can be constructed in such a way that the Lagrangian density L lies in 3+1 dimensional spacetime and respects the (3+1)d Poincar'e symmetry, while the effect of SDR is fully governed by a properly defined integral measure d ρ . In such a scenario, scalar, spinor and vector fields are linear representations of (3+1)d Lorentz group SO (3 , 1) (up to a gauge transformation for gauge fields). Practically, this is similar to the conventional dimensional reduction regularization method [15], which maintains the 4d Lorentz symmetry and continues physics to n < 4 . Thus, our model is free from Lorentz-violation constraints in the cosmic ray observations and collider experiments at the tree level. In addition, our present study focuses on the scattering of longitudinal weak bosons, and their amplitudes are equivalent to that of the corresponding Goldstone bosons at high energies according to the equivalence theorem [16]. The amplitudes of scalar Goldstone bosons are cleanly defined in general dimension n . Manipulations of scalar fields do not involve contracting or counting Lorentz indices, and thus do not rely on details of realizing the SDR. In general n -dimensional spacetime, the action functional should remain dimensionless. Hence the Lagrangian has the mass-dimension [ L ] = n , and the gauge coupling g has mass-dimension [ g ] = (4 -n ) / 2. We can always define a new dimensionless coupling ˜ g and transfer the mass-dimension of g to another mass-parameter. Since the gauge coupling g in (2) becomes super-renormalizable for n < 4 and thus insensitive to the UV, it is natural to scale g by the W mass M W [which is the only dimensionful parameter of the Lagrangian (2) in 4d], with ˜ g being dimensionless. The value of ˜ g is given by that of g at n = 4. We will concentrate on the tree-level analysis in this study, so the coupling ˜ g is a scale-independent constant. In fact, the scaling (5) is well justified for more reasons. We may easily wonder why we could not use the UV cutoff Λ UV in the scaling of g as a replacement of the infrared mass-parameter M W of the theory. This is because in spacetime dimension n < 4, the gauge coupling g is super-renormalizable with positive mass-dimension [ g ] = (4 -n ) / 2 > 0 . Such a super-renormalizable coupling must be insensitive to the UV cutoff of the theory, contrary to a non-renormalizable coupling with negative mass-dimension and thus naturally suppressed by negative powers of the UV cutoff Λ UV ( e.g., in n > 4 or in association with certain higher-dimensional operators). It is easy to imagine that for a super-renormalizable theory in dimension n < 4 , if its coupling g were scaled as g = ˜ g Λ (4 -n ) / 2 U V , it would even make tree-level amplitude UV divergent and blow up as Λ UV →∞ ; this is clearly not true. On the other hand, it is well-known that a non-renormalizable coupling g with negative mass-dimension [ g ] ≡ -p < 0 should be scaled as g = ˜ g/ Λ p U V , and thus the tree-level amplitude naturally approaches zero when Λ UV →∞ , as expected. Since spacetime dimension flows to n = 2 in the UV limit, we observe that the theory (2) is well-behaved at high energies. This is because all gauge couplings of the Lagrangian (2) in n < 4 dimensions becomes super-renormalizable, and the gauge boson propagators scale as 1 /p 2 in high momentum limit under the R ξ gauge-fixing. So we only concerns about gauge boson mass-terms in (2) or (3), which is the origin of nonrenormalizability and unitarity violation in 4d. But, in our construction the spacetime dimension flows to n = 2 in high energy limit where (3) just describes a 2d gauged nonlinear sigma model and is renormalizable, as is well known. We also note that in n < 4 dimensions gauge bosons can acquire masses via new mechanisms other than the Higgs mechanism. For instance, in the 2d Schwinger model [17], radiative corrections to the vacuum polarization from a massless-fermion loop generate a nonzero photon mass, m γ = e √ π . Also, the 3d ChernSimons term induces a topological mass for the corresponding gauge field [18], m cs = κe 2 . Hence, it is natural to have an explicit mass-term of vector boson in a lower dimensional field theory. We will further demonstrate below that such a mass-term is indeed harmless in a Higgsless SM with the TeV scale SDR, and the unitarity of high energy longitudinal WW scattering is ensured.", "pages": [ 2, 3, 4 ] }, { "title": "4. Longitudinal Weak Boson Scattering under SDR", "content": "As a simple illustration of the unitarization mechanism of longitudinal WW scattering under the SDR, we first present the analysis in the SM without Higgs boson (called the Higgsless SM and denoted by HLSMSDR). It is noted that such a scenario can be consistent with the current LHC data since the 125 GeV new boson may be something else, such as a dilaton-like particle [6]. The effect of SDR is most clearly seen in this case. After this, we will further extend this mechanism to the Higgsful SM under the SDR (including a Higgs boson and called the HFSM-SDR), in the next section. The minimal 4d Higgsless SM violates unitarity at TeV scale, because the SM Higgs boson plays the key role to unitarize the bad high energy behaviors of the longitudinal WW scattering. For instance, without Higgs boson, the amplitude of W + L W -L → Z 0 L Z 0 L has non-canceled E 2 term, where E cm is the c.m. energy. This bad E 2 behavior leads to unitarity violation at TeV scale. In contrast, for the conventional 4d SM, this E 2 term is exactly canceled by the contribution of the s -channel Higgsexchange, which is the key to ensure the SM unitarity. In lower dimensions, the longitudinal amplitudes remain the same as in 4d. But, we observe that the form of partial wave expansion changes, due to the phase-space reduction for final state. Hence, the E 2 -cancellation described above is no longer essential for ensuring the unitarity. This is an essential feature of the unitarization mechanism through SDR, i.e., the WW scattering amplitudes remain unitary at high energies under SDR, even without a Higgs boson. To be explicit, we recall that unitarity condition for S -matrix arises from probability conservation, SS † = S † S = 1. This leads to T † T = 2 /Ifractur m T , where T is defined via S = 1 + i T , and is related to the amplitude T via T = (2 π ) n δ n ( p f -p i ) T with p i ( p f ) the total momentum of the initial (final) state. For 2 → 2 scattering, T depends only on the c.m. energy E cm and scattering angle θ . Thus, in this case we can always expand T ( E cm , θ ) in terms of partial waves a /lscript ( E cm ) for n > 3 dimensions, with λ n = 2(16 π ) n/ 2 -1 Γ( n 2 -1), ν = 1 2 ( n -3), N ν /lscript = π Γ( /lscript +2 ν ) 2 2 ν -1 /lscript !( /lscript + ν )Γ 2 ( ν ) , and C ν /lscript ( x ) is the Gegenbauer polynomial of order ν and degree /lscript . This partial wave expansion holds for n > 3 because the eigenfunctions of rotation generators (namely the Gegenbauer function) are not well defined below n = 3 . The appearance of factor E 4 -n cm in the expansion of T is expected, since the S -matrix of 2 → 2 scattering has a massdimension 4 -n in n dimensions, and the partial wave amplitude a /lscript is dimensionless by definition. Then, we can derive unitarity conditions for the elastic and inelastic partial waves, ∣ ∣ /Rfractur e a el /lscript ∣ ∣ /lessorequalslant ρ e 2 , ∣ ∣ a el /lscript ∣ ∣ /lessorequalslant ρ e , and ∣ ∣ a inel /lscript ∣ ∣ /lessorequalslant √ ρ i ρ e / 2, where ρ e ( ρ i ) is a symmetry factor of final state in 2 → 2 elastic (inelastic) scattering, and equals 1! (2!) for the final state particles being nonidentical (identical) [19]. With these, we perform the coupled channel analysis for electrically neutral channels. There are two relevant initial/final states, | W + L W -L 〉 and 1 √ 2 | Z 0 L Z 0 L 〉 , and the corresponding amplitudes form a 2 × 2 matrix, Thus, we derive the s -wave amplitude from the matrix (8) in n -dimensions and extract the maximal eigen- value after the diagonalization, Although the partial wave expansion (7) holds for n > 3 , we can make analytical continuation of (9) as a function of dimension n to the full range 2 /lessorequalslant n /lessorequalslant 4. Here, we perform the analytic continuation on the complex plane of spacetime dimension n , while the one-to-one mapping between n and µ is only defined within the real interval 2 /lessorequalslant n /lessorequalslant 4. In Fig. 1, we present the unitarity constraint for the standard model without a Higgs boson, under Eq. (1) with γ = 1 . 5, where we have varied the transition scale Λ UV = (4 , 5 , 6) TeV. The shaded yellow region is excluded by the unitarity bound | a max 0 | /lessorequalslant 1. The s -waves of HLSM-SDR always have a rather broad 'lump' around 1 . 5 -5TeV and then fall off quickly, exhibiting desired unitary high energy behaviors. For comparison, we also show the results of the 4d SM with a 600 GeV Higgs boson, and the naive 4d Higgsless SM which breaks unitarity at E cm /similarequal 1 . 74TeV. Next, we compute the cross sections for W + L W -L → Z 0 L Z 0 L and W + L W + L → W + L W + L , as shown in Fig. 2. The 4d unitarity condition for inelastic cross sections is, σ inel /lessorequalslant 4 πρ e E -2 cm . We derive the generalized form in n -dimensions, where λ n and N ν 0 are defined below Eq. (7). Fig. 2 demonstrates how the SDR works as a new mechanism to successfully unitarize the high energy behaviors of cross sections without invoking extra hypothesized particle (such as the SM Higgs boson). Furthermore, the new predictions of the HLSM-SDR are universal and show up in all WW scattering channels. This is an essential feature of our model and will be crucial for discriminating the HLSM-SDR from all other models of the EWSB at the LHC. Here we note that in n -dimensions the cross section σ has its mass-dimensions equal [ σ ] = 2 -n , while the experimentally measured cross section σ exp always has mass-dimension -2, as the detectors record events in 4d. So we need to convert the theory cross section σ under the SDR to σ exp , where the extra mass-dimensions of σ should be scaled by the involved energy scale E cm of the reaction, σ exp = σE n -4 cm . As a final remark, it was found [19] in 4d that varying the phase space may strongly alter the unitarity limit. Ref. [19] observed that the enlarged phase space of 2 →N scattering (due to properly increasing the number N of gauge bosons in the final state) will enhance the cross section and result in a new class of much stronger unitarity bounds for all light SM fermions. Interestingly, the current study just shows the other way around: for 2 → 2 scattering, reduction of the phase space of final states from decreasing the spacetime dimension n can significantly reduce the partial wave amplitudes and cross sections, leading to the unitarity restoration.", "pages": [ 4, 5, 6, 7 ] }, { "title": "5. Weak Boson Scattering in Higgsful SM with SDR", "content": "In this section, we extend the new mechanism in Sec. 4 to the Higgsful SM with SDR (HFSM-SDR). In this construction, the quantum-gravity-induced SDR at TeV scales provides a natural solution to the hierarchy problem that plagues the Higgs boson in the conventional 4d SM. For TeV-scale SDR, it is expected that new physics effects induced by the quantum gravity will show up in the low energy effective theory. So, the Higgs boson can have anomalous couplings with WW and ZZ gauge bosons, and thus behaves as non-SM-like. In unitary gauge, we can write down the leading anomalous gauge interactions of the Higgs boson [13, 14] from the effective Lagrangian (4), where the anomalous couplings ∆ κ, ∆ κ ' = 0 represent new physics. Besides the hierarchy problem, the conventional 4d SM also suffers constraints from the Higgs vacuum instability and the triviality of Higgs self-coupling. If such a 4d SM would be valid up to Planck scale, then the SM Higgs boson mass is bounded within the range[20], 133 GeV /lessorsimilar M h /lessorsimilar 180GeV. Hence, a Higgs mass outside this window will indicate a non-standard Higgs boson in association with new physics. The Higgs boson in our present model under the TeV-scale SDR has anomalous couplings induced from quantum gravity and thus behaves as non-SM-like. With the recent LHC data [5], model-independent fits already put some interesting constraints on the Higgs /negationslash anomalous couplings. Using the fitting result of [21], we find that for M h = 125GeV, the ∆ κ in (11) is bounded within the range, ∆ κ = 0 . 2 +0 . 4 -0 . 5 . /negationslash In the conventional 4d SM with (11), it was found [14] that the WW scattering has non-canceled large E 2 behavior and will eventually violate unitarity at TeV scale. But in our new model, the TeV-scale SDR can always unitarize WW scattering and predicts different behaviors for cross sections, as shown in Fig. 3 for γ = 1 . 5 . In Fig. 3(a)-(b), we study WW scattering process for probing a non-SM Higgs boson with mass M h = 125 GeV [5] and sample anomalous couplings ∆ κ = ± 0 . 3 . We find that the cross sections under SDR unitarization (middle colored curves) have sizable excesses above the 4d SM with a 125 GeV Higgs boson (∆ κ = 0, flat black curve). Then, they fall off in the 2 -4TeV region, consistent with the corresponding unitarity limits. Fig. 3 also shows that for the usual non-unitarized 4d SM with nonzero anomalous coupling ∆ κ = 0, the cross section (upper black curve) rapidly increases and eventually violates unitarity around E cm = 2TeV for this scattering channel. Before concluding this section, we would like to clarify the validity range of our effective theory of the SDR. This validity range lies between the WW ( ZZ ) threshold (around 160 -180GeV) and the UV-cutoff Λ UV = O (5TeV). It is clearly shown in our Fig. 2 and Fig. 3, where the relevant scattering energy E cm (for our model to be discriminated from the 4d-SM and 4d-HLSM at the LHC) is always within 0 . 2 -3 TeV, which is significantly below 4 TeV /lessorequalslant Λ UV . Moreover, within this energy region 0 . 2 -3TeV (relevant to the LHC test), we can explicitly derive the dimensional flow from Eq. (1), n /similarequal 3 . 98 -3 . 07 , (under the typical input of Λ UV = 5TeV and γ = 1 . 5), which is significantly above n = 2 . 1 This clearly shows that for our effective theory study we do not need to invoke any detailed UV dynamics at or above Λ UV .", "pages": [ 7, 8 ] }, { "title": "6. Conclusions", "content": "We have studied the exciting possibility for the onset of spontaneous dimensional reduction (SDR) at TeV scales. We demonstrated that the TeV-scale SDR can play a key role to unitarize longitudinal weak boson scattering. We have constructed an effective theory of the SM under the SDR, either without a Higgs boson or with a light non-standard Higgs boson. In the first construction, it nonlinearly realizes the electroweak gauge symmetry and its spontaneous breaking. The model becomes manifestly renormalizable at high energies by power counting. We found that the non-canceled E 2 contributions to the WW scattering amplitudes are unitarized by the SDR at TeV scales (Fig. 1), and the scattering cross sections exhibit different behaviors (Fig. 2). This will be probed at the LHC. Here the recent observation of a 125 GeV boson at the LHC (8 TeV) could be something else, such as a dilaton-like particle [6]. In passing, we note that the unitarity of WW scattering in generic 4d technicolor theories was recently studied in Ref. [22]. For the second construction of the Higgsful SM with SDR, we studied the WW scattering with a light non-standard Higgs boson of mass 125 GeV. It has effective anomalous couplings with gauge bosons as induced from the TeV-scale quantum gravity effects [cf. Eq. (11)]. Fig. 3(a)-(b) showed that under the SDR, the cross section of W + L W -L → Z 0 L Z 0 L process with anomalous Higgs couplings has distinctive invariant-mass distributions from the naive 4d SM Higgs boson over the energy regions around 0 . 2 -3TeV. This will be definitively probed by the next LHC runs at 14 TeV collision energy with higher luminosity. For future works, it is useful to further develop a method for quantizing field theories with SDR and compute the sub-leading effect of loop corrections in fractional spacetime [12], which should have better UV behavior than the usual 4d SM and thus is expected to agree even better with the precision data. This is fully beyond the current scope and will be further explored in future works. A systematical expansion of our study in the present Letter is given elsewhere [23]. As the final remark, our effective theory construction is also partly motivated by the asymptotic safety (AS) scenario of quantum general relativity (QGR) 'a la Weinberg [24][25]. In the AS scenario, the theory is originally defined in (3+1)d, while solving the exact renormalization group equation of QGR points to nontrivial UV fixed point, under which the graviton two-point function exhibits effective two-dimensional UV behavior[25]. Here, the SDR is reflected in anomalous scalings of the fields, as well as physical variables like the spacetime curvature. Such anomalous scalings share the similarity with our effective theory construction, while the field contents are still defined in (3+1)d and respect the (3+1)d Lorentz symmetry. Our effective theory is a simplified formulation at low energy, so it does not rely on any detailed UV dynamics of the AS scenario. It is interesting to further study the quantitative connection between the SDR and the AS scenario. We also note that the Hoˇrava-Lifshitz model [26] of quantum gravity can provide a concrete field-theoretical realization of SDR with UV-completion, which has relatively tractable Lagrangian. Thus, the various scaling properties in our effective theory are expected to arise from the formulation of the Hoˇrava-Lifshitz model. We will consider these two interesting scenarios for future works.", "pages": [ 8, 9 ] }, { "title": "Acknowledgments", "content": "We are grateful to Gianluca Calcagni, Steven Carlip and Dejan Stojkovic for discussing the spontaneous dimensional reduction, to Daniel Litim for discussing the asymptotic safety, and to Petr Hoˇrava for discussing the Hoˇrava-Lifshitz gravity. We thank Francesco Sannino and Chris Quigg for discussions during their visits to Tsinghua HEP Center. This work was supported by National NSF of China (under grants 11275101, 11135003, 10625522, 10635030) and National Basic Research Program (under grant 2010CB833000).", "pages": [ 9 ] }, { "title": "References", "content": "[5] G. Aad et al., [ATLAS Collaboration], Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214 [hep-ex]]. S. Chatrchyan et al., [CMS Collaboration], Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235 [hep-ex]].", "pages": [ 9 ] } ]

[ { "title": "A Novel Mechanism to Generate FFLO States in Holographic Superconductors", "content": "James Alsup ∗ Computer Science, Engineering and Physics Department, The University of Michigan-Flint, Flint, MI 48502-1907, USA", "pages": [ 1 ] }, { "title": "Eleftherios Papantonopoulos †", "content": "Department of Physics, National Technical University of Athens, GR-15780 Athens, Greece", "pages": [ 1 ] }, { "title": "George Siopsis ‡", "content": "Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996-1200, USA (Dated: November 8, 2018) We discuss a novel mechanism to set up a gravity dual of FFLO states in strongly coupled superconductors. The gravitational theory utilizes two U (1) gauge fields and a scalar field coupled to a charged AdS black hole. The first gauge field couples with the scalar sourcing a charge condensate below a critical temperature, and the second gauge field provides a coupling to spin in the boundary theory. The scalar is neutral under the second gauge field. By turning on an interaction between the Einstein tensor and the scalar, it is shown that, in the low temperature limit, an inhomogeneous solution possesses a higher critical temperature than the homogeneous case, giving rise to FFLO states. PACS numbers: 11.25.Tq, 04.70.Bw, 71.45.Lr, 71.27.+a The AdS/CFT correspondence, which was discovered in string theory, has opened up a broad avenue for the exploration of condensed matter systems at strong coupling. By using a holographic principle, these systems (described by gauge field theories) are mapped onto weakly coupled gravitational systems of one additional dimension, in which physical quantities can be computed. This holographic principle (gauge theory / gravity duality) has been applied to the study of conventional and unconventional superfluids and superconductors [1], Fermi liquids [2], and quantum phase transitions [3]. The high-Tc superconductors, such as cuprates and iron pnictides, are examples of unconventional superconductors which exhibit competing orders that are related to the breaking of the lattice symmetries. This breaking introduces inhomogeneities and a study of the effect of inhomogeneity of the pairing interaction in a weakly coupled BCS system [4] suggests that inhomogeneity might play a role in high-Tc superconductivity. In an effort to explain this behavior a 'striped' superconductor was proposed [5]. Holographic striped superconductors were discussed in [6] where a modulated chemical potential was introduced and it was shown that below a critical temperature superconducting stripes develop. Properties of the striped superconductors and backreaction effects were studied in [7, 8]. Striped phases were also found in electrically charged RN-AdS black branes that involve neutral pseudo-scalars [9]. Inhomogeneous phases also appear when a strong external magnetic field coupled to the spins of the conduc- tion electrons is applied to a high-field superconductor. This results in a separation of the Fermi surfaces corresponding to electrons with opposite spins (for a review see [10]). If the separation is too high, the pairing is destroyed and there is a transition from the superconducting state to the normal one (paramagnetic effect). An intriguing new state of matter at the transition point was proposed by Fulde and Ferrell [11] and Larkin and Ovchinnikov [12] (the FFLO state) but it has not been found experimentally so far. This state is characterized by a space modulated order parameter, corresponding to an electron pair having nonzero total momentum. A way to understand the formation of the FFLO phase in a superconductor-ferromagnetic system (S/F) is to use the generalized Ginzburg-Landau expansion. In order to describe the paramagnetic effect in the presence of a strong external magnetic field, the usual | ψ | 4 -GinzburgLandau functional has to be modified with coefficients in the functional which depend also on the magnetic field. In this case, the ( B,T ) phase diagram exhibits a different behavior indicating that the minimum of the functional does not correspond to a uniform state, and a spatial variation of the order parameter decreases the energy of the system. To describe such a situation, it is necessary to add a higher-order derivative term in the expansion of the Ginzburg-Landau functional (for a detailed account see [13]). There are several studies of the behavior of holographic superconductors in the presence of an external magnetic field. Non-trivial spatially dependent solutions have been found, like the droplet [14] and vortex solutions with integer winding number [15-17]. An analytic study on holographic superconductors in an external magnetic field was carried out in [18]. In a model resulting from a consistent truncation of type IIB string theory, anisotropic solutions at low temperature were found [19], showing similarity between the phase diagrams of holographic superfluid flows and those of ordinary superconductors with an imbalanced chemical potential. A holographic superconducting model with unbalanced fermi mixtures at strong coupling was discussed in [20]. The charge and spin transport properties of the model were studied, but the phase diagram did not reveal the occurrence of FFLO-like inhomogeneous superconducting phases. In our recent work [21], we proposed a gravity dual of FFLO states in strongly coupled superconductors. The gravity sector consisted of two U (1) gauge fields and a scalar field. The first gauge field had a non-zero scalar potential term which was the source of the charge condensate in the boundary theory through its coupling to the scalar field. The second U (1) gauge field corresponded to an effective magnetic field acting on the spins in the boundary theory. The scalar field was neutral under the second U (1) gauge field. We looked first at the behavior of the system at or above the critical temperature. The Eintein-Maxwell system was solved by a dyonic black hole with electric and magnetic charges, as in [20]. At the critical temperature, the system underwent a second-order phase transition and the black hole acquired hair. To find the critical temperature, we worked in the grand canonical ensemble and solved the scalar equation in the background of the dyonic black hole. It was found that the system possessed inhomogeneous solutions for the scalar field, which however always gave a transition temperature lower than the maximum transition temperature (i.e. critical temperature) of the homogeneous solution. Therefore the homogeneous solution was always dominant. Next, we turned on an interaction term of the magnetic field to the scalar field of the generalized GinzburgLandau gradient type (in a covariant form). The scalar field equation was modified and the resulting inhomogeneous solutions gave a transition temperature which was higher than the one of the homogeneous solutions. We attributed this behavior of the system to the appearance of FFLO states. We noted that the appearance of the FFLO states was more pronounced as T c /µ → 0, and the magnetic field of the second U (1) gauge group was large. In this letter, we propose a novel mechanism for the generation of the gravity dual of FFLO states in the low temperature limit. In our previous work we showed that, in order to generate the FFLO phase, we needed a direct coupling of the magnetic field to the scalar field. We will show that this interaction term can be effectively generated through the coupling of the Einstein tensor to the scalar field. The reason is that since the electromagnetic fields backreact on the metric, the Einstein tensor has encoded the information of these fields. As before, the bulk theory consists of two U (1) gauge fields and a scalar field. The first gauge field has a nonzero scalar potential term and the second U (1) gauge field corresponds to a chemical potential (imbalance) for spin. The scalar field is neutral under the second U (1) gauge field. Note, too, the second U (1) is self-dual under /vector E ↔ /vector B and alternatively the boundary theory can be understood in terms of a magnetic field instead of the chosen spin chemical potential. The interaction between the Einstein tensor and the scalar field is most often seen in scalar-tensor theories. The interest stems from the galilean symmetry of the system where the action is invariant under shifts of field derivatives by a constant vector. Thereby the higherderivative theory has only second order equations of motion [22]. It was shown that this term acts as an effective cosmological constant and produces an early entrance into a quasi-de Sitter stage as well as a smooth exit [23]. Cosmic evolution for vanishing cosmological constant has also been investigated in [24]. The coupling has been realized in string cosmology from an effective heterotic action, up to α ' corrections [25] and also in N = 1 four-dimensional new-minimal supergravity theories [26]. Moreover, interest away from cosmology has developed as the interaction has been used to study phase transitions for vanishing cosmological constant [27] and effects on conventional holographic superconductors employing anti-de Sitter space [28]. The presence of this term modifies the scalar field equation and the resulting inhomogeneous solutions give a transition temperature which is higher than the homogeneous solutions. We attribute this behavior to FFLO states. Note that as before, the appearance of the FFLO states is more pronounced as T c /µ → 0 and the gauge field of the second U (1) gauge group is near its maximum value. Consider the action S = where F AB = ∂ A A B -∂ B A A , F AB = ∂ A A B -∂ B A A are the field strengths of the U (1) potentials A A and A A , respectively. We set L = 8 πG = 1. The Einstein-Maxwell equations, admit a solution which is a four-dimensional AdS black hole of two U (1) charges, with the horizon radius set at z = 1. The two sets of Maxwell equations admit solutions of the form, respectively, and with corresponding field strengths having non-vanishing components for electric fields in the z -direction, respectively, Then from the Einstein equations we obtain The Hawking temperature is In the limit µ, δµ → 0 we recover the Schwarzschild black hole. Next, we consider a scalar field φ , of mass m , and U (1) 2 charge ( q, 0), coupled to the Einstein tensor. The action is where D A = ∂ A + iqA A and G AB is the Einstein tensor. ξ is the new coupling constant determining the strength of the interaction between the scalar field and the Einstein tensor. We also included a convenient ξ -dependent factor in the mass term. We shall consider the range of ξ for which the factor is positive, Firstly, we consider the conventional case setting ξ = 0. The asymptotic behavior (as z → 0) of the scalar field is For m 2 ≥ -5 / 4, there is only one normalizable mode. However, for the range, -9 / 4 ≤ m 2 < -5 / 4, there are two allowable choices of ∆, leading to two distinct physical systems. As we lower the temperature, an instability arises and the system undergoes a second-order phase transition with the black hole developing hair. This occurs at a critical temparture T c which is found by solving the scalar wave equation in the above background, with the metric function h given in (7) and the electrostatic potential A t in (4). Although the wave equation (13) possesses ( x, y )-dependent solutions, the symmetric solution dominates and the hair that forms has no ( x, y ) dependence. To see this, let us introduce ( x, y )-dependence and consider a static scalar field which is an eigenstate of the twodimensional Laplacian, For example, if φ varies sinusoidally in the x -direction, φ ∼ e iQx , then τ = Q 2 and the modulation is realized in the boundary CFT through the order parameter 〈O〉 ∼ e iQx . It is also possible for φ to be rotationally symmetric in the ( x, y ) plane, φ ∼ J 0 ( √ τ ( x 2 + y 2 )). For τ = 0, we recover the homogeneous solution. Upon factoring out the ( x, y ) dependence, where Y ( x, y ) is an eigenfunction of the two-dimensional Laplacian with eigenvalue -τ (eq. (14)), the scalar field is represented by ψ ( z ) and the wave equation becomes Before we proceed with a discussion of solutions, notice that there is a scaling symmetry This means that the system possesses a scale which we have fixed for simplicity of notation. This arbitrary scale is often taken to be the radius of the horizon r + , after changing coordinates to Since we fixed the scale, we should only be reporting on scale-invariant quantities, such as T/µ , δµ/µ , τ/µ 2 , etc. It is also convenient to introduce the scale-invariant parameter to describe the effect of the chemical potential imbalance. We shall be working in the grand canonical ensemble at fixed chemical potentials µ and δµ (or β ). The ensemble is defined uniquely by specifying the parameters q and ∆. One can then vary τ , which parametrizes the solutions, to study the behavior of the system. Since we fixed the scale, we shall solve the wave equation (16) for fixed values of the scale-invariant parameters τ/µ 2 and β , while demanding regularity at the horizon and ψ ∼ z ∆ + at the boundary. Thus, we obtain µ = µ 0 as an eigenvalue. More precisely, we obtain µ/r + as an eigenvalue, where r + is the scale in the system. Since the chemical potential µ is fixed (grand canonical ensemble), the solution of the wave equation (16) fixes the scale ( r + = r +0 , so that µ 0 → µ/r +0 ) , and therefore the transition temperature T 0 below which a mode with the given τ may develop. We obtain which is of the same form as a Reissner-Nordstrom black hole with effective chemical potential The maximum transition temperature is the critical temperature T c of the system. As we cool down the system in its normal state, the transition temperature T 0 = T c is reached first and the mode with the corresponding τ is the first to develop. In the homogeneous case, τ = 0, the maximum transition temperature is obtained for β = 0. In this case, we recover the Reissner-Nordstrom black hole. As we increase β , the temperature (20) decreases. The scalar wave equation is the same as its counterpart in a Reissner-Nordstrom background, but with effective charge so that q eff µ eff = qµ . It is known [1] that the instability [29] occurs for all values of q eff , including q eff = 0, if ∆ ≤ ∆ ∗ , where ∆ ∗ = ∆ + for m 2 = -3 2 , or explicitly, For ∆ ≤ ∆ ∗ , β can increase indefinitely. The transition temperature T 0 for τ = 0 has a minimum value as a function of β , and as β →∞ , T 0 diverges. For ∆ > ∆ ∗ , q eff has a minimum value at which the transition temperature vanishes and the black hole attains extremality. This is found by considering the limit of the near horizon region [30, 31]. One obtains At the minimum ( T 0 = 0), µ 2 eff = 12, and β attains its maximum value, This limit is reminiscent of the Chandrasekhar and Clogston limit [32] in a S/F system, in which a ferromagnet at T = 0 cannot remain a superconductor with a uniform condensate. where Note that the boundary behavior is unaltered from (12). The coupling to the Einstein tensor alters the near horizon limit of the theory so that /negationslash In the inhomogeneous case ( τ = 0), the above argument still holds with the replacement m 2 → m 2 + τ . The effect of this modification is to increase the minimum effective charge to and thus decrease the maximum value of β (25). We always obtain a transition temperature which is lower than the corresponding transition temperature (for same β ) in the homogeneous case ( τ = 0). It follows that the critical temperature is the transition temperature of the homogeneous mode, and the latter dominates the condensate. Notice also that τ ≤ τ max , where the maximum value is attained when q min = q (so that β max = 0). We deduce from (26), The supporting numerical results are shown in figure 1. /negationslash Now let us consider the effect of coupling to the Einstein tensor by setting ξ = 0. The wave equation is modified to For ξ < 1 / 6, it is easily seen from (29) that f ± (1) > 0, so that the effective mass increases with τ , as in the conventional ξ = 0 case. The minimum effective charge is found from the nearhorizon geometry in the zero temperature (extremal) limit. Using µ 2 eff = 12 and f ± (1) = 1 -3 ξ ± 3 ξ , we obtain to be compared with (26). Finally, τ has a maximum value found by setting q min = q in (31), Thus, for ξ < 1 / 6, even though the results differ numerically from the case ξ = 0, they are not qualitatively different. The maximum transition temperature (i.e., the critical temperature of the system) is always attained for τ = 0 (homogeneous case). As ξ approaches the critical value 1 6 , the maximum value of τ diverges. As we increase ξ past the critical value, i.e., for (with ξ still satisfying (10)), the range of τ extends to infinity ( τ max is infinite), and the behavior of the system changes qualitatively . For ξ above the bound (33), the minimum charge decreases for τ > 0, and therefore the maximum value of β (25) increases compared to the value in the homogeneous case ( τ = 0). Thus, there is a neighborhood near zero temperature in which the inhomogeneous solution has higher transition temperature than the homogeneous one. As we increase τ , the corresponding transition temperature increases. This expected behavior is also seen numerically. As we keep increasing τ , we are no longer in the zero temperature limit and geometrical considerations near the horizon are no longer applicable. Thus, although the effective mass (30) keeps decreasing below the AdS 2 BF bound, the latter is no longer relevant, and the wave equation possesses acceptable solutions for all τ . Although we can no longer argue analytically, we analyzed the behavior of the system numerically. As τ increases, the corresponding transition temperature keeps increasing. The maximum transition temperature, which would be identified with the critical temperature of the system, is attained asymptotically as τ →∞ (recall that there is no maximum value of τ for ξ > 1 / 6). The value of the critical temperature is found by analytically solving the wave equation in the limit τ → ∞ . It is easy to see by considering an expansion around the horizon that we ought to have f -= 0. We deduce thus determining the asymptotic transition (critical) temperature to be which is dependent solely upon our coupling constant ξ . To find the critical temperature numerically (and confirm the analytic prediction (35)), we fix the chemical potentials µ and δµ (or β ) and numerically solve the wave equation (28) for all allowed values of τ . Figure 2 displays the transition temperatures of various modes for small values β . The plots attain their maximum at the homogeneous mode, τ = 0, and therefore the homogeneous solution is dominant. The low temperature region is probed with larger values of β . Our results are plotted in figure 3. The homogeneous solution possesses a transition temperature that is below the majority of non-zero τ and hence the inhomogeneous solutions dominate. As we increase τ , the corresponding transition temperature increases and approaches the asymptotic value (35), as expected. The asymptotic value (critical temperature) is an upper bound for the transition temperatures of the various modes. Figure 4 displays the transition temperature numerically calculated for ξ = . 2 for select values of τ . The point where the inhomogeneous solution becomes dominant is found at the crossing between τ = 0 and the large τ asymptotic (critical) temperature seen in the body of the figure. In a condensed matter system with an order parameter possessing wavenumber Q , the lattice spacing a is related by Q ∼ 1 /a [10]. In our system, effectively a → 0, which corresponds to τ = Q 2 → ∞ . Therefore, we expect the critical temperature to correspond to τ → ∞ . It would be desirable to include lattice effects so that the critical temperature corresponds to a large but finite value of τ [33, 34]. An effective way of accomplishing this is by including higher order terms in the Lagrangian. Let us introduce a cutoff that suppresses large momentum ( τ ) modes in the Einstein coupling term. To do this covariantly, introduce the derivative operator where F is the field strength of the second U (1) potential, and D A is the gauge derivative (see eq. (9)). Then modify the action for the scalar field (9) to The function S ( x ) is chosen so that S (0) = 1 and S ( x ) → 0, as x →∞ . We also introduced a new (small) parameter α . It is convenient to choose The wave equation (28) is modified to and the functions f ± (eq. (29)) are modified to Notice that in the limit τ → ∞ , we have f ± → 1, so for large τ , the solutions approach those in the standard case ξ = 0, in which there is a maximum allowed value of τ (eq. (27)). The supporting numerics are shown in figure 5, where a maximum transition temperature (critical temperature) at finite τ may be clearly seen. In conclusion , we have developed a gravitational dual theory for the FFLO state of condensed matter. The gravitational theory consists of two U (1) gauge fields and a scalar coupled to a charged AdS black hole. The first gauge field produces the instability for a condensate to form, while the second controls chemical potential associated with spin. In the absence of an interaction of the Einstein tensor with the scalar field, the system possesses dominant homogeneous solutions for all allowed values of the spin chemical potential. In the presence of the interaction term, at low temperatures, the system is shown to possess a critical temperature for a transition to a scalar field with spatial modulation as opposed to the homogeneous solution. It is desirable to fully understand the interplay between the different modes once below the critical temperature. This will require a non-linear analysis of the Einstein- [12] A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)]. Maxwell-scalar equations. Additionally, the dependence of the critical temperature on the modulation wavenumber, which is intertwined with the presence of a lattice is an intriguing aspect. Work in these directions is in progress.", "pages": [ 1, 2, 3, 4, 5, 6, 7 ] } ]

[ { "title": "Hyun Min Lee", "content": "School of Physics, KIAS, Seoul 130-722, Korea.", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider the renormalization group(RG) improved inflaton potential in unitarized Higgs inflation where the original Higgs inflation is unitarized by the addition of a real singlet scalar of sigmamodel type. The sigma field coupling to the Higgs, which is introduced to reproduce a large non-minimal coupling of the Higgs below the sigma scalar threshold, also improves the Standard Model vacuum stability due to the RG running. Furthermore, the same sigma field coupling determines the reheating temperature or the number of efoldings. Considering the uncertainties in the number of efoldings in the model, we show that the loop-corrected spectral index and tensor-to-scalar ratio are consistent with nine-year WMAP and new Planck data within 1 σ .", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Theoretical problems in the standard Big Bang cosmology such as horizon, homogeneity, flatness and relics have motivated to introduce the early period of cosmic inflation [1] by the addition of a scalar field, the so called inflaton, for the vacuum energy to dominate the universe. The initial condition for the large scale structure is set by the quantum fluctuation of the inflaton during inflation so the resultant post-inflation cosmology is then well described. The recent nine-year WMAP data [2] shows the evidence for a tilt in the primordial spectrum at the 5 σ level, constraining the inflation models with more precision. Furthermore, the first results on the measurement of cosmological parameters by PLANCK [3] have been released and have given strong constraints on inflation models [4,5]. Thus, we are entering the era of precision cosmology to probe the remnants of the cosmic inflation, as has been the case in the Standard Model(SM) of particle physics. Higgs inflation [6] has drawn much attention from both particle physics and cosmology communities, as a Higgs-like boson has been discovered recently at the Large Hadron Collider [7] and the model links between the SM and the inflation period in a minimal way. The Higgs doublet has a large non-minimal coupling to gravity [8] so the Higgs boson plays a role of the inflaton at large field values. However, there is a drawback in the original Higgs inflation due to the unitarity problem [9]. Although the unitarity cutoff during inflation is larger than the one in the vacuum [10], an inflationary plateau beyond the unitarity cutoff is still not justified under the perturbative expansion [11, 12]. Therefore, an extra degree of freedom at the unitarity scale is introduced to restore the unitarity [11] or an appropriate counterterm without extra degrees of freedom is required to cancel the dangerous interactions coming from the non-minimal coupling [13]. As inflation occurs at the high energy scales, it is important to perform the renormalization group(RG) running from low energy until the inflation scale, to compare with the low energy parameters [14,15]. In particular, the stability of the SM vacuum requires the Higgs quartic coupling to remain positive at high scales [16, 17]. In the original Higgs inflation, the standard RG analysis is trustable only up to the unitarity cutoff in the vacuum while the inflation occurs at the Higgs field values higher than the unitarity cutoff. Although the effect of the RG running between the unitarity cutoff and the inflation scale has been assumed to be small, the unknown dynamics restoring the unitarity below the unitarity cutoff could spoil the inflationary plateau and/or the inflationary predictions. In this paper, we consider the unitarized Higgs inflation proposed in Ref. [11], where a singlet scalar field of sigma-model type is introduced to restore the unitarity. In this model, unitarity is preserved all the way to the Planck scale and gives rise to the effective action of the Higgs inflation below the sigma scalar threshold. In the full theory, even if the Higgs doublet has a non-minimal coupling of order one, a large non-minimal coupling of the sigma field makes inflation possible along a new flat direction, which is a linear combination of the Higgs and sigma fields. Since the sigma field values for inflation are smaller than the Planck scale, the perturbative expansion is believed to be valid above the sigma scalar threshold too. We obtain the one-loop RG improved inflaton potential with the RG equations modified in the presence of a large non-minimal coupling for the sigma field. The vacuum stability can be improved by the RG running due to the mixing coupling between the Higgs and sigma fields, provided that the SM vacuum is guaranteed to be stable at the sigma scalar threshold. There are uncertainties in the number of efoldings because the reheating temperature depends on the sigma coupling to the Higgs. Taking this result into account, we show that the loop-corrected spectral index, controlled by the same sigma coupling, and the tensor-to-scalar ratio, are consistent with the nine-year WMAP data within 1 σ . The paper is organized as follows. We first begin with the description of the unitarized Higgs inflation and discuss the effective theories at low energy and during inflation. Then, we compute the one-loop Coleman-Weinberg corrections for the RG-improved inflaton potential and show how the vacuum stability is improved in this model. Next we present the results of the spectral index, the tensor-to-scalar ratio, the running of the spectral index, etc. In the next section, we give a brief discussion on the reheating temperature in relation to the number of efoldings. Finally conclusions are drawn. There is one appendix containing the RG equations applicable to the energy scales above the sigma field threshold.", "pages": [ 2, 3 ] }, { "title": "2 Inflation with non-minimal coupling and unitary Higgs", "content": "In order to solve the unitarity problem, we require extra dynamical degrees of freedom to restore unitarity without ruining the flat plateau. A UV complete model with a real singlet scalar of sigma-model type was proposed in [11] and it has been shown that extra singlet coupling could also solve simultaneously the vacuum instability problem in the SM [12,18]. The Jordan-frame Lagrangian of the model is where M,ω and v are mass parameters with v glyph[lessmuch] M,ω (so that the σ field is heavy) and ξ, ζ are positive non-minimal couplings with ξ glyph[greatermuch] ζ . The large nonzero vev of σ , 〈 σ 〉 glyph[similarequal] ω , is crucial to make the unitarity cutoff Λ UV larger. It is straightforward to find that where the Planck mass is now M 2 Pl = M 2 + ξω 2 , and we measure the contribution of the σ vev by the ratio r = ξω 2 /M 2 Pl , which in general can take values from 0 to 1. One can see how the cutoff is pushed up to rM Pl for moderate values of r glyph[greaterorsimilar] 1 /ξ . In the following discussion, for simplicity, we consider a simplified verison of the unitarized Higgs inflation where the tree-level Einstein term and the non-minimal coupling for the Higgs doublet is absent, M = 0 and ζ = 0, in Jordan frame. Then, the Jordan-frame action in unitary gauge with H = (0 , φ ) T / √ 2 is where and ω ≡ M P √ ξ is chosen to reproduce the Jordan-frame action of the Higgs inflation with a positive non-minimal coupling ξ h = -λ Hσ λ σ ξ for λ Hσ < 0, after integrating out the σ field by σ 2 = -λ Hσ λ σ φ 2 + ω 2 . The mass of the σ field in the vacuum is given by where ¯ σ denotes the canonically normalized field. The COBE constraint precisely fixes the sigma mass in the vacuum to be M ¯ σ ≈ 10 13 GeV [12]. Below the sigma mass scale, the effective action in Jordan frame is where the effective non-minimal coupling ξ eff and quartic coupling λ eff for the Higgs are matched to the fundamental couplings as Thus, the Higgs quartic coupling λ H can be larger than the SM value inferred from the Higgs mass, helping to ensure the vacuum stability at large field values when the sigma field is lighter than the instability scale Λ I in the SM [12]. Thus, we are forced to the Higgs masses for which Λ I > 10 13 GeV. This requires m h > 125 GeV (at 90% CL in M t from the kinematical top mass at the Tevatron and α s ) [12], which is marginally compatible with the Higgs-like boson discovered by ATLAS and CMS [7]. But, we note that the vacuum stability bound on the Higgs mass depends on the top pole mass, which still has a large uncertainty as suggested from the top pair production cross section measurements at the Tevatron [19,20]. In this work, we assume that the SM vacuum is stable within the uncertainties of the top pole mass at the sigma field threshold and consider the possibility that the loop corrections of the sigma field help improve the vacuum stability. Performing a Weyl scaling of the metric, we obtain the Einstein-frame action from eq. (3) as follows, Redefining the fields by σ ≡ ω e χ/ √ 6 M P and ˜ φ ≡ ωφ/σ = φe -χ/ √ 6 M P , the above action becomes with We note that the kinetic terms for the sigma and Higgs fields is of sigma-model type, with the coset space described by SO (1 , 5) /SO (5). Taking | σ | glyph[greatermuch] ω , the Einstein-frame potential approximates the potential for ˜ φ , Thus, for λ Hσ < 0, the potential has two minima at ˜ φ = ± √ -λ Hσ λ H ω ≡ ± ˜ φ 0 . Therefore, after stabilizing ˜ φ at one of the minima, we obtain the flat potential for χ as Therefore, the sigma field drives a slow-roll inflation while the Higgs field is stabilized at a large VEV during inflation. The difference from a single-field inflation with non-minimal coupling is that the Higgs field contributes a large vacuum energy during inflation and participates in the reheating process as will be discussed in the later section. Here, we note that a positive vacuum energy during inflation is obtained for λ H > λ 2 Hσ λ σ . Thus, the vacuum stability condition becomes the condition for the positive inflaton vacuum energy so it has not been improved at tree level, as compared to the SM, where the corresponding condition from the matching scale at the sigma mass scale is λ eff = λ H -λ 2 Hσ λ σ > 0. Since | ˜ φ 0 | glyph[lessmuch] M P for ξ glyph[greatermuch] 1, the kinetic mixing term is ignored and both χ and ˜ φ are canonical scalar fields. Heneceforth we set M P = 1.", "pages": [ 3, 4, 5 ] }, { "title": "3 Effective inflaton potential", "content": "We consider the one-loop Coleman-Weinberg corrections in unitarized Higgs inflation for the effective potential for inflation and discuss the effect of the sigma-field couplings on the vacuum stability.", "pages": [ 5 ] }, { "title": "3.1 One-loop inflaton potential", "content": "First, ignoring the contribution coming from the inflaton, we get the one-loop ColemanWeinberg potential as where the 'effective' masses for the heavy mode ˜ φ , Goldstone bosons, W and Z bosons and top quark are given in order as From eq. (14), we find that all logarithms contain the corrections to the effective quartic couplings for ˜ φ while only the Higgs portal term gives rise to the correction to the 'physical' mass of ˜ φ evaluated at ˜ φ = 0 as follows, with We note that the loop mass has the same functional form for χ as for the tree-level mass and it can be absorbed by renormalizing the tree-level mass parameter. In the end of inflation, the inflaton rolls down to the minimum of the potential at χ glyph[similarequal] 0, so the large mass terms for ˜ φ vanish. Using the equation of motion for ˜ φ , the Goldstone boson masses vanish while the effective masses of the rest become Here, we note that all the masses are of the same form as in the SM without non-minimal coupling but with the Higgs being replaced by ˜ φ ( χ ). After plugging the above masses in eq. (14), we obtain the one-loop corrected inflaton potential renormalized at µ = ˜ φ ( χ ) with eq. (23) as follows, where the effective vacuum energy during inflation ˆ V 0 is given by where all the running couplings are evaluated at µ = ˜ φ . Consequently, the effective inflaton potential is determined by the Higgs quartic coupling λ H , the extra quartic couplings, λ σ , λ Hσ , the SM gauge couplings and the top Yukawa coupling.", "pages": [ 6, 7 ] }, { "title": "3.2 Sigma-field coupling and vacuum stability", "content": "As discussed before, positivity of the tree-level inflaton potential requires λ H > λ 2 Hσ /λ σ , which is the same as the vacuum stability condition in the SM, λ eff > 0, from eq. (8). However, the effect of the running quartic couplings and the threshold corrections to the effective vacuum energy may make the vacuum energy larger. Thus, the vacuum stability can be guaranteed during inflation, once ensured at the sigma mass scale. From appendix A, the RG equation for λ eff ≡ λ H -λ 2 Hσ /λ σ with δλ ≡ λ 2 Hσ /λ σ is Thus, due to the positive contribution coming from the sigma-field couplings in the RG equation, the vacuum instability scale gets higher. On the left of Fig. 1, we depict the running of the effective Higgs quartic coupling above the sigma field threshold at M σ = 10 13 GeV, depending on the tree-level shift in the effective Higgs quartic coupling. We also show on the right of Fig. 1 that the quartic couplings producing a sizable shift δ = 0 . 3 in the effective Higgs quartic coupling remain perturbative all the way to the Planck scale.", "pages": [ 7 ] }, { "title": "4 Corrections to the inflationary observables", "content": "In the one-loop improved inflaton potential, the effective vacuum energy has threshold corrections coming from the heavy modes of non-inflaton fields coupled to the inflaton field. Since the threshold corrections depend on the running couplings, in turn, the inflaton field value, they can give extra contributions to the spectral index. By using the RG equations given in appendix A, we consider the threshold corrections to the spectral index and other inflationary observables in this section. Furthermore, we discuss the reheating temperature and the predicted number of efoldings in the model, depending on the mixing coupling between the Higgs and sigma fields.", "pages": [ 8 ] }, { "title": "4.1 Spectral index and tensor-to-scalar ratio", "content": "The slow-roll parameters are The slow-roll conditions are glyph[epsilon1] glyph[lessmuch] 1 and | η | glyph[lessmuch] 1. The first condition('slowly varying') corresponds to making the Hubble parameter during inflation proximate to constant while the second condition comes from the slowly varying condition plus 3 H ˙ χ = -V ' ('slowroll approximations'). Then, the spectral index and the tensor-to-scalar ratio are then evaluated at the horizon exit, according to the following, When we compute the slow-roll parameters, we take the tree-level vacuum energy to be dominant over the Coleman-Weinberg correction. Then, the inflaton potential is given by the tree-level one with the running couplings: where the couplings depend on the canonical inflaton field χ through the renormalization condition, µ = ˜ φ ( χ ) with eq. (23). In Fig. 2, we show the RG scale dependence of the inflaton vacuum energy. We note that from the renormalization condition µ = ˜ φ with eq. (23), the RG scale during inflation is of order | λ Hσ | M P / ( λ H √ ξ ), which is about 10 16 GeV for λ H ∼ λ Hσ from the COBE normalization as will be discussed later in this section. Then, the field derivatives of the potential are where use is made of ∂ ln µ/∂χ = 1 √ 6 e -2 χ/ √ 6 / (1 -e -2 χ/ √ 6 ) for µ = ˜ φ in the chain rule for ∂ ˆ V 0 ∂χ = ∂ ˆ V 0 ∂ ln µ ∂ ln µ ∂χ . Therefore, we get the slow-roll parameters as where use is made of the RG equations in appendix A to get So, the loop corrections to the slow-roll parameters are determined by δλ , λ H and the gauge and top Yukawa couplings. When we make use of the effective Higgs quartic coupling at the matching scale by λ eff = λ H -δλ , there is only one unknown parameter, λ H or δλ . Here, we note that the running effect of the non-minimal coupling ξ is suppressed by 1 /ξ . We also note that the second derivative of the vacuum energy is given by where β i /λ i glyph[lessmuch] 1 is assumed in the last inequality. Thus, we ignore the second derivative terms with respect to ln µ in the slow-roll parameters. We can compute the total number of e-foldings as where We note that the χ dependence of A in the integrand can be ignored, because δλ i = β i δ ln µ ∼ β i 1 √ 6 e -2 χ/ √ 6 δχ is exponentially suppressed during inflation. Here, e 2 χ f / √ 6 = 2 | A | / √ 3 corresponds to the field value at which glyph[epsilon1] = 1 when the slow-roll dynamics ends. So, we need | A | > √ 3 2 from e 2 χ f / √ 6 > 1. On the other hand, from eq. (40), we also get e 2 χ i / √ 6 = 2 3 | A | (2 N + √ 3) at the horizon exit. Since we can determine λ H in terms of λ eff and δλ at the sigma threshold and the top pole mass determines y t with some uncertainties, the loop corrections eventually depend on the unknown δλ for a given λ eff at the sigma field threshold. The same parameter δλ controls the vacuum stability above the sigma field threshold as discussed in section 3.2. Then, we can rewrite the slow-roll parameters in terms of the number of e-foldings, Therefore, the spectral index becomes For instance, for the tree-level potential with δλ = 0, we obtain n s = 0 . 9647 -0 . 9670 for N = 59 -62. For comparison, the measured spectral index from nine-year WMAP with eCMB, BAO and H 0 is given by n s = 0 . 9608 ± 0 . 0080 [2]. Furthermore, the recent Planck data combined with the WMAP large-angle polarization, indicates a more precise value of the spectral index as n s = 0 . 9603 ± 0 . 0073 [4]. Thus, the spectral index obtained in the model is consistent with Planck data, within 1 σ . On the left of Fig. 3, we show the dependence of the spectral index on the number of efoldings and the loop corrections as a function of the tree-level shift in the Higgs quartic coupling, δλ , and find that the obtained spectral index is consistent with the current observation within 1 σ . In the unitarized Higgs inflation, the reheating temperature is sensitive to the Higgs component of the inflaton, which is determined by the mixing coupling λ Hσ between the Higgs and sigma fields. The spectral index varies by 0 . 002, depending on the number of efoldings, N = 59 -62, in our model, as will be discussed in the next section. We also note that for the number of efoldings being fixed, the loop corrections make | ∆ n s | less than 0 . 001, so the effect of the loop corrections is much smaller than the one of the number of efoldings. However, the expected sensitivity at PLANCK in measuring the spectral index is ∆ n s = ± 0 . 004 [3] so it might not be possible to measure the reheating temperature or the loop corrections precisely by PLANCK yet. On the other hand, for N = 62(59), the tensor-to-scalar ratio is given by Thus, the result is consistent with the current limit, r < 0 . 13 at 95% C. L., from WMAP + eCMB + BAO + H 0 [2] and r < 0 . 11 at 95% C. L. from Planck data [4]. We note that the tensor-to-scalar ratio is insensitive to the loop corrections as the glyph[epsilon1] slow-roll parameter runs little as seen from eq. (42). Furthermore, the COBE normalization of the power spectrum constrains the inflation parameters from the following quantity evaluated at horizon exit, From ∆ 2 R = (2 . 464 ± 0 . 072) × 10 -9 [2], for the number of efoldings N = 60, we obtain the following constraint, The running effect of the quartic couplings can be traded off for a different value of ξ . As shown in Fig. 1, the effective Higgs quartic couplings can run to a larger value by order of magnitude. Then, the necessary non-minimal coupling for the COBE normalization becomes larger than what we would have obtained from the tree-level value of λ eff . For instance, for λ eff = 0 . 01 at the sigma field threshold, the needed non-minimal coupling would be of order 10 3 but the RG running makes λ eff order of 0 . 1 so that ξ ∼ 10 4 . In Fig. 4, we show the ratio of loop-level to tree-level values of ξ as a function of the RG scale. From λ H = λ eff + λ 2 Hσ /λ σ , we note that for λ 2 Hσ glyph[greatermuch] λ eff λ σ , the COBE normalization condition (47) with eq. (7) becomes the same as in the original Higgs inflation, ξ eff / √ λ eff glyph[similarequal] 47000. This is the case when the vacuum instability scale in the SM is just above the sigma scalar threshold.", "pages": [ 8, 9, 10, 11, 12, 13 ] }, { "title": "4.2 Running of spectral index", "content": "The running of the spectral index is given by with Using and eqs. (42), (43), and ignoring the higher derivative terms with respect to ln µ , we obtain Consequently, for the tree-level potential with δλ = 0, we obtain dn s /d ln k = 4 . 1 × 10 -4 for N = 60, which is consistent with the Planck+WMAP constraints, dn s /d ln k = -0 . 013 ± 0 . 009 at 68% C. L. within 1 . 5 σ [4]. On the right of Fig. 3, we show the dependence of the running of the spectral index on the number of efoldings and the loop corrections. We find that the loop corrections contribute to the running by ± 2 × 10 -5 , which is too small to be observed.", "pages": [ 13 ] }, { "title": "5 Reheating temperature and number of efoldings", "content": "From eq. (11), we rewrite the scalar potential in Einstein frame as Here, we ignored the weak-scale Higgs mass term. At the end of inflation, e -2 χ f / √ 6 M P = √ 3 / 2 and ˜ φ 2 f = ˜ φ 2 0 (1 -√ 3 / 2) with ˜ φ 2 0 = -λ Hσ λ H ω 2 . Therefore, both ˜ φ and χ carry the potential energies of order ω 4 for the quartic couplings of order unity at the onset of reheating. By expanding the potential around χ = 0, we get the reheating dynamics to be a hybrid type which has both quadratic and quartic potentials with the mixing term, as follows, The dynamics of the reheating process is much involved, so we just consider how the reheating temperature depends on the mixing coupling λ Hσ between the Higgs and sigma fields, without going into the details. After the biggest cosmological scale, k -1 = H -1 0 = 3000 h -1 Mpc, leaves the horizon, the number of efoldings is where use is made of a he H he = a 0 H 0 in the second equality. Assuming that slow-roll inflation is followed promptly by matter domination and consequently by radiation domination, we have √ Here, g ∗ = 106 . 75 is the effective number of degrees of freedom in the SM and the inflaton vacuum energy at the end of inflation is given by V end glyph[similarequal] (4 . 8 × 10 15 GeV) 4 from the COBE normalization. When ˜ φ f glyph[lessmuch] ω in the end of inflation, i.e. | λ Hσ | glyph[lessmuch] λ H , the inflaton is just the sigma field with the inflaton mass being given by m χ = √ ( λ σ / 3) M P /ξ . In this case, from eqs. (10) and (53), the relevant interaction terms for reheating in the action in Einstein frame are where where f i are the canonically normalized SM fermions. Then, the inflaton decays into a pair of ˜ φ 's dominantly by both the gravitational kinetic interaction and the Higgs-portal term, with the decay rate, So, for 2 | λ Hσ | √ 3 /λ σ glyph[lessmuch] m χ /M P glyph[similarequal] 1 . 2 × 10 -5 , where use is made of m χ = 2 . 9 × 10 13 GeV from the COBE normalization, the inflaton decay is dominated by the gravitational interaction. Otherwise, it is the sigma field coupling that dominantly determines the inflaton decay rate. From Γ χ = H rh = √ ρ rh / ( √ 3 M P ) with eq. (57), the reheating temperature is given by where the number of degrees of freedom in the Higgs doublet is taken into account by N s = 4. For instance, for λ σ ∼ λ H ∼ 1 and | λ Hσ | ∼ 0 . 01, we would get the reheating temperature as T rh ∼ 10 12 GeV. For a sizable | λ Hσ | ∼ λ H , we get ˜ φ f ∼ ω , so the reheating process is similar to the SM Higgs inflation, because the Higgs carries a large fraction of the inflaton energy in the end of inflation and reheats the SM particles by the Higgs interactions. In Fig. 5, we show the scalar fields on the left and the scalar potential on the right as a function of time during the reheating. We note that there are multiple zero crossings of the Higgs within a single oscillation of the χ field, because m ˜ φ glyph[greatermuch] m χ even during the reheating. In this case, the SM particles can be also produced non-perturbatively in the preheating stage [22] due to the parametric resonance in the presence of the Higgs interactions so the reheating temperature can be higher than the one obtained from the perturbative decay as in the original Higgs inflation. The details of preheating in our model is beyond the scope of this work, so instead we quote T rh = (3 -15) × 10 13 GeV in Higgs inflation [21] as the maximum reheating temperature possible. Consequently, depending on the size of λ Hσ , the reheating temperature T rh varies between 4 . 4 × 10 9 GeV and 1 . 5 × 10 14 GeV. Therefore, using eq. (55), we obtain the number of efoldings in the following range, We have used the above result as the representative values for the number of efoldings in the analysis of the previous sections.", "pages": [ 14, 15, 16 ] }, { "title": "6 Conclusion", "content": "We have revisited the unitarized Higgs inflation with a real scalar singlet of sigma-model type, from the perspective of the loop corrections. As the relevant energy scales including the inflation are much below the unitarity cutoff of the model, the Planck scale, we can use the perturbative expansion to estimate the loop corrections to the quartic couplings of the model and in turn calculate the inflationary observables. Since the mixing coupling between the sigma and Higgs fields is required to reproduce the effective large non-minimal coupling for the Higgs doublet below the sigma scalar threshold, the model has been regarded as a UV completion of the Higgs inflation in a sense of the effective action. The extra coupling of the new dynamical scalar contributes to the RG running of the Higgs quartic coupling, improving the vacuum stability. Furthermore, the same sigma coupling determines the RG-improved inflaton potential and controls the loop corrections to the spectral index, etc, being consistent with the nine-year WMAP data. There is an uncertainty in reheating temperature or the number of efoldings in the model, because the Higgs component of the inflaton varies depending on the sigma coupling. We conclude that the loop corrections to the spectral index in the model are under control within the uncertainties in the reheating temperature.", "pages": [ 16 ] }, { "title": "Acknowledgments", "content": "The author thanks Fedor Bezrukov for the early discussion on the loop corrections and the reheating dynamics in Higgs inflation and Wan-Il Park for useful comments.", "pages": [ 16 ] }, { "title": "Appendix A: Renormalization group equations", "content": "We take into account the effects of a non-minimal coupling to gravity of the sigma and Higgs fields through suppression factors in the RGE [15]. The one-loop RG evolution of the scalar quartic couplings above the sigma-field threshold is governed by and The RG equations for non-minimal couplings are The suppression factors, c σ and c h , are given in terms of the Weyl rescaling factor Ω 2 = ξσ 2 /M 2 P as c σ = Ω -2 ( ∂χ/∂σ ) -2 = 1 / (6 ξ ) glyph[lessmuch] 1 and c h = Ω -2 ( ∂φ/∂h ) -2 = 1, where χ and φ are canonical fields. Therefore, the running of λ H is SM-like, as loops containing the sigma scalar are suppressed for c σ glyph[lessmuch] 1 and Higgs loops are the same as in the SM. The suppressed sigma-field loops can help to keep the non-minimal coupling ζ of the Higgs small under the RG evolution. Using the RG equations (A.1)-(A.3), we also get the RG equation for λ eff ≡ λ H -λ 2 Hσ /λ σ , with δλ ≡ λ 2 Hσ /λ σ , as The two-loop RG equations for the gauge and Yukawa couplings are", "pages": [ 16, 17 ] } ]

[ { "title": "Observing Higgs boson production through its decay into γ -rays: A messenger for Dark Matter candidates", "content": "Nicol'as Bernal, 1 C'eline Bœhm, 2, 3 Sergio Palomares-Ruiz, 4,5 Joseph Silk, 6 and Takashi Toma 2 1 Bethe Center for Theoretical Physics and Physikalisches Institut, Universitat Bonn, Nußallee 12, D-53115 Bonn, Germany 2 Institute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE, UK 3 LAPTH, U. de Savoie, CNRS, BP 110, 74941 Annecy-Le-Vieux, France. 4 Centro de F'ısica Te'orica de Part'ıculas (CFTP), Institut o Superior T'ecnico, Universidade T'ecnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal 5 Instituto de F'ısica Corpuscular (IFIC), CSIC-Universitat de Val'encia, Apartado de Correos 22085, E-46071 Valencia, Spain 6 UMR7095 - Institut d'Astrophysique de Paris - 98 bis boulevard Arago - 75014 Paris (Dated: today) In this Letter, we study the γ -ray signatures subsequent to the production of a Higgs boson in space by dark matter annihilations. We investigate the cases where the Higgs boson is produced at rest or slightly boosted and show that such configurations can produce characteristic bumps in the γ -ray data. These results are relevant in the case of the Standard Model-like Higgs boson provided that the dark matter mass is about 63 GeV, 109 GeV or 126 GeV, but can be generalised to any other Higgs boson masses. Here, we point out that it may be worth looking for a 63 GeV line since it could be the signature of the decay of a Standard Model-like Higgs boson produced in space, as in the case of a di-Higgs final state if m χ /similarequal 126 GeV. We show that one can set generic constraints on the Higgs boson production rates using its decay properties. In particular, using the Fermi-LAT data from the galactic center, we find that the dark matter annihilation cross section into γ + a Standard Model-like Higgs boson produced at rest or near rest cannot exceed 〈 σ v 〉 ∼ a few10 -25 cm 3 / s or 〈 σ v 〉 ∼ a few10 -27 cm 3 / s respectively, providing us with information on the Higgs coupling to the dark matter particle. We conclude that Higgs bosons can indeed be used as messengers to explore the dark matter mass range.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "On-going searches at the LHC have been rewarded by one of the greatest particle physics discoveries that could possibly be made in such a machine, namely the finding of a seemingly new fundamental scalar or pseudo-scalar particle [1-4]. At present measurements of the couplings of this new boson to Standard Model (SM) particles along with the absence of charged particles tend to suggest that this is a SM Higgs boson. However this remains to be proven. While such a discovery certainly validates our understanding of the origin of particle masses, it also constrains the types of theories that could be proposed to go beyond the Standard Model (BSM). For instance, some of the simplest Supersymmetric (SUSY) models which have been proposed in the literature tend to predict a mass for the Higgs boson that is smaller than the measured value mH /similarequal 125 -126 GeV [5] and are therefore likely to be ruled out. Moreover, the good agreement between the measured branching ratios and those expected in the SM (apart perhaps for the two-photon channel) enables one to set a stringent constraint on the Higgs invisible decay width and to constrain theories in which the Higgs is strongly coupled to the dark matter (DM) candidate ( χ ) [6]. Nevertheless, the information collected so far at the LHC is not sufficient to exclude the possibility that this new boson has a BSM origin. In fact, some non-minimalistic SUSY extensions were shown to predict a 'light' Higgs boson with essentially indistinguishable characteristics from those expected within the SM (the remainder of the spectrum in this model being typically beyond the scale accessible at LHC) [7]. Hence, at present the origin of this new boson remains an open question and one needs more clues to determine whether this Higgs boson candidate has a SM origin or not. Examining its 'dark' coupling using other tools than the LHC could be one way to proceed. In this Letter, we propose to exploit this discovery together with recent astrophysical data to constrain the Higgs boson production cross section in some specific annihilating DM scenarios. We shall focus on the SM-like boson with a mass of 126 GeV, but our analysis can be extended to any Higgs boson candidate. Now that a SM-like Higgs (or a new) boson has been discovered and its main characteristics are well determined, one can make use of its decay properties (and in particular the photon spectrum subsequent to the Higgs boson decay) to determine whether it has been produced by DM in our galactic halo, for instance. Observing the decay of a Higgs boson produced at rest (or slightly boosted) in space would indeed be suggestive of new physics and provide a new window on long-lived neutral particles. The scheme that we have in mind is the production of one or two Higgs bosons by DM annihilations, although an analogous exercise can be done for decaying DM, with similar qualitative arguments for DM masses a factor of 2 higher. Once a Higgs boson is produced, it is expected to decay immediately, thereby generating γ -rays. If the associated flux is large enough, this could lead to anomalous features in the γ -ray spectrum (in particular, an excess of photons at some specific energies with respect to the background expectations) which can be searched for. Note that in what follows we will only focus on the γ -ray emission from the galactic centre, but our analysis could be extended to other regions of the Milky Way as well as the emission arising from DM annihilations in dwarf galaxies. The γ -ray signature associated with a SM-like Higgs boson decay in our galaxy is expected to be a smooth continuum spectrum due to the Higgs decay into SM particles [8]. However, here we show that if the Higgs boson is produced at rest, its decay into two gamma ( H → γγ ) could lead to a potentially detectable monochromatic line at E γ ∼ 63 GeV in addition to the continuum, even though the associated branching ratio is very suppressed with respect to other channels. The corresponding signal in an experiment such as FermiLAT should be a bump around E γ = mH / 2 (that is E γ ∼ 63 GeV for a SM-like Higgs boson) and possibly a broad γ -ray excess at lower energies, depending on the ratio between the line and the continuum. Here we show that it is worth looking for such a line in γ -ray data, as it could be a mean to probe specific annihilating DM scenarios. In particular, in the case of a SM-like Higgs boson, one could probe DM masses of about m χ /similarequal 63 GeV (for χχ → H γ ), m χ /similarequal 109 GeV (for χχ → HZ ) or m χ /similarequal 126 GeV (for χχ → HH ) 1 . In Section II we discuss the production of the SM-like Higgs boson at rest in DM annihilations. After reviewing the possible DM annihilation processes which can create one (or two) Higgs boson(s) in the final state, we study the detectability of the signature of a Higgs boson decay with the Large Area Telescope (Fermi-LAT) on board the Fermimission and discuss the implications for DM scenarios. We also comment on the slightly boosted Higgs boson in Section III and conclude in Section IV.", "pages": [ 1, 2 ] }, { "title": "II. HIGGS BOSON PRODUCED AT REST BY DM ANNIHILATIONS", "content": "In order to produce a Higgs boson in space and at rest, the DM mass and spin must have specific values. Quantitative statements depend on how many Higgs bosons are produced in the final state. In the case of DM annihilations into two SM-like Higgs bosons, the DM mass must be about m χ /similarequal mH /similarequal 126 GeV (regardless of its spin). If on the contrary, DM annihilations produce only one SM-like Higgs boson plus a photon in the final state, the DM mass must be about m χ /similarequal mH / 2 /similarequal 63 GeV (assuming that it has a spin-1/2 or spin-1) while it should be about 109 GeV if it produces a Higgs boson plus a Z boson in the final state (assuming a spin-0,1/2 or spin-1). In what follows, we will focus on these three specific cases, as they lead to the production of SM-like Higgs bosons at rest but, of course, an analogous analysis can be done for heavier (presumably BSM) Higgs bosons. We now point out some general Higgs boson production mechanisms which could prevail for DM candidates with a mass m χ /similarequal 63 GeV, 109 GeV and 126 GeV. Examples of relevant Feynman diagrams are given in Fig. 1.", "pages": [ 2 ] }, { "title": "A. Production mechanisms for m χ /similarequal 126 GeV", "content": "DM candidates with a mass slightly greater than 126 GeV can produce two Higgs bosons at rest or near rest in the final state either through box diagrams or, if DM is directly coupled to the Higgs, through tree-level process (see Fig. 1). In a SUSY framework for example, two Higgs bosons can be produced via box diagrams involving, e.g., charginos and W boson or quarks and squarks from the third generation [10]. Disregarding for the moment the possible velocity-squared dependence which arises due to the Majorana nature of the neutralino, these diagrams are expected to be relatively suppressed with respect to other annihilation channels which occur at tree-level (such as for example neutralino annihilations into b ¯ b or W + W -via a t-channel sbottom or chargino exchange respectively). However they could still be sizable if the Higgs boson has large couplings to the particles in the box or if there is a large mass degeneracy between the neutralino and the chargino ( χ ± ) for example (if we consider the χ ± -W ∓ box diagram [10-13]). Alternatively, the DM could also pair annihilate into two Higgs bosons through a pseudoscalar Higgs boson s-channel exchange. If, in particular, the mass of the pseudo-scalar is about twice the DM mass, one expects a large resonant interaction and potentially a large diHiggs boson production. In both cases however, one also expects a large DM pair annihilation rate into two γγ , ZZ , Z γ , H γ , HZ leading to extra γ -ray lines. In many scenarios, these process are related, thus giving interesting constraints on the model. However, large branching ratios into γγ , ZZ , Z γ , H γ , HZ could be detrimental to the searches for a 63 GeV line. For example, in 'conventional' BSM scenarios such as SUSY, the di-photon final state is supposed to be slightly larger than the di-Higgs production (notably because it is not phase-space suppressed). Since the di-photon final state relies on charged loop diagrams, one therefore expects a large production of charged particles from the DM pair annihilations at tree-level which poses a problem for the detectability of the 63 GeV line. Indeed, if the contribution from annihilations into b -quarks is significant, it is likely that the line at 63 GeV would be totally swamped by the continuum γ -ray emission resulting from the b hadronization, fragmentation and subsequent decay, with an endpoint energy equal to the DM mass, m χ /similarequal 126 GeV. There are several ways out, nevertheless. For example, if the charged particles which contribute to the direct photon emission (loop-suppressed) are all heavier than the DM [14], the DM pair annihilation into such particles is not kinematically allowed, thus enabling the di-Higgs final state to be visible. In SUSY, this means that one would have to suppress the t-channel sbottom exchange diagram and perhaps introduce a singlet-like heavy Higgs boson mostly coupled to very heavy charged particles [14]. Alternatively, there could be scenarios where the di-photon and di-Higgs final states are produced by enhanced box diagrams but in which the sbottom exchanges are very suppressed so that the production of b -quarks at treelevel is suppressed. In scenarios with a SM-like Higgs boson and no extra pseudo-scalar boson, the tree-level production of b -quarks is expected to be velocity-suppressed. If potential loop/box process, susceptible to imply b -quarks at tree-level, are also suppressed by the introduction of very heavier mediators, the detectability of the 63 GeV line originating from enhanced box diagrams could be significant. We also note that in models such as the NMSSM where one can have both a very heavy ( A ) and very light ( a ) pseudoscalar Higgs bosons, the requirement of having a resonant A exchange if m χ /similarequal 126 GeV (i.e., mA = 2 m χ /similarequal 252 GeV) implies that one could also produce at tree-level the Aa final state, with A produced at rest. The decay of the A into two photons could then generate a line at 126 GeV which could be confused with the direct (resonant) DM pair annihilations into two photons. The dominance of one process over the other would mostly depend on the mass difference | mA -2 m χ | and the strength of the coupling of the neutralino to the Higgs boson, which itself is constrained by the width of the invisible Higgs decay channel [15-17]. Such an ambiguity in the origin of a possible line at E /similarequal 126 GeV in this framework could be of interest in the context of the 130 GeV and 111 GeV bumps observed in the Fermi-LAT data [18-25]. For candidates with this mass ( m χ /similarequal 126 GeV), the condition of predicting a 63 GeV line from a SM-like Higgs boson produced at rest guaranties a final state with two SM-like Higgs bosons. However should such a line be seen, one would have to disentangle it from the direct annihilations of DM particles with a mass of m χ /similarequal 63 GeV into two photons. Also it may be challenging to disentangle the di-Higgs boson final state from the H γ final state. These aspects will be discussed in the next section. Note that all the final states mentioned above have already been considered in detail in the literature for generic DM masses (see, e.g., Refs. [8, 10, 11]). However, to our knowledge, the γ -ray signature expected from a Higgs boson decay produced by a ∼ 126 GeV DM candidate has not been studied explicitly 2 . Many authors have exploited the presence of a single photon in DM pair annihilation final states as a γ -ray signature [8, 12, 27-33]. However, the possibility of these prompt photon lines being accompanied by additional lines due to Higgs production at rest has not been pointed out. To our knowledge, the fact that the DM pair annihilation into two photons could be simply confused with a Higgs boson (not necessarily SM-like) production, when mH /similarequal 2 m χ , has not been mentioned in the literature yet.", "pages": [ 2, 3 ] }, { "title": "B. Production mechanisms for m χ /similarequal 63 GeV", "content": "Due to their mass, candidates with m χ /similarequal 63 GeV can only produce one SM-like Higgs boson at rest in the final state. The DM spin is then fixed by the nature of the second particle in the final state. The exact final state can also enable one to determine the Higgs boson production mechanism. For example, the H γ final state implies that the Higgs boson production must be a loop-suppressed process since the DM is assumed to be neutral and cannot produce a photon in the final state without coupling to charged particles (unless one considers 'dipole' DM [34]). Usually one exploits the presence of a single photon in the final state to look for such a process (see, e.g., Ref. [8]). However, the corresponding direct γ -ray line would appear at very low energy, namely E γ = m χ ( 1 -m 2 H / ( 4 m 2 χ )) /lessmuch 1 GeV, to which Fermi-LAT might still be sensitive. Hence the only line that is experimentally accessible comes from the Higgs decay at 63 GeV. Nevertheless, observing such a line may not unambiguously point towards the production of a Higgs boson: DM pair annihilations into γγ could also produce a monochromatic line at the same energy as the Higgs boson decay if the DM mass is about 63 GeV. Hence, there could be some confusion about the origin of the line, even though such a detection would definitely point towards new physics. In some models, this possible confusion could be solved by simply comparing the expected cross sections in different channels. For example, in scenarios with photon mixing [35], the Zd s-channel exchange into γ H would be larger than the γγ final state, so a signal at 63 GeV could be expected. However there could be tricky situations. For example, if m χ /similarequal 63 GeV, both the χχ → γγ and χχ → H γ process are expected to be very large if they are realized through a Higgs portal, i.e., χχ → H → γγ , H γ . The kinematic condition to see a line at m χ /similarequal mH / 2 /similarequal 63 GeV indeed immediately implies that the H exchange is resonant. Hence, both final states should be copiously produced. If H is the SM Higgs boson, the magnitude of χχ → γγ versus χχ → H γ is fixed by the ratio of the t -t -γ versus the t -t -H couplings and the phase space factor. Thus, for a SM-like Higgs boson produced very close to rest, the phase space factor eventually suppresses a bit the H γ final state. Yet, ultimately one should detect the sum of the two contributions. Note that the importance of the χχ → γγ and χχ → H γ processes through the SM-like Higgs portal ultimately depends on the mass difference ∆ = 2 m χ -mH , as well as the χ -χ -H coupling. The latter can be tuned (in fact reduced) to compensate for the smallness of ∆ , in order to avoid too large a resonant annihilation effect, although it cannot be arbitrarily large. The maximum value of the χχ → H → γγ cross section is actually set indirectly by the ATLAS and CMS experiments. The associated cross section is maximal when both ∆ becomes smaller than the Higgs boson decay width ( Γ H ) and the χ -χ -H coupling is maximal. Both are being measured at LHC through the Higgs visible and invisible decay width [36]. A too large χ -χ -H coupling would make the Higgs decay invisible and be in conflict with SM predictions. The above discussion assumes that the DM pair annihilation through the Higgs portal cross section is not velocitydependent. However, if they turn out to be suppressed and box diagrams are more important, models with kinetic mixing might again lead to a larger value of the cross section for the H γ final state (with respect to the γγ final state). Would such a line be seen, it would remain to be determined whether it originates from a SM-like Higgs boson decay into two photons or a model of the type discussed above. However, when m χ /similarequal 63 GeV, the DM pair annihilations into any other channel would produce a γ -ray spectrum with energies E γ < m χ . Hence the line at ∼ 63 GeV would not be buried under the continuum spectrum unlike what could occur for m χ > 63 GeV, as discussed in the previous subsection.", "pages": [ 3, 4 ] }, { "title": "C. Production mechanisms for m χ /similarequal 109 GeV", "content": "When the DM mass is about 109 GeV, the χχ → HZ process can occur (for both bosonic and fermionic DM) via a tchannel DM exchange diagram (if DM can couple directly to the Higgs) or a s-channel Z exchange diagram. This process can also occur through box diagrams. For such a value of the DM mass, both the Z and SM-like H bosons are produced close to rest and should lead to distinctive signatures. In addition to the 63 GeV line from the SM-like Higgs boson decay, there could be a line at ∼ 109 GeV coming from the DM annihilations into two photons. Associated with this case, there could also be a line at ∼ 72 GeV from the direct photon in the H γ final state if this channel is not suppressed. The dominance of one over the other one depends again on the couplings and exact process, while their visibility essentially depends on the background at these energies. Note that γ -ray line at ∼ 109 GeV from direct annihilation into two photons could be consistent with the possible line detected at 111 GeV [20, 21] and could be used to constrain the DM interactions.", "pages": [ 4 ] }, { "title": "D. Additional remarks", "content": "The results displayed in the next section hold independently of whether the new particle discovered at CERN is the Higgs boson or not. Since the observed branching ratios are compatible with the SM Higgs predictions (within 2 σ ), our conclusion regarding whether one can see a monochromatic line at ∼ 63 GeV should remain identical. Some of the Higgs production mechanisms that we discuss in this paper may be associated with a large spin-independent elastic scattering cross section with a nucleon and could be ruled out by DM direct detection experiments. In particular if the DM has a mass in the GeV-TeV range, its interactions could be severely constrained by the XENON100 exper- iment [37, 38]. Since this requires to specify a model and we intend to set model-independent constraints, we assume that the underlying DM particle model is compatible with the results from the latest direct detection experiments. However, for concrete models such a compatibility has to be checked.", "pages": [ 4 ] }, { "title": "E. Detectability of the line emission and continuum", "content": "The γ -ray emission subsequent to Higgs production typically occurs from the Higgs boson decay into, e.g., γγ , b ¯ b , etc. Since all the channels have very well-known branching ratios, the γ -ray flux can be predicted quite accurately (albeit astrophysical uncertainties). Predictions depend on the photon energy spectrum dN γ / dE γ associated with the Higgs boson decay. Typically, for a Higgs boson of about 126 GeV produced at rest, one expects a smooth spectrum (due to dominant decay into b ¯ b ) plus a monochromatic line due to H → γγ [39-41]. In the SM, (for mH = 126 GeV) the Higgs boson decay into γγ is suppressed by a factor of ∼ 4 × 10 -3 with respect to the b ¯ b final state [41], so one may think that the γ -ray line is hidden by the continuum. However, channels such as b ¯ b emit photons at lower energies than E = mH / 2 (owing to final state radiation, hadronization, fragmentation and decay). As a result, even though the flux associated with the monochromatic line is meant to be suppressed, in principle it could be distinguishable from the continuum emission. In order to compute the dN γ / dE γ spectrum, we use PYTHIA 6.4 [42], where we set the branching ratio for H → γγ to 2 . 28 × 10 -3 [41] 3 . The result is displayed in Fig. 2. Clearly, the monochromatic line appears to be distinguishable from the smooth spectrum, even though it is suppressed. Now, we estimate the associated flux from DM annihilations (an analogous analysis could be performed for decaying DM) around the galactic center and compare it to the current Fermi-LAT data. We will assume a generic DM candidate, with a thermal average of the annihilation cross section times the relative velocity of 〈 σ v 〉 ≡ 〈 σ v DMDM → H +( γ , Z , H ) 〉 = 3 × 10 -26 cm 3 / s, where in each case we consider that the only annihilation channel is HH , H γ or HZ . The differential flux of prompt γ -rays generated from DM annihilations in the smooth DM halo from a direction within a solid angle ∆Ω is given by [28] where dN γ / dE γ is the differential γ -ray yield, η is a symmetry factor which for Majorana DM is equal to 1 and 1/2 if DM is not a self-conjugate particle, ρ ( r ) is the DM density profile and r is the distance from the galactic center. The spatial integration of the square of the DM density profile is performed along the line of sight within the solid angle of observation ∆Ω . More precisely, r = √ R 2 /circledot -2 sR /circledot cos ψ + s 2 , and the upper limit of integration is s max = √ ( R 2 MW -sin 2 ψ R 2 /circledot )+ R /circledot cos ψ , where ψ is the angle between the direction of the galactic center and that of observation and R /circledot is the distance from the Sun to the galactic center. Being the contributions at large scales negligible, the choice of the size of the Milky Way halo, R MW is not crucial. Thus, the flux of DM annihilations can be written as with the dimensionless quantity J ( ψ ) defined as where for the distance from the Sun to the galactic center and for the local DM density we use R /circledot = 8 . 25 kpc and ρ /circledot = 0 . 386 GeV/cm 3 , respectively [43]. Although for some DM density profiles, the integration of J ( ψ ) in the solid angle of observation can be done analytically [44], here we consider an Einasto profile [45], for which there is no analytical solution, and compute it numerically. This density profile is parametrized as where rs = 20 kpc is a characteristic length. Following Refs. [51-55], we consider a 20 o × 20 o squared region centred on the galactic center, for which ∫ J ( ψ ) d Ω = 20 . 5sr. In Fig. 3 we compare the expected flux from this region and compare it with the Fermi-LAT data. To obtain the measured flux, we take the Fermi -LAT data obtained from August 4, 2008 to October 1, 2012. We extract the data from the Fermi Science Support Center archive [56] and select only events classified as CLEAN . We use a zenith angle cut of 105 · to avoid contamination by the Earth's albedo and the instrument response function P7CLEAN V6 . In the upper panel of Fig. 3 we show the γ -ray spectra for three different annihilation channels, H γ (upper red line), HZ (black dotted line) and HH (orange line), in which the Higgs is produced very close to rest. The DM mass for each case is m χ = 63 GeV, 109 GeV and 126 GeV, respectively. As can be seen from the plot, the fluxes for the three cases are very similar, but the H γ final state is slightly more visible than the two others 4 , mainly because of the lower value of the DM mass in this case. Since the flux scales linearly with the cross section, these lines emerge from the γ -ray background when the associated production cross section is greater than 〈 σ v 〉 ∼ 2 . 5 ( 5 ) × 10 -25 cm 3 / s for H γ ( HH ), thereby ruling out a Higgs boson production cross section larger than this value. This can be seen from the lower panel of Fig. 3, where we show the value of 〈 σ v 〉 for which the signal would be equal to the observed background. Interestingly enough, for the case of DM annihilations into H γ or HH , producing Higgs at rest, the γ -ray line from the very suppressed H → γγ channel (see Fig. 2), is expected to provide a more restrictive limit than the dominant continuum. The limits that we sketch are very conservative as they assume no background from astrophysical sources. A dedicated search for Higgs boson decay lines would require to account for the background modeling and to optimize the detection window [18-25]. However here we simply want to illustrate the potential detectability of these lines. Note that our limits are in agreement with the detailed Fermi-LAT searches of γ -ray lines [57]. These were obtained by the Fermi-LAT analysis for m χ /similarequal 63 GeV and χχ → γγ can be directly compared to the ones presented here for χχ → H γ and m χ /similarequal 63 GeV. While the Fermi-LAT limit is 〈 σ v 〉 ∼ 3 × 10 -28 cm 3 / s (cf. Fig. 15 in Ref. [57]), we obtain 〈 σ v 〉 ∼ 2 . 5 × 10 -25 cm 3 / s, the ∼ 10 -3 difference coming from the branching ratio for H → γγ . Similarly, for the case of χχ → HH and m χ /similarequal 126 GeV, the limit obtained from the γ -ray line from Higgs decay is just a factor of 2 weaker than that for χχ → H γ and m χ /similarequal 63 GeV (explained as a factor of 2 in favour of HH due to having two Higgs bosons and a factor of 4 in favour of H γ due to the factor of two in the DM mass). In DM models where there is a correlation between the di-photon and H γ , HZ and/or HH final states, the ratio of the flux associated with the prompt γ -ray line to that of the Higgs boson decay line can be used to test the model. In particular when m χ /similarequal 126 GeV, one expects the following ratio φγγ σ . In the absence of evidence for a specific DM model and a precise correlation between these two final states, searching for the Higgs decay line could allow us to obtain a constraint on the DM-Higgs boson interactions. The main difficulty associated with these searches consists in removing the astrophysical background sources but these searches are worthwhile, as they could reveal new physics and point towards models with multiple scalar and pseudo-scalar Higgs bosons with large DM-Higgs couplings, for example.", "pages": [ 4, 5, 6 ] }, { "title": "III. BOOSTED HIGGS AND MULTIPLE HIGGS BOSONS SCENARIOS", "content": "We can now investigate the case of boosted Higgs production and multiple Higgs scenarios.", "pages": [ 6 ] }, { "title": "A. Boosted Higgs boson", "content": "The Higgs boson decay line considered in the previous section is now replaced by a broad excess which shows up as a less prominent feature. For χχ → HH , this box-shaped part of the spectrum is a particular case of those studied in Ref. [58]. However, in the cases discussed here, this broad excess is accompanied by a smooth spectrum from the Higgs decay into all other possible channels plus a possible line due to prompt photon emission in the H γ final state. These features are illustrated in Fig. 4, where the γ -ray spectrum due to Higgs decay for a Higgs boson ( mH = 126 GeV) produced with an energy EH /similarequal 130 GeV is depicted. Over the continuum from the other Higgs decay channels, a bump at ∼ 60 GeV, corresponding to the Higgs boson decay into two photons, can still be distinguished. Below 10 GeV, the continuum is two orders of magnitude (or more) brighter than the line, so the limit on the Higgs boson production, for DM masses for which the Higgs boson is boosted, is actually obtained from the continuum rather than from the broad excess at E γ ∼ 60 GeV. This can be seen in Fig. 5, which is analogous to Fig. 3, but now for m χ = 81 GeV ( H γ ), 111 GeV ( HZ ) and 130 GeV ( HH ), such that, for all these cases, the produced Higgs has an energy close to 130 GeV. For the H γ final state, note that there is a γ -ray line emitted at 32 GeV, in addition to the box-shaped spectrum at E γ ∼ 5080 GeV and the continuum from Higgs decays. This line originates from the prompt γ in the final state and provides the most stringent bound on Higgs boson production cross section. Actually, in the case of χχ → H γ , the prompt γ -ray is always in the energy window accessible by Fermi-LAT if the Higgs is not produced very close to rest. Using the FermiLAT data for this annihilation channel and for m χ /similarequal 81 GeV, we obtain a limit of about 〈 σ v 〉 /lessorsimilar 4 × 10 -27 cm 3 / s. This is comparable to the γ -ray line limits obtained by Fermi-LAT for χχ → γγ with m χ /similarequal 32 GeV, that is 〈 σ v 〉 /lessorsimilar 2 × 10 -28 cm 3 / s (cf. Fig. 15 in Ref. [57]), after correcting the χχ → H γ cross section limit by a factor of ( 1 / 2 )( 32 / 81 ) 2 to account for the fact that there is only one prompt photon in the H γ final state with respect to γγ and that the DM mass is different.", "pages": [ 6, 7 ] }, { "title": "B. Multiple Higgs bosons scenarios", "content": "In minimal SUSY models, in addition to a SM-like Higgs, one expects a heavier CP-even Higgs ( H 2) and a heavier CPodd Higgs (A). If the heavier CP-odd Higgs boson mass is about 2 m χ , annihilations into γγ through CP-odd Higgs portal could be resonant and produce a line at m χ . In fact, this process has been proposed to explain the bump at 130 GeV in the Fermi-LAT data [59-61]. In these configurations, the A γ and H 2 γ final states might be possible too, leading to the production of a CP-odd Higgs boson on-shell or slightly boosted CP-even H 2 if mH 2 /similarequal mA . These final states should be slightly suppressed with respect to the γγ final states due to the phasespace suppression factor, but would still contribute to the γ -ray data at E γ = m χ . In the NMSSM, final states such as Aa and H 2 a may be possible too, with a a second pseudo-scalar Higgs boson which can be light and A , H 2 two heavy Higgs bosons [62]. Such final states could lead to the production of Higgs bosons produced at rest when 2 m χ /similarequal mA , H 2 + ma and could be resonant when ma /lessmuch mA . The same process could be in fact relevant for low DM mass scenarios such as those discussed in Ref. [7].", "pages": [ 7 ] }, { "title": "IV. CONCLUSIONS", "content": "In this Letter, we have considered the γ -ray signatures from the decay of a Higgs boson produced in our galactic halo from DMannihilations. We have considered, in particular, the case where the Higgs boson is SM-like (with a mass of 126 GeV and SM branching ratios) and showed that the Higgs boson production cross section for annihilating DM particles with masses m χ /similarequal 63 GeV, 109 GeV and 126 GeV (Higgs produced very close to rest), cannot exceed 〈 σ v 〉 ∼ few × 10 -25 cm 3 / s. The limit is in fact mostly driven by the γ -ray line from H → γγ . These results can be trivially generalised to other Higgs boson masses (as relevant in BSM models with multiple Higgs bosons and Higgs mass spectrum such as the NMSSM) leading to different DM scenarios. We have also considered the case of a slightly boosted Higgs boson and shown that the associated signature would exhibit a broad (box-shaped) γ -ray excess. However, the continuum associated with the other Higgs boson decay modes and to the second particle in the final state would lead to a brighter γ -ray emission, which can be used to constrain the Higgs boson production cross section. Focusing in particular on the H γ final state for a SM-like Higgs boson produced with an energy EH = 130 GeV, we find that the Higgs boson production cross section cannot exceed ∼ 4 × 10 -27 cm 3 / s. Therefore, we have obtained a simple estimate for the limit on the Higgs boson production cross section that is independent of any other DM annihilation channels and demonstrates that performing Higgs boson decay line searches could be useful to probe the Higgs boson dark couplings (i.e., couplings to DMparticles). This must be compared to the limits set on the invisible Higgs boson decay branching ratios obtained by using LHC measurements (cf., for example, Ref. [17]), but the two approaches (collider and indirect detection searches) are complementary.", "pages": [ 7, 8 ] }, { "title": "Acknowledgments", "content": "We would like to thank M. Cirelli, M. Dolan, G. G'omezVargas and C. McCabe for useful discussions. NB is sup- ported by the DFG TRR33 'The Dark Universe'. CB and SPR thank the Galileo Galilei Institute for Theoretical Physics for its hospitality. JS and CB are supported by the ERC advanced grant 'DARK' at IAP, Paris. SPR is partially supported by the Portuguese FCT through CERN/FP/123580/2011 and CFTPFCT UNIT 777, which are partially funded through POCTI (FEDER) and by the Spanish Grant FPA2011-23596 of the MINECO.TTis supported by the European ITN project (FP7PEOPLE-2011-ITN, PITN-GA-2011-289442-INVISIBLES). Numerical computation in this work was partially carried out at the Yukawa Institute Computer Facility. (2012), 1006.0477.", "pages": [ 8, 9 ] } ]

[ { "title": "Entropy and temperatures of Nariai black hole", "content": "Myungseok Eune 1, ∗ and Wontae Kim 1, 2, 3, † 1 Research Institute for Basic Science, Sogang University, Seoul, 121-742, Korea 2 Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea 3 Department of Physics, Sogang University, Seoul 121-742, Korea (Dated: October 30, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "The statistical entropy of the Nariai black hole in a thermal equilibrium is calculated by using the brick-wall method. Even if the temperature depends on the choice of the time-like Killing vector, the entropy can be written by the ordinary area law which agrees with the Wald entropy. We discuss some physical consequences of this result and the properties of the temperatures. Keywords: Black Hole, Hawking temperature", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "It has been claimed that the entropy of a black hole is proportional to the surface area at the event horizon [1], and then the Schwarzschild black hole has been studied through the quantum field theoretic calculation [2]. One of the convenient methods to get the entropy is to use the brick-wall method, which gives the statistical entropy satisfying the area law of the black hole [3]. Then, there have been extensive applications to various black holes [4-26]. In fact, another way to obtain the entropy is to regard the entropy of black holes as the conserved Noether charge corresponding to the symmetry of time translation [27]. For the Einstein gravity, the Wald entropy is always given by the A H / (4 G ), where A H and G are the surface area at the event horizon and the Newton's gravitational constant, respectively. Actually, there are many extended studies for the entropy as the Noether charge in the general theory of gravity including the higher power of the curvature [28-33]. The fact that the cosmological constant seems to be positive in our universe deserves to study the Schwarzschild black hole on the de Sitter background, which can be easily realized in the form of the Schwarzschild-de Sitter (SdS) spacetime. It has the black hole horizon and the cosmological horizon, and the observer lives between them. In this spacetime, the temperature of the black hole is different from the temperature due to the cosmological horizon when 0 < M < 1 / (3 √ Λ), where M and Λ are the mass of the black hole and the cosmological constant, respectively. Therefore, we are in trouble to study the thermodynamics of the system since it is not in thermally equilibrium due to the different temperatures. Nevertheless, there are several studies for the entropy through the improved brick-wall method for the SdS black hole [34-36] and the Kerr-de Sitter black hole [37]. In order to avoid the difficulty due to the nonequilibrium state of the SdS black hole, they have considered two thin-layers near the black hole horizon and the cosmological horizon, and then calculated the entropy for each thin layer. On the other hand, for the special limit of M = 1 / (3 √ Λ) in the SdS spacetime, the two horizons are coincident in the Schwarzschild coordinate. However, the Nariai metric is obtained through the coordinate transformation to avoid the coordinate singularity, where the two horizons are still separated [38-41]. The Nariai spacetime is in thermally equilibrium since the black hole and the cosmological horizon give the same temperatures. Thus, we can treat the whole Nariai spacetime as one thermodynamical system. However, even in spite of thermodynamic equilibrium, there few thermodynamic studies. Moreover, one can define two kinds of temperatures for the Nariai black hole: the Bousso-Hawking temperature and the Hawking temperature since there exist two different normalizations of timelike Killing vectors [39]. In this paper, we would like to study the statistical entropy of the Nariai black hole by using the brick-wall method. In section II, we introduce the SdS spacetime and the Nariai spacetime, and define two kinds of temperatures based on the different normalizations of the Killing vectors. We will also apply the Wald formula tor the Nariai black hole in order to get the entropy without resort to normalizations of the Killing vector. In section III, the entropy will be calculated by using the brick-wall method. Although the energy and the temperature depend on the normalization of the time-like Killing vector, the normalization-independent statistical entropy can be obtained, which is compatible with the Wald entropy. Finally, summary and discussion are given in section IV.", "pages": [ 2, 3 ] }, { "title": "II. TEMPERATURES AND WALD ENTROPY IN NARIAI BLACK HOLE", "content": "Let us start with the four-dimensional Einstein-Hilbert action with the cosmological constant Λ, which is given by The equation of motion obtained from the action (1) becomes The static and spherically symmetric solution of Eq. (2) is written as with Hereafter, we will consider only the Schwarzschild-de Sitter spacetime with Λ > 0. For 0 < M ≤ 1 / (3 √ Λ), it has two horizons of the black hole horizon r b and the cosmological horizon r c . In this case, the metric function (4) can be neatly written as f ( r ) = (1 -r b /r )[1 - (Λ / 3)( r 2 + r b r + r 2 b )] = ( r -r b )( r c -r )( r + r b + r c ) / [ r ( r 2 b + r b r c + r 2 c )]. For M = 0, it has only the cosmological horizon with r c = √ 3 / Λ. The symmetry of time translation in the SdS spacetime can be described by a timelike Killing vector, which is written as where γ t is a normalization constant. In the standard normalization, γ t is obtained from the condition to satisfy ξ µ ξ µ = -1 at the asymptotically flat Minkowski spacetime. For instance, its value usually becomes γ t = 1 for a Schwarzschild metric. In the SdS spacetime, there is no asymptotically flat region, so that we should consider the reference point r g where the gravitational acceleration vanishes due to the balance between the forces of the black hole by the mass and the cosmological horizon by the cosmological constant. Thus, we can choose the normalization constant in Eq. (5) to satisfy ξ µ ξ µ = -1 at that reference point r g , which yields where the reference point can be found from f ' ( r g ) = 0 and is explicitly given by r g = (3 M/ Λ) 1 / 3 . Now, the surface gravities κ b and κ c on the black hole horizon and the cosmological horizon are written as respectively. Then, the temperatures along with the normalization (6) are calculated as which are called the Bousso-Hawking temperatures [39]. This temperature can be also obtained from ˜ ξ = ∂/∂ ˜ t when the time is rescaled as ˜ t = t √ f ( r g ), where ξ µ ξ µ = -1 is satisfied at r = r g . On the other hand, in the Euclidean geometry, the Hawking temperature agrees with the inverse of the period of the Euclidean time to avoid a conical singularity at the horizon. Setting the Euclidean time τ to τ = it , the Euclidean line element of Eq. (3) is written as From Eq. (9), the Hawking temperatures for the black hole horizon and the cosmological horizon become respectively, which agree with the temperatures obtained from the Killing vector (5) with the normalization constant γ t = 1. Note that the Hawking temperature (10) is definitely different from the Bousso-Hawking temperature (8). For the scaled Euclidean time given by ˜ τ = i ˜ t , the Bousso-Hawking temperatures are obtained. Similar argument can be done for the Nariai black hole by taking the limit of M = 1 / (3 √ Λ) in Eq. (4) so that the two horizons are coincident in the Schwarzschild coordinate. In this degenerate case with r b = r c , the metric (3) should be transformed to an appropriate coordinate system because it has the coordinate singularity and becomes inappropriate. Near the degenerate case, the mass can be written as [38-40] where the degenerate case can be obtained by taking /epsilon1 = 0. One can define the new time and the radial coordinate ψ and χ by In terms of the new coordinates (12), the line element (3) is written in the form of up to the first order in /epsilon1 . For the case of /epsilon1 = 0, Eq. (13) is called the Nariai metric, which is given by In this coordinate system, the back hole horizon and the cosmological horizon correspond to χ = 0 and χ = π , respectively, where the proper distance between the two horizons is given by π/ √ Λ which is not zero. From now on, we will study this Nariai black hole which is actually real geometry to describe thermal equilibrium since the horizon temperature is the same with the cosmological temperature. However, there are two kinds of temperatures depending on the definitions of the normalization of the Killing vector. With Eqs. (11) and (12), the Killing vector (5) becomes to the leading order in /epsilon1 . Using Eq. (7), the Bousso-Hawking temperature is calculated as This can be also obtained from the Killing vector ˜ ξ = ∂/∂ ˜ ψ at the coordinate system with the rescaled time ˜ ψ = ψ/ √ Λ. As expected, the temperature of the black hole horizon is the same with that of the cosmological horizon. On the other hand, one can also get the Hawking temperature from the Euclidean metric (14) by setting the Euclidean time as ψ E = iψ . Then, the Euclidean Nariai metric can be written as In order to avoid a conical singularity at the two horizons, the period of the Euclidean time for the black hole horizon or the cosmological horizon are chosen as 2 π , respectively. Then the Hawking temperatures are given by which corresponds to the surface gravity obtained from the Killing vector ∂/∂ψ using Eq. (7). It is interesting to note that the Hawking temperature is constant as long as the Nariai condition M = 1 / (3 √ Λ) is met. Moreover, it can be easily checked that the BoussoHawking temperature (16) is obtained from the condition to avoid a conical singularity at the horizons for the scaled Euclidean time ˜ ψ E = ψ E / √ Λ. In order to find the Wald entropy of the Nariai spacetime, one should consider a diffeomorphism invariance with the Killing vector ξ µ which is associated with the conservation law of ∇ µ J µ = 0 [27-32], for which the Noether potential J µν can be defined by J µ = ∇ ν J µν . If a Lagrangian is written in the form of L = L ( g µν , R µνρσ ), then the Noether potential is given by [31, 32] where For a timelike Killing vector, the Wald entropy [27] is expressed by where κ and h µν are the surface gravity and the induced metric on the hypersurface Σ of a horizon, respectively. And /epsilon1 µν is defined by where n µ is the outward unit normal vector of Σ. The proper velocity u µ of a fiducial observer moving along the orbit of ξ µ is given by u µ = α -1 ξ µ with α ≡ √ -ξ µ ξ µ . For the Nariai metric (14), the Killing vector is given by where γ is a normalization constant, which will be not specified in this section. From the norm of the Killing vector, we obtain α = γ sin χ/ √ Λ and u µ = ξ µ /α = -δ ψ µ sin χ/ √ Λ. The outward unit normal vectors of the black hole horizon and the cosmological horizon are calculated as n µ = (1 / √ Λ) δ χ µ and n µ = -(1 / √ Λ) δ χ µ , respectively. Then, the nonzero components of Eq. (22) are /epsilon1 ψχ = -/epsilon1 χψ = ± sin χ/ (2Λ), where the upper sign and the lower sign correspond to the black hole horizon and the cosmological horizon, respectively. Now, for the action (1), we obtain which leads to Inserting Eq. (23) into Eq. (7), we can obtain κ b,c = γ . Then, from Eq. (21), the Wald entropy is given by where A b and A c are the areas of the black hole horizon and the cosmological horizon, respectively. The total area given by the two horizons becomes A = 8 π/ Λ since A b = A c = 4 π/ Λ. Eventually, the entropy (26) can be rewritten as which also agrees with the Bekenstein-Hawking entropy. After all, we obtained the Wald entropy expressed by the expected area law, which is independent of the normalization of the Killing vector.", "pages": [ 3, 4, 5, 6, 7, 8 ] }, { "title": "III. ENTROPY FROM BRICK-WALL METHOD", "content": "In the Nariai black hole governed by the line element (14), the black hole temperature is the same with the cosmological temperature as seen from Eqs. (16) and (18), which imply that the net flux is in fact zero. Thus, the thermal equilibrium can be realized in this special configuration, which is different from the non-equilibrium SdS black hole. In order to calculate the statistical entropy in this thermal background [3], we will consider a quantum scalar field in a box surrounded by the two horizons. The Klein-Gordon equation for the scalar field is written as where m is the mass of the scalar field. By using the WKB approximation with Φ ∼ exp[ -iωψ + iS ( χ, θ, φ )] under the Nariai metric (14), the square module of the momentum is obtained as where k χ = ∂S/∂χ , k θ = ∂S/∂θ , and k φ = ∂S/∂φ . Then, the number of quantum states with the energy less than ω is calculated as where V p denotes the volume of the phase space satisfying k 2 + m 2 ≤ 0. For simplicity, we take the massless limit of m 2 = 0. As seen from (30), the number of states diverges at the horizons of χ = 0 , π , so that we need the UV cutoff at χ = h b and χ = π -h c . The UV cutoff parameters h b and h c are assumed to be very small. Then, the free energy is given by Then, the entropy becomes The proper lengths for the UV parameters are given by which leads to h b,c = √ Λ ¯ h b,c . Then, Eq. (32) is written as within the leading order of ¯ h b,c . When we perform the WKB approximation with the line element (14), the coordinate ψ plays a role of the time. The corresponding Killing vector is given by ξ = ∂/∂ψ and β in Eq. (35) should be taken as the inverse of the Hawking temperature (18). Then, the entropy is obtained as where /lscript P ≡ √ G /planckover2pi1 /c 3 is the Plank length. If the cutoff is chosen as ¯ h b,c = /lscript P / √ 90 π like the case of the Schwarzschild black hole [3], the entropy (36) is remarkably written as where the total area is defined by A = A b + A c for convenience. Then, it agrees with one quarter of the horizon area of the Bekenstein-Hawking entropy. From the viewpoint of the renormalization [42], the total entropy can be written as the sum of the Wald entropy (27) and the quantum correction of Eq. (36). If we consider the bare gravitational coupling constant in the classical entropy, the divergent part can be easily absorbed in the gravitational constant.", "pages": [ 8, 9, 10 ] }, { "title": "IV. DISCUSSION", "content": "By using the brick-wall method for the Nariai spacetime, we obtained the BekensteinHawking entropy which is proportional to the area of the horizon. In the brick-wall method, β was not the inverse of the Bousso-Hawking temperature but the inverse of the Hawking temperature. The reason why is that the time is chosen as ψ and the standard form of the corresponding Killing vector is given by ∂/∂ψ . If we consider the scaled time ˜ ψ = ψ/ √ Λ, the Killing vector is given by ξ = ∂/∂ ˜ ψ = √ Λ ∂/∂ψ , which yields the Bousso-Hawking temperature (16). Then, the WKB approximation in the brick-wall method should be performed for the scalar field in the form of Φ ∼ exp[ -i ˜ ω ˜ ψ + iS ( χ, θ, φ )] = exp[ -i ˜ ωψ/ √ Λ+ iS ( χ, θ, φ )]. This indicates that the energy in Eq. (29) becomes ˜ ω = ω √ Λ and we can easily show that β H ω = β BH ˜ ω . In the calculation of the free energy (31), ω in the integrand should be replaced by ˜ ω/ √ Λ and the integration should be performed for ˜ ω . Then, we can obtain the same entropy with Eq. (36) based on the Bousso-Hawking temperature. Therefore, the entropy is always written as the area law of the Wald entropy, whereas the temperature and the energy depend on the choice of the time, that is, the normalization of the timelike Killing vector. The final comment is in order. As for the Bousso-Hawking temperature, it can be regarded as a Tolman temperature [43]. It was defined at the vanishing surface gravity where it is the counterpart of the asymptotically Minkowski space in the asymptotically flat black holes. The Bousso-Hawking temperature can be derived from the definition of the Tolman temperature of T loc = T H / √ g ψψ = √ Λ / (2 π sin χ ) where T H = 1 / (2 π ). If we move the observer, for instance, to the black hole horizon of χ = 0 or to the cosmological horizon of χ = π , the temperature goes to infinity. In particular, at the middle point of χ = π 2 , it produces the Bousso-Hawking temperature. So the Bousso-Hawking normalization of Killing vector is compatible with the Tolman temperature. So, we can identify the Bousso-Hawking temperature with the Tolman temperature at the reference point.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "We would like to thank Y. Kim for exciting discussion. M. Eune was supported by National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology) (NRF-2010-359-C00007). W. Kim are supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number 2005-0049409, and the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2012-0002880).", "pages": [ 11 ] } ]

[ { "title": "d ≥ 5 magnetized static, balanced black holes with S 2 × S d -4 event horizon topology", "content": "Burkhard Kleihaus, Jutta Kunz and Eugen Radu Institut fur Physik, Universitat Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany August 8, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We construct static, nonextremal black hole solutions of the Einstein-Maxwell equations in d = 6 , 7 spacetime dimensions, with an event horizon of S 2 × S d -4 topology. These configurations are asymptotically flat, the U (1) field being purely magnetic, with a spherical distribution of monopole charges but no net charge measured at infinity. They can be viewed as generalizations of the d = 5 static dipole black ring, sharing its basic properties, in particular the presence of a conical singularity. The magnetized version of these solutions is constructed by applying a Harrison transformation, which puts them into an external magnetic field. For d = 5 , 6 , 7, balanced configurations approaching asymptotically a Melvin universe background are found for a critical value of the background magnetic field.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "A remarkable property of black rings is the existence of regular configurations with gauge dipoles that are independent of all conserved charges. This strongly contrasts with the picture valid in d = 4 black hole physics, and implies a violation of the 'no hair' conjecture and of the black hole uniqueness. These aspects are clearly illustrated by the d = 5 black ring found by Emparan in [1], which was the first example of a black object that is asymptotically flat, possesses a regular horizon and is the source of a dipolar gauge field. This exact solution of the Einstein-Maxwell-dilaton equations has an event horizon of S 2 × S 1 topology. The U (1) field is purely magnetic, being produced by a circular distribution of magnetic monopoles 1 . Then the ring creates a dipole field only, with no net charge measured at infinity 2 . Similar to the vacuum case [2], the generic dipole rings (in particular the static ones) are plagued by conical singularities. The balance is achieved for a critical (nonzero) value of the angular momentum only. It is clear that the dipole ring solution in [1] should have generalizations in more than five dimensions. However, the analytic construction of these solutions seems to be intractable within a nonperturbative approach. Some progress in this direction has been achieved by using the blackfold approach. There the central assumption is that some black objects, in certain ultra-spinning regimes, can be approximated by very thin black strings or branes curved into a given shape, see [5], [6], [7]. Ref. [8] has found in this way generalizations of the dipole black ring for several topologies of the horizon, in particular for the ring case, S 1 × S d -3 . However, the blackfold approach has some limitations; for example, black holes with no black membrane behavior cannot be described within this framework. A different approach for the construction of d ≥ 5 black objects with a nonspherical topology of the horizon has been proposed in Ref. [9], [10]. The solutions are found in this case nonperturbatively, by solving numerically the Einstein equations with suitable boundary conditions. A number of new solutions have been constructed in this manner, in particular recently Ref. [11] has given numerical evidence for the existence of balanced spinning vacuum black rings in d ≥ 6 dimensions beyond the blackfold limit, and analyzed their basic properties. In this work we propose to construct new static nonextremal black objects with a S 2 × S d -4 topology of the event horizon in d = 6 and 7 dimensions, by extending the results in [9] to the case of Einstein-Maxwell theory. These solutions can be viewed as higher dimensional generalizations of the d = 5 static dipole ring in [1], the magnetic field being analogous to a dipole, with no net charge measured at infinity. However, in the absence of rotation, these configurations have a conical singularity which provides the force balance that allows for their existence for any d ≥ 5. However, as discussed in [12], the conical singularity of the d = 5 static dipole ring can be removed by 'immersing' it in a background gauge field. In this work we show that this holds for d > 5 solutions as well. By applying a magnetic Harrison transformation, the conical singularities disappear for a critical value of the background magnetic field. The resulting configurations describe d > 5 balanced black holes with a horizon of S 2 × S d -4 topology, in a Melvin universe background.", "pages": [ 1, 2 ] }, { "title": "2.1 The ansatz and equations", "content": "We consider the Einstein-Maxwell theory in d -spacetime dimensions, defined by the following action the corresponding equations of motion being The solutions in this work are static and axisymmetric configurations, with a symmetry group R t × U (1) × SO ( d -3) (where R t denotes the time translation). Following the Appendix C of [10], we take the following metric ansatz: where d Ω 2 d -4 is the unit metric on S d -4 , the range of θ is 0 ≤ θ ≤ π/ 2 and ψ is an angular coordinate, with 0 ≤ ψ ≤ 2 π . Also, r and t correspond to the radial and time coordinates, respectively. We shall see that for the solutions in this work, the range of r is 0 < r H ≤ r < ∞ ; thus the ( r, θ ) coordinates have a rectangular boundary well suited for numerics. For any value of d , the U(1) potential has a single component, It is of interest to mention that the model admits a dual formulation, with an 'electric' version of (2.1), with where ˜ F = /starF = dB is a ( d -2)-form field strength (then the only nonvanishing components of the ( d -3)form potential B are B Ω t ). However, in this work we shall restrict to the magnetic description within the Einstein-Maxwell theory. An appropriate combination of the Einstein equations, E t t = 0 , E r r + E θ θ = 0, E ψ ψ = 0, and E Ω Ω = 0, yields the following set of equations for the functions f 1 , f 2 , f 3 and f 0 : From the Maxwell equations, it follows that the magnetic potential A ψ is a solution of the equation In the above relations, we have defined ( ∇ U ) · ( ∇ V ) = ∂ r U∂ r V + 1 r 2 ∂ θ U∂ θ V, and ∇ 2 U = ∂ 2 r U + 1 r 2 ∂ 2 θ U + 1 r ∂ r U. The remaining Einstein equations E r θ = 0 , E r r -E θ θ = 0 yield two constraints. Following [13], we note that setting E t t = E ϕ ϕ = E r r + E θ θ = 0 in the identities ∇ µ E µr = 0 and ∇ µ E µθ = 0, we obtain the CauchyRiemann relations ∂ θ ( √ -gE r θ ) + ∂ ¯ r ( √ -g 1 2 ( E r r -E θ θ ) ) = 0 , ∂ ¯ r ( √ -gE r θ ) -∂ θ ( √ -g 1 2 ( E r r -E θ θ ) ) = 0 , (with r 2 ∂/∂r = ∂/∂ ¯ r ). Thus the weighted constraints satisfy Laplace equations, and the constraints are fulfilled, when one of them is satisfied on the boundary and the other at a single point [13]. We close this part by remarking that the solutions in this work can also be studied by using Weyl-like coordinates, with ds 2 = ¯ f 1 ( ρ, z )( dρ 2 + dz 2 ) + f 2 ( ρ, z ) dψ 2 + f 3 ( ρ, z ) d Ω 2 d -4 -f 0 ( ρ, z ) dt 2 , and A = A ψ ( ρ, z ) dψ . The general transformation between ( ρ, z ) and ( r, θ ) coordinates is given in Ref. [10]. Indeed, the vacuum limit of the solutions in this work ( A ψ ≡ 0) was studied in Ref. [9] by employing the ( ρ, z )-coordinates. The metric Ansatz (2.3) in terms of ( r, θ ) allows, however, for a better numerical accuracy.", "pages": [ 2, 3 ] }, { "title": "2.2.1 Boundary conditions", "content": "The equations (2.6) are solved subject to a set of boundary conditions which results from the requirement that the solutions describe asymptotically flat black objects with a regular horizon of S 2 × S d -4 topology 3 . We assume that as r → ∞ , the Minkowski spacetime background (with ds 2 = dr 2 + r 2 ( dθ 2 +cos 2 θdψ 2 + sin 2 θd Ω 2 d -4 ) -dt 2 ) is recovered, while the gauge potential vanishes. This implies Also, we impose the existence of a nonextremal event horizon, which is located at a constant value of the radial coordinate, r = r H > 0. There we require The boundary conditions at θ = π/ 2 are The absence of conical singularities requires also r 2 f 1 = f 2 on that boundary. The boundary conditions for θ = 0 are more complicated, since they encode the non-trivial topology of the horizon. The idea here is that for some interval r H ≤ r < R , we have for the metric the same conditions as for θ = π/ 2, the asymptotic behaviour f 2 ∼ cos 2 θ , f 3 ∼ sin 2 θ being recovered for r > R (with R > r H an input parameter). Therefore, for r H < r < R , we impose For r > R we require instead Although the constants R,r H which enter the above relations have no invariant meaning, they provide a rough measure for the radii of the S d -4 and S 2 spheres, respectively, on the horizon. Also, we shall see that the parameter Ψ fixes the local charge of the solutions.", "pages": [ 3, 4 ] }, { "title": "2.2.2 Global quantities", "content": "The metric of a spatial cross-section of the horizon is Since, from the above boundary conditions, the orbits of ψ shrink to zero at θ = 0 and θ = π/ 2 while the area of S d -4 does not vanish anywhere, the topology of the horizon is S 2 × S d -4 (in fact, for all nonextremal solutions in this work, f 2 ( r H , θ ) ∼ sin 2 2 θ while f 1 ( r H , θ ) and f 3 ( r H , θ ) are strictly positive and finite functions). The event horizon area is given by The Hawking temperature as computed from the surface gravity or by requiring regularity on the Euclidean section, is where V d -4 is the area of the unit sphere S d -4 . where the constraint equation E θ r = 0 guarantees that the Hawking temperature is constant on the event horizon. At infinity, the Minkowski background is approached. The total mass of the solutions is given by [14] (where the integral is taken over the ( d -2) -sphere at spatial infinity and k = ∂/∂t ) and can be read from the asymptotic expression for f 0 , Using Gauss' theorem, the Einstein equations and the boundary conditions (2.8)-(2.12), one finds from (2.16) the following Smarr-type relation Here Q is the 'local' magnetic charge which enters the thermodynamics 4 as defined by evaluating the magnetic flux over the S 2 sphere around the horizon, and Φ is the thermodynamical conjugate variable to Q , As expected, in the absence of rotation, all these black objects with S 2 × S d -4 horizon topology are plagued by conical singularities. As one can see from the boundary conditions, in this work we have chosen 5 to locate the conical singularity at θ = 0, r H < r < R , where we find a conical excess ∣ ∣ such that 1 16 π ∫ F 2 √ -gd d -1 x = 2Φ Q . Therefore, following [15], we interpret the solutions as describing a spherical S d -4 distribution of monopole charges, though with a zero net charge (see also [16]). This can be interpreted as the higher dimensional analogue of a 'strut' ( e.g. a membrane for d = 5), preventing the collapse of the configurations. Although the presence of a conical singularity is an undesirable feature, it has been argued in [17], [18], that such asymptotically flat black objects still admit a thermodynamical description. Moreover, when working with the appropriate set of thermodynamical variables, the BekensteinHawking law still holds, while the parameter δ enters the first law of thermodynamics. Without going into details, we mention that the conjugate extensive variable to δ is where Area is the space-time area of the conical singularity's world-volume. For the line-element (2.3), the line element of the two dimensional surface spanned by the conical singularity is which implies", "pages": [ 4, 5 ] }, { "title": "3.1 The d = 5 static dipole black ring", "content": "The static dipole black ring is usually written in ring or in Weyl coordinates, where it takes a relatively simple form. In what follows we shall write it within the ansatz (2.3), (2.4), which results in rather complicated expressions. However, this helps us to make contact with the numerical solutions found for d > 5. In the ( r, θ )-coordinates, the metric functions f i in the line element (2.3) are given by (note that for d = 5, the sphere Ω d -4 reduces to a circle): where the magnetic potential (written in a gauge such that A ψ ( θ = π/ 2) = 0) being In the above relations we note with R 4 = √ ( r 4 H + R 4 R 2 -r 4 + r 4 H r 2 cos 2 θ ) 2 + ( r 4 -r 4 H ) 2 r 4 sin 2 2 θ . Also, f (0) i are the functions which enter the line element of the d = 5 static vacuum black ring, with and where are the mass, temperature and area of the vacuum static black ring solution, and where R 3 = √ (1 + R 4 r 4 -2 R 2 r 2 cos 2 θ )(1 + r 8 H r 4 R 4 -2 r 4 H r 2 R 2 cos 2 θ ). This solution has three parameters, r H , R (which were introduced in the previous section) and w , which is fixed by the value of the magnetic potential at θ = 0 , r H < r < R via (note that 0 ≤ w < 1): A direct computation shows that this is indeed a solution of the Einstein-Maxwell equations. Also, one can see that c 1 → 1 , c 2 → 1 and A ψ → 0 as w → 0, this corresponding to the vacuum black ring limit. The computation of the quantities of interest for this solution is a straightforward application of the general formalism in Section 2.2.2. In the nonextremal case, one can write the following suggestive expressions: with 0 ≤ U < ( R 2 -r 2 H ) 2 / (4 R 2 r 2 H ) a free parameter 6 . The static dipole rings have a conical excess while the expression of the corresponding conjugate extensive variable A cannot be written in closed form. The basic properties of the d = 5 non-extremal solution turn out to be generic and will be discussed in the following subsection. Here we mention only that the extremal solutions are found by taking the limit r H → 0 in the relations (3.1)-(3.6). They have a relatively simple form with T ( ± ) = R 1 ± R 2 , R 1 = √ r 4 + R 4 -2 r 2 R 2 cos 2 θ and w = Q / √ Q 2 +3 R 2 . The horizon of the extremal solutions has zero area, since the length of the S 1 direction vanishes there, g ψψ → 0. Their mass and and potential are given by M = 3 π Q R 2 4( Q + √ Q 2 +3 R 2 ) , Φ = 3 πR 2 2( Q + √ Q 2 +3 R 2 ) , their conical excess is δ = -4 π (4 Q 3 +9 R 2 ) ( -Q + √ Q 2 +3 R 2 ) 3 .", "pages": [ 5, 6, 7 ] }, { "title": "3.2.1 Remarks on the numerics", "content": "Higher dimensional generalizations of the d = 5 nonextremal solution (3.1)-(3.6) are found by replacing in the five dimensional line element the S 1 direction which is not associated with the magnetic potential, with the line element of a round ( d -4)-sphere, while preserving at the same time the basic properties of the metric functions and of the magnetic potential. Since no closed form solution is available in this case, the set of five coupled non-linear elliptic partial differential equations (2.6), (2.7) is solved numerically, subject to the boundary conditions (2.8)-(2.12). The numerical scheme we have used is identical with that described at length in [10] and thus we shall not enter into details. We mention only that in practice we have worked with a set of 'auxiliary' functions F i defined via 7 where f (0) i are 'background' functions corresponding to the d = 5 static vacuum black ring as given by (3.5). These 'background' functions f (0) i are used to fix the topology of the horizon and to absorb the coordinate divergencies of the functions f i . The 'auxiliary' functions F i are smooth and finite everywhere such that they do not lead to the occurence of new zeros of the functions f i (therefore the rod structure of the solutions remains fixed by f (0) i [10]). However, F i encode the effects of changing the spacetime dimension from d = 5 and also of introducing the local charge Q . In our approach, the input parameters are the value d of the spacetime dimension, the event horizon radius r H , the radius R of the S d -4 sphere, and the value of the local charge Q ( i.e. the parameter Ψ in the boundary conditions (2.11)). The physical parameters are encoded in the values of the functions f i (and their derivatives) on the boundary of the integration domain. For example, the mass parameter M is computed from the asymptotic form (2.17) of the metric function g tt = -f 0 , the Smarr relation (2.18) being used to verify the accuracy of the solutions.", "pages": [ 7 ] }, { "title": "3.2.2 Properties of the solutions", "content": "To obtain nonextremal Einstein-Maxwell solutions with S 2 × S d -4 horizon topology, one starts with the vacuum configurations in [9] and turns on the parameter Ψ which enters the boundary conditions for the magnetic potential. The iterations converge, and, in principle, repeating the procedure it is possible to obtain solutions with arbitrary values of Q . We have started with a test of the numerical scheme, by recovering in this way the d = 5 static dipole black rings. Afterwards, new solutions in d = 6 , 7 dimensions have been studied in a systematic way. Solutions with d > 7 should also exist; however, we did not try to find them and their study may require a different numerical method. We mention that, for all solutions, we have verified that the Kretschmann scalar stays finite everywhere 8 . The central result in this work is that the d = 5 static nonextremal dipole ring has higher dimensional generalizations with a S 2 × S d -4 horizon topology. Moreover, the properties of the five dimensional solutions are generic, being recovered for d > 5. Let us start with a discussion of the solutions' features for a fixed value of the magnetic charge Q . Perhaps the most important feature is that all d ≥ 5 solutions have conical singularities. Thus we have found it convenient to take the relative conical excess δ/ ( δ -2 π ) as the control parameter and to consider the following dimensionless quantities 9 , the scale being fixed here by M : with p 1 = (( d -2 16 π ) d -2 V d -2 ) 1 d -3 , p 2 = 1 d -3 ( 2 2( d -1) π d -2 ( d -2) V d -2 ) 1 d -3 two coefficients which have been chosen such that a H = 1 , t H = 1 corresponds to the Schwarzschild-Tangherlini black hole. In terms of the dimensionless ratio r H /R , the solutions interpolate between two limits (although these regions of the parameter space are difficult to approach numerically). For R →∞ and r H , Q nonvanishing, the radius on the horizon of the S d -4 -sphere increases and asymptotically it becomes a ( d -4) -plane, while δ → 0. After a suitable rescaling 10 , one finds the magnetically charged black brane solution A = -Q (1 + cos 2 θ ) dψ, where This corresponds to a magnetically charged Reissner-Nordstrom black hole uplifted to d -dimensions, i.e. with ( d -4)-flat directions. The limit r H /R → 1 is somehow more subtle, since the conical excess diverges, δ →-∞ , and the magnetic field vanishes. As can be seen in Figure 1, the Schwarzschild-Tangherlini black hole with an S d -2 horizon topology is recovered in this limit. This can be understood by studying the d = 5 exact solution. There, as R → r H one finds c 1 → 1+ O ( R -r H ) , c 2 → 1+ O ( R -r H ) while A ψ ∼ O ( R -r H ) 2 ( i.e. a vanishing charge), with the limiting expressions f 1 = f 2 / ( r 2 cos 2 θ ) = f 3 / ( r 2 sin 2 θ ) = (1+ r 2 H /r 2 ) 2 , f 0 = ( r 2 -r 2 H ) 2 / ( r 2 + r 2 H ) 2 . Some results illustrating these aspects are shown in Figure 1 (note that we have found similar results for other values of Q as well). It seems that similar to the d = 5 case, the extremal solutions are found in the limit r H → 0, for nonvanishing R and Q . However, we could not approach this limit and the numerical construction of the extremal solutions would require a different numerical scheme, with another set of 'background' functions. This holds also for the d = 5 solutions, in which case it can be understood by noticing that the behaviour of the metric functions f 1 , f 3 as r → r H ( i.e. f 1 ∼ 1 /r 2 , f 2 ∼ r 2 ) is not compatible with the boundary conditions (2.9). We conjecture that the picture found for d = 5 is generic and the extremal solutions will always possess a horizon with vanishing area. A different situation which can be studied numerically is to keep fixed the radii r H and R and to vary the value of the local charge Q . Interestingly, turning on a magnetic field increases the absolut value of the conical excess, see Figure 2 (left). For fixed r H , R , the values of the magnetic potential, horizon area, the parameter A and the mass increase with Q , while the temperature decreases.", "pages": [ 8, 9, 10 ] }, { "title": "4 Balanced black holes with S 2 × S d -4 event horizon topology in a Melvin universe background", "content": "The occurrence of conical singularities is not an unusual feature in general relativity. However, sometimes this pathology can be cured by placing the solutions in an external field (see e.g. [20]-[22]). This was the case for the d = 5 static dipole ring [12] and also for extremal solutions in [15], which could be balanced by 'immersing' them in a background gauge field, via a magnetic Harrison transformation. Unsurprisingly, this works also for the configurations considered in this work. The magnetic Harrison transformation can be summarized as follows (see e.g. [23]). Let us consider a solution of the Einstein-Maxwell equations of the form with ∂/∂y a Killing vector. Then the configuration with solves also the Einstein-Maxwell equations (with B an arbitrary parameter). The Harrison transformation (4.2) applied with respect to the Killing vector ∂/∂ψ results in the following line element and the new magnetic potential One can see that the new line element preserves some of the basic properties of the B = 0 seed configuration. The horizon is still located at r = r H and has an S 2 × S d -4 topology, since the qualitative behaviour of the metric functions at θ = 0 , π/ 2 remains unchanged (note that Λ > 0 everywhere). However, the geometry is distorted and the asymptotic behaviour is very different. As r →∞ , the solution becomes which is a higher dimensional generalization of the d = 4 Melvin magnetic universe [24]. A direct calculation shows that the horizon area and the temperature of the new solutions (4.4), (4.5) are not affected by the external magnetic field, coinciding with the corresponding quantities of the B = 0 seed configurations. Moreover, by employing the same approach as in [25], [12], it is straightforward to show that the mass of the new solutions, as defined with respect to the Melvin universe background, still preserves the expression found in the asymptotically flat case 11 . The configurations with generic values of B possess again a conical singularity at θ = 0, r H < r < R . However, this conical singularity vanishes for a critical value of the magnetic field, The dimensionless quantity b c = B c Q is shown in Figure 2 (right) as a function of the parameters a H and R/r H for d = 5 , 6 , 7 solutions. One can see that b c diverges as the Schwarzschild limit is approached.", "pages": [ 10, 11 ] }, { "title": "5 Conclusions", "content": "In this work we have shown numerical evidence that the vacuum static black holes with S 2 × S d -4 horizon topology discussed in [9] admit nonextremal generalizations in Einstein-Maxwell theory. These new solutions have a dipolar magnetic field, which is created by a spherical S d -4 distribution of monopoles. They also share the basic properties of the d = 5 static dipole ring and possess conical singularities, which, in the absence of rotation, prevent the black objects to collapse. Of course, on general grounds, one expects the d > 5 new solutions in this work to possess rotating generalizations and thus to achieve balance for a critical value of the angular momentum. Unfortunately, the explicit construction of such solutions proves a very difficult numerical problem, see the discussion in [10]. However, as discussed in the second part of this work, these static black objects with a S 2 × S d -4 topology of the horizon can be held in equilibrium by switching on a magnetic field with an appropriate strength. To the best of our knowledge, this is the first explicit construction of d > 5 static and balanced black objects which are regular on and outside an event horizon of non-spherical topology 12 . However, the magnetic field does not vanish asymptotically, such that the background spacetime corresponds in this case to a d -dimensional Melvin universe. Therefore the construction of asymptotically flat, static balanced black objects with a non-spherical horizon topology remains an open problem. Our preliminary results indicate that the solutions in this work can be generalized to include a dilaton. In this ways, they could be uplifted to higher dimensions and interpreted in a string theory context. Moreover, we expect that all static configurations with a non-spherical horizon topology discussed in [10] would admit generalizations with a dipolar magnetic field. Although the asymptotically flat static solutions will possess conical singularities, the interaction with an external magnetic field would balance them. Acknowledgements.We gratefully acknowledge support by the DFG, in particular, also within the DFG Research Training Group 1620 'Models of Gravity'.", "pages": [ 11 ] } ]

[ { "title": "A new phase of scalar field with a kinetic term non-minimally coupled to gravity", "content": "Amir Ghalee Department of Physics, Tafresh University, P. O. Box 39518-79611, Tafresh, Iran We consider the dynamics of a scalar field non-minimally coupled to gravity in the context of cosmology. It is demonstrated that there exists a new phase for the scalar field, in addition to the inflationary and dust-like (reheating period) phases. Analytic expressions for the scalar field and the Hubble parameter, which describe the new phase are given. The Hubble parameter indicates an accelerating expanding Universe. We explicitly show that the scalar field oscillates with timedependent frequency. Moreover, an interaction between the scalar field in the new phase and other fields is discussed. It turns out that the parametric resonance is absent, which is another crucial difference between the dynamics of the scalar field in the new phase and dust-like phase. PACS numbers: 98.80.Cq", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "To solve the flatness problem and the horizon problem in cosmology, Alan Guth introduced the inflation paradigm [1]. The simple model, which is a single scalar field (inflaton) with minimally coupling to gravity, is described by an action where κ 2 ≡ 8 πG . Although model (1) with quadratic potential, V ( ϕ ) = 1 2 m 2 ϕ 2 , is consistent with the cosmic microwave background [4], many models have been proposed to produce the inflation era [3]. Regarding the motivations, many attempts are devoted to derive the inflation era by more fundamental principles or find connections between inflaton and other fields that are used in particle physics(e.g. Higgs field) [3, 5, 6]. An intuitive picture of the dynamics of inflaton is as follows: the inflaton field rolls slowly down its potential (inflationary period), eventually the field(s)oscillates around the minimum of its potential and decays into light particles (reheating period). It has been assumed that the reheating period takes place just after the inflationary period. Also, since the inflaton has lost its energy, it behaves like dust matter in the reheating period. For the quadratic potential, this intuitive picture is supported by analytic methods [5], ( see [8] for the quartic potential). Usually model builders use the stated scenario for their models ( see [9] for a different scenario). Then, they set limits on the parameters of the models. So, one can divide the dynamics of inflaton in two phases; inflationary phase and matter dominate phase(near minimum of the potential). In this work, we show that in the following effective action where G µν is Einstein's tensor, the scalar field has a new phase. Germani et.al [6], proposed the above action with quartic potential, λϕ 4 , in the context of the new Higgs inflation. The cosmological perturbation of the model and reheating period of the model have been studied in [10] and [11] respectively. This paper is organized as follows: in § II we briefly review the model and obtain equations, then we qualitatively discuss why the scalar field of model (2), has a new phase for any typical potential, V ( ϕ ). In § III analytic solutions for the Hubble parameter and the scalar field are provided for the quadratic potential. The solutions describe the new phase of the scalar field. In § IV we investigate an implication of the new phase by considering an interaction between a relativistic field and the scalar field. To introduce reader to the method that is used in this paper, we re-derive solution of the scalar field of the model (1) in reheating period in Appendix A, which is identical with the solution in the textbook [5]. Appendix B is devoted to some properties of the Fresnel integrals.", "pages": [ 1 ] }, { "title": "II. EVIDENCES FOR EXISTENCE OF THE NEW PHASE", "content": "To obtain field equation and the Friedman equation of the model in FRW background metric, we use the ADM formalism with the following metric ansatz [12] By inserting this expression into (2) we have [6, 10] by varying the action(4) with respect to the laps N and ϕ and setting N to 1, we obtain κ 2 , The behavior of the scalar field can be divided into two regimes [13] Equations (6a) and (6b) are the same as equations which are derived from (1) and studied in textbooks [5, 7]. Since we want to compare other regime with this regime, we quote the main results. The second term on the left hand side of (6b), which is always positive, acts as a dissipative force. Therefore, the scalar filed rolls toward the minimum of the potential, and it behaves as a dust matter. For the simplest potential, V ( ϕ ) = 1 m 2 ϕ 2 , it has been shown that [5] 2 where g (0) is an arbitrarily constant. It is important to note that expressions in (7) are only valid for H ( t ) < m , so higher order terms, have been neglected [5]. where w eff is the effective equation of state The inflationary phase of this regime, w eff → -1, was studied in [6, 10]. Note that during the inflationary phase, the second term on the left hand side in (8b) is positive, and acts as a dissipative force. So, again, the scalar filed rolls toward the minimum of the potential , but at H = 2 / 3 t , it is vanished and, after some time, its sign may be changed and act as a driving force. Another point is that the right hand side of (8b), depends on H , which is not constant in this period. This behavior, shows that we need a different analysis for this regime.", "pages": [ 1, 2 ] }, { "title": "III. EXPLICIT SOLUTIONS FOR THE QUADRATIC POTENTIAL", "content": "In this section we consider the simple potential V ( ϕ ) = 1 2 m 2 ϕ 2 , then we seek solutions for the equations. We use the so-called averaging method [14], which is used in study of nonlinear differential equations. In Appendix A, we use this method to re-derive (7). Let us define a new time variable τ as It is worth to mention that this step is only a trick to solve the equations, when we obtain the solutions we'll back to the original time variable, t . Combining Eqs. (8a),(8b) and (10), with the quadratic potential, gives where prime denotes the differential with respect to τ . In order to use the method of averaging, we define the following relations [14] where f ( τ ) is an arbitrary function. Differentiation of the righthand side of (12a) must be equals to the righthand side of (12b). Hence we have Substitution of (12a) and (12b) into (11) results in H ' and f ' can be found from Equations (13) and (13) as So far we have not used any approximation for H , and f ( τ ). In the method of averaging, equations in (15) are replaced by averaged expressions. For this goal, note that if we keep H and f fixed [15], the righthand side of (15) are π -periodic in τ , so we can average over τ . By applying the stated procedure, one can show that (16) where < · · · > ≡ ∮ ( · · · ) dτ/π denotes an average over τ (but H and f are fixed). So, from Eqs. (15) and (16), we have The averaged equations are very easy to solve: We can use formula (10) and (18) to obtain explicit relation between τ and t as By using Eqs. (18) and (19) we have so, the effective equation of state becomes Moreover, from (12a) and (19), we also obtain where H ( t ) is given by (20). The new phase of the scalar field is described by (20) and (22). According to (21), for this new phase we have w eff < -1 / 3, so, the Universe is driven to accelerated expansion phase by the scalar field. Expression (22), shows that the scalar field oscillates with time-dependent 'frequency', as indicated in Fig.1. Both of the properties are more different than the dust-like phase, which is described by (7).", "pages": [ 2, 3 ] }, { "title": "IV. AN IMPLICATION", "content": "In this section, to show an implication of properties of the new phase, we will consider an interaction between a matter field, χ , and the scalar field. The decay of the scalar field into a relativistic field can be described by [5] It has been shown that if the interaction takes place during the dust-like phase, the parametric resonance instability is occurred [5, 16]. Notice that this result is obtained if the expansion of the universe is neglected. The stated assumption, seems to be a reasonable condition, at least for the first approximation, if the rate of interaction is too fast compared to the Hubble expansion time, see discussions about this note in [5]. As mentioned in [5], what one actually finds from present of the parametric resonance, is that the perturbative analysis is rather misleading, during dust-like phase of the inflaton. Here,our aim is to quest for the parametric resonance, during the new phase of the scalar field. If the other scalar field, χ , is decomposed into Fourier modes as then using (22) and (23), the following equation for the Fourier modes is obtained where Now, following [5], we will neglect the expansion of space. So, If we define the following variables it thus follows that To obtain an approximate solution for χ k , we expand χ k as The expression (29) is valid, if χ 1 k has not terms that grow without bound as t → ∞ . So, it is necessary to crosscheck this condition after we obtain an explicit expression for χ 1 k . Substituting (29) into (28), and keeping all terms up to second-order in ε , yields The first equation in (30) gives where b 1 and b 2 are the integration constants. Using (31) and the following variables the second equation in (30) can be solved as Where Q 1 , Q 2 , are the integration constants, and C ( x ), S ( x ) are the Fresnel integrals ( see Appendix B). From properties of the Fresnel integrals[17], we conclude that expression (33) has not any terms that grow without bound as t →∞ . So, the parametric resonance is absent in (25). Therefore, we expect that the perturbation approximation for χ , which is given by (29), is assured for all times. If we used (25) instead of (6b), we would have terms that grow without bound as t →∞ , which are sources of the parametric resonance during dust-like phase of the scalar field.", "pages": [ 3, 4 ] }, { "title": "ACKNOWLEDGMENTS", "content": "I would like to thank S. Vasheghani Farahani for read the manuscript. I am grateful for helpful discussions with H. Mohseni Sadjadi, P. Goodarzi.", "pages": [ 4 ] }, { "title": "Appendix A", "content": "In this appendix, to show the power of the method that is used in this paper, we re-derive (7), by the averaging method. By introducing β ≡ mt , and V ( ϕ ) = 1 2 m 2 ϕ 2 , equations (6a) and (6b) can be written to give Now, consider the following relations where g ( β ) is an arbitrary function. Differentiation of the righthand side of (A2a) must be equals to the righthand side of (A2b). Hence we have Substitution of (A2a) and (A2b) into (A1) yield By algebraic manipulations, H ' and g ' can be found from Equations (A3) and (A4) as These expression for H , and g ( β ) are exact. The advantage of the averaging method is that an approximation to the solution of (A5) can be obtained by replacing (A5) with its averaged equations as follows: if we keeping H and g fixed, the righthand side of (A5) are π -periodic in β . By noting that where <> is used to indicate that we average over β , the averaged equations can be solved as where g(0) is an arbitrary constant. Using (A2a) and (A7), we have where H ( t ) is given by (A7). According to the averaging method, the above expressions for Hubble parameter and the scalar field are valid for H ( t ) < m . The results are the same as those obtained in [5], which are also valid for H ( t ) < m .", "pages": [ 4, 5 ] }, { "title": "Appendix B", "content": "The Fresnel integrals are defined by [17] A series expansion for x < 1 gives Therefore, lim x → 0 C ( x ) = lim x → 0 S ( x ) = 0. Asymptotic expansion of the integral are given by Hence, lim x →∞ C ( x ) = lim x →∞ S ( x ) = 1 / 2.", "pages": [ 5 ] } ]

[ { "title": "Effects of massive photons from the dark sector on the muon content in extensive air showers", "content": "Jan Ebr ∗ , Petr Neˇcesal Institute of Physics of the Academy of Sciences of the Czech Republic, Na Slovance 1999/2, 18221 Prague 8, Czech Republic", "pages": [ 1 ] }, { "title": "Abstract", "content": "Inspired by recent astrophysical observations of leptonic excesses measured by satellite experiments, we consider the impact of some general models of the dark sector on the muon production in extensive air showers. We present a compact approximative expression for the bremsstrahlung of a massive photon from an electron and use it within Monte Carlo simulations to estimate the amount of weakly interacting photon-like massive particles that could be produced in an extensive air shower. We find that the resulting muon production is by many orders of magnitude below the average muon count in a shower and thus unobservable. Keywords: dark matter, bremsstrahlung, extensive air shower, muon production", "pages": [ 1 ] }, { "title": "1. Motivation", "content": "The relatively recent observations of excess lepton fluxes from space, as measured by PAMELA [1] and ATIC [2] have motivated large interest in models of dark matter annihilation that could explain these data, while staying in agreement with other existing astrophysical evidence. For the ultra-high energy cosmic ray (UHECR) experiments, neither these low-energy fluxes, nor their hypothetical parent particles are directly observable. But the common feature of such models is that they need to add new physics to increase the production of leptons with respect to hadrons in the current universe. Email addresses: [email protected] (Jan Ebr), [email protected] (Petr Neˇcesal) While this production is mainly targeted at electrons and positrons, many of such processes also lead to extra production of muons. Some evidence of disagreement in muon production in extensive air showers initiated by high-energy cosmic rays with the predictions of Monte Carlo simulations has been given already by the DELPHI [3], ALEPH [4] and L3 [5] experiments at LEP and it has been repeatedly reported by the Pierre Auger Observatory [6]. In both cases, the data indicate that the current interaction models may significantly underestimate the number of muons produced. It is then only natural to ask, whether some of these extra muons could be accounted for if some of the above mentioned new physics is incorporated into the Monte Carlo simulations. Instead of considering every single model of the dark sector that has ever been proposed (for a very recent review of models with specific references, see Ch. 4 of [7]), we turn to the work [8], which is rather general. There it is argued that, considering not only the above-mentioned excess, but also general cosmological observations and direct dark matter searches, it is not unreasonable to expect the TeV-scale dark matter to be accompanied by a relatively light particle with mass around 250 MeV and some weak coupling to ordinary matter. This idea is further corroborated in [9] for the special case of such a particle being essentially a massive photon that couples to ordinary matter via kinetic mixing suppressed by a small factor of the order of /epsilon1 ≈ 10 -2 -10 -3 . We call this particle a 'dark photon' for brevity as the factor /epsilon1 effectively appears in any vertex that includes both a standard model particle and a dark photon, thus making it difficult to detect by electromagnetic interactions. This scheme is not only backed by a compelling theoretical motivation, but also relatively simple to implement as a first look into the topic, yet reasonably general; thus we focus on it in the rest of the paper. In the following article [10] the authors show that this model leads to the prediction of specific collider signatures in the form of 'lepton jets' stemming from the prediction of the TeV-scale particles in hadronic collisions. These events are too rare to have any effect on extensive air showers, as there are only hundreds of sufficiently high-energy hadronic interactions in a single air shower, which itself is a rather rare event - the total luminosity in UHECRs is simply too small for even Standard model electroweak effects to have any impact on observable data, even more so for exotics. Nevertheless, this model is still interesting because the dark photon, being coupled to ordinary electric charge, albeit weakly, can be produced via bremsstrahlung from electrons. The amount of electromagnetic interactions of photons and leptons in each shower is by many orders of magnitude larger than that of hadronic interactions and so it is not immediately obvious what size of cross-section for the dark photon bremsstrahlung (investigated in the next section) is needed to produce observable effects. The attractive feature of a massive photon is that it can decay into a pair of a charged particle and its antiparticle. Additionally, the dark photon is 'dark', that is, it has limited interactions with ordinary matter. Thus, dark photons in the relevant range of masses will almost always decay instead of producing a pair in the electromagnetic field of an atom in the air. In the case of pair production, almost all produced pairs are electron-positron as the cross-section falls with the fourth power of the lepton mass, whereas the decay of the dark photon proceeds democratically into every kinematically possible final state, save for threshold effects. Thus, for mass of the dark photon m γ ∈ (212 , 280) MeV, for every dark photon produced, there is on average one muon added to the shower. For higher masses, pion final states are possible, but muon production is still sizeable.", "pages": [ 1, 2, 3 ] }, { "title": "2. Dark photon bremsstrahlung", "content": "The problem of bremsstrahlung of a massless photon from a lepton interacting with an atomic target in quantum electrodynamics is a well-known one and, to the leading order, it is exhaustively described in [11]. Interestingly, we did not find an expression for the bremsstrahlung of a massive photon in any literature, so we had to derive one ourself. Elementary as it may seem, the calculation is actually quite tedious. Thus, even though it is technically possible to just add a photon mass into the equations in [11] and proceed, this would be a major task and prone to errors. Instead we note the work [12] where it is shown that similar results can be derived using the computationally much simpler Weizsacker-Williams approximation, where the 2 → 3 problem is reduced to a 2 → 2 Compton scattering times a factor determined by kinematics and the scattering target. Schematically where P i is the initial four-momentum of the target, P 1 and P 2 are the initial and final four-momenta of the lepton, k is the four-momentum of the produced (massive or not) photon, α = e 2 / 4 π is the fine structure constant and χ is a factor that involves the form-factors of the target, which is independent of the 2 → 2 process. The subscript t = t min denotes that the 2 → 2 process is evaluated using a particular kinematic set up. In [13], this approximation is used for the bremsstrahlung of a massive axion. While the resulting formula is not directly applicable to the production of a massive photon, most of the work is actually done. The difference is only in the matrix element for the 2 → 2 process, which is a well-known function. The difficult part, which is the kinematics, is exactly the same for any massive particle, while being vastly different from the case of a massless photon. From eq. (7) of [13] we observe that the kinematics for the 2 → 2 process can be worked out so that where |M| is the absolute value of the invariant matrix element for the 2 → 2 process averaged over initial state polarisations and summed over final state polarisations. For massive photons, where we can safely neglect the electron mass if we are interested in dark photons capable of decaying into two muons. This approximation was numerically checked against the full result and the agreement is better than a fraction of a per cent for m γ = 250 MeV in almost the whole range of x , while the length of the formula is significantly reduced. Using eq. (1) and kinematics, we deduce that the bremsstrahlung cross-section is where x is the fraction of E 1 carried by the produced dark photon and θ is its production angle with respect to the incoming electron in the laboratory frame. Here we must keep the electron mass non-zero not only because of the behaviour for x → 1 but also because the γ -factor of the electron can be huge. To proceed with the angular integration we must specify the χ -factor. Again, we take it from [13]. In the 'complete screening' limit it can be written as where Z is the atomic number of the target. 'Complete screening' refers to an approximation valid when - an explicit numerical calculation again shows, that it is well justified. To make the production of a 250 MeV dark photon even possible, the γ -factor squared of the electron has to be in the order of at least 10 5 and thus the scattering is strongly suppressed for large angles. The suppression is in fact strong enough that we can extend the integral in θ to infinity, yielding a much more compact analytical result, namely Here we keep only the electron masses that cannot be neglected by any means. Our ultimate goal is to compare this expression with the well-known formula for massless photon bremsstrahlung, that is Note that in the massless case, the term proportional to Z + Z 2 is often neglected, whereas in the massive case, it is the dominant contribution to the cross-section. The x → 0 divergence for massless photons is obviously removed by the photon mass and the expression for massive photons is peaked at 1. What is more important, for interesting values of m γ , the suppression of the cross-section by the photon mass is huge for almost every x , with a typical value of m γ as shown in Figure 1.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3. Simulations", "content": "As the cross-section for dark photon production is very small, we can incorporate this effect into full Monte Carlo simulations of the air showers in a very simple way: for each bremsstrahlung event in the shower we give either the ratio of eqs. (7) and (8) using the kinematics of the particular scattering, or zero when a dark photon could not be emitted. Then we sum these values over the whole shower, resulting in an approximate mean number of massive photons that would be produced in the given shower. This scheme is neglecting the influence of dark photon production on the rest of the shower and exact energy conservation, but we assume that the production rate is small compared to the overall number of particles in the shower and thus such correction will be very small. We will check the validity of this assumption after presenting the results. Specifically, we use the CORSIKA [14] program (version 6.900) for detailed simulation of extensive air showers. The EGS4 routines are used for the electromagnetic cascade, as they produce the necessary data for individual particles, as opposed to the analytical NKG package. The EGS4 also incorporates the LPM (Landau-Pomeranchuk-Migdal) effect [15]. The Earth's magnetic field and altitude were adjusted to the Pierre Auger Observatory site. Hadronic interactions were primarily treated by GHEISHA [16] (low-energy) and QGSJET II [17] (high-energy) models and simulations were carried out with primary proton and iron particles at primary energies 10 18 eV, 10 19 eV and 10 20 eV respectively, at zenith angle 38 · as the most common arrival direction for a detector located at a flat surface. The azimuth is generated randomly. To estimate the effect of the choice of hadronic models, we compare the results at one chosen primary particle type (proton) and energy (10 19 eV) with simulations using either FLUKA [18] for the lowenergy hadronic interactions or EPOS 1.99 [19] or SIBYLL 2.1 [20] for the high-energy interactions. Together we carried out 9 sets of simulations with different settings with 100 simulations in each set.", "pages": [ 6, 7 ] }, { "title": "4. Results", "content": "To give concrete numbers, we choose the mass of the photon to be 250 MeV as a favourable value for the muon production. From Fig. 1 one can see that the production of dark photons is larger for smaller values of photon mass, but when we want to consider muon production, the obvious lower limit is 212 MeV (twice the muon mass) - we choose a value slightly higher to avoid dealing with threshold effects; incidentally it is the value considered as likely in [8] based on astrophysical data. In Table 1 we present the average numbers of dark photons (per shower) produced in simulations with GEISHA and QGSJETII with different primary energies and particle type. The differences between individual simulations are a combination of physical fluctuations of the interactions, known as the 'shower-to-shower fluctuations' [21], and of the effect of the particle thinning [22]. The distributions of these fluctuations are not Gaussian - in fact, they follow approximately log-normal distributions and thus the mean values and their uncertainties are heavily influenced by the tails of the distributions. To give the reader a better idea of the fluctuations, we have indicated the range of values in which the central 68 % (corresponding to 1 σ for the normal distribution) of each distribution lies, instead of just the standard deviation. The increase of the fluctuations between simulations (and thus of the uncertainty of the mean value) with energy is related to the effect of thinning which is set relatively to the primary energy and thus it is a stronger effect at higher energy (otherwise the computing times would be prohibitively large). While the distribution of the results from individual simulations is not normal, we assume that the distribution of the means of different samples of a given size approaches the normal distribution and thus we estimate the error of the mean as the standard deviation of each sample divided by square root of the number of simulations. In Table 2 we present for one particular energy and primary particle a comparison between results obtained using different low- and high-energy hadronic interaction models. All these results have to be further multiplied by the square of the suppression factor /epsilon1 introduced in Sect. 1 as there is one vertex with dark photon and normal matter in the Feynman diagram for Bremsstrahlung and the matrix element is squared in the cross-section. The exact value of /epsilon1 is to some extent a free parameter of the model, thus we present the results without it. Note that to avoid direct detection and cosmological constraints, this factor has to be of the order /epsilon1 2 ≈ 10 -4 or even lower. A dark photon with m γ = 250 MeV decays to a pair of muons almost exactly in 50 % of cases, so there is on average one muon ( µ + or µ -) produced per one dark photon.", "pages": [ 7, 8, 9 ] }, { "title": "5. Discussion", "content": "Both the composition of the cosmic rays at ultra-high energies and the correct choice of a hadronic interaction model are currently unknown. Nevertheless we observe that, at our comparison energy of 10 19 eV, all the values obtained for different choices of both composition and hadronic model are compatible with each other within less than 2 standard errors. 1 Thus our analysis is largely independent of these unknown inputs. The amount of dark photons produced is dependent on the primary energy, but this result is expected as even with a relatively simple model [23] it can be shown that the overall number of particles in a shower scales approximately linearly with primary energy. For comparison with our values, note that total number of muons on the ground level for a single shower in our set of simulations is of the order of 10 7 - 10 9 (depending mainly on primary energy and composition). Even at the highest energy observed in cosmic rays (10 20 eV) and using the maximal value of /epsilon1 2 = 10 -4 , we predict less than one muon originating from a dark photon to be produced in 1000 showers. As the flux of primary particles of such energy is very small (about 0.01 particle per km 2 per year [24]) even the current largest UHECR detector, The Pierre Auger Observatory, has not yet observed that many events at this energy (in 2011 they reported less than a hundred events above 5 × 10 19 eV [25]). The situation is slightly better at lower energies: for 10 18 eV (again using the maximal value of /epsilon1 ) our results permit only one muon from a dark photon in approximately 30000 showers, but as the energy spectrum is very steep, the flux is much larger, ∼ 100 primary particles per km 2 per year. Still, it is obvious that the influence of possible dark photon production on the muon content in extensive air", "pages": [ 9 ] }, { "title": "showers is extremely small.", "content": "From the previous discussion it also follows that the assumption in our method (that the effect of the dark photon production on the simulation of the rest of the shower is small) is justified, as in the vast majority of showers, there are no dark photons produced at all.", "pages": [ 10 ] }, { "title": "6. Conclusions", "content": "Motivated by recent observational and theoretical development in describing the possible dark matter in our Universe on one side and the discrepancy between the observed muon number in extensive air showers with simulations on the other, we have conducted a study where we tried to see if a particular feature of the former (the production of dark photons) could help the fix the discrepancy in the latter. We found out that even for the most favourable values of parameters that can be feasibly adopted ( m γ = 250 MeV and /epsilon1 2 = 10 -4 ) and for various energies, primary particles and interaction models, the muon production in the EAS caused by dark photon decay is negligible. This result is valid for any massive photon-like particle with an interaction that is governed by the standard quantum electrodynamics, modified only by the suppression factor /epsilon1 . As an useful by-product, we presented a closed-form expression for the bremsstrahlung of a massive photon from a lepton in the framework of the Weizsacker-Williams approximation.", "pages": [ 10 ] }, { "title": "Acknowledgements", "content": "We are grateful to Jan ˇ R'ıdk'y for the idea to consider these models in extensive air showers and for his valuable advice. The contribution is prepared with the support of Ministry of Education, Youth and Sports of the Czech Republic within the project LA08016 and with the support of the Charles University in Prague within the project 119810.", "pages": [ 10 ] } ]

[ { "title": "Sabine Hossenfelder", "content": "Nordita, Roslagstullsbacken 23, 106 91 Stockholm, Sweden", "pages": [ 1 ] }, { "title": "Abstract", "content": "It is generally believed that quantum gravity is necessary to resolve the known tensions between general relativity and the quantum field theories of the standard model. Since perturbatively quantized gravity is non-renormalizable, the problem how to unify all interactions in a common framework has been open since the 1930s. Here, I propose a possibility to circumvent the known problems with quantizing gravity, as well as the known problems with leaving it unquantized: By changing the prescription for second quantization, a perturbative quantization of gravity is sufficient as an effective theory because matter becomes classical before the perturbative expansion breaks down. This is achieved by considering the vanishing commutator between a field and its conjugated momentum as a symmetry that is broken at low temperatures, and by this generates the quantum phase that we currently live in, while at high temperatures Planck's constant goes to zero. Keywords: Quantum gravity, Semi-classical gravity", "pages": [ 1 ] }, { "title": "1. Why Quantize Gravity?", "content": "The gravitational interaction stands apart from the other interactions of the standard model by its refusal to be quantized. This still missing theory of quantum gravity is believed necessary to complete our understanding of nature. Strictly speaking, quantizing gravity is not the problem - gravity can be perturbatively quantized. The problem is that the so quantized theory is perturbatively nonrenormalizable and cannot be understood as a fundamental theory. It breaks down at high energies when quantum gravity would be most interesting. The attempt to find a theory of quantum gravity has lead to many proposals, but progress has been slow. Absent experimental evidence, reasons for the necessity of quantum gravity are theoretical, most notably: These points have all been extensively studied and discussed in the literature. The most obvious way out seems to be a non-perturbative theory, and several attempts to construct one are under way. It is worthwhile for the following to identify the problems with coupling a classical to a quantum field. The one problem, as illuminated by Hannah and Eppley is that the fields would have different uncertainty relations, and their coupling would require a modification of the quantum theory. Just coupling them as they are leads to an inconsistent theory. The beauty of Hannah and Eppley's tought argument is its generality, but that is also its shortcoming, because it does not tell us how a suitable modification of quantum theory could allow such a coupling to be consistent. The second problem is that it is unclear how, mathematically, the coupling should be realized, as the quantum field is operator-valued and the classical field is a function on space-time. One possible answer to this is that any function can be identified with an operator on the Hilbert space by multiplying it with the identity. However, the associated operators would always be commuting, so they are of limited use to construct a geometrical quantity that can be set equal to the operator of the stress-energy-tensor (SET) of the quantum fields. Another way to realize the coupling is to extract a classical field from the operator of the SET by taking the expecation value. The problem with this approach is that the expectation value may differ before and after measurement, which then conflicts with local conservation laws in general relativity. Coupling the classical field to the SET's expectation value is thus usually considered valid only in the approximation when superpositions carry negligible amounts of energy. These difficulties can be circumvented by changing the quantization condition in such a way that gravity can be perturbatively quantized at low energies, but at energies above the Planck energy - energies so high that the perturbative expansion would break down - it becomes classical and decouples from the matter fields. The mechanism for this is making Planck's constant into a field that undergoes symmetry breaking and induces a transition from classical to quantum. In three dimensions, Newton's constant is G = ¯ hc / m 2 Pl , so if we keep mass units fix, G will go to zero together with ¯ h , thus decoupling gravity. It should be emphasized that the ansatz proposed here does not renormalize perturbatively quantized gravity, but rather replaces it with a different theory that however reproduces the perturbative quantization at low energies by construction. We will show in the following how this change of the quantization condition addresses the above listed three problems. In the approach considered here, Planck's constant is treated as a field. Since the normal, fixed, value of Planck's constant (denoted ¯ h 0 ) appears as a vacuum expectation value, we could divide the field by this vacuum expectation value to obtain a dimensionless quantity. We will not do this in the following, because it would make the interpretation less intuitive. It should be pointed out though that the approach can be reformulated in terms of a dimensionless field because we do have access to a dimensionful constant that serves as reference. In the following we set c = 1 but keep ¯ h and Boltzmann's constant k B . The signature of the metric is (+ , -, -, -) .", "pages": [ 1, 2, 3 ] }, { "title": "2. Quantization by Spontaneous Symmetry Breaking", "content": "Consider a massless real scalar field f ( x , t ) with canonically conjugated momentum p f ( x , t ) . Second quantization can be expressed through the equal time canonical commutation relations or, equivalently, by the commutation relations for annihilation and creation operators. If ¯ h = 0, they commute and one is dealing with a classical field. The dimension of a scalar field is most easily found by noting that the kinetic term ¶ n f¶ n f in the Lagrangian should have dimension of an energy density, so that the integral over space-time has the dimension of an action. One has a freedom here whether a constant is in front of this term. If one is dealing with a quantum theory, one often puts an ¯ h 2 there because then each derivative together with an ¯ h gives a momentum. Since we eventually want to make contact to a classical theory, we will not put any ¯ h 's in front of the kinetic term, but instead chose the classical convention from the start on. In three spatial dimensions this mean then that the dimension of the Lagrangian is [ E / L 3 ] , where E denotes energy and L denotes length. So the dimension of f is [ E 1 / 2 L -1 / 2 ] , and that of the conjugated momentum p f is [ E 1 / 2 L -3 / 2 ] . The Lagrangian is without additional factors, and p f = ¶ L / ¶ ˙ f as usual. With this convention, what is in the quantum theory referred to as the 'mass term' has actually the form f 2 / l 2 ∗ , where l ∗ is a length scale. This is because the unquantized field is not a priori associated with particles that could be assigned a mass. With that dimensional consideration, let us now look at the quantization prescription from a new perspective. The requirement that the fields commute, f ( x ) p ( y ) = p ( y ) f ( x ) , can be understood as a symmetry, where we find the classical theory as the symmetric, commuting, phase and quantum mechanics in the phase with broken symmetry where the fields do not commute. In the familiar language of spontaneous symmetry breaking we parameterize Here, ¯ h 0 is the normal (measured) value of Planck's constant, m ∗ is a constant of dimension mass, and a ( x , t ) is a real scalar field. With a Fourier-transform, one can express this in terms of annihilation and creation operators for f of the form where ( w k , glyph[vector] k ) is the wavevector, w 2 k = | glyph[vector] k | 2 , and ˜ a is the Fourier transform of a If a is constant, then ˜ a reduces to the usual delta-function. For the operations leading to the above expressions, we have merely used the definition of scalar products that are not affected by the modification of the quantization condition. We have distributed the dimensionful constant so that a has no prefactor in the path integral, ie the dimension of a is [ 1 / L ] . We have further chosen to keep a mass scale fixed rather than a length scale for the reason indicated in the introduction. We can then write ¯ h ( x , t ) = ¯ h 2 0 a / m ∗ for Planck's constant, which is now a field. The field a itself is quantized according to the same prescription as f up to constants, ie [ a ( x , t ) , p a ( y , t )] = i ¯ h 0 a ( x , t ) d 3 ( x -y ) . If we expand a in annihilation and creation operators, these too obey the commutation relation (5) with a different prefactor. Now we add a kinetic term for a and a symmetry breaking potential, so that the transition amplitude in the path integral is exp ( -i ˆ S ) , where the hat indicates a dimensionless quantity, and The minima of the potential are at a = ± m ∗ / h 0 , and we live in the vacuum with ¯ h = ¯ h 0 . (In order to render the kinetic term of f symmetric under a sign change of a , one would have to also change the sign of m ∗ along with a . This makes the example of the scalar field somewhat unappealing, but we will see below that this is not necessary for the case of gravity, which is what we are actually interested in.) As mentioned in the introduction, to obtain a dimensionless quantity whose value is meaningful regardless of units, one could now normalize the field ¯ h to ¯ h 0 . However, this would make the interpretation of the appearing quantities less intuitive, so we will not do this redefinition here. The previously made statement that the failure of the field and its momentum to commute represents a breaking of symmetry should be understood as a rephrasing of the breaking of symmetry in the ground state of the above potential. This whole exercise can be summarized by saying that we've chosen the dimensions of the fields so that in the path integral the field a , which determines Planck's constant, appears in the usual form. Thus, we are left with an action of two scalar fields, where the symmetry breaking part works as normally. The only unusual is the quantization condition for the the fields. Symmetry breaking then happens with a drop of temperature because the minima of the potential change with the temperature [3]. In the limit T glyph[greatermuch] T c = 2 m ∗ k B , the finite temperature corrected potential receives additional terms where k B is Boltzmann's constant, n is the number-density of the field, and the dots indicate terms of higher order in ¯ h 0 . Note that they are indeed higher order in ¯ h 0 and not in ¯ h because there's no additional ¯ h in the potential. The first correction term can be interpreted as a temperature-dependent mass term for the a -field that counteracts the negative mass term in the potential. The second correction term is the free energy of a massless spin-0 boson. To understand what happens if the symmetry is broken, we first note that in principle the temperature dependence of the potential is not a quantum effect, it is an in-medium effect. However, Planck's constant appears in statistical mechanics as the normalization constant for the measure of momentum space. Taking it to zero does not create a classical limit, but instead ill-defined quantities. One frequently considers limits in which Planck's constant is small compared to some other quantity of dimension action. That is formally similar to changing Planck's constant, but has a very different physical meaning. We don't want to look at the limit where Planck's constant is small compared to some other value, we want to change it. This is not a process that statistical thermodynamics in its standard form deals with. Eg the free energy of the bosonic gas that appears in the potential is normally written ( kT ) 4 / ¯ h 3 , which seems to diverge with ¯ h to zero. But to obtain a meaningful limit, we have to keep a physically meaningful quantity fixed. As the above notation already suggests, we opt to keep the number density fixed when varying a but not any other variable, ie This does of course not mean the number density is constant. To begin with it depends on the temperature. What Eq (8) expresses is that there is no additional temperature-dependence in n from the variation of a . In other words, we identify n =( kT / ¯ h 0 ) 3 . There is another way to look at this limit. Planck's constant is the measure in phase space, and can be understood as a product of a length for the coordinate space and an energy for momentum space. We have introduced a mass scale in the beginning, which we keep fix, so then the unit for the length has to go to zero with Planck's constant, which is why the number-density would diverge in these units. This unphysical behavior is cured by choosing a constant unit of length instead, which has the same effect as putting ¯ h 0 in the denominator of n . The potential for the quantized scalar field further receives corrections from quantum fluctuations which leads to the Coleman-Weinberg effective potential. In the usual case for a f 4 -interaction, the loop corrections become large for small f and therefore quantum fluctuations can break the classical symmetry. Since in our case the potential for the field a takes the usual form, the same can happen here. It is less clear what happens with the loop corrections in the high-temperature limit when the modification of the quantization condition becomes relevant. As mentioned above, what we actually want to unquantize is not a f 4 theory, but gravity. In this case, the normalized action is Other matter fields can be added like the scalar field example. This looks very much like Brans-Dicke, but has a different kinetic term and the symmetry breaking potential. The most relevant difference though is the quantization condition. (Note that the first term does not change sign under a →-a .) In summary, for this to work, one needs the normalized action of the form (6) together with the quantization postulate (3). Since the quantization postulate is essentially a constraint on the fields, one could alternatively add it with a Lagrange multiplier to the Lagrangian. That might not be particularly elegant, but the point is here just to show that it can be done.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3. How unquantization addresses the problems with quantizing gravity", "content": "The coupling constant of gravity to matter in the perturbative quantization is √ G = √ ¯ h / m Pl and thus goes to zero with ¯ h → 0, provided the Planck mass is held fix. This limit was previously discussed from a different perspective in [4, 5] as being of interest as a non-quantum relic of quantum gravity in the limit ¯ h , G → 0. Here, we suggested to not only consider this a corner of the parameter space, but a limit that is actually realized at high energies. The weakening of gravity at high energies can also be found in scenarios where gravity is asymptotically safe. The difference between both cases is most easily shown by a figure, see Fig 1. Let us now revisit the motivations for the need for quantum gravity: a classical field coupled to a quantized field. There is a phase when both are quantized and coupled, and one when both are unquantized and decoupled. One might say that fundamentally the fields are neither classical nor quantum in the same sense that water is fundamentally neither liquid nor solid. It remains to be addressed what happens to the divergence of the perturbative expansion of perturbatively quantized gravity. This is necessary to understand what happens in a highly energetic scattering event where the mean field approximation that is made use of for the symmetry restoration with temperature is not applicable. We first note that at high energies the operator that contributes the negative mass term to the potential of a becomes less relevant than the quartic term, so at least asymptotically the symmetry should be restored in the sense that the vacuum expectation value of ¯ h goes to zero. With the field rescaling that we have used here (see also [8]), the perturbative expansion is also an expansion in ¯ h , but it is not quite as easy as this for the following reason. Consider an S -matrix expansion within the above model. The expansion of the S -matrix works as usual, but the modification comes into play when we look at a transition amplitude of that S -matrix with some interaction vertices. It is evaluated by using the commutation relations repeatedly until annihilation operators are shifted to the very right side, acting on the vacuum, which leaves c -numbers (or the Feynman rules respectively). Now, if Planck's constant is a field, every time we use the commutation relation, we get a power of a and the respective factors of the constant h 0 and m ∗ . So, in the end we do not have powers of the vacuum expectation value of ˜ a , but the expectation value of powers of ˜ a . We thus need to take a second step, that is using the commutation relations on ˜ a itself. But exchanging any two terms in the expansion of ˜ a will only generate one new ˜ a from the commutator (and the respective powers of ¯ h 0 ). One can thus get rid of the expectation value of powers, so that in the end we will have a series in h 0 and vacuum expectation values of ˜ a . If a goes to zero only for infinitely large energies, then one cannot tell if the series is finite without further investigation of its convergence properties. However, if we consider the symmetry breaking potential to be induced by quantum corrections at low order, the transition to full symmetry restoration may be at a finite value of energy. In this case then, the quantum corrections which would normally diverge would cleanly go to zero, removing this last problem with the perturbative quantization of gravity. We have discussed here a model based on a modification of the quantization condition that we have interpreted as a procedure of unquantization. This modification should however primarily be understood as a motivation for the model, and as a guide for the interpretation of its effects. It is possible that the approach can be reformulated in other ways that offer a different interpretation of the behavior. The solution proposed here has the potential to address a long standing problem in theoretical physics. To be successful in that however, a closer investigation is required. There is, most importantly, the question of experimental contraints from coupling the scalar field that is Planck's constant to gravity, which might lead to a modification of general relativity and observable consequences. It also remains to be seen if a concrete example for the symmetry breaking can be constructed in which it can be shown explicity, and beyond the general argument for such a possibility given above, that the perturbation series converges. The betafunction of the model and its relation to the case of Asymptotically Safe Gravity is of key interest here. And while the avoidance of the Big Bang and black hole singularities are the most relevant cases for our universe, it remains to be seen if a more widely applicable statement can be derived that addresses the singularity theorems in general.", "pages": [ 7, 8, 9, 10 ] }, { "title": "Acknowledgements", "content": "I thank Cole Miller, Roberto Percacci and Stefan Scherer for helpful discussions.", "pages": [ 10 ] } ]

[ { "title": "New Geometric Transition as Origin of Particle Production in Time-Dependent Backgrounds", "content": "Sang Pyo Kim ∗ Department of Physics, Kunsan National University, Kunsan 573-701, Korea † and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: June 16, 2021) By extending the quantum evolution of a scalar field in time-dependent backgrounds to the complex-time plane and transporting the in-vacuum along a closed path, we argue that the geometric transition from the simple pole at infinity determines the multi-pair production depending on the winding number. We apply the geometric transition to Schwinger mechanism in the time-dependent vector potential for a constant electric field and to Gibbons-Hawking particle production in the planar coordinates of a de Sitter space. PACS numbers: 12.20.-m, 04.62.+v, 03.65.Vf The interaction of quantum field with a background gauge field or spacetime can produce particles as a nonperturbative quantum effect. Schwinger mechanism is pair production by a constant electric field, which provides enough energy to separate charged pairs from the Dirac sea [1]. Hawking radiation is the emission of particles from vacuum fluctuations, which are separated by the horizon of a black hole [2]. Recently Hawking radiation has been interpreted as quantum tunneling of virtual pairs near the horizon of the black hole [3]. Also it has been known for long that an expanding spacetime produces particles [4] and de Sitter (dS) radiation has a thermal distribution [5]. In quantum field theory the out-vacuum of a quantum field may differ from the in-vacuum through the interaction with a background field and may be expressed as multi-particle states of the in-vacuum via the Bogoliubov transformation [6]. In the in-out formalism the scattering matrix between the in-vacuum and the out-vacuum determines the probability for the in-vacuum to remain in the out-vacuum, which in turn is given by the pair-production rate for bosons or fermions [7]. The exact vacuum persistence and pair production requires the solution of the quantum field in the background field. With limited knowledge of exact solutions, approximation scheme proves a practical approach or provides an intuitive understanding for pair production. Various approximation schemes have been proposed for Hawking radiation [8]. Each Fourier mode of a scalar field in the time-dependent vector potential for a constant electric field and in the planar coordinates of a dS space describes the scattering problem over a potential barrier. In the phase-integral method Schwinger pair-production rate is determined by the action integral among quasi-classical turning points in the complex plane of time or space [9, 10]. In particular, the action integral has been proposed as the contour integral in the complex plane for Schwinger mechanism [9] and for Hawking radiation [11]. Furthermore, the Stokes lines and anti-Stokes lines for more than one pair of quasi-classical turning points distinguish boson and fermion pair production [10] and the dimensionality of particle production in dS spaces [12]. The complex analysis has been used in connection with particle production [13, 14], the instanton action [15, 16], and the worldline instanton [17, 18]. In this paper we propose the geometric transition of the Hamiltonian in the complex-time plane as a new interpretation of particle production in time-dependent backgrounds. It has been observed that a time-dependent Hamiltonian can have the geometric transition in the complex-time plane [19-21]. In the functional Schrodinger picture each Fourier mode of quantum field in a time-dependent background has the Hamiltonian with time-dependent frequency and/or mass. We argue that the evolution of the in-vacuum along a complex closed path of non-zero winding number leads to the geometric transition from the simple pole at infinity and results in particle production, in strong contrast with the trivial real-time evolution without level-crossings. The evolution of the in-vacuum along a path in the complex-time plane is reminiscent of the closed-time path integral in the in-in formalism [7]. A complex scalar field with mass m and charge q in a constant electric field in the (d+1)-dimensional Minkowski spacetime has the Fourier-decomposed, time-dependent Hamiltonian [in units of c = /planckover2pi1 = 1] where Here k ⊥ and k ‖ are the transverse and longitudinal momenta and the vector potential is A ‖ ( t ) = -Et , which provides a time-dependent background. The complex scalar field is equivalent to two real scalar fields: one for particle and the other for antiparticle. In the planar coordinates of the (d+1)-dimensional dS space where H HC is the Hubble constant, a massive real scalar field has the time-dependent Hamiltonian where In the functional Schrodinger picture, the quantum state obeys the time-dependent Schrodinger equation For the purpose of this paper, it is sufficient to consider the time-dependent oscillator defined along the real-time axis as In the real-time evolution, the annihilation and creation operators diagonalize the Hamiltonian as Thus, an initial state evolves by the evolution operator, which can be given by the time-ordered integral or product integral [22] In terms of the number states (9), the evolution operator can be further written as [21] where H D ( t ) and Φ( t ) denote the diagonal matrix and the column vector, respectively, and A ( t ) is the induced vector potential Here and hereafter overdots denote derivatives with respect to the real or complex time. Hence ˆ U ( t 0 , t 0 ) from t 0 to any future time t and back to t 0 along the real-time axis becomes unity since H D ( t ) and A ( t ) do not have any singularity due to ω ( t ) > 0, which is the case of charged scalars in a time-dependent vector potential or real scalars in a dS space. Note that the path from t 0 to t 0 along the real-time axis is a loop of zero-winding number, which will be denoted as C (0) ( t 0 ) with the base point t 0 . In other words, ˆ U ( C (0) ( t 0 )) = I in the complex-time plane and the scattering amplitude between the in-vacuum and the transported in-vacuum is unity along the real-time axis The in-in formalism thus becomes trivial as long as the real time is concerned. However, a time-dependent Hamiltonian, provided that it has a level-crossing in the complex-time plane, leads to the geometric transition amplitude, which is responsible for an exponential decay of the initial state and the transition to other states [19, 20]. In a similar manner, let us extend the Hamiltonian H ( t ) to the complex-time plane and assume that H ( z ) is analytic and the orthonormality 〈 m,z | n, z 〉 = δ mn holds. In a properly chosen Riemann sheet in the complex-time plane, the frequencies (2) and (5) have two branch points of the form /negationslash where f ( z ) is an analytic function. Though the complex frequency (16) has level-crossings in the whole complex plane, we cut two branch lines from z 0 and z ∗ 0 as shown in Fig. 1, which make ω ( z ) analytic in the proper Riemann sheet. Defining ˜ H ( z ) := H D ( z ) -A T ( z ), we notice that the matrix-valued ˜ H ( z ) does not commute with ˜ H ( z ' ) in general for z = z ' unless ω 2 ( z ' ) ˙ ω ( z ) = ω 2 ( z ) ˙ ω ( z ' ). Then, without level-crossings, all the time-ordered integrals of ˆ H ( z ) in the complex-time plane along closed paths of the same winding number and with the same base point t 0 on the real-time axis are equal to each other [22] Hence the time integral in the complex-time plane is independent of paths and depends only on the homotopy class of winding numbers. In fact, the path in the left panel of Fig. 1 has the winding number 1 and is equivalent to another path C (0) ( t 0 ) along the real-time axis plus the loop C (1) (0) of the winding number 1 encircling z = 0 while the path C (3) ( t 0 ) in Fig. 2 has the winding number 3. Thus, since the time integral along C (0) ( t 0 ) in the real-time axis is unity, the time integral along a curve C ( t 0 ) consisting of C (0) ( t 0 ) and C ( n ) (0) of winding number n around z = 0 becomes However, in the lowest order of the Magnus expansion [23], the residue theorem holds for the matrix-valued ˜ H ( z ) [22] where Res[ ˜ H ( z 1 )] is the residue of the simple pole z 1 at infinity [24]. In fact, the frequencies (2) and (5) do have the simple pole at infinity, so the scattering amplitude between the in-vacuum and the transported in-vacuum is approximately given by where the factor of 1 / 2 for the vacuum state (13) is taken into account and n is the winding number. The dynamical phase does not contribute to the scattering amplitude since it returns to t 0 and the frequency does not have any finite simple poles. The exponentially decaying scattering amplitude implies transitions to excited states, that is, particle production. The residue at infinity is found by the large z -expansion of the complex frequency (16) In the first case of charged scalars in the constant field, the proper Riemann sheet is the entire complex plane with branch cuts as shown in Fig. 1. Then the magnitude square of the scattering amplitude is the Schwinger pair-production rate for n -pairs of charged particles and antiparticles. It is analogous to the multiinstanton actions for pair production in the Coulomg gauge for static electric fields [15]. In the second case of real scalars in the dS space, the proper Riemann sheet -π/H HC < Im t ≤ π/H HC and a conformal mapping e H HC t = z may be chosen, in which the scattering amplitude square is the Boltzmann factor for Gibbons-Hawking radiation in the dS space. In summary, we showed that the geometric transition from a simple pole at infinity in the complex-time plane could explain particle production in a constant electric field and in a dS space. The Fourier-decomposed Hamiltonian for charged scalars in the constant electric field and for real scalars in dS space is infinite number of oscillators with time-dependent frequencies and/or mass. In the real-time evolution, any state prepared at an initial time that evolves into a future time and returns to the initial time remains in the same state with a trivial phase factor since there is no level-crossing, so the scattering amplitude between the in-vacuum and the transported in-vacuum is unity. However, we argued that the evolution along a closed path in the complex-time plane obtains a new geometric transition coming from the residue of the simple pole at the infinity, which differs from the geometric transition coming from level-crossings. Further, the scattering amplitude between the in-vacuum and the in-vacuum transported along a path of winding number n leads to production of n -pairs. Finally, a few comments are in order. First, it is worth to note that the geometric transition for the in-vacuum, though a consequence of nonstationarity of the Hamiltonian, resolves the factor of two puzzle for tunneling interpretation of Hawking radiation. The factor of two puzzle was explained in different ways [25-28] and by including the temporal contribution under the coordinate transformation from the embedding geometry [29-31]. The scattering amplitude between the in-vacuum and the transported in-vacuum takes the vacuum energy into account, which counts only a half of the energy quanta. Second, the massless limit of eq. (23) gives a unity scattering amplitude square between the in-vacuum and the transported in-vacuum in the complex plane as in eq. (15). In fact, under the conformal mapping z = e H HC t the infinity has a double pole and thus does not contribute to the residue, which implies that the probability for the transported in-vacuum to remain in the in-vacuum is unity. The result is consistent with no production of massless particles in dS spaces in the in-out formalism, but there are subtle issues in the massless limit [32, 33]. Third, the geometric transition can be generalized to a frequency that has many level crossings and finite simple poles, which is the case of generic time-dependent vector potentials and global coordinates of dS spaces and the Friedmann-Robertson-Walker spacetime [34]. Then the homotopy classes of paths are classified by finite simple poles inside the loops. The homotopy classes may have something to do with the Stokes phenomenon for particle production [10, 12], which is beyond the scope of this paper. Another issue not pursued in this paper is the stimulated pair production from an initial particle state.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "Acknowledgments", "content": "The author thanks Eunju Kang for drawing figures. This paper was initiated during the Asia Pacific School on Gravitation and Cosmology (APCTP-NCTS-YITP Joint Program) in Jeju, Korea in 2013, benefited from helpful discussions at the Workshop on Gravitation and Numerical Relativity (APCTP Topical Program) and completed at Yukawa Institute for Theoretical Physics, Kyoto University. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2012R1A1B3002852).", "pages": [ 5 ] } ]

[ { "title": "Dynamics of Potentials in Bianchi Type Scalar-Tensor Cosmology", "content": "M. Sharif ∗ and Saira Waheed † Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan.", "pages": [ 1 ] }, { "title": "Abstract", "content": "The present study investigates the nature of the field potential via new technique known as reconstruction method for the scalar field potentials. The key point of this technique is the assumption that Hubble parameter is dependent on the scalar field. We consider Bianchi type I universe in the gravitational framework of scalar-tensor gravity and explore the general form of the scalar field potential. In particular, this field potential is investigated for the matter contents like barotropic fluid, the cosmological constant and Chaplygin gas. It is concluded that for a given value of Hubble parameter, one can reconstruct the scalar potentials which can generate the cosmology motivated by these matter contents. Keywords: Scalar-tensor theory; Scalar field; Field potentials. PACS: 98.80.-k; 04.50.Kd", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The reality of cryptic dominant component of the universe distribution labeled as dark energy (DE) and its resulting phenomena of cosmic acceleration has become a center of interest for the researchers. The existence of this unusual sort of DE is supported by the observational results of many astronomical experiments like Supernova (Ia) [1, 2], Wilkinson Microwave Anisotropy Probe (WMAP) [3] and Sloan Digital Sky Survey (SDSS) [4], galactic cluster emission of X-rays [5], large scale-structure [6] and weak lensing [7]. These experiments reveal the present day cosmic acceleration by evaluating the luminosity distance relation of some type of objects known as standard candles. They also lead to the conclusion that our universe is nearly flat. In order to resolve these issues, numerous attempts are made which can be categorized on the basis of the used technique. Basically, two approaches have been reported in this context: the modification in the matter configuration of the Lagrangian density and the modification in the whole gravitational framework described by the action. The Chaplygin gas [8] and its modified forms [9], cosmological constant [10], tachyon fields [11], quintessence [12], viscosity effects [13] and k-essence [14] etc. are some DE candidates belonging to the first category. The second approach includes examples of modified theories like f ( R ) gravity [15], Gauss-Bonnet gravity [16], f ( T ) theory [17], f ( R, T ) gravity [18] and scalar-tensor theories [19]. The study of scalartensor theories in the subject of cosmology has a great worth due to its vast applications and success [20]. The complete history of the universe from the early inflationary epoch to the final era of cosmic expansion can successfully be discussed by using scalar field as DE candidate [21]. Basically, the alternating gravitational theories are proposed by the inclusion of some functions or terms as a possible modification of Einstein gravity that cannot be derived from the fundamental theory. This raises a question about the appropriate choice of these functions by checking their cosmological viability. However, the process of reconstruction provides a way for having a cosmologically viable choice of these functions. Such a procedure has been adopted by many researchers [22]-[31]. The reconstruction procedure is not a new technique as it has a long history for the reconstruction of DE models. In order to have a better understanding of this technique, we may refer the readers to study some interesting earlier papers [32]. Basically, this technique enables one to find the form of the scalar field potential as well as scalar field for a particular value of the Hubble parameter in terms of scale factor or cosmic time. It is worth investigating the nature of scalar field potential in the context of scalar-tensor theories. Using reconstruction approach, the nature of the field potential for a minimally coupled scalar-tensor theory has been discussed [22]. The scalar potentials for tachyon field [23] as well as for solutions involving two scalar fields [24] have been reconstructed through this technique. This is also extended to the modified gravitational frameworks including non-minimal coupled scalar-tensor theories [25], Gauss-Bonnet gravity [26], F ( T ) theory [27] and the non-local gravity model [28]. Kamenshchik et al. [29] used this technique to reconstruct the scalar field potential for FRW universe in the induced gravity and discussed it for some types of matter distribution which can reproduce cosmic evolution. The same authors [30] used superpotential approach to reconstruct the field potential for FRW model in a non-minimally coupled scalar-tensor gravity and explored its nature for different cases like de Sitter and barotropic solutions describing the cosmic evolution. In this paper, we discuss the nature of the field potential using the reconstruction procedure for locally rotationally symmetric (LRS) Bianchi type I (BI) universe model. The paper is organized as follows. In the next section, we provide a general discussion of this technique and explore the form of scalar field potential. Section 3 is devoted to study the field potentials using the barotropic fluid, the cosmological constant and the Chaplygin gas as matter contents. In the last section, we discuss and conclude the results.", "pages": [ 1, 2, 3 ] }, { "title": "2 General Formulation of the Field Potential", "content": "The scalar-tensor gravity is generally determined by the action [31] where U is the coupling of geometry and the scalar field, V is the selfinteracting potential, R is the Ricci scalar and ω is the interaction function. We can discuss different cases of scalar-tensor theories by taking different values of U ( φ ). When both U, ω are constants, the above action yields the Einstein-Hilbert action with quintessence scalar field, for U = φ with ω = ω 0 , ω ( φ ), it corresponds to simple Brans-Dicke (BD) and the generalized BD gravity with scalar potential, respectively. For U ( φ ) = 1 2 γφ 2 , where γ is any non-zero constant and constant ω , it leads to the action of the induced gravity. Anisotropic and spatially homogeneous extension of flat FRW model, BI universe with the expansion factors A and B is given by the metric [33] and the respective Ricci scalar is The average scale factor a ( t ), the universe volume V , the directional Hubble parameters ( H 1 along x direction while H 2 along y and z directions) and the mean Hubble parameter are given by /negationslash In order to deal with highly non-linear equations, we take a physical assumption for the scale factors, i.e., A = B m ; m = 0 , 1 [34]. This condition is originated from the fact that in a spatially homogeneous model, the normal congruence to homogeneous expansion corresponds to the proportionality of the shear scalar σ and the expansion scalar θ , in other words, the ratio of these quantities σ θ is constant. This condition has been used by many researchers for the discussion of exact solutions [35]. The above condition further yields the relations ˙ A A = m ˙ B B and A A = m B B + m ( m -1) ˙ B 2 B 2 , consequently the Ricci scalar takes the form For BI universe model, we have √ -g = B ( m +2) and the respective pointlike Lagrangian density constructed by partial integration [36] of the above action (when ω = ω 0 , where ω 0 is an arbitrary constant) is given by where we have neglected the boundary terms. In order to formulate the corresponding field equations, we use the Euler-Lagrange equations which describe the dependent field equation for the BI model and the evolution equation of scalar field. Thus we have The energy relation (conserved quantity) [37] for the Lagrangian density (4) can be written as E L = ˙ B ∂ L ∂ ˙ B + ˙ φ ∂ L ∂ ˙ φ - L that yields the independent field equation for BI universe (when substituted equal to zero) When m = 1, these equations reduce to the case of FRW universe [29]. For the special choice of U , we evaluate the scalar potential in terms of scale factor, directional Hubble parameter and scalar field. We consider the directional Hubble parameter as a function of scale factor or cosmic time by taking different cases of matter contents. The scalar field is found as a function of scale factor or cosmic time and then the scale factor as a function of scalar field by inverting the obtained expression. Finally, we evaluate the Hubble parameter in terms of scalar field and hence the form of scalar potential. We shall explore the nature of the potential that can generate the cosmic evolution described by these matter contents. Equation (7) yields or equivalently, which provides Using this equation in Eq.(6), it follows that We investigate two cases for the coupling function U , i.e., when U = U 0 , where U 0 is a non-zero constant and U ≡ U ( φ ). In the first case, Eq.(9) becomes /negationslash /negationslash For the scalar field in terms of scale factor B , we have where prime indicates derivative with respect to scale factor, yielding solution where c 1 is a constant of integration. One can solve this integral for particular values of the Hubble parameter. In the second case, we consider U ≡ U ( φ ) (a non-minimal coupling of geometry and scalar field). Equation (9) can be written for scalar field in terms of scale factor and directional Hubble parameter as This equation is discussed for two particular choices of U . When U = φ , i.e., the simple BD gravity, it follows that For the case of induced gravity described by U ( φ ) = 1 2 γφ 2 , Eq.(11) yields These two equations are difficult to solve analytically unless the function H 2 ( B ) is given. For the sake of simplicity, we introduce a new variable x ≡ φ ' φ which yields φ '' φ = x ' + x 2 and hence Eq.(13) turns out to be Further, we assume x ≡ 2 γ ω 0 +4 γ f ' f , where f is an arbitrary function of the scale factor B . Also, x = φ ' φ thus integration leads to φ = f 2 γ/ ( ω 0 +4 γ ) . Using this value of x in Eq.(14), we obtain We see that Eq.(12) is difficult to transform in x by the above transformation. If we consider the scalar field as a constant then Eq.(8) yields the scalar potential V = 2(1 + m ) UH 2 2 , 0 , where H 2 2 , 0 is constant directional Hubble parameter. Multiplying the Klein-Gordon equation (6) both sides with this value of V , we obtain the scalar potential which is obviously a constant (as V 0 and φ are constants). When ω ≡ ω ( φ ), the field equations (5) and (7) remain the same except that the constant ω 0 is replaced by ω ( φ ) while Eqs.(6) becomes Solving the field equations (5), (7) and (16), we have the same expressions as Eqs.(10), (12) and (15) except ω 0 is replaced by ω ( φ ). In the following, we discuss Eqs.(10), (12) and (15) separately to construct potential.", "pages": [ 3, 4, 5, 6, 7, 8 ] }, { "title": "3 Potential Construction", "content": "Now we discuss the scalar field potential by taking three different matter contents.", "pages": [ 8 ] }, { "title": "3.1 Barotropic Fluid", "content": "First we consider the barotropic fluid (a particular case of the perfect fluid) with equation of state (EoS), p = kρ, 0 < k < 1 , where p and ρ are pressure and density, while k is the EoS parameter. In order to find the evolution of Hubble parameter due to barotropic fluid, we consider the Einstein field equations for BI universe model as where we have used the condition A = B m and also combined the two dependent field equations. The integration of the energy conservation equation yields ρ = ρ 0 B -(1+ k )( m +2) , where ρ 0 is an integration constant. Consequently, the directional Hubble parameters are found to be where the integration constant is taken to be zero. The evolution of Hubble parameter is H ' 2 ( B ) H 2 ( B ) = -(1+ k )( m +2) 2 B . The corresponding deceleration parameter turns out to be positive, i.e., q = -1 + 3( k +1) 2 which is consistent with the barotropic fluid. Using these values in Eq.(10), we obtain where φ 0 is a non-zero integration constant. This shows that the constant coupling of geometry and scalar field, i.e., U = U 0 for the barotropic fluid leads to the logarithmic form of scalar field which further corresponds to expanding or contracting scalar field versus scale factor B on the basis of sign. Consequently, the scale factors turn out to be We see that the scale factors are of exponential form which indicate rapid cosmic expansion for the expanding scalar field. The corresponding field potential is This is of power law nature and indicates inverse power law behavior for m> 0 as 0 < k < 1. For the variable ω ( φ ), we consider the ansatz ω ( φ ) = ω 0 φ n ; n > 0 so that the scalar field takes the following form where c 1 is an integration constant and c 2 = 2(1+2 m )(1+ k ) U 0 ω 0 . For the sake of simplicity, we take c 1 = 0 and hence the scalar field becomes Thus the scale factors in exponential form are Consequently, the potential turns out to be which contains the product of inverse power law and logarithmic functions of the scale factor. For U = φ , Eq.(12) takes the form When ω = ω 0 or ω ( φ ) = ω 0 φ n , the solution to this differential equation is quite complicated and cannot provide much insights. However, if we take m = -1 / 2 and ω = ω 0 , then this leads to /negationslash where c 3 and c 4 are integration constants and ω 0 = 0. The respective scale factors are and the corresponding scalar field potential turns out to be We can conclude that the scalar field is described by logarithmic function and the scale factors are of exponential nature which yields expansion for increasing scalar field while the potential turns out to be of power law nature. Now we discuss the induced gravity case and evaluate the function f by using the Hubble parameter and its evolution in Eq.(15) which leads to whose solution is where c 5 and c 6 are arbitrary constants while c 7 and c 8 are given by The corresponding scalar field is φ ( B ) = ( c 5 B r 1 + c 6 B r 2 ) 2 γ ω 0 +4 γ which is clearly of power law nature. Since it is difficult to invert this expression for the scale factor B in terms of φ , so we take either c 5 = 0 or c 6 = 0, which leads to either or We see that the scale factors are also of power law nature and show expanding or contracting behavior depending upon the values of the involved parameters. The scalar field potential (8) then turns out to be This may be of positive or inverse power law nature depending upon the values of parameters. For variable ω , the analytical solution of Eq.(15) is not possible. However, the corresponding numerical solution can be found by using the initial conditions f (1) = 0 . 67 and f ' (1) = 1 . 95 and is given by the polynomial interpolation where we have taken m = 2 , γ = 0 . 25 , k = 0 . 5 and ω = 0 . 9 φ 2 . The corresponding scalar field is φ ( B ) = ( f ( B )) 2 γ ω 0 +4 γ , yielding the form of the field potential in polynomial form which represents positive power law nature. Here the scalar field is in polynomial form which cannot be inverted for scale factor B . We have plotted the potentials given by Eqs.(19), (20), (22) and (24) versus scale factor B as shown in Figure 1 . It is found that in all cases, the scalar field potentials are positive decreasing functions except for the plot (c) which has a signature flip from positive to negative with the increase in scale factor (this graph corresponds to the negative value of m ). We can conclude that for a positive behavior of the field potential (which is physically acceptable), we should take positive range of m .", "pages": [ 8, 9, 10, 11, 13 ] }, { "title": "3.2 Cosmological Constant", "content": "In this case, we take p = -ρ and hence the energy density becomes a constant, i.e., ρ = ρ 0 . The corresponding directional Hubble parameters and its evolution are given by The deceleration parameter turns out to be a dynamical quantity q = -(1 + 1 2( m +2)ln( B ) ). It is interesting to mention here that in our case, the directional Hubble parameters are dependent on the scale factor B (due to anisotropy) whereas in the case of FRW universe, the Hubble parameter is independent of the scale factor, i.e., it turns out to be constant. We use these values in the previously discussed three cases, i.e., U = U 0 , φ and U = 1 2 γφ 2 . Equation (10) provides ( φ ' ) 2 = (1+2 m ) U 0 ω 0 ( m +2) 1 B 2 ln( B ) whose integration leads to φ ( B ) = ± √ -2 ln( B ) c 10 + c 9 , where c 9 is an integration constant while c 10 = 2(1+2 m ) U 0 ω 0 ( m +2) . This leads to the scale factor as an exponential function of the scalar field B ( φ ) = exp( -1 / 2 c 10 ( φ -c 9 )). Likewise, for ω = ω 0 φ n , the scalar field is found to be where c 11 is an integration constant while c 10 is the same as above. Using these values in Eq.(8), the field potential can be determined which would include the product terms of scale factor and logarithmic function. In the case of simple BD gravity, Eq.(12) is not easy to solve for both cases ω = ω 0 and ω = ω 0 φ n . However, the corresponding numerical solutions can be constructed in a similar way as we have discussed in the previous case. The scalar field as well as the potentials constructed, in this way, would be of polynomial nature. For m = -1 / 2, it leads to φ '' + ω 0 2 φ ' = 0 and hence where c 12 and c 13 are integration constants. The field potential corresponding to these values can be obtained from Eq.(8) which would be of power law nature. For the case of induced gravity, Eq.(15) provides Solving this equation, we have the solution in terms of Kummer functions where c 14 and c 15 are integration constants. Since φ = f 2 γ/ ( ω 0 +4 γ ) , consequently the scalar field potential can be determined (it would be a lengthy expression in Kummer function). For ω 0 = -4 γ , the solution is where c 16 and c 17 are integration constants. The corresponding potential can be determined by using the value of the scalar field φ = f 2 γ ω 0 +4 γ in Eq.(8). It would include the integral term and hence cannot be categorized as power law, exponential or logarithmic form.", "pages": [ 13, 14, 15 ] }, { "title": "3.3 Chaplygin Gas", "content": "Finally, we consider the Chaplygin gas EoS as DE candidate which is defined by p = -C ρ , where C is some positive constant. In order to discuss the potential, we use the above EoS parameter in the energy conservation equation and then integration leads to ρ ( B ) = ( C + c 18 B -2( m +2) ) 1 / 2 , where c 18 is an integration constant. Using this value in Eq.(17), it follows that whose evolution yields where p 1 = 4 C 1 / 2 m +3 (1 -m 2 + m +4 1+2 m ) and p 2 = c 18 ( m +2)( m +3) √ C (1 + m 2 + m +4 1+2 m ). For the constant coupling of scalar field and geometry ( U = U 0 ) with ω = ω 0 , we have Thus we can determine the field potential that can generate the cosmic evolution of Chaplygin gas matter (it would be in integral form). For ω = ω 0 φ n , the scalar field is Clearly, it is not possible to have an explicit expression for scalar field in terms of scale factor B and hence the form of the respective field potential cannot be determined. For simple BD gravity with ω = ω 0 and ω = ω 0 φ n , we could not find analytical solutions but numerical solutions can be constructed in a similar pattern as we have discussed earlier. For induced gravity, analytical solution is only possible if we take p 2 = 0, which further implies the same cases as we have found in the cosmological constant case (as H ' 2 H 2 = 1 2 B ln( B ) ).", "pages": [ 15, 16 ] }, { "title": "4 Summary and Discussion", "content": "This paper investigates scalar field potentials by a new technique known as the reconstruction technique for the field potentials. We have applied this technique to BI universe model in the context of general scalar-tensor theory. The general form of the field potential without assigning any values of U, V and H 2 has been explored. We have also discussed two particular cases of U , i.e., when it is a constant and U = U ( φ ). In both cases, the field potential depends upon the scale factor B , the scalar field and the directional Hubble parameter H 2 . Further, we have taken two cases for ω , i.e., ω = ω 0 and ω = ω 0 φ n . It is found that an explicit form of the field potential cannot be found in terms of scale factor unless we choose some particular value of the Hubble parameter. For this purpose, we have taken the evolution of Hubble parameter motivated by the barotropic fluid, the cosmological constant and the Chaplygin gas matter contents. In literature [33, 38], four types of scalar field potentials have usually been discussed, i.e., the positive and inverse power laws, the exponential and the logarithmic potentials while other forms are multiple of these four types. For the barotropic fluid, the potential can be found but it is not possible for the simple BD gravity. We have also observed that for constant U , the scalar fields are logarithmic functions for both ω = ω 0 and ω = ω 0 φ n , while the scale factors are of exponential nature. Also, for simple BD gravity with m = -0 . 5 and ω = ω 0 , the scale factors are exponential functions while for the induced gravity, they turn out to be of power law form. In order to examine their behavior, we have plotted the field potentials versus scale factor B as shown in Figure 1 . It is concluded that the field potentials are positive and decrease to zero except for the case of simple BD gravity where we have taken negative value of m . We may conclude that for positive field potential, we should impose the condition m > 0. We have also discussed a numerical approach (polynomial interpolation) for the cases where no analytical solution exists. Likewise, for the cosmological constant candidate of DE with constant coupling function U , we can determine the form of the field potential without taking any condition for both ω , however in other cases, we have to impose some certain conditions. In the case of Chaplygin gas matter contents, the scalar field potential can be discussed only for ω = ω 0 with U = U 0 . However, in other cases, either the explicit analytical solution is not possible or we have the same expression of the field potential as in the case of cosmological constant. It would be worthwhile to investigate the form of the field potential for the exponential form of coupling function of scalar field and geometry. This procedure may lead to some interesting results when the chameleon mechanism is taken into account in the framework of scalar-tensor gravity.", "pages": [ 16, 17 ] } ]

[ { "title": "Reconcile the AMS-02 positron fraction and Fermi-LAT/HESS total e ± spectra by the primary electron spectrum hardening", "content": "Qiang Yuan a,b and Xiao-Jun Bi b a Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, P.R.China b Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, P.O.Box 918-3, Beijing 100049, P.R.China (Dated: October 10, 2018) The recently reported positron fraction up to ∼ 350 GeV by AMS-02 seems to have tension with the total electron/positron spectra detected by Fermi and HESS, for either pulsar or dark matter annihilation/decay scenario as the primary positron sources. In this work we will show that the tension will be removed by an adjustment of the primary electron spectrum. If the primary electron spectrum becomes harder above ∼ 50 GeV, similar as the cosmic ray nuclei spectrum, the AMS02 positron fraction and Fermi/HESS data can be well fitted by both the pulsar and dark matter models. This result may be suggestive of a common origin of the cosmic ray nuclei and the primary electrons. Furthermore, this study also implies that the properties of the extra sources derived from the fitting to the AMS-02 data should depend on the form of background. PACS numbers: 96.50.S-", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The AMS-02 collaboration reported the very precise measurement of the positron fraction e + /e ± with energies up to 350 GeV recently [1]. The positron fraction shows a continous increase up to ∼ 100 GeV, which is consistent with the previous PAMELA result [2, 3] and is lower than that measured with Fermi-LAT [4]. A flattening of the positron fraction above ∼ 100 GeV is revealed by the AMS-02 data, for the first time. The AMS-02 data implies that there is excess of positrons above tens of GeV compared with the standard cosmic ray (CR) background, and the amount of excess positrons should be less than previously estimated according to the PAMELA data. Several works appears to explain the AMS-02 data with pulsars [5] or dark matter (DM) scenarios [6-8]. A thorough study of the properties of the extra positron sources, including the astrophysical one like pulsars and the dark matter (DM) scenario, based on the AMS02 data and the electron (or e ± ) spectra measured by PAMELA [9], Fermi-LAT [10, 11] and HESS [12, 13], was given shortly after the publication of the AMS-02 data [14] (Paper I). It was found that there was difficulty to fit the AMS-02 positron data and the Fermi-LAT/HESS total electron spectra simultaneously, either in the pulsar scenario or in the DM scenario. The results seem to imply that there might be tension between the AMS-02 data and the Fermi-LAT/HESS data, in the present theoretical framework. This conclusion has been confirmed by other studies 1 [16, 17]. One possible reason leading to the tension is the constraint on the electron injection parameters by the pure electron spectrum by PAMELA. If the PAMELA data are not included, the primary electron spectrum has larger free space and the AMS-02 data and Fermi data can be easier to be fitted simultaneously, as shown in some recent works to explain the AMS-02 result [5-8]. Several possibilities to reconcile these two data sets were discussed in Paper I, including multiple components of the extra sources and the existence of spectral hardening of the primary electron spectrum. The idea to introduce a spectrum hardening of primary electrons to fit the data was also raised in [16, 18]. The spectrum hardening was stimulated by the observed spectral hardening of the nuclei spectra in recent years by several collaborations 2 [19-21]. A unified spectral hardening at rigidity R ∼ 200 GV (or E k ∼ 200 GeV/n) was measured precisely by PAMELA or CREAM [20, 21]. If there is a spectral hardening of the CR nuclei spectra, it is natural to expect a similar hardening of the primary electron spectrum. Models to explain the spectral hardening include the multi-component sources [22-26], non-linear acceleration of the particles [27], or the propagation effect [28, 29]. In [18] the authors pointed out that if there was a spectral hardening of the primary electron spectrum, there would be a less steep increase (or decrease) of the positron fraction above ∼ 200 GeV. In the work we investigate in detail whether to involve such a spectral hardening of the primary electron spectrum can help eliminate the tension between the AMS-02 data and the Fermi-LAT/HESS data. We employ the CosRayMC tool developed in [30] to fit the observational data within the high dimensional parameter space. The GALPROP package 3 [31] to calculate the propagation of the charged CRs has been embedded in the Markov Chain Monte Carlo (MCMC) algorithm, which is well known to be efficient for the survey of high-dimensional correlated parameter space [32], in CosRayMC. The diffusion-reacceleration propagation frame is adopted, and the major propagation parameters are D 0 | R 0 =4GV = 5 . 94 × 10 28 cm 2 s -1 , δ = 0 . 377, v A = 36 . 4 km s -1 and z h = 4 . 04 kpc [33]. The goodness of fit, constraints and implication of the model parameters are discussed. In the next Section we simply describe the models to fit the data. The results are presented in Sec. III. In Sec. IV we give the discussions and conclusions.", "pages": [ 1, 2 ] }, { "title": "II. MODEL", "content": "In this section we describe the major aspects of the theoretical models to reproduce the electron/positron data briefly. The injection spectra of the primary protons (heavier nuclei are less important in this study) and electrons are both assumed to be broken power-law functions with respect to momentum p where p br , 1 represents the low energy break, p br , 2 is the high energy break to be responsible for the spectral hardening, γ 0 , γ 1 and γ 2 are the spectral indices in different momentum ranges. We also employ the log-parabolic function to describe the spectral hardening of the electrons, i.e., q ( p ) ∝ ( p/p e br , 1 ) -γ 1 + γ 2 log( p/ MeV) for p > p e br , 1 . In this case p e br , 2 is not used. The absolute fluxes of protons and electrons are determined through normalizing the propagated fluxes to normalization factors A p and A e . The background positrons are expected to be produced through the collision of CR nuclei with the interstellar medium (ISM) during the propagation. The parameterization of pp collision in [34] is employed to calculate the secondary production of positrons and electrons. Similar as done in Paper I, we further introduce a free factor c e + to adjust the absolute fluxes of the secondary positrons and electrons to fit the data. Such a factor may represent the uncertainties of the hadronic interactions, propagation models, the ISM density distributions, and the nuclear enhancement factor from heavy elements. In the PAMELA era it was found that the background contribution are not enough to explain the observed positron fraction and total e ± data [35, 36]. Therefore the extra sources of e ± beyond the traditional CR background are introduced to explain the data. We will base on the same theoretical framework to fit the AMS-02 data in the work, assuming continuously distributed pulsars or the DM annihilation/decay to be the extra sources of e ± . The injection spectrum of e ± from pulsars is assumed to be power-law with an exponential cutoff where A psr is the normalization factor, α is the spectral index and p c is the cutoff momentum. The spectral index α is limited in the range 1 . 4 to 2 . 2 according to the γ -ray observations of pulsars [37]. The spatial distribution of pulsars is taken to be the cylindrically symmetric form given in [38] where R /circledot = 8 . 5 kpc is the distance of the solar location to the Galactic center, z s ≈ 0 . 2 kpc is the characteristic height of the Galactic disk. As for DM scenario (taking annihilation as illustration), we focus on the leptonic two-body annihilation channels µ + µ -and τ + τ -, as implied according to the PAMELA and Fermi-LAT/HESS data of the electrons/positrons and the antiprotons [39-41]. The positron/electron production function from DM annihilation is (assumed to be Majorana particles) where m χ is the mass of DM particle, 〈 σv 〉 is the velocity weighted annihilation cross section, dN/dp is the yield spectrum for one annihilation of a pair of DM particles, and ρ ( r ) is the DM density profile. The spatial profile of DM energy density is taken to be Navarro-Frenk-White (NFW, [42]) distribution with parameters r s = 20 kpc and ρ s = 0 . 26 GeV cm -3 . For low energy particles we further employ a simple force field approximation to take into account the solar modulation effect [43]. Since the operation period of PAMELA and AMS-02 is close to the solar minimum, the modulation potential is required to be smaller than 1 GV. Note, however, the low energy part of the positron fraction measured by PAMELA and AMS-02 might not be reproduced with such single solar modulation model, and more complicated charge-sign dependent modulation effect is necessary [44, 45].", "pages": [ 2, 3 ] }, { "title": "III. RESULTS", "content": "We first determine the parameters of the proton injection spectrum through fitting to the PAMELA [20] and CREAM [21] data. For CREAM data we include 10% systematic uncertainties as discussed in [21]. The high energy break p p br , 2 is fixed to be 230 GeV as suggested by the PAMELA data. The best fitting parameters of the proton spectrum are: γ 0 = 1 . 80, γ 1 = 2 . 42, γ 2 = 2 . 33, p p br , 1 = 12 . 3 GeV, and solar modulation potential φ = 495 MV. The normalization of the propagated proton flux at 100 GeV is A p = 4 . 55 × 10 -9 cm -2 s -1 sr -1 MeV -1 . Comparison of the best fitting spectrum of protons with the observational data is shown in Fig. 1. We see very good agreement between the calculated spectrum and the data. The minimum χ 2 value is about 24 for 72 degree of freedom (dof). Since the observational period of protons by PAMELA is almost the same with that of electrons by PAMELA and positrons by AMS-02, we should expect a common modulation amplitude for these particles (besides the charge-sign dependent effect). Therefore we employ a prior on the modulation potential φ = 500 ± 53 MV comes from the fit of the proton data. We first fix the electron second break energy p e br , 2 at 230 GeV, same as that of protons. The best fitting results of the positron fraction and electron spectrum are shown in Fig. 2. The fitting parameters and the χ 2 value are presented in Table I. Compared with the case without spectral hardening of the primary electrons (Paper I), the fitting is indeed improved. The χ 2 value decrease from ∼ 280 to 235 with one more parameter. However, the overall fitting is still not satisfactory. We can see from Fig. 2 that when AMS-02 data are well reproduced, the model expectation is lower than the Fermi data, which is similar with the findings in Paper I. We then relax the break momentum of the electrons and redo the fit. In this case we find the improvement is significantly, as shown in Fig. 3. The parameters are also given in Table I. The minimum χ 2 value over dof is about 1 . 0, which implies a rather good fitting. However, the break momentum p e br , 2 is required to be about 45 GeV, which is significantly smaller than that of the nuclei. The difference of the spectral indices below and above p e br , 2 is about 0 . 3. As a comparison, such a value is measured to be ∼ 0 . 2 for protons and ∼ 0 . 3 for Helium [20]. Note, for the fit of proton spectrum in a wider energy range, as shown in Fig. 1, the spectral difference is only about 0 . 1. It is also possible that the spectral hardening is not a break but a smooth hardening instead, as shown in many models [23, 27]. We may use a log-parabolic function to approximate the smooth hardening of the electron spectrum. Fig. 4 presents the results of the fit with logparabolic shape of the primary electron spectrum. We find the fit is also improved, with the minimum χ 2 value slightly larger than that with p e br , 2 free. The fitting parameters are given in Table I. From above we see that including a spectral hardening of the primary electron spectrum, both the PAMELA, AMS-02 and Fermi data can be well fitted with a single component of the extra sources. It is a natural expectation that there is a hardening in the primary electron spectrum, given the observed hardening of the CR nuclei. However, the position of the break might be different from that of nuclei. It is a problem needs to be further understood theoretically. If the hardening of the primary electron spectrum can be confirmed, it would be important to understand the origin and acceleration of the Galactic CRs. Since AMS-02 could measure the pure electron spectrum with much higher precision that PAMELA, we give the expected pure electron spectra in Fig. 5 for the above three cases of the hardening. The future AMS-02 data may test the existence and detailed shape of the primary electron spectrum. Finally we discuss the DM annihilation as the sources of the e ± . The annihilation final states are assumed to be µ + µ -and τ + τ -. The primary electron spectrum is parameterized with Eq. (1), and p e br , 2 is allowed to be free in the fit. The fitting results are shown in Figs. 6 and 7, for µ + µ -and τ + τ -final states respectively. The fitting parameters are compiled in Table II. It is shown that the DM models can give comparable fittings to the data compared with pulsars. The break momentum of the primary electrons, p e br , 2 is also similar with that derived in the pulsar scenario, and is smaller than p p br , 2 ≈ 230 GeV. The 1 σ and 2 σ favored regions on the m χ -〈 σv 〉 parameter plane are given in Fig. 8. For µ + µ -channel DM with mass 0 . 8 -1 . 5 TeV is favored, while for τ + τ -channel the mass is obtained to be 2 -4 TeV. The boost factor of the annihilation cross section compared with the natural value to give the proper relic density is about hundred to thousand. Such results do not differ much from the ones obtained through fitting the PAMELA positron fraction and the Fermi/HESS e ± data [41]. The exclusion limits on the DM annihilation into µ + µ -and τ + τ -pairs by γ -rays from the Galactic center (thin lines, [52]) and the dwarf galaxies (thick lines, [53]) are also plotted in Fig. 8. The results show that for the τ + τ -channel the Fermi γ -rays always give very strong constraints on the annihilation cross section. The constraints for the µ + µ -channel is weaker. The Galactic center γ -rays tend to exclude the parameter space to explain the e ± excesses. However it may suffer from the uncertainties of the density profile of DM in the halo center. The more robust limits from the dwarf galaxies can not exclude the favored parameter region.", "pages": [ 3, 4, 5 ] }, { "title": "IV. CONCLUSION AND DISCUSSION", "content": "The study of the highly precise data of positron fraction in CRs reported by AMS-02, as well as the pure electron spectrum measured by PAMELA and the total e ± spectra measured by Fermi and HESS, shows that it is difficult to use a single component of the extra sources to explain the e ± excesses (Paper I, [14]). In this work we show that an additional break of the primary electron spectrum can improve the fit significantly. The best fitting break momentum is about 40 -50 GeV and the spectral difference γ 2 -γ 1 is ∼ 0 . 3 -0 . 4. As a comparison, the break momentum of protons is about 230 GeV, and the spectral difference is ∼ 0 . 1. The hardening behavior of the electron spectrum is different from that of nuclei, which makes the understanding of the fine structures of the CR spectra non-trivial. The different behaviors of the spectral hardening between nuclei and electrons are probably due to the fact that high energy electrons should come from nearby regions, and less number of relevant sources leads to larger fluctuations of the electron spectrum than that of nuclei. It is also possible that, if one or several nearby sources are responsible to the spectral hardening, the accelerated electron-to-proton ratio is higher for these sources. In the presence of a hardening of the primary electron spectrum, both the pulsar and DM scenarios can give comparable fit to the data. However, the DM scenario are strongly constrained by the γ -rays, especially for the tauon final state. We would like to point out that it will be equivalent to take the harder part of the electron spectrum and to drop the constraints from the PAMELA electron data. In such ways both the AMS-02 positron fraction and Fermi total e ± spectrum can be fitted simultaneously. The AMS-02 will measure the electron spectrum with high precision in the near future. Whether there is a hardening in the electron spectrum or a lower e ± total spectrum than Fermi will soon be answered by AMS-02.", "pages": [ 5 ] }, { "title": "Appendix A: Results of the background positrons and electrons", "content": "For the convenience of use we tabulate the fluxes of the background positrons and electrons calculated with the best fitting parameters of the pulsar models in Table III. For DM models the results have little difference. Note for background positrons an additional factor c e + as given in Table I needs to be multiplied. Here the local interstellar fluxes are given. If one wants to better reproduce the low energy electron spectrum, the solar modulation with modulation potential given in Table I is necessary.", "pages": [ 5 ] }, { "title": "Acknowledgments", "content": "This work is supported by 973 Program under Grant No. 2013CB837000 and the National Natural Science Foundation of China under Grant Nos. 11075169, 11105155. 1304.1791. a Normalization at 25 GeV in unit of cm -2 s -1 sr -1 MeV -1 . 1303.0530.", "pages": [ 5, 7 ] } ]

[ { "title": "Non-local massive gravity", "content": "Leonardo Modesto 1, ∗ and Shinji Tsujikawa 2, † 1 Department of Physics & Center for Field Theory and Particle Physics, Fudan University, 200433 Shanghai, China 2 Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan (Dated: July 11, 2018) We present a general covariant action for massive gravity merging together a class of 'nonpolynomial' and super-renormalizable or finite theories of gravity with the non-local theory of gravity recently proposed by Jaccard, Maggiore and Mitsou (Phys. Rev. D 88 (2013) 044033). Our diffeomorphism invariant action gives rise to the equations of motion appearing in non-local massive massive gravity plus quadratic curvature terms. Not only the massive graviton propagator reduces smoothly to the massless one without a vDVZ discontinuity, but also our finite theory of gravity is unitary at tree level around the Minkowski background. We also show that, as long as the graviton mass m is much smaller the today's Hubble parameter H 0 , a late-time cosmic acceleration can be realized without a dark energy component due to the growth of a scalar degree of freedom. In the presence of the cosmological constant Λ, the dominance of the non-local mass term leads to a kind of 'degravitation' for Λ at the late cosmological epoch.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The construction of a consistent theory of massive gravity has a long history, starting from the first attempts of Fierz and Pauli [1] in 1939. The Fierz-Pauli theory, which is a simple extension of General Relativity (GR) with a linear graviton mass term, is plagued by a problem of the so-called van Dam-Veltman-Zakharov (vDVZ) discontinuity [2]. This means that the linearized GR is not recovered in the limit that the graviton mass is sent to zero. The problem of the vDVZ discontinuity can be alleviated in the non-linear version of the Fierz-Pauli theory [3]. The non-linear interactions lead to a well behaved continuous expansion of solutions within the so-called Vainshtein radius. However, the nonlinearities that cure the vDVZ discontinuity problem give rise to the so-called Boulware-Deser (BD) ghost [4] with a vacuum instability. A massive gravity theory free from the BD ghost was constructed by de Rham, Gabadadze and Tolley (dRGT) [5] as an extension of the Galileon gravity [6]. On the homogenous and isotropic background, however, the selfaccelerating solutions in the dRGT theory exhibit instabilities of scalar and vector perturbations [7]. The analysis based on non-linear cosmological perturbations shows that there is at least one ghost mode (among the five degrees of freedom) in the gravity sector [8]. Moreover it was shown in Ref. [9] that the constraint eliminating the BD ghost gives rise to an acausality problem. These problems can be alleviated by extending the original dRGT theory to include other degrees of freedom [1012] (like quasidilatons) or by breaking the homogeneity [13] or isotropy [14, 15] of the cosmological background. Recently, Jaccard et al. [16] constructed a nonlocal theory of massive gravity by using a quadratic action of perturbations expanded around the Minkowski background. This action was originally introduced in Refs. [17, 18] in the context of the degravitation idea of the cosmological constant. The resulting covariant nonlinear theory of massive gravity not only frees from the vDVZ discontinuity but respects causality. Moreover, unlike the dRGT theory, it is not required to introduce an external reference metric. Jaccard et al. [16] showed that, on the Minkowski background, there exists a scalar ghost in addition to the five degrees of freedom of a massive graviton, by decomposing a saturated propagator into spin-2, spin-1, and spin-0 components. For the graviton mass m of the order of the today's Hubble parameter H 0 , the vacuum decay rate induced by the ghost was found to be very tiny even over cosmological time scales. The possibility of the degravitation of a vacuum energy was also suggested by introducing another mass scale µ much smaller than m . In this paper we propose a general covariant action principle which provides the equations of motion for the non-local massive gravity [16] with quadratic curvature terms. The action turns out to be a bridge between a class of super-renormalizable or finite theories of quantum gravity [19-25] and a diffeomorphism invariant theory for a massive graviton. The theory previously studied in Refs. [19-25] has an aim to provide a completion of the Einstein gravity through the introduction of a non-polynomial or semipolynomial entire function (form factor) without any pole in the action. In contrast, the non-local massive gravity studied in this paper shows a pole in the classical action making it fully non-local. However, the Lagrangian for massive gravity can be selected out from the theories previously proposed [19-25] once the form factor has a particular infrared behavior. The non-local theory resulting from the covariant Lagrangian is found to be unitary at tree level on the Minkowski background. Moreover, the theory respects causality and smoothly reduces to the massless one without the vDVZ discontinuity. We will also study the cosmology of non-local massive gravity on the flat Friedmann-Lemaˆıtre-RobertsonWalker (FLRW) background in the presence of radiation and non-relativistic matter 1 . Neglecting the contribution of quadratic curvature terms irrelevant to the cosmological dynamics much below the Planck scale, the dynamical equations of motion reduce to those derived in Ref. [16]. We show that, as long as the graviton mass m is much smaller than H 0 , the today's cosmic acceleration can be realized without a dark energy component due to the growth of a scalar degree of freedom. Our paper is organized as follows. In Sec. II we show a non-local covariant Lagrangian which gives rise to the same equation of motion as that in non-local massive gravity with quadratic curvature terms. We also evaluate the propagator of the theory to study the tree-level unitarity. In Sec. III we study the cosmological implications of non-local massive gravity in detail to provide a minimal explanation to dark energy in terms of the graviton mass. We also discuss the degravitation of the cosmological constant induced by the non-local mass term. Conclusions and discussions are given in Sec. IV. Throughout our paper we use the metric signature η µν = diag(+1 , -1 , -1 , -1). The notations of the Riemann tensor, the Ricci tensor, and Ricci scalar are R µ νρσ = -∂ σ Γ µ νρ + . . . , R µν = R σ µνσ and R = g µν R µν , respectively.", "pages": [ 1, 2 ] }, { "title": "II. SUPER-RENORMALIZABLE NON-LOCAL GRAVITY", "content": "Let us start with the following general class of nonlocal actions in D dimension [19-25], where κ = √ 32 πG ( G is gravitational constant), | g | is the determinant of a metric tensor g µν , /square is the d'Alembertian operator with /square M = /square /M 2 , and M is an ultraviolet mass scale. The first two lines of the action consist of a finite number of operators multiplied by coupling constants subject to renormalization at quantum level. The functions h 2 ( z ) and h 0 ( z ), where z ≡ -/square M , are not renormalized and defined as follows for general parameters ˜ a n and ˜ b n , while The form factor V ( z ) -1 in Eq. (3) is made of two parts: (i) a non-local operator ( /square + m 2 ) / /square which goes to the identity in the ultraviolet regime, and (ii) an entire function e H( z ) without zeros in all complex planes. Here, m is a mass scale associated with the graviton mass that we will discuss later when we calculate the two-point correlation function. H( z ) is an entire function of the operator z = -/square M , and p γ +N+1 ( z ) is a real polynomial of degree γ +N+1 which vanishes in z = 0, while N = ( D -4) / 2 and γ > D/ 2 is integer 2 . The exponential factor e H( z ) is crucial to make the theory super-renormalizable or finite at quantum level [19-25]. Let us expand on the behaviour of H( z ) for small values of z : For the most simple choice p γ +N+1 ( z ) = z γ +N+1 , H( z ) simplifies to In particular lim z → 0 H( z ) = 0. We will expand more about the limit of large z in Sec. II B, where we will explicitly show the power counting renormalizability of the theory. is the incomplete gamma function [19].", "pages": [ 2 ] }, { "title": "A. Propagator", "content": "In this section we calculate the two point function of the gravitational fluctuation around the flat space-time. For this purpose we split the g µν into the flat Minkowski metric η µν and the fluctuation h µν , as Writing the action (1) in the form S = ∫ d D x L , the Lagrangian L can be expanded to second order in the graviton fluctuation [33] where A µ = h µν ,ν , φ = h (the trace of h µν ), F µν = A µ,ν -A ν,µ and the functionals of the D'Alembertian operator α ( /square ) , β ( /square ) are defined by The d'Alembertian operator in Eq. (9) must be conceived on the flat space-time. The linearized Lagrangian (9) is invariant under infinitesimal coordinate transformations x µ → x µ + κξ µ ( x ), where ξ µ ( x ) is an infinitesimal vector field of dimensions [ ξ ( x )] = [mass] ( D -4) / 2 . Under this shift the graviton field is transformed as h µν → h µν -ξ µ,ν -ξ ν,µ . The presence of this local gauge invariance requires for a gauge-fixing term to be added to the linearized Lagrangian (9). Hence, if we choose the usual harmonic gauge ( ∂ µ h µν = 0) [21, 34] the linearized gauge-fixed Lagrangian reads where the operator O is made of two terms, one coming from the linearized Lagrangian (9) and the other from the gauge-fixing term (11). Inverting the operator O [33], we find the following two-point function in the momentum space (with the wave number k ), where we omitted the tensorial indices for O -1 . The operators { P (2) , P (1) , P (0) , ¯ P (0) } , which project out the spin-2, spin-1, and two spin-0 parts of a massive tensor field, are defined by [33] where ω µν = k µ k ν /k 2 and θ µν = η µν -k µ k ν /k 2 . These correspond to a complete set of projection operators for symmetric rank-two tensors. The functions α ( k 2 ) and β ( k 2 ) are achieved by replacing /square → -k 2 in the definitions (10). By looking at the last two gauge-invariant terms in Eq. (13), we deem convenient to introduce the following definitions, Through these definitions, the gauge-invariant part of the propagator greatly simplifies to", "pages": [ 3 ] }, { "title": "B. Power counting super-renormalizability", "content": "The main properties of the entire function e H( z ) useful to show the super-renormalizability of the theory are the following, where we assumed p γ +N+1 ( z ) = z γ +N+1 . The first limit tells us what is the leading behaviour in the ultraviolet regime, while the second limit confirms that the next to the leading order goes to zero faster then any polynomial. Let us then examine the ultraviolet behavior of the theory at quantum level. According to the property (18), the propagator and the leading n -graviton interaction vertex have the same scaling in the high-energy regime [see Eqs. (2), (4), (15), (17), and (18)]: In Eq. (20) the indices for the graviton fluctuation h µν are omitted and h i ( -/square M ) is one of the functions in Eq. (2). From Eqs. (19) and (20), the upper bound to the superficial degree of divergence is In Eq. (21) we used the topological relation between vertexes V , internal lines I and number of loops L : I = V + L -1, as well as D = 2N+4. Thus, if γ > D/ 2, then only 1-loop divergences survive and the theory is super-renormalizable. Only a finite number of constants is renormalized in the action (1), i.e. κ , ¯ λ , a n , b n together with the finite number of couplings that multiply the operators O ( R 3 ) in the last line of Eq. (1). We now assume that the theory is renormalized at some scale µ 0 . Therefore, if we set in Eq. (2), the functions (15) and (16) reduce to Thus, in the momentum space, only a pole at k 2 = m 2 occurs in the bare propagator and Eq. (17) reads The tensorial structure of Eq. (24) is the same as that of the massless graviton and the only difference appears in an overall factor 1 / ( k 2 -m 2 ). If we take the limit m → 0, the massive graviton propagator reduces smoothly to the massless one and hence there is no vDVZ discontinuity. Assuming the renormalization group invariant condition (22), missing the O ( R 3 ) operators in the action (1), and setting ¯ λ to zero, the non-local Lagrangian in a D dimensional space-time greatly simplifies to On using the function α ( /square ) = 2( V ( /square ) -1 -1) / ( κ 2 /square ), the Lagrangian (25) can be expressed as If we are interested only in the infrared modifications of gravity, we can fix H( -/square M ) = 0. This condition restricts our class of theories to the non-local massive gravity.", "pages": [ 3, 4 ] }, { "title": "C. Unitarity", "content": "We now present a systematic study of the tree-level unitarity [33]. A general theory is well defined if 'tachyons' and 'ghosts' are absent, in which case the corresponding propagator has only first poles at k 2 -m 2 = 0 with real masses (no tachyons) and with positive residues (no ghosts). Therefore, to test the tree-level unitarity, we couple the propagator to external conserved stress-energy tensors, T µν , and we examine the amplitude at the poles [35]. When we introduce the most general source, the linearized action (12) is replaced by where g is a coupling constant. The transition amplitude in the momentum space is Since the stress-energy tensor is conserved, only the projectors P (2) and P (0) will give non-zero contributions to the amplitude. In order to make the analysis more explicit, we expand the sources using the following set of independent vectors in the momentum space [33, 35-37]: where /vector/epsilon1 i are unit vectors orthogonal to each other and to /vector k . The symmetric stress-energy tensor reads where we introduced the notation X ( µ Y ν ) ≡ ( X µ Y ν + Y µ X ν ) / 2. The conditions k µ T µν = 0 and k µ k ν T µν = 0 place the following constraints on the coefficients a, b, d, e i , f i [33]: where k 2 := k 2 0 -/vector k 2 . The conditions (31) and (32) imply while the condition (33) leads to Introducing the spin-projectors and the conservation of the stress-energy tensor k µ T µν = 0 in Eq. (28), the amplitude results where T := η µν T µν . The residue at the pole k 2 = m 2 reads If we assume the stress-tensor to satisfy the usual energy condition, then the following inequality follows Using the conditions (35), (36), and (40) in Eq. (39), we find that ∣ for D ≥ 3. This shows that the theory is unitary at tree level around the Minkowski background. As we see in Eq. (38) the contribution to the residue from the spin-0 operator P (0) is negative, but the spin-2 operator P (2) provides a dominant contribution with a positive sign of Res A ∣ ∣ k 2 = m 2 . Hence the presence of the spin-2 mode is crucial to make the theory unitary.", "pages": [ 4, 5 ] }, { "title": "D. Equations of motion", "content": "Let us derive the equations of motion up to curvature squared operators O ( R 2 ) and total derivative terms [17, 38-41]. The action of our theory is S = ∫ d D x L , where the Lagrangian is given by Eq. (25). The variation of this action reads where we omitted the argument -/square M of the form factor V -1 . We also used the relations ∇ µ g ρσ = 0, ∇ µ G µν = 0, and The action is manifestly covariant in general. Hence its variational derivative (the left hand side of the modified Einstein equations) exactly satisfies the Bianchi identity Taking into account the energy-momentum tensor T µν , the equation of motion at the quadratic order of curvatures reads Except for the very high-energy regime the quadratic curvature terms should not be important in Eq. (45). Neglecting the O ( R 2 µν ) terms and setting e H( -/square M ) = 1 in Eq. (45), it follows that which is the same equation as that studied in Ref. [16] in the context of non-local massive gravity with the graviton mass m . If we apply Eq. (46) to cosmology, the d'Alembertian is of the order of /square ∼ d 2 /dt 2 ∼ ω 2 , where ω is the characteristic frequency of a corresponding physical quantity. Provided ω /greatermuch m the term m 2 /square -1 G µν in Eq. (46) is suppressed relative to G µν , so that the Einstein equation G µν /similarequal 8 πGT µν is recovered. In order to realize the standard radiation and matter eras, it is expected that m should not be larger than H 0 . At the late cosmological epoch, the effect of the non-local term m 2 /square -1 G µν can be important to modify the dynamics of the system. If we take the derivative of Eq. (46) by exerting the operator /square , it follows that This equation is invariant under the symmetry which realizes the Afshordi-Smolin idea [42] for the degravitation of the cosmological constant. Equation (47) does not admit exact de Sitter solutions. There exist deSitter solutions characterized by G dS µν = 8 πGρ eff Λ g µν for the modified model in which the operator /square in Eq. (47) is replaced by /square + µ 2 , where µ is a small mass scale [16]. If the energy-momentum tensor on the right hand side of Eq. (47) is given by T (Λ) µν = ρ Λ g µν , we obtain the effective cosmological constant ρ eff Λ = ρ Λ µ 2 / ( m 2 + µ 2 ). For µ much smaller than m , it follows that ρ eff Λ /lessmuch ρ Λ . In the limit µ → 0, the effective cosmological constant disappears completely. The crucial point for the above degravitation of ρ Λ is that both /square G dS µν and and /square T (Λ) µν vanish at de Sitter solutions. For the background in which the matter density ρ varies (such as the radiation and matter eras), the two d'Alembertians in Eq. (47) give rise to the contributions of the order of ω 2 . In other words, the above degravitation of ρ Λ should occur at the late cosmological epoch in which ω drops below µ [16]. A detailed analysis given in Sec. III shows that, even for ρ Λ = 0 and µ = 0, a late-time cosmic acceleration occurs on the flat FLRW background. This comes from the peculiar evolution of the term m 2 /square -1 G µν in Eq. (46), by which the equation of state smaller than -1 can be realized. Even in the presence of the cosmological constant, the non-local term eventually dominates over ρ Λ at the late cosmological epoch. In the following we focus on the theory based on the field equation (46), i.e., µ = 0.", "pages": [ 5, 6 ] }, { "title": "III. COSMOLOGICAL DYNAMICS", "content": "We study the cosmological dynamics on the fourdimensional flat FLRW background characterized by the line element ds 2 = -dt 2 + a 2 ( t )( dx 2 + dy 2 + dz 2 ), where a ( t ) is the scale factor with the cosmic time t . Since we ignore the O ( R 2 µν ) terms and set H( -/square M ) = 0 in Eq. (45), our analysis can be valid in the low-energy regime much below the Planck scale. We introduce a tensor S µν satisfying the relation by which the second term on the left hand side of Eq. (46) can be written as m 2 /square -1 G µν = m 2 S µν . In order to respect the continuity equation ∇ µ T µν = 0 of matter, we take the transverse part S T µν of the symmetric tensor S µν , that is, ∇ µ S T µν = 0. Then, Eq. (46) can be written as We use the fact that S µν can decomposed as [16, 43] where the vector S µ has the time-component S 0 alone in the FLRW background, i.e., S i = 0 ( i = 1 , 2 , 3). From Eq. (51) we have where u ≡ S 0 0 and v ≡ S i i , and a dot represents a derivative with respect to t . In the presence of the matter energy-momentum tensor T µν = ( ρ, a 2 Pδ ij ), the (00) and ( ii ) components of Eq. (50) are where H = ˙ a/a . Taking the divergence of Eq. (51), it follows that 2 ∇ µ S µν = ∇ µ ( ∇ µ S ν + ∇ ν S µ ). From the ν = 0 component of this equation we obtain The (00) and ( ii ) components of Eq. (49) give which can be decoupled each other by defining In summary we get the following set of equations from Eqs. (53)-(57): From Eqs. (59)-(61) one can show that the continuity equation ˙ ρ + 3 H ( ρ + P ) = 0 holds. For the matter component we take into account radiation (density ρ r , pressure P r = ρ r / 3), non-relativistic matter (density ρ m , pressure P m = 0), and the cosmological constant (density ρ Λ , pressure P Λ = -ρ Λ ), i.e., ρ = ρ r + ρ m + ρ Λ and P = ρ r / 3 -ρ Λ . Each matter component obeys the continuity equation ˙ ρ i +3 H ( ρ i + P i ) = 0 ( i = r, m, Λ). In order to study the cosmological dynamics of the above system, it is convenient to introduce the following dimensionless variables where r H ≡ H ' /H , and a prime represents a derivative with respect to N = ln( a/a i ) ( a i is the initial scale factor). From Eq. (59) it follows that We define the density parameter of the dark energy component, as Ω DE ≡ Ω Λ + Ω NL . From Eqs. (59) and (60) the density and the pressure of dark energy are given respectively by Then, the dark energy equation of state w DE = P DE /ρ DE can be expressed as From Eq. (60) the quantity r H = H ' /H obeys by which the effective equation of state of the Universe is known as w eff = -1 -2 r H / 3. On using Eqs. (60)-(63) and the continuity equation of each matter component, we obtain the following differential equations In Eq. (71) the derivative of r H is given by Let us first study the case in which the cosmological constant is absent (Ω Λ = 0). We assume that m is smaller than the today's Hubble parameter H 0 , i.e., m /lessorsimilar H 0 . During the radiation and matter dominated epochs the last term in Eq. (68) should be suppressed, so that r H /similarequal -3 / 2 -Ω r / 2 is nearly constant in each epoch. Integrating Eqs. (69) and (70) for constant r H ( > -3) and neglecting the decaying modes, we obtain where c 1 and c 2 are constants. During the radiation era ( r H = -2) these solutions reduce to U = c 1 and V = c 2 e ( √ 33 -1) N/ 2 -1 / 2, while in the matter era ( r H = -3 / 2) we have U = 2 N + c 1 and V = c 2 e ( √ 137 -3) N/ 4 -3 / 8. Since V grows faster than U due to the presence of the term -8 V in Eq. (70), it is a good approximation to neglect U relative to V in the regime | V | /greatermuch 1. The field S is amplified by the force term on the right hand side of Eq. (71). Meanwhile the homogeneous solution of Eq. (71) decays for r H = -2 and -3 / 2. Then, for | V | /greatermuch 1, the field S grows as which behaves as S /similarequal (5 √ 33 + 3) c 2 e ( √ 33 -1) N/ 2 / 136 during the radiation era and S /similarequal 3(5 √ 137 + 17) c 2 e ( √ 137 -3) N/ 4 / 784 during the matter era. From Eq. (67) the dark energy equation of state reduces to w DE /similarequal ( V + 4 S ) / (3 V -4 S ' + 4 Sr H ). Using the above solutions, we obtain from which w DE /similarequal -1 . 791 in the radiation era and w DE /similarequal -1 . 725 in the matter era. This means that the dark component from the non-local mass term comes into play at the late stage of cosmic expansion history. Indeed, there exists an asymptotic future solution characterized by Ω NL = 1 with constant r H . In this regime we have r H /similarequal -3 / 2 -m 2 / (8 H 2 )( U -V -4 S ) in Eq. (68). Meanwhile, if r H is constant, the mass term m does not appear in Eqs. (69)-(71), so that the solutions (75)-(78) are valid, too. Since w DE is equivalent to w eff = -1 -2 r H / 3 in the limit Ω NL → 1, it follows that Since r H is constant, the growth rates of the Hubble parameter squared H 2 are the same as those of V and S . Hence we have the super-inflationary solution H ∝ a √ 57 / 6 -1 / 2 approaching a big-rip singularity. However the above study neglects the contribution of the O ( R 2 ) terms, so inclusion of those terms can modify the cosmological dynamics in the high-curvature regime. In order to confirm the above analytic estimation, we numerically integrate Eqs. (69)-(73) with the initial conditions U = U ' = 0, V = V ' = 0, and S = S ' = 0 in the deep radiation era. In Fig. 1 we plot the evolution of w DE and w eff as well as the density parameters Ω NL , Ω m , Ω r versus the redshift z r = 1 /a -1. Clearly there is the sequence of radiation (Ω r /similarequal 1, w eff /similarequal 1 / 3), matter (Ω m /similarequal 1, w eff = 0), and dark energy (Ω NL /similarequal 1, w eff /similarequal -1 . 5) dominated epochs. We identify the present epoch ( z r = 0) to be Ω NL = 0 . 7. As we estimated analytically, the dark energy equation of state evolves as w DE /similarequal -1 . 791 (radiation era), w DE /similarequal -1 . 725 (matter era), and w DE /similarequal -1 . 506 (accelerated era). Notice that, even with the initial conditions V = V ' = 0, the growing-mode solution to Eq. (70) cannot be eliminated due to the presence of the term -2 r H . Taking into account the decaying-mode solution to V in the radiation era, the coefficient c 2 of Eq. (76) corresponding to V = V ' = 0 at N = 0 (i.e., a = a i ) is c 2 = √ 33 / 132 + 1 / 4 /similarequal 0 . 29. Up to the radiation-matter equality ( a = a eq ), the field evolves as V /similarequal 0 . 29( a/a i ) ( √ 33 -1) / 2 . Since V is proportional to a ( √ 137 -3) / 4 in the matter era, it follows that for a eq < a < 1. In the numerical simulation of Fig. 1 the initial condition is chosen to be a i = 2 . 6 × 10 -7 with a eq = 3 . 1 × 10 -4 . Since the cosmic acceleration starts when the last term in Eq. (68) grows to the order of 1, we have m 2 V 0 / (8 H 2 0 ) ≈ 1, where V 0 is the today's value of V . Using the analytic estimation (80), the mass m is constrained to be m ≈ 10 -7 H 0 . In fact, this is close to the numerically derived value m = 1 . 5 × 10 -7 H 0 . Thus, the mass m is required to be much smaller than H 0 to avoid the early beginning of cosmic acceleration. If the onset of the radiation era occurs at the redshift z r larger than 10 15 , i.e., a i /lessorsimilar 10 -15 , the analytic estimation (80) shows that the mass m needs to satisfy the condition m /lessorsimilar 10 -17 H 0 to realize the successful cosmic expansion history. If we consider the evolution of the Universe earlier than the radiation era (e.g., inflation), the upper bound of m should be even tighter. On the de Sitter background ( ˙ H = 0) we have r H = 0, in which case the growth of V can be avoided for the initial conditions V = V ' = 0. However, inflation in the early Universe has a small deviation from the exact de Sitter solution [44] and hence the field V can grow at some extent due to the nonvanishing values of r H . For the theoretical consistency we need to include the O ( R 2 ) terms in such a high-energy regime, which is beyond the scope of our paper. /negationslash", "pages": [ 6, 7, 8 ] }, { "title": "B. Ω Λ = 0", "content": "In the presence of the cosmological constant with the energy density ρ Λ , the cosmological dynamics is subject to change relative to that studied in Sec. III A. During the radiation and matter eras we have 3 H 2 /greatermuch | ( m 2 / 4)( U + 3 V -4 ˙ S 0 ) | in Eq. (59) and hence 3 H 2 /similarequal 8 πGρ . In order to avoid the appearance of ρ Λ in these epochs, we require the condition ρ Λ /lessorsimilar 3 H 2 0 / (8 πG ). The non-local mass term finally dominates over the cosmological constant because the equation of state of the former is smaller than that of the latter. If the condition 8 πGρ Λ /greatermuch | ( m 2 / 4)( U +3 V -4 ˙ S 0 ) | is satisfied today , the non-local term comes out in the future. The case (a) in Fig. 2 corresponds to such an example. In this case, the dark energy equation of state is close to -1 up to z r ∼ -0 . 9. It then approaches the asymptotic value w DE = -1 . 506. In the case (c) the transition to the asymptotic regime w DE = -1 . 506 occurs even earlier (around z r ∼ 100). For smaller values of ρ Λ , the dominance of the nonlocal term occurs earlier. In the case (b) of Fig. 2 the energy densities of the non-local term and the cosmological constant are the same orders today (Ω NL = 0 . 36 and Ω Λ = 0 . 34 at z r = 0). In this case the dark energy equation of state starts to decrease only recently with the today's value w DE = -1 . 39. Observationally it is possible to distinguish between the three different cases of Fig. 2. In the limit that ρ Λ → 0, the evolution of w DE approaches the one shown in Fig. 1. For smaller ρ Λ the graviton mass m tends to be larger because of the earlier dominance of the non-local term. For the cases (a), (b), and (c), the numerical values of the mass are m = 8 . 37 × 10 -9 H 0 , m = 1 . 22 × 10 -7 H 0 , and m = 1 . 46 × 10 -7 H 0 , respectively. In Fig. 3 we plot the evolution of Ω NL and Ω Λ for the initial conditions corresponding to the case (b) in Fig. 2. After Ω NL gets larger than Ω Λ today, Ω NL approaches 1, while Ω Λ starts to decrease toward 0. This behavior comes from the fact that, after the dominance of Ω NL , the terms on the left hand side of Eq. (59) balance with each other, i.e., 3 H 2 +( m 2 / 4)( U +3 V -4 ˙ S 0 ) /similarequal 0. Then the cosmological constant appearing on the right hand side of Eq. (59) effectively decouples from the dynamics of the system. This is a kind of degravitation, by which the contribution of the matter component present in the energy density ρ becomes negligible relative to that of the non-local term.", "pages": [ 8, 9 ] }, { "title": "IV. CONCLUSIONS AND DISCUSSIONS", "content": "In this paper we showed that the field equation of motion in the non-local massive gravity theory proposed by Jaccard et al. [16] follows from the covariant non-local Lagrangian (25) with quadratic curvature terms. This is the generalization of the super-renormalizable massless theory with the ultraviolet modification factor e H( -/square M ) . Expanding the Lagrangian (25) up to second order of the perturbations h µν on the Minkowski background, the propagator of the theory can be expressed in terms of four operators which project out the spin-2, spin-1, and two spin-0 parts of a massive tensor field. The propagator (24) smoothly connects to that of the massless theory in the limit m → 0 and hence there is no vDVZ discontinuity. We also found that the theory described by (25) is unitary at tree level, by coupling the propagator to external conserved stress-energy tensors and evaluating the residue of the amplitude at the pole ( k 2 = m 2 ). In the presence of a conserved energy-momentum tensor T µν , the non-local equation of motion following from the Lagrangian (25) is given by Eq. (45). In the lowenergy regime much below the Planck scale the quadratic curvature terms can be negligible relative to other terms, so that the equation of motion reduces to (46) for H( -/square M ) = 0. We studied the cosmological dynamics based on the non-local equation (46) in detail on the flat FLRW background. The tensor field S µν , which satisfies the relation (49), can be decomposed into the form (51). In order to respect the continuity equation ∇ µ T µν = 0 for matter, the transverse part of S µν needs to be extracted in the second term on the left hand side of Eq. (46). Among the components of the vector S µ in Eq. (51), the three vector S i ( i = 1 , 2 , 3) vanishes because of the symmetry of the FLRW space-time. In addition to the vector component S 0 , we also have two scalar degrees of freedom U = S 0 0 + S i i and V = S 0 0 -S i i / 3. /negationslash While the above property of the non-local massive gravity is attractive, the evolution of w DE smaller than -1 . 5 during the matter and accelerated epochs is in tension with the joint data analysis of SNIa, CMB, and BAO [45]. In the presence of the cosmological constant Λ (or other dark energy components such as quintessence), the dark energy equation of state can evolve with the value close to w DE = -1 in the deep matter era (see Fig. 2). In such cases the model can be consistent with the observational data. In the asymptotic future the non-local term dominates over the cosmological constant, which can be regarded as a kind of degravitation of Λ. Among these dynamical degrees of freedom, the scalar field V exhibits instabilities for the cosmological background with ˙ H = 0. Even in the absence of a dark energy component, a late-time accelerated expansion of the Universe can be realized by the growth of V . In order to avoid an early entry to the phase of cosmic acceleration, the graviton mass m is required to be very much smaller than the today's Hubble parameter H 0 . We showed that the equation of state of this 'dark' component evolves as w DE = -1 . 791 (radiation era), w DE = -1 . 725 (matter era), and w DE = -1 . 506 (accelerated era), see Fig. 1. Recently, Maggiore [46] studied the modified version of the non-local massive gravity in which the second term on the left hand side of Eq. (46) is replaced by m 2 ( g µν /square -1 R ) T , where T denotes the extraction of the transverse part. In this theory the -2 ˙ H term on the right hand side of Eq. (63) disappears, in which case the growth of V can be avoided for the initial conditions V = ˙ V = 0 (i.e., decoupled from the dynamics). Since the growth of the fields U and S 0 is milder than that of the field V studied in Sec. III A, w DE evolves from the value slightly smaller than -1 during the matter era to the value larger than -1 [46]. It will be of interest to study whether such a theory can be consistently formulated in the framework of the covariant action related to the super-renormalizable massless theory. While we showed that the theory described by the covariant Lagrangian (25) is tree-level unitary on the Minkowski background, it remains to see what happens on the cosmological background. This requires detailed study for the expansion of the Lagrangian (25) up to sec- ond order in cosmological perturbations about the FLRW background. We leave such analysis for future work.", "pages": [ 9, 10 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "L. M. and S. T. are grateful to Gianluca Calcagni for the invitations to 1-st i-Link workshop on quantum gravity and cosmology at which this project was initiated. S. T. is supported by the Scientific Research Fund of the JSPS (No. 24540286) and financial support from Scientific Research on Innovative Areas (No. 21111006). [arXiv:1107.2403 [hep-th]]; L. Modesto, arXiv:1305.6741 [hep-th]. [hep-th/0206188].", "pages": [ 10, 11 ] } ]

[ { "title": "Jose Luis Bl'azquez-Salcedo 1 , Jutta Kunz 2 , Francisco Navarro-L'erida 3", "content": "1 Dept. de F'ısica Te'orica II, Ciencias F'ısicas Universidad Complutense de Madrid, E-28040 Madrid, Spain 2 Institut fur Physik, Universitat Oldenburg Postfach 2503, D-26111 Oldenburg, Germany Dept. de F'ısica At'omica, Molecular y Nuclear, Ciencias F'ısicas Universidad Complutense de Madrid, E-28040 Madrid, Spain", "pages": [ 1 ] }, { "title": "Abstract", "content": "Extremal rotating black holes in Einstein-Maxwell theory feature two branches. On the branch emerging from the Myers-Perry solutions their angular momentum is proportional to their horizon area, while on the branch emerging from the Tangherlini solutions their angular momentum is proportional to their horizon angular momentum. The transition between these branches occurs at a critical value of the charge, which depends on the value of the angular momentum. However, when a dilaton is included, the angular momentum is always proportional to the horizon area.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Although in D = 4 dimensions the Kerr-Newman solution represents the unique family of stationary asymptotically flat black holes of Einstein-Maxwell (EM) theory, the corresponding D > 4 charged rotating black holes have not been obtained in closed form yet. Only certain subsets are known: the generalization of the static black hole to higher dimensions pioneered by Tangherlini [1], and the rotating vacuum black holes, obtained by Myers and Perry (MP) [2]. Other subsets could be constructed perturbatively [3, 4, 5, 6, 7] and numerically [8, 9]. 1 Nevertheless, if additional fields and/or interactions are allowed into the theory, exact higher dimensional charged rotating black holes can be obtained by solution generating techniques. For example, in the simplest Kaluza-Klein (KK) case, a boost is done to the D +1 embedding of the uncharged D -dimensional MP black holes along the extra dimension. The result is a charged D -dimensional black hole in Einstein-Maxwell-dilaton (EMd) theory. The dilaton coupling constant h for this solution has a particular value, which we denote h KK , that depends on the dimension D [10]. To generate rotating EMd black hole solutions with other values of the coupling constant h , currently perturbative or numerical techniques must be used. D -dimensional stationary black holes possess, in general, N independent angular momentum J i associated with N orthogonal planes of rotation [2], where N is the integer part of ( D -1) / 2, corresponding to the rank of the rotation group SO ( D -1). As a result, we can distinguish between oddD and evenD black holes, where the latter have an unpaired spatial coordinate [2]. In the particular case in which all N angular momenta are equal in magnitude, the EMd equations simplify considerably, yielding, for odd dimensions, cohomogeneity-1 equations from which the angular dependence can be extracted analytically. Hence, the equations reduce to a more tractable system of ordinary differential equations. When the N angular momenta are of equal magnitude, J = | J i | , it is interesting to note that, for extremal MP black holes, the angular momentum J and the horizon area A H are proportional: J = √ 2( D -3) A H . This is a special case of a more general type of relations for MP black holes in terms of the non-degenerate inner and outer horizon areas of non-extremal black holes [11], and was pointed out in 4 dimensions before [12, 13, 14, 15, 16, 17, 18]. In the case of charged black holes, the relation for the product of the horizon areas can be typically written as a sum between the squares of the angular momentum and some power of the charge [11, 12, 13, 14, 15, 16, 17, 18, 19]. In this paper we study this kind of relations between area, angular momentum and charge for extremal EM and EMd black holes with equal angular momentum. We construct the global solutions numerically and local solutions in the near horizon formalism. The EM case is special since two different branches of charged extremal solutions exist. One branch emerges from the uncharged MP black holes, and the other branch emerges from the static Tangherlini black holes. The area relations are different on each branch: the first branch retains the proportionality between the angular momentum and the area of the MP 3 solutions. Thus the area of these charged black holes is independent of the charge. In constrast, the second branch exhibits a proportionality between the angular momentum and the horizon angular momentum, while the charge enters into the area relation yielding A 2 H = C 1 J 2 Q -3 / 2 + C 2 Q 3 / 2 , where C 1 and C 2 are some constants and Q is the electric charge. However, as soon as the dilaton is coupled, the branch structure changes, and only a single branch similar to the first branch of the EM case - is found. Again along this branch the proportionality between the angular momentum and the area persists for all extremal solutions. We will proceed by first presenting the D = 5 results and then discussing their generalization to odd D > 5 dimensions.", "pages": [ 1, 2 ] }, { "title": "2. 5D EMd near horizon solutions", "content": "In 5 dimensions, the EMd action can be written as where R is the curvature scalar, φ the scalar dilaton field, h the dilaton coupling constant and F µν = ∂ µ A ν -∂ ν A µ the field strength tensor, where A µ denotes the gauge vector potential. The units have been chosen so that 16 πG = 1, G being Newton's constant. If we set h = 0, the pure EM action is recovered, while h KK = √ 2 3 is the KK value. For cohomogeneity-1 solutions the isometry group is enhanced from R × U (1) 2 to R × U (2), where R represents time translations. This symmetry enhancement allows to factorize the angular dependence and thus leads to ordinary differential equations. Following the near horizon formalism [20, 21], we now obtain exact near horizon solutions for these extremal EM and EMd black holes. In terms of the left-invariant 1-forms σ 1 = cos ψd ¯ θ +sin ψ sin ¯ θdφ , σ 2 = -sin ψd ¯ θ +cos ψ sin ¯ θdφ , and σ 3 = dψ + cos ¯ θdφ , the near horizon metric can be written as where we have defined 2 θ = ¯ θ , φ 2 -φ 1 = φ , φ 1 + φ 2 = ψ , θ ∈ [0 , π/ 2], ( ϕ 1 , ϕ 2 ) ∈ [0 , 2 π ]. The horizon is located at r = 0, which can always be achieved via a transformation r → r -r H . Note, that the metric is written in a corotating frame. The metric corresponds to a rotating squashed AdS 2 × S 3 spacetime, describing the neighborhood of the event horizon of an extremal black hole. The corresponding Ansatz for the gauge potential in the co-rotating frame reads The dilaton field is simply given by Φ = u . The parameters k , v i , q i and u are constants, and satisfy a set of algebraic relations, which can be obtained, according to [20, 21], in the following way. Evaluating the Lagrangian density √ -g L for the near horizon geometry (2) and integrating over the angular coordinates yields the function f , from which the field equations follow. In particular, the derivatives of f with respect to the parameters vanish except for where J is the total angualar momentum and Q is the charge. From these equations a set of algebraic relations for the near horizon expressions (2), (3) is obtained. The entropy function is obtained by taking the Legendre transform of the above integral with respect to the the parameter k , associated with both angular momenta, J 1 = J 2 = J , and with respect to the parameter q 1 , associated with the charge Q , Then the entropy associated with the black holes can be calculated by evaluating this function at the extremum, S = E extremal . For the discussion of the solutions we need to consider the EM and the EMd case separately. In the EM case, i.e. for h = 0, the system of equations yields two distinct solutions, depending on two parameters. These two solutions of the near horizon geometry have been found independently by Kunduri and Lucietti in [22]. 2 Here we now calculate the charges and entropies associated with these two branches. The solution containing the MP limit, and thus the first branch, has v 2 = 4 v 1 , v 3 = 2 -q 2 2 v 1 , q 1 = 0, k = 1 2 and while the solution containing the Tangherlini limit, and thus the second branch, has v 2 = 4 v 1 , v 3 = 1 4 k 2 +1 , q 1 = -(2 k +1)(2 k -1) √ 3 2 √ | v 1 | 4 k 2 +1 , q 2 = -2 √ 3 k √ | v 1 | 4 k 2 +1 and where J H is the horizon angular momentum. The two solutions match at k = 1 / 2, where q 1 = 0. At this critical point the angular momentum can be written as Thus we have the surprising result that along the first branch, the proportionality of the angular momentum and the area known for the MP black holes, continues to hold in the presence of charge until the critical point is reached. In contrast, on the second branch we have proportionality of the angular momentum and the horizon angular momentum. /negationslash In the case of the EMd black holes, only one solution is found. It can be obtained by replacing q 2 → q 2 e -hu , ¯ Q = Qe hu in the first branch solution of the EM case. Hence, as long as h = 0, the angular momentum and the area are always proportional, independent of h and Q . In particular, this includes the KK case, where the full solution is known analytically.", "pages": [ 2, 3 ] }, { "title": "3. 5D EMd black hole solutions", "content": "We now need to consider the full solutions, which we obtain by numerical integration. For the metric we employ the parametrization for the gauge potential we use while the dilaton field is described by the function Φ( r ). The resulting set of coupled ODEs then consists of first order differential equations for a 0 and n , and second order differential equations for f , m , n , ω , a k and Φ. The equation for a 0 allows to eliminate this function from the system. To obtain asymptotically flat solutions, the metric functions should satisfy the following set of boundary conditions at infinity, f | r = ∞ = m | r = ∞ = n | r = ∞ = 1, ω | r = ∞ = 0. For the gauge potential we choose a gauge such that a 0 | r = ∞ = a ϕ | r = ∞ = 0. For the dilaton field we choose φ | r = ∞ = 0, since we can always make a transformation φ → φ -φ | r = ∞ . In isotropic coordinates the horizon is located at r H = 0. An expansion at the horizon yields f ( r ) = f 4 r 4 + f α r α + o ( r 6 ), m ( r ) = m 2 r 2 + m β r β + o ( r 4 ), n ( r ) = n 2 r 2 + n γ r γ + o ( r 4 ), ω ( r ) = ω 1 r + ω 2 r 2 + o ( r 3 ), a 0 ( r ) = a 0 , 0 + a 0 ,λ r λ + o ( r 2 ), a k ( r ) = a k, 0 + a k,µ r µ + o ( r 2 ), Φ( r ) = Φ 0 +Φ ν r ν + o ( r 2 ). Interestingly, the coefficients α , β , γ , λ , µ and ν are non-integer. Only ω has an integer expansion. To construct the solutions numerically, we employ a compactified radial coordinate, x = r/ ( r + 1). We then reparametrize the metric in terms of the functions f = ˆ fx 2 , m = ˆ m , n ( r ) = ˆ n , ω ( r ) = ˆ ω (1 -x ) 2 , a k = ˆ a k /x 2 , and Φ = ˆ Φ /x 2 to properly deal with the non-integer coefficients in the horizon expansion, eliminating possible divergences in the integration of the functions. We employ a collocation method for boundary-value ordinary differential equations, equipped with an adaptive mesh selection procedure [23]. Typical mesh sizes include 10 3 -10 4 points. The solutions have a relative accuracy of 10 -10 . The estimates of the relative errors of the global charges and other physical quantities are of order 10 -6 . Fig. 1 exhibits the ratios J/A H and J/J H versus the charge Q/M for extremal 5 D EM ( h = 0) and KK ( h = h KK ) black holes. It clearly reveals the two branches of the extremal EM solutions, together with their matching point. This is in constrast to the single branch of the EMd solutions, shown here for the KK case. /negationslash We exhibit in Fig. 2 the domain of existence of the EM and EMd black holes for dilaton coupling constants h = 0, 2, 0.5 and h KK . Here we display the area A H /M 3 / 2 versus the charge Q/M for extremal and static 5 D black holes. All black holes of the respective theories can be found within these boundaries. Again we note the different structure for the EM case. The EM static extremal solution has finite area, whereas for h = 0 the static extremal solution is singular with vanishing area.", "pages": [ 3 ] }, { "title": "4. EMd black holes in odd D > 5", "content": "In a straightforward generalization the near horizon solutions can be constructed for arbitrary odd dimensions D > 5. In the EM case we retain two branches of solutions, the MP branch with and the Tangherlini branch with In the EMd case the near horizon solutions possess only a single branch corresponding to the first branch, with J = √ 2( D -3) A H . We have performed the respective set of numerical calculations in 7 D and in 9 D , and obtained results that are analogous to the 5 D case.", "pages": [ 3, 4 ] }, { "title": "5. Comparison with other theories", "content": "Let us now compare these results with those of two theories whose extremal black holes also exhibit a branch structure with two distinct branches: the rotating dyonic black holes of 4-dimensional KK theory [24], and the 5-dimensional black holes of Einstein-Maxwell-ChernSimons (EMCS) theory (minimal D = 5 supergravity) [25]. In the first example the 4-dimensional black holes are characterized by their mass M , angular momentum J , electric charge Q and magnetic charge P . In the extremal case, only three of these charges are independent and two distinct surfaces, S and W , are found. The restriction to P = Q then yields two distinct branches. The S branch, J > PQ , emerges from the extremal Kerr solution, and presents all the normal characteristics of charged rotating solutions, such as an ergo-region and non-zero angular velocity. On the other hand, the W branch, J < PQ , possesses no ergo-region and has vanishing horizon angular velocity, although the angular momentum of the black holes along this branch does not vanish. At the matching point of both branches, J = QP , the horizon area is zero and the configuration is singular. Nevertheless, the area-angular momentum relation for these extremal solutions can be written as Note, that the electric and magnetic charges are entering the relation for both branches, and that the only difference in the area relation is an overall sign in the expression, depending on whether we are on the S ( J > PQ ) or on the W ( J < PQ ) branch. The second example exhibits rather analogous features. Here we consider 5-dimensional black holes in EMCS theory for the supergravity value of the CS coupling constant, λ = 1 (in an appropriate parametrization). In the extremal case, when both angular momenta possess equal magnitude, the black holes are parametrized by the angular momentum J and the charge Q . Again two branches of extremal black holes are present. The first branch has J 2 > -4 3 √ 3 π Q 3 and is the ordinary branch with an ergoregion, while the second branch has J 2 < -4 3 √ 3 π Q 3 and is ergo-region free with vanishing horizon angular momentum. The area-angular momentum relation for both branches reads At the matching point of both branches the horizon area is again zero and the solution is singular, and again there is a change of sign in the area-angular momentum relation depending on the branch. Thus in these cases, both charge and angular momentum are entering the area relation. Moreover, the relations (14) and (15) are in accordance with the general expressions obtained in [11], which also depend on both, the charges and the angular momenta.", "pages": [ 4 ] }, { "title": "6. Further remarks", "content": "It is interesting to note that for the extremal rotating black holes in EM theory with equal angular momenta, a branch structure with two distinct branches is found, where for one of the branches - the one emerging from the MP solution - the area is independent of the charge of the configuration. Along this branch of solutions, the area remains proportional to the angular momentum and the charge is not entering the relation. This is different from other charged black holes considered before. However, once the critical extremal EM solution 3 is passed, the charge enters again into the area relation, yielding the expression where C 1 = 3 1 / 4 π √ 2 in our normalization. In contrast to the two branches of global extremal EM black hole solutions, the two branches of EM near-horizon solutions do not end at the critical solution. Thus a study of only near-horizon solutions is insufficient to clarify the domain of existence of extremal solutions, as was first observed for the extremal dyonic black holes of D = 4 GaußBonnet gravity [26]. /negationslash Interestingly, in the general EMCS theory (with CS coupling constant λ = 1 [27, 28]), there appear even more than two branches of extremal black holes for sufficiently large CS coupling [29]. As in the case discussed above, however, the area of these branches of rotating charged black holes always depends on both, the charge and the angular momentum. /negationslash Whereas the branch structure of these extremal black holes is very intriguing, their relation with the corresponding near horizon solutions is surprising as well. In particular, a given near horizon solution can correspond to i) more than one global solution, ii) precisely one global solution, or iii) no global solution at all. It would be interesting to perform an analogous study for the general EMd theory (with dilaton coupling constant h = h KK [30]), since the analogy between the known black holes of both theories suggests that a similar more complex branch structure would be present for sufficiently large dilaton coupling.", "pages": [ 4, 5 ] }, { "title": "Acknowledgements.-", "content": "We would like to thank B. Kleihaus and E. Radu for helpful discussions. We gratefully acknowledge support by the Spanish Ministerio de Ciencia e Innovacion, research project FIS2011-28013, and by the DFG, in particular, the DFG Research Training Group 1620 'Models of Gravity'. J.L.B was supported by the Spanish Universidad Complutense de Madrid.", "pages": [ 5 ] } ]

[ { "title": "Theory of optomechanics: Oscillator-field model of moving mirrors", "content": "Chad R. Galley, 1, 2 Ryan O. Behunin, 3 and B. L. Hu 4, 5 1 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91125, USA 2 Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91106, USA 3 Center for Nonlinear Studies and Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico 87545, USA 4 Joint Quantum Institute and Maryland Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, USA 5 Institute for Advanced Study and Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (Dated: November 29, 2021) In this paper we present a model for the kinematics and dynamics of optomechanics [1] which describe the coupling between an optical field, here modeled by a massless scalar field, and the internal (e.g., determining its reflectivity) and mechanical (e.g., displacement) degrees of freedom of a moveable mirror. As opposed to implementing boundary conditions on the field we highlight the internal dynamics of the mirror which provides added flexibility to describe a variety of setups relevant to current experiments. The inclusion of the internal degrees of freedom in this model allows for a variety of optical activities of mirrors from those exhibiting broadband reflective properties to the cases where reflection is suppressed except for a narrow band centered around the characteristic frequency associated with the mirror's internal dynamics. After establishing the model and the reflective properties of the mirror we show how appropriate parameter choices lead to useful optomechanical models such as the well known Barton-Calogeracos model [2] and the important yet lesser explored nonlinear models (e.g., Nx coupling) for small photon numbers N , which present models based on side-band approximations [3] cannot cope with. As a simple illustrative application we consider classical radiation pressure cooling with this model. To expound its theoretical structure and physical meanings we connect our model to field-theoretical models using auxiliary fields and the ubiquitous Brownian motion model of quantum open systems. Finally we describe the range of applications of this model, from a full quantum mechanical treatment of radiation pressure cooling, quantum entanglement between macroscopic mirrors, to the backreaction of Hawking radiation on black hole evaporation in a moving mirror analog.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Optomechanics deals with the interaction of light with mechanical systems. (For an introductory review see e.g., [1] and references therein.) Though old in name it is relatively new in content - optomechanics has a history at least as old as radiation pressure [4]. At the quantum level, optomechanics can be traced at least as far back as Casimir [5], who showed that there is an attractive force between two conducting plates from the change of ground state energy in the presence of boundary conditions, and to Casimir and Polder [6] who calculated the force on an atom near an ideal mirror. The last decade has seen intense interest in several areas that are all under the umbrella of optomechanics. To name one such area: the dynamical Casimir effect [7] where a moving object, be it a moving mirror in vacuum, a contracting gas bubble in a fluid (e.g., sonoluminescence as advocated in [8]), a time-varying magnetic flux bias threading a SQUID terminating a coplanar transmission line [9], or even the spacetime (see below). Optomechanics is of renewed current interest because of at least three new developments. The first relating to nanotechnology [10], where miniature mechanical motion can be transduced or manipulated with high precision by capacitive coupling or optical control, or in nano-scale wave guides where radiation pressure effects become important, e.g. leading to large tailorable photon-phonon couplings which give rise to a large enhancement of stimulated Brillouin scattering [11]. The second pertains to quantum information, where information stored in atoms and photons can interface with mechanical devices [12, 13]. The third pertains to the use of atoms as optical elements [14]. Historically, the gravitation physics community also has explored mirror-field interactions in several ways. For example, cosmological particle creation in the early universe (studied by Parker and Zel'dovich in the 60's-70's [15, 16]) is a form of the dynamical Casimir effect since it arises from the parametric amplification of vacuum fluctuations by the expansion of the universe. Another example was the use of a uniformly accelerated mirror as an analog model of Hawking-Unruh effects developed by Davies and Fulling [17, 18]) [19, 20]. It should be noted that these effects are different from cosmological particle creation as both the black hole and the uniformly accelerated detector/mirror have event horizons while the former in general does not (the de Sitter and anti- de Sitter universes being notable exceptions). Yet another example, Forward following the suggestion by Joseph Weber [21] proposed using laser interferometers for the detection of gravitational waves, which has since ushered in today's large-scale and international ground-based gravitational wave detection effort [22-24]. These gravitational wave detectors are probably the best illustration of the reverse function of optomechanics since in this case an impinging gravitational wave displaces the mirrors in the interferometer and the laser beam in the optical arms picks up the corresponding signal. In terms of practical applications, an optomechanical process that is actively pursued now is mirror cooling by radiation pressure (see e.g., [25]). Optomechanics also provides an excellent means for probing foundational issues in quantum physics. Sample studies include: 1) Reaching beyond the standard quantum limit using superconducting [26] and nanoelectromechanical [27] devices; 2) Schemes for the improvement of signal-to-noise ratio in gravitational wave interferometer detectors [28-31] based on earlier theoretical work of Braginsky and Khalil [32], Caves [4, 33], Unruh [34], Kimble, Thorne, et al. [3]; 3) Quantum superposition and entanglement of macroscopic objects such as between a mirror and the field [35] and between two mirrors [36]; and 4) Gravitational decoherence, both in its possible limitation to the precision of atom interferometry [37] and as a justification for a modified quantum theory [38]. Theoretical development for optomechanics also began quite early, most notably in the classic papers of Moore [39], Fulling and Davies [17], Jaekel and Reynaud [40], Barton and Calogeracos [2], Law [41], Dodonov and Klimov [42], Schutzhold, et al. [43] who took a canonical Hamiltonian approach, and Hu and Matacz [44], Golestanian and Kardar [45, 46], Wu and Lee [47], Fosco, Lombardo and Mazzitelli [48] who took a path integral approach. There is also a lineage of work on relativistically moving mirrors as analog models of the Hawking effect (see e.g., [49-51]). However, many theoretical aspects remain untouched or were treated loosely (some even erroneously). In view of the momentous recent advances in optomechanics, we find it timely and necessary to construct a more solid and complete theoretical framework of moving mirrors interacting with a quantum field. Our goal is to come up with models and theories capable of treating all of the problems listed above yet conceptually simple and theoretically systematic enough to be viable and useful. Admittedly not a simple task [52], we will delineate different aspects as we progress. Suffice it to mention that this first series of papers present the basic models and theories of optomechanics both for a closed (this paper) and open (sequel paper in this series) [53] system dynamics of moving mirrors in a quantum field. These models can be used to treat the broad class of problems related to the dynamical Casimir effect, among other things. The second series includes the back-action of quantum fields on the mirror, which is needed for treating mirror cooling (for earlier work see references in [54]), the results therein could be applied to the related topics of quantum friction [55-58] and vacuum viscosity [16, 46, 59]. A third series will focus on basic issues in quantum information, making use of the stochastic equations derived in a following paper for moving mirrors interacting with a quantum field, specifically on quantum decoherence, superposition, and entanglement of mirrors and field. A different vein using similar techniques but staged in curved spacetimes is the moving mirror analog of backreaction of Hawking radiation on the evolution of a black hole (for background see references in e.g., [60]). In this first paper we tend to the first order of business, namely, that of developing a useful microscopic model for any number of mirrors interacting with a field. We consider a massless scalar field in one spatial dimension for simplicity. The more realistic electromagnetic field in three spatial dimensions can be treated with a slight modification in the form of mirror-field coupling, known as the minimal coupling (see the appendix of [61]). In most prior considerations for the primary functions of a mirror its reflective properties (say, by the AMO community) and the boundary conditions it imposes on the ambient field (say, by the field theory community), i.e., its amplitude has to vanish at the location of the mirror, are considered in a disjoint manner. The advantage of the present model is that it avoids the necessity for considering boundary conditions (e.g., a la Fulling and Davies). Only upon elimination of the explicit dependence of the internal degrees of freedom of the mirror would the field equations require careful attention to boundary conditions. We then consider the kinematics of mirror motion , which also has an effect on the field. For example, the motion can parametrically amplify the field modes, including its vacuum fluctuations, which results in particle creation (in the field theory language) or 'motion-induced'/'acceleration' radiation (in the atom-optics language). From practical experience physical mirrors have surfaces possessing 'light' (as opposed to 'heavy') degrees of freedom that interact with externally incident radiation in such a way as to maintain the appropriate boundary conditions that depend on the material composition of the mirror. Physical mirrors are transparent to sufficiently high frequency components of the field because the mirror's internal degrees of freedom are not energetic enough to (strongly) couple to field modes with arbitrarily high frequencies. For field modes with frequencies far below this cut-off frequency, known as the plasma frequency, the mirror becomes nearly perfectly reflecting. In this paper, we treat the mirror motion as that of a particle with mass M and corresponding to the center of mass of the mirror. To account for the mirror's reflectivity, we model the mirror's 'light' internal degree of freedom as a simple harmonic oscillator with mass m and natural frequency Ω. This internal variable q ( t ) is taken to couple bilinearly to the massless scalar field at the mirror's location. Because this model involves the mutual interaction of the internal oscillator, the field, and the center of mass motion of the mirror we shall call this model an mirroroscillator-field (MOF) model for optomechanical applications. Further details and properties of the MOF model are given in Section II. In Section II we demonstrate the mirror's ability to reflect and transmit incident radiation and to perfectly reflect or transmit radiation upon judicious choices for the parameter values of the internal oscillator. We also compare our model with two commonly used models/descriptions for mirrors: 1) The model of Barton and Calogeracos (BC) [2] (described in Section II B) for partially transmitting mirrors; and 2) The auxiliary field approach of Golestanian and Kardar [45, 46]. We also show that our model extends the BC model to nonadiabatic regimes of the internal oscillator dynamics. In Section III we turn our attention to a moving mirror by extending our model to allow for arbitrary motion, relativistic or non-relativistic. In Section IV we describe the MOF model for multiple moving mirrors and focus our attention on how our model appropriately describes multiple reflections and transmissions of radiation incident on a cavity. Hence, the MOF model is also applicable to multiple-mirror systems in general and to a cavity, in particular, which should be useful for laboratory related studies. In Section V we apply the MOF model to describe (classical) mirror cooling by radiation pressure and indicate the role of the mirror's internal oscillator. In Section VI we show how the bilinear coupling in the MOF model relates to the phenomenological model of moving mirrors wherein the radiation pressure acts on the mirror through the number of incident photons times the position of the mirror (which we refer to as an Nx -type coupling). In Section VII we show how the MOF model of N moving mirrors is related to models of quantum Brownian motion (QBM) involving N harmonic oscillators coupled to a bath of oscillators. The available and exact master equations for the latter model will facilitate, among other things, our later studies of entanglement between two mirrors, a prototype problem in macroscopic quantum phenomena as described above. Finally, in Section VIII we summarize our findings and mention further work in progress toward the construction of a more complete theory of optomechanics.", "pages": [ 2, 3 ] }, { "title": "II. A MIRROR AT REST MODELED BY A BILINEAR OSCILLATOR-FIELD COUPLING", "content": "In this section we introduce a model for a mirror at rest interacting with a scalar field. Our system consists of a mirror with mass M that we treat as being point-like so that, when allowed to move, its trajectory is described by coordinates Z ( t ). The 'light' degrees of freedom, which are responsible for the reflective function of the mirror, is modeled as an internal oscillator q ( t ) with mass m glyph[lessmuch] M and natural frequency Ω. For brevity, we will refer to this internal mirror oscillator as a mirosc . Modeling the 'light' degrees of freedom by a simple harmonic oscillator is functionally similar to the idealization of the internal degrees of freedom of an atom as a 'two-level' system when considering the atom's optical activities (such as spontaneous and stimulated emission) when interacting with a field via a resonant type of coupling [62] from a 'harmonic atom' coupling with a bosonic bath when multiple level activities become important. Lastly, we take the mirosc to couple to the external (possibly quantum) scalar field Φ( t, x ) in a manner that is linear in both quantities (i.e., bilinearly coupled). Taken together, we will refer to this model categorically as an mirror-oscillator-field (MOF) model for optomechanical applications. Different oscillatorfield couplings in this model give rise to different models familiar in optics. However, we will always be considering a bilinear coupling in this paper. We shall show below how this model can describe, with appropriate choices of parameters, a range of perfectly and imperfectly reflecting mirrors. We also show how it relates to the model of Barton and Calogeracos (BC) [2] used in the quantum optics community and to the auxiliary field model used more in the field-theory community [45, 46].", "pages": [ 3, 4 ] }, { "title": "A. Reflectivity of a mirror modeled by a bilinear oscillator-field coupling", "content": "To demonstrate that the MOF model described above actually possesses the ability to reflect incoming modes it is sufficient to put the mirror at rest at the origin so that the action is given by where η αβ = diag(1 , -1) is the metric of 1+1 dimensional Minkowski space-time. In units where glyph[planckover2pi1] = c = 1 the coupling constant λ has dimensions of (mass) -2 = (length) 2 . The equations of motion are obtained by varying (2.1) in the usual way, Let a plane wave with frequency ω be incident on the mirror from the left ( L ) so that the field is given by where R ( ω ) and T ( ω ) are the frequency-dependent reflection and transmission coefficients, respectively. For the steady-state evolution of the mirosc-field system we can take q to oscillate with the same frequency as the incident radiation so that where the amplitude A is determined from (2.3) to be The field is continuous at the location of the mirror, Φ ωL ( t, 0 + ) = Φ ωL ( t, 0 -), and the discontinuity of the spatial derivative is found by integrating (2.2) over a vanishingly small interval encompassing the mirror's position, Using these conditions and the equations of motion in (2.2) and (2.3), the reflection and transmission coefficients are given in terms of the three mirosc parameters ( m, Ω , λ ) by There are three ways that the particle can perfectly reflect incident radiation: 1) In the strong oscillator-field coupling limit, λ → ∞ ; 2) When the oscillator is resonantly excited by the monochromatic radiation, Ω = ω , (independently of the values of λ and m ); and 3) In the limit that the mass of the mirosc vanishes, m → 0. In all three cases, the reflection and transmission coefficients are R ( ω ) = -1 and T ( ω ) = 0, respectively. Likewise, perfect transmission can be attained in three manners: 1) In the limit of vanishingly small oscillator-field coupling, λ → 0; 2) When the mirosc frequency is arbitrarily large, Ω →∞ ; and 3) When the mirosc mass is arbitrarily large, m →∞ . In all three cases, R ( ω ) = 0 and T ( ω ) = 1. The reflection and transmission coefficients possess interesting features that depend on the parameters of the theory ( m, Ω , λ ). The squared complex norm of the reflection coefficient from (2.8) is where we have introduced the dimensionless quantity y = ω/ Ω. To characterize the dependence of the reflection coefficient on the frequency ω of the incident field mode we observe that the local minima and maxima occur for y min = 1 / √ 3 and y max = { 0 , 1 } , respectively. The maximum at ω = 0 is an artifact of the monopole coupling between the field and the mirosc. For a dipole coupling the reflection coefficient vanishes at ω = 0. The reflection coefficient equals 1 at both maxima and the value of | R | 2 at y min is where we define the plasma frequency Ω p for the partially transmitting mirror to be and the index r p ≡ Ω / Ω p = 4 m Ω 3 / (3 3 / 2 λ 2 ). We can use this minimum in the reflected radiation to indicate when the two maxima of | R | 2 are sufficiently separated and distinguishable. While this is subjective we take | R ( y min ) | 2 = 1 / 2 to be our defining requirement, which fixes r p = 1. The implication is that if r p glyph[greatermuch] 1 then the reflection coefficient is sharply peaked about ω = Ω. Under this condition, the parameters of the mirosc can be tuned to selectively reflect incident radiation in a narrow bandwidth centered on ω = Ω, which occurs when the mirosc is resonantly excited (or very nearly so) by the incident field. Whenever the mirosc parameters are such that r p glyph[lessmuch] 1 then the local minimum at y min = 1 / √ 3 is close to 1 and the mirosc reflects modes over a broader frequency bandwidth. Furthermore, the mirror will reflect more than half of the incident radiation so long as the frequency of the field is less than ∼ Ω to a good approximation. Hence, if the mirosc mass is made smaller or the oscillator-field coupling constant λ is made larger then more modes will be reflected more strongly by the mirror. Fig. (1) shows the basic features of the mirror's scattering properties studied in this section and contains plots of the reflection coefficient | R ( y ) | 2 for r p equal to, much larger, and much smaller than 1.", "pages": [ 4, 5, 6 ] }, { "title": "B. Relation to common mirror models and approaches", "content": "The bilinearly-coupled oscillator-field model introduced above possesses interesting physical limits that relate to two well-known and commonly used mirror models. The first model arises when the mirosc evolves adiabatically with the field and gives rise to the model of Barton and Calogeracos (BC) for a partially transmitting mirror. The second model arises when the mass of the mirosc becomes arbitrarily small, in which case the mirosc serves the role of an auxiliary field that relates to the path integral approaches of [45, 46, 48], which describe a quantum field interacting with a perfectly reflecting mirror(s). In Section VI we also relate the MOF formulation of optomechanics to a commonly used model which describes the effects of radiation pressure by invoking a phenomenological coupling between the number of photons impinging the mirror and the mirror's position.", "pages": [ 6 ] }, { "title": "1. Barton-Calogeracos model", "content": "The BC model has been used quite often in quantum optics and it is worth summarizing its primary properties before showing how it can be derived from our MOF model. Much of BC's attention focuses on quantizing the nonrelativistic limit of the theory where the mirror velocity is much smaller than c . We do not present their results here but refer the reader to the original papers of [2] for further details. The action for a mirror at rest in the BC model is, in 1+1 dimensions of space-time, where γ is related to the plasma frequency of the mirror [2]. Extremizing the action gives the equations of motion The reflection and transmission of a normal mode of the field incident on the mirror from the left ( x < 0) is We demand that the field be continuous across the mirror Φ ωL ( t, 0 + ) = Φ ωL ( t, 0 -) and that its derivative satisfy where R ( ω ) and T ( ω ) are the frequency-dependent reflection and transmission coefficients, reflectively, with the property that | R | 2 + | T | 2 = 1. This jump condition follows from integrating the field equations across the mirror's position at x = 0. Together with the field equation these conditions imply that As the parameter γ becomes arbitrarily large we see that the reflection becomes perfect and the incoming phase of the field changes by π radians The ability of the BC model to reproduce perfect and imperfect reflection comes from using the quadratic interaction Φ 2 ( t, 0). With this specific coupling to the mirror the jump condition across the origin (2.16) is linear in Φ at the mirror, which is vital for obtaining the normal mode in (2.15). The MOF model in (2.1) can be related, under appropriate conditions, to the BC model. Observe from (2.3) that if q ( t ) evolves adiabatically with time, then the mirosc follows the time-development of the field at the mirror's position Substituting this approximation for the oscillator variable into the scalar field equation (2.2) gives Comparing with (2.14) we recover the model of BC by identifying γ with the parameters of the mirosc and hence to the plasma frequency of the MOF model Therefore, in the limit that the mirosc changes adiabatically the MOF model yields the BC model. An equivalent way of connecting to the BC model is to take the mass of the mirosc to zero, m → 0 but keep the quantity m Ω 2 ≡ κ constant in this limit, which requires the mirosc natural frequency to approach infinity, Ω →∞ . In this limit, the mirosc also follows the time-development of the field The identification with the BC model then follows the same steps as in the previous paragraph and, in particular, one finds that γ = λ 2 / (2 κ ). It is worth pointing out that the massless limit m → 0 here does not imply that the mirror is perfectly reflecting as in the previous section. This is because of the additional requirement that m Ω 2 = κ remain constant. In fact, the reflection coefficient (2.8) in this limit becomes and the mirror becomes perfectly reflecting when λ →∞ . Through the identification in (2.23) we may attach heuristic physical interpretations to m , Ω (or κ ) and λ . In [ ? ], Barton and Calogeracos observe that their model is equivalent to a jellium sheet of zero width, i.e., a surface of vanishing thickness having a surface current density generated by the motion of small charge elements with charge density n s . If these elements have charge n s e per unit area and mass n s m e per unit area then BC find Identifying these microscopic variables to those in our MOF model via (2.23) gives the following relationship This suggests identifying the mirosc field coupling as a charge per unit area, λ → n s e , and κ as a mass per unit area, κ → n s m e / (4 π ). That is, λ can be viewed as a surface charge density and κ = m Ω 2 as a surface mass density. This interpretation may be useful for developing a similar MOF model for a mirror in 3+1 dimensions.", "pages": [ 6, 7 ] }, { "title": "2. Models using auxiliary fields", "content": "The MOF model reduces to another well-known description of mirrors if we take the limit m → 0. In this limit our model describes a perfectly reflecting mirror, as discussed earlier, and the action (2.1) becomes The key point is that the mirosc possesses no dynamics in this limit. Thus, the quantity ψ ( t ) ≡ λq ( t ) possesses no dynamics of its own and can be regarded as an auxiliary field . 2 m Ω In the path-integral formulation of the quantum theory, the massless mirosc limit gives rise to the following generating functional [63] Then, noting that the path integral over ψ ( t ) is just the Fourier representation of the Dirac delta functional, it follows that the generating functional describes a quantum scalar field constrained to vanish at the location of the mirror (only those field configurations that vanish at x = 0 will contribute to the path integral). The vanishing of the field at the location of the mirror is equivalent to the perfect reflection of an incident field [45, 46]. Our bilinearly-coupled MOF model (2.1) has successfully reproduced two models describing the interactions of a field with a mirror at rest: 1) The partially transmitting BC mirror model when q ( t ) evolves adiabatically; and 2) an auxiliary field approach that enforces the field to vanish at the mirror when the mass of the mirosc is vanishingly small. In turn, these two models can be related to each other. Specifically, noting that the delta functional above can be approximated by a narrow Gaussian it follows that (2.31) becomes which is increasingly more accurate for larger values of γ . Hence, BC falls out from the generating functional approach if we smear the delta functional constraint that enforces the field to vanish on the surface of the mirror. Likewise, using the action for the BC model in the generating functional formalism gives the perfect reflection limit when γ → ∞ . See Fig. (2) for the relationships among these theories.", "pages": [ 7, 8 ] }, { "title": "III. A MOVING MIRROR IN THE MOF MODEL", "content": "As pointed out in the Introduction the physics is quite different in the two cases when the mirror is moving relativistically compared to the case when it is moving slowly. The former relates to cosmological particle creation and radiation emitted from black holes or in uniformly accelerated detectors in the Hawking-Unruh effects while the latter is closer to accessible laboratory situations such as mirror movements caused by the passing of gravitational waves in interferometer detectors and mirror cooling from the field's back-action in the form of radiative pressure and quantum friction. The MOF model presented here provides a unified framework for treating both, albeit very different, situations. For cases when the mirror motion is prescribed such as coplanar waveguides terminated by a SQUID [9, 64], or when the mirror possesses non-trivial reflective properties [14] our model can meet the needs of current experiments by providing a rich set of reflective properties and a tractable formalism capable of providing analytical insight. For systems where the mirror motion is dynamically determined by the mutual interaction of the mirror's center of mass, it's internal motion, and the field our model provides a computational ease. This simplification results from the fact that boundary conditions are not imposed on the field from the outset but determined by a selfconsistent elimination of the mirror's internal motion. This facilitates the derivation of equations of motion for the mirror's mean position which will be adopted in Sec. V to describe classical radiation pressure cooling, and in later papers in this series to provide a fully quantum mechanical treatment of mirror cooling and the mirror-analog of the black hole back-reaction.", "pages": [ 8, 9 ] }, { "title": "A. Action formulation", "content": "Allowing the mirror to move requires the addition of an extra term describing its motion along the worldline Z µ ( λ ) where λ is an affine parameter and µ = 0 , 1. The physics must remain invariant under any reparameterization of the mirror's worldline λ → λ ( ¯ λ ), which requires modifying the action (2.1) for a static mirror in the following way where an overdot denotes differentiation with respect to the worldline parameter, U µ ( λ ) = ˙ Z µ ( λ ) is the 2-velocity of the mirror, dλ √ U α U α = dτ is the invariant proper time element as measured by an observer on the worldline, and indices with Greek letters are raised and lowered with the Minkowski metric η αβ = diag(1 , -1). The field still couples bilinearly to the mirosc via the last term of the action so that the reflective properties studied in the previous section are retained by the model. The corresponding Euler-Lagrange equations of motion are easily found to be where we conveniently have chosen to parameterize the worldline by the proper time τ at this point since then U α U α = 1 and U α ˙ U α = 0, which help simplify the expressions. The quantity M eff ( τ ) in (3.4) is an effective mass for the mirror and is given by Notice that the effective mass has contributions from the rest mass of the mirror ( M ), the energy of the oscillator ( m ˙ q 2 / 2 + m Ω 2 q 2 / 2), and the interaction energy of the mirror-oscillator-field system ( -λq Φ( Z )). In other words, the effective mass is the rest mass of the mirror plus the total internal energy of the mirosc. The structure of (3.2) and (3.4) is reminiscent of a field coupled to a scalar point charge, which here is played by the time-dependent mirosc amplitude q ( t ). In 3 + 1 dimensions, such a system exhibits a radiation reaction force on the charge proportional to the third time derivative of the particle's position and exhibits the infamous class of unphysical runaway solutions in the absence of any external influences. Below, we show that no such unphysical solutions manifest in our MOF model here. To show this, we first solve the field equation in (3.2), which gives where we ignore the homogeneous solution and where the retarded Green's function in 1+1 spacetime dimensions is where σ is half of the squared distance between x α and x ' α as measured by the straight line (i.e., a geodesic) connecting them, namely, The derivative of the field evaluated on the worldline is then To evaluate the integrals in (3.10) we will need to determine the behavior of δ ( σ ) and ∂ ν ( τ -τ ' ) when τ ' ≈ τ . This follows by expanding (3.8) around s ≡ τ ' -τ near zero, giving From (3.8) it follows that since the mirror's worldline is time-like then σ ( Z µ ( τ ) , Z µ ( τ ' )) is always positive except at τ ' = τ where it vanishes. Hence, the delta function in the first line of (3.10) receives a contribution only at coincidence, when τ ' = τ . Likewise, the δ ( τ -τ ' ) in the second line of (3.10) gives support to the integral at coincidence. where we have used the identities U α U α = 1, U α ˙ U α = 0, and U α U α = -˙ U α ˙ U α , which are valid in the proper time parameterization of the worldline. Therefore, writing the delta function in (3.10) as a delta function of s and then expanding (3.11) for s near zero gives In addition, the second integral in (3.10) is proportional to The important point to note is that the first integral in (3.10) is potentially divergent. However, we will show now that no divergence actually manifests. To see this, we observe that (3.8) implies ∂ ν σ ( x α , x ' α ) = x ν -x ' ν , which, when evaluated on the worldline and expanding around s equal zero, yields Note also that the above equation implies that [ ∂ ν s ] s =0 = -U ν ( τ ) since from (3.11) it follows that ∂ ν σ = s∂ ν s + O ( s 3 ). The integral in (3.10) thus becomes Evaluating the integral over s and using 2 θ ( -s ) = 1 -sgn( s ) we find that which is finite. In addition, the derivative of the field above, which is proportional to U ν , is contracted with η µν -U µ U ν in (3.4) to get the force on the mirror, thereby giving zero . Hence, the equation of motion for the mirror's worldline from (3.4) is simply and the mirror moves inertially. The reason for this trivial dynamics is because the field is not generated by any external sources and because we have ignored the initial configuration of the field (i.e., homogeneous solutions to the field equation (3.2)). Both of these types of sources will impart a non-trivial dynamics for the mirror's motion.", "pages": [ 9, 10 ] }, { "title": "B. Hamiltonian formulation", "content": "Here, we provide a Hamiltonian formulation of the MOF model. To do this, we find it convenient to parameterize the worldline by the coordinate time t wherein the action (3.1) becomes where U ( t ) = dZ ( t ) dt and from which the Lagrangian is To derive the Hamiltonian H we first identify the conjugate momenta, where the effective mass in terms of the conjugate momenta is The Legendre transformation of (3.19) yields the Hamiltonian after some algebra For completeness, we give Hamilton's equations of motion ( t ) (3.28) which can be shown to be equivalent to the Euler-Lagrange equations in (3.2)-(3.4). As discussed in the previous section, an external source will be needed to generate non-trivial forces on the mirror. Depending on the application, it may be more convenient to work in a reference frame wherein the interaction between the field and the mirror's worldline decouple from each other so that the mirror always remains at rest at the origin. A transformation to such a non-inertial frame is advocated in [2] and may be useful for canonically quantizing the MOF model. However, we will not pursue this representation here.", "pages": [ 11 ] }, { "title": "C. A slowly moving mirror in the MOF model", "content": "Under all laboratory conditions to date the speed of the mirror is small compared to c and justifies developing the non-relativistic limit of the mirror-oscillator-field model. For example, it was recently demonstrated that film bulk acoustic resonators (FBARs) [65] as large as ≈ 0 . 5mm can be mechanically oscillated up to 3GHz. The corresponding speed of the FBAR (having a modulation depth of 10 -8 ) is only v ≈ 4 . 4m/s, which is much smaller than c . Thus, for laboratory applications, the non-relativistic limit of the MOF action in (3.1) is entirely appropriate. The relativistic Lagrangian (3.19) expanded in powers of ˙ Z glyph[lessmuch] 1 and retaining the lowest order contributions in the velocity yields where we have dropped the term depending solely on the constant mass of the mirror M and V ( Z ) describes the potential energy of the mirror's motion. The related Hamiltonian follows from a Legendre transform of (3.31) and is found to be The equations of motion are easily derived from (3.31) or (3.32) so we do not give them here.", "pages": [ 12 ] }, { "title": "IV. MULTIPLE MOVING MIRRORS IN THE MOF MODEL", "content": "In the previous sections we introduced a model for a mirror whose scattering and reflective properties are described by an oscillator, the mirosc, coupled bilinearly to the field. In this section we extend the MOF model to include multiple spatially separated partially transmitting mirrors that interact mutually via the field. The Lagrangian for N moving mirrors (possibly relativistically) with masses M a ( a = 1 , . . . , N ) can be written as and the Euler-Lagrange equations of motion follow straightforwardly and are simply given by Eqs. (3.2)-(3.4) with all mirosc and worldline parameters and variables receiving a subscript a to label the mirror. For completeness and for later use, the corresponding Hamiltonian is where the effective mass of the mirror has the same interpretation as before (i.e., mirror rest mass plus total internal energy) except now the total internal energy includes the energy of all N mirosc's and their interaction energies with the field, In the non-relativistic limit, the Lagrangian and the Hamiltonian are In the remainder of this section, we investigate the scattering properties of incident radiation on two mirrors at rest. The equations of motion for the two-mirror MOF model are ( a = 1 , 2) Let a monochromatic plane wave of frequency ω be incident from the left so that The part of the mode ψ ωL ( x ) can be found using the linearity of the field equation from which the superposition principle allows us to write the contributions from multiple reflections and transmissions off of and through both mirrors as The geometric series can be summed for | R 1 R 2 | < 1 whereby To find the reflection and transmission coefficients in terms of the incident frequency ω we assume that the mirosc is in a steady-state evolution and oscillates at the same frequency of the radiation so that The field is continuous at the locations of each mirror and the discontinuity of the spatial derivative is to be consistent with the source of the field equation The mirosc amplitudes A 1 , A 2 satisfy the mirosc equations of motion so that we have six equations for the six unknowns { R a , T a , A a } (note the subscript a = 1 , 2). Thus, the reflection and transmission coefficients are and the amplitude of oscillation for the miroscs are One can check that the identities | R a | 2 + | T a | 2 = 1 are indeed satisfied. The incident field mode (4.9) can then be written as ext When the mirror at x = 0 is perfectly transmitting and the mirror at x = L is perfectly reflecting the field mode is which vanishes as x → L , as expected. In the complementary case when the mirror at x = 0 is perfectly reflecting the field mode incident from the left is as also expected. Hence, the MOF model describes the partially reflecting and transmitting properties of two, and generally more, mirrors.", "pages": [ 12, 13, 14 ] }, { "title": "V. CLASSICAL MIRROR COOLING WITH THE MOF MODEL", "content": "In this Section, we show how the MOF model can be used to describe mirror cooling within a completely classical context. In a following paper, we discuss quantum effects in mirror cooling using the MOF model [54]. The setup is as follows. Consider a cavity formed by two mirrors. We take one of the mirrors to be fixed at the origin and perfectly reflecting so that the (classical scalar) field satisfies Dirichlet boundary conditions, Φ( t, 0) = 0. As this fixed and perfectly reflecting mirror will, by assumption, possess no dynamics then we will model the second mirror by the MOF model. This second mirror possesses a mirosc internal degree of freedom and will be free to move in response to the forces imparted by the field. The motion of this second mirror is assumed to be small relative to the size of the cavity, L , and to move on a time-scale much longer than all other time scales in the problem. The partial reflectivity of the second mirror allows, for example, a laser field, generated by an external source J ext ( x α ), to couple to the cavity.", "pages": [ 14 ] }, { "title": "A. Arbitrary bilinear coupling strength", "content": "The MOF Lagrangian for the system described in the previous paragraph is given by Eq. (3.19) where we have included an external source J ext ( x α ) for the field and the second mirror (the dynamical one) has coordinates x = L + Z ( t ) and moves within a harmonic potential with natural frequency Ω 0 . The Euler-Lagrange equations for the field, the mirosc, and the coordinates of the movable mirror are Our first step will be to solve (5.2) for the field and eliminate its appearance in the remaining equations of motion. Assuming that there is no initial field present [66], so that Φ is generated by J ext and by interactions with the remaining degrees of freedom, then the solution to (5.2) is Φ( x α ) = ∫ d 2 x G ( x α ; x ' α ) J ( x ' α ) + λ ∫ dt G ( x α ; t , L + Z ( t )) q ( t ) (5.5) where the retarded Green's function G ( x α ; x ' α ) for the field subject to Dirichlet boundary conditions at the fixed mirror is given by Note that if x = x ' = L > 0 then ' ' ' ' ' Substituting (5.5) into the remaining equations (5.3) and (5.4) gives where is the propagated external source for the field. Next, we solve for the mirosc variable, q ( t ). At this point we can take advantage of the assumption that Z ( t ) glyph[lessmuch] L so that the typical amplitude of the mirror's motion is much smaller than the size of the cavity. This implies we can write the solution for the oscillator perturbatively as q = q 0 + q 1 + · · · where q n = O ( Z n ). The equation of motion for the leading order mirosc dynamics is The solution to (5.11) is given by (again, ignoring homogeneous solutions) where the kernel D ( τ ) is found to be The equation of motion for the first order perturbative correction to the mirosc dynamics is The right side of (5.14) simplifies somewhat since (5.6) implies that and so (5.14) can be written as Thus, the solution to (5.16) is given by Next, we expand the equation of motion for the worldline to leading order in Z ( t ) to find where F [ Z ( t )] accounts for the external forces and backreaction from the cavity field and mirosc, and is given by where In addition, we can also derive the explicit form for q 0 ( t ) given the expression for the external source Using these expressions we shall evaluate the time-average of (5.18) over the pump period, T = 2 π/ Ω D . The time-average of the terms independent of q 1 ( t ) in (5.19) are easily evaluated. However, the time-average of the terms We find that the general motion of the mirror as influenced by the cavity field is described by a delay integro-differential equation. The backreaction terms above will be shown to lead to several effects. First, the driven field will build up in amplitude inside the cavity formed by the perfect mirror and the mirror-oscillator. This will lead to a spatially varying radiation pressure and a shift in the frequency of the mirror's mechanical motion. Next, depending on the equilibrium position of the mirror the cavity field can either accept from or donate energy to the mirror's motion arising from retardation effects (see e.g. [1] for a detailed explanation of cooling due to retardation). Finally, nonMarkovian effects will be present which show how the mirror's motion in the past influences its future movements, these effects are accounted for in time-delayed and integral terms.", "pages": [ 14, 15, 16 ] }, { "title": "B. The weak-coupling limit", "content": "As an example application of these equations we will explore mirror cooling in the weak coupling limit i.e. λ 2 / ( m Ω 3 ) glyph[lessmuch] 1. For many systems of physical interest there exists a large separation between the values of the cavity frequency and the oscillation frequency Ω 0 , which allows for a multiple time-scale analysis. In the following we will assume that the cavity frequency, the mirosc's frequency Ω, and the pump frequency Ω D are all much larger than the frequency of the mirror's mechanical motion Ω 0 . Under these circumstances we may time-average the mirror's equation of motion in (5.31) over the pump period 2 π/ Ω D . Since the mirror's mechanical motion is very slow compared to this pumping time-scale its trajectory can be safely factored out of any time-averaging integrals so that Let the external source of the field be given by where 〈〈·〉〉 denotes the time average. so that where we take the initial time to be at t = t i and at the end of the calculation take the limit t i →-∞ . Performing the spacetime integration gives where The first two spatial derivatives of F ext evaluated at x = L are similarly evaluated giving in (5.18) containing q 1 ( t ) requires some elaboration. First, we write q 1 ( t ) in (5.17) with Z ( t ) and D ( t -t ' ) replaced by their Fourier transforms, Evaluating the t ' integral and then integrating over ω gives In the weak coupling limit, λ 2 glyph[lessmuch] m Ω 3 , the effect of the cavity field on the (forced) mirror motion is sufficiently small that the mirror will continue to oscillate at a frequency nearly equal to Ω 0 . Consequently, we expect Z ( ν ) to be sharply peaked for frequencies ν ∼ Ω 0 . Since L is inversely proportional to the cavity period and ν ∼ Ω 0 then it follows that νL glyph[lessmuch] 1 and Ω D glyph[greatermuch] ν . Therefore, expanding all terms but Z ( ν ) in the integrand of (5.29) for ν near zero [ ? ] Lastly, since the mirror moves on a time scale ( ∼ 1 / Ω 0 ) much longer than the round trip travel time of light in the cavity ( ∼ 2 L ) then Ω 0 L glyph[lessmuch] 1 and we may expand all delay terms about their instantaneous values so that, for example, Z ( t -2 L ) = Z ( t ) -2 L ˙ Z ( t ) + O ((Ω 0 L ) 2 ). Using (5.30) one may then easily compute the time-average of the terms depending on q 1 ( t ) in (5.19). Putting everything together, and remembering to expand the delay terms as discussed above, we find that (5.18) becomes which is simply the equation for a forced, damped harmonic oscillator with mass M , frequency [Ω 2 0 -∆Ω 2 ( L )] 1 / 2 , and damping coefficient Γ( L ). Notice that the latter two quantities depend explicitly on the length of the cavity. After time-averaging the explicit form for the radiation pressure is given by the frequency shift is given by and the damping coefficient is given by Fig. 3 shows the force on the mirror due to the resulting radiation pressure F rad ( L ) (solid line) and the damping constant Γ( L ) (dashed line) as a function of the movable mirror's unperturbed position L from the static mirror at the origin. The parameter values chosen for these plots are given in the corresponding figure caption. We observe that when the cavity is pumped by an external source, the field energy inside builds up and results in a force from radiation pressure F rad ( L ) that varies depending on the size of the unperturbed cavity. The gradient of the radiation pressure and the Markov approximation of the integral terms, i.e. those terms containing q 1 ( t ), leads to a shift in the oscillation frequency of the mirror's center of mass motion. These optical spring effects are quantified by the term ∆Ω( L ) and changes depending on L . The bottom panel of Fig. 3 shows the fractional change in the mirror's natural frequency [1 -∆Ω 2 ( L ) / Ω 2 0 ] 1 / 2 , which can become imaginary precisely where the real part goes to zero in that plot. An important point to note here is that our weak coupling approximation is valid when | Ω 0 -∆Ω( L ) | glyph[lessmuch] Ω D . For the values indicated in the figure caption and with L = 3 . 295ms we see that the mirror's modified natural frequency satisfies √ Ω 2 0 -∆Ω 2 ≈ 3 . 64Hz << Ω D , which is consistent with the weak coupling approximation. It is also important to mention that the mirror's motion can become unstable when the mirror is to the right of the resonance, namely, the mirror's spring constant, i.e. K = M (Ω 2 -∆Ω 2 ( L )), becomes negative as shown in Fig. 3. For the parameter values given in the caption of Fig. 3, the moving mirror's motion is damped, the top right panel in Fig. 3, and exemplifies the 'cooling' aspect of this classical system to dissipate its input energy into the cavity field.", "pages": [ 16, 17, 18 ] }, { "title": "VI. REDUCTION OF MOF MODEL TO MODELS WITH NX-COUPLING", "content": "In the previous section we used the MOF model in the classical regime to describe the damped motion of one mirror of a cavity forced by interactions with an external (laser) field. In the corresponding quantum theory of mirror cooling, one usually models the interaction between the mirror and the field by the radiation pressure ∼ ˆ N ˆ x where ˆ N is the number operator of quanta (photons) impinging on the mirror's surface and ˆ x is the position operator of the mirror [1, 4, 33]. We will refer to this interaction as ' Nx -coupling.' The basic motivation for this type of interaction can be easily understood by considering the Hamiltonian for a single cavity mode of the form, H cav ∼ ω cav ( L ) a † a , where a ( a † ) is the annihilation (creation) operator for field quanta and ω cav ( L ) is the frequency of a cavity mode of size L . Since the frequency of the cavity modes scales as the inverse cavity length ω cav ( L ) ∼ 1 /L , when we allow the cavity length to vary by a small amount x the frequency is perturbed to leading order as ω cav ( L + x ) ≈ ω cav ( L )(1 -x/L + ... ). For small cavity length changes the Hamiltonian becomes H cav ≈ ω cav ( L )(1 -x/L ) a † a In this section, we show how the quantum MOF model relates to models with Nx -coupling. In doing so, we highlight the assumptions that must be made to connect the two models. We thereby demonstrate that the MOF model should be an improvement of the oft used background field approximation for the cavity field [3, 32] In particular, the MOF model should be very useful for studying optomechanical systems having low numbers of cavity photons where quantum effects can become quite interesting and important. Consider a cavity formed by two mirrors. As in Section V, we take the first mirror (at x = 0) to be fixed for all time and perfectly reflecting so that the field satisfies Dirichlet boundary conditions at the origin. We assume the second mirror to be partially transmitting and dynamical with small perturbations to its equilibrium position at x = L > 0. The second mirror will be described by the MOF model. Recall the Hamiltonian (3.32) for a slowly moving mirror where we have included an external source J ext ( x α ) for the field. We shall show, by making a number of assumptions, that the interaction component of the above Hamiltonian can be reduced to the Nx -coupling. The internal physics of the mirror for many standard radiation pressure cooling calculations is accounted for phenomenologically through the introduction of a cavity quality factor which accounts for the dissipation of field energy from within the cavity. In distinction, the MOF model accounts for the detailed information of the mirror's internal dynamics. We first will solve for the mirosc to find its effect on the mirror's motion. In this way we trade the microscopic information about the mirror for a macroscopic description in terms of the mirror's susceptibility, which will establish the link between Nx -coupling and the MOF model. The Heisenberg equations of motion for the field (3.2) and the mirosc (3.3) variables are We can eliminate the field's explicit dependence on the mirosc q ( t ) by solving (6.4) and plugging the solution into the wave equation (6.3). In the regime where the mirosc evolves adiabatically so that | q | glyph[lessmuch] | Ω 2 q | the approximate solution to (6.4) is given by where the second term on the right side is a correction to the leading order, instantaneous solution and is due to the fact that the full mirosc dynamics is generally non-Markovian. This can be seen from the general solution of (6.4) where q h ( t ) is the homogeneous solution and g ret ( t ; t ' ) is the retarded Green's function for the mirosc More specifically, the mirosc receives contributions from the past as implied by the integral in (6.6). However, the approximation (6.5) is valid if the mirosc degree of freedom is 'light' thus responding nearly instantaneously to external influences. Substituting the approximate mirosc solution (6.5) into the wave equation (6.3) gives the effective dynamics for the field Notice that (6.8) can be derived from the following effective Lagrangian where · · · denotes the higher order terms in (6.5). The interaction Hamiltonian corresponding to the above effective Lagrangian is found to be Assuming that Z ( t ) glyph[lessmuch] L , we may expand the effective interaction Hamiltonian in (6.10) to find where Notice that the leading order interaction Hamiltonian is independent of Z ( t ) so that it exerts no force on the movable mirror. In fact, one can group H (0) eff int with the free Hamiltonian for the field that, when taken together, describes the free evolution of the field in a cavity where one mirror is fixed at x = 0 and perfectly reflecting and the other mirror is fixed at x = L but partially transmitting. The remaining terms in the effective interaction Hamiltonian describe the perturbative response of the second mirror to its coupling with the field and vice versa. To leading order in Z ( t ), we can express the field in terms of a homogeneous solution via the cavity's normal modes and in terms of the external source J ext , where H . c . is the Hermitian conjugate of the preceding terms, u k ( x ) are the normal modes of the cavity and satisfy such that u k (0) = 0 since the mirror at x = 0 is perfectly reflecting. The retarded Green's function here satisfies with Dirichlet boundary conditions at the origin G cav ret ( ω k ; x, 0) = 0 and G cav ret ( ω k ; 0 , x ' ) = 0. Also, N k is chosen so that [Φ( t, x ) , Π( t, x ' )] = i glyph[planckover2pi1] δ ( x -x ' ) for x and x ' greater than zero. These commutation relations require a k and a † k to be annihilation and creation operators, respectively. For the following we focus entirely on the component of the interaction Hamiltonian coming from the field inside the cavity. The field outside of the cavity gives rise to a constant and position independent radiation pressure that only yields a shift in the equilibrium position of the mirror at x = L . If the cavity is pumped by a laser beam with a frequency slightly detuned from one of the cavity resonances and if the cavity quality factor is large then the cavity field, represented as a mode sum, can be approximated well by a single mode. Expressing the field in terms of the fundamental cavity resonance we find, at linear order Z ( t ), that the interaction Hamiltonian (6.13) is given by where · · · refers to corrections arising from time derivatives of the field appearing in (6.13) and For many systems of interest the frequency of the fundamental cavity mode is much larger than the typical frequency of the mirror's motion (i.e. Ω 0 /ω k glyph[lessmuch] 1). Under such conditions the mirror's position changes adiabatically over many oscillations of the cavity field allowing a time average (denoted by double angled brackets) of the effective interaction Hamiltonian Here, T is the period of the cavity's fundamental mode and N is a large integer such that (2 π ) / Ω 0 glyph[greatermuch] NT . Since Z ( t ) is approximately constant over the entire integration range it can be taken outside of the time-average giving This step is equivalent to taking the rotating wave approximation. The key point is that the first term on the right side is a classical radiation pressure originating solely from the external source while the second term is a quantum mechanical radiation pressure and is, in fact, the Nx -coupling. Before concluding this section, we collect the main assumptions used in relating the MOF model to the phenomenological radiation pressure interaction Hamiltonian. The assumptions are as follows: Under these assumptions we have shown that the effective interaction between the mirror and the cavity field is given by an Nx -coupling. It is possible that the Nx -coupling can be obtained using a different setup and assumptions. However, our purpose here is not to elucidate all the ways that the Nx -coupling can be derived from the MOF model but rather to show that it can be derived from a microphysics model of a moving mirror.", "pages": [ 18, 19, 20, 21 ] }, { "title": "VII. MIRROR-OSCILLATOR-FIELD (MOF) MODEL AND QUANTUM BROWNIAN MOTION", "content": "In this Section we shall establish a connection between the MOF model for N moving mirrors and N harmonic oscillators interacting with a bath of harmonic oscillators that constitute an environment for the N oscillators. The latter system has a long and well-developed history for providing a simple model with which to study quantum Brownian motion (QBM). Hence, if a relationship between the MOF model and QBM exists then one should be able to exploit the results of many previous studies (regarding decoherence, (dis)entanglement, fluctuation-dissipation relations, etc.) to apply towards moving mirror systems. We show here that such a relationship does indeed exist.", "pages": [ 21 ] }, { "title": "A. Static mirrors and QBM", "content": "Consider a mirror at rest that is fixed at Z ( t ) = 0 for all time. The MOF Hamiltonian for this configuration follows from (3.32) It is well known that a field can be represented as a continuum of harmonic oscillators, some of which have arbitrarily large natural frequencies. However, such large frequencies are not usually physically relevant (and often lead to divergences that must be properly handled with well-established renormalization techniques) so that one can simply impose a cut-off frequency Λ, which has the effect of ensuring that all calculated quantities are finite [67]. The mode decomposition of the field is If we restrict the field to the interior of a 1-dimensional (but large) volume V then the normal modes of the field are simply so that the time dependence of the k th mode has the following representation in terms of creation and annihilation operators In terms of this mode decomposition, the Hamiltonian is Notice that the coupling constant λ in the last term can be grouped with the mode function u σ k (0) to give an effective coupling constant that depends on the particular mode C σ k ≡ λu σ k (0). Therefore, the Hamiltonian for this system is which is precisely the Hamiltonian for a harmonic oscillator q ( t ) coupled to an environment composed of a bath of harmonic oscillators { ϕ σ k ( t ) } . In other words, the MOF model for a mirror at rest can be related to quantum Brownian motion where the field provides the environment that the mirosc interacts with. QBM has a long history and is well-studied so that results already found in that literature can be applied directly to the interaction of a field with a static mirror via the MOF model. For example, the master equation is exactly known for this system [68] and so one can study its behavior near the perfectly-reflecting limit where λ →∞ or, equivalently, m → 0 as well as in a non-zero temperature regime. A similar result holds for N mirrors held at rest at positions x = L a with a = 1 , . . . , N . It is straightforward to see that the corresponding Hamiltonian, when decomposing the field into harmonic oscillators, is where the effective bilinear coupling constant is C σ ka ≡ λu σ k ( L a ). Therefore, N static mirrors in the MOF model correspond to N harmonic oscillators (mirosc variables) coupled to a bath of oscillators (the field). For N = 2 oscillators coupled to a general environment, the exact master equation has been derived in [69] and thus can be used to provide a different perspective and new insights in the description of a field coupled to two partially transmitting mirrors via the MOF model.", "pages": [ 21, 22 ] }, { "title": "B. Slowly moving mirrors and QBM", "content": "Turn next to find the relationship between slowly moving mirrors in the MOF model and quantum Brownian motion. Let us first consider one mirror since the result for N mirrors will generalize in an obvious way. Assume that the mirror is in an externally generated potential V ( x ), such as a harmonic trap. Then the Hamiltonian in (7.7) is where we have included the worldline variable to the Hamiltonian. Notice that from a QBM perspective, the effective coupling constant acquires a time dependence since the mode function is now time dependent, u σ k ( Z ( t )). However, if the potential V ( x ) restricts the motion of the mirror to be only small perturbations from its equilibrium position at x = 0 then we may expand the mode function about the origin so that the interaction term above becomes Therefore, the Hamiltonian (7.10) is equal to an unperturbed Hamiltonian, given by the 1-harmonic oscillator QBM Hamiltonian in (7.8), plus an interaction Hamiltonian that describes perturbations due to the small displacement of the mirror that arise from interactions between the field oscillators and the mirosc, Hence, one can compute the perturbations of, for example, the exact master equation for 1-harmonic oscillator QBM to study the behavior of a movable, partially transmitting mirror. Notice that if V ( Z ) = M Ω 2 0 Z 2 ( t ) / 2 then (7.12) describes a nonlinearly coupled QBM system where the mirosc and the mirror's position are the two oscillators in an open system that couples to the bath provided by the field oscillators. The nonlinearity is only in the mirror's position (i.e., from the O ( Z 2 ) terms above) but the mirosc and the field oscillators still couple to each other bilinearly. The generalization to N mirrors should be obvious with the Hamiltonian describing the system being where the unperturbed position of the a th mirror is at x = L a . In particular, one can compute the perturbations of, for example, the exact master equation for 2-harmonic oscillator QBM [69] to study entanglement, decoherence, etc., of a cavity with movable, partially transmitting mirrors.", "pages": [ 22, 23 ] }, { "title": "VIII. SUMMARY AND FURTHER DEVELOPMENTS", "content": "In this paper we constructed a microphysics model of moving mirrors interacting with a quantum field. The novel ingredient we introduced is a harmonic oscillator (a 'mirosc') model describing the internal degrees of freedom of the mirror that couples to the incident radiation thereby providing a mechanism for the dynamical interplay of the mirror-field system. Since the field can transfer (receive) energy and momentum to (from) the mirosc the collection of them serves the function of a partially reflecting or transmitting mirror. We showed that this mirror-oscillator-field (MOF) system can perfectly reflect or perfectly transmit radiation depending on the values of the mirosc mass m , natural frequency Ω, and coupling strength λ to the field. Perfect reflection can be attained in three ways: 1) m → 0; 2) λ → ∞ ; and 3) Frequency ω of an incident wave is equal to the mirosc natural frequency Ω. Limits 1) and 2) exhibit perfect reflection (or nearly so) among a broad frequency bandwidth whereas limit 3) strongly reflects modes with frequencies near Ω because of a resonant excitation of the mirosc. The MOF model reduces to several commonly used models of moving mirrors in a quantum field. We showed that when the mirosc variable q ( t ) evolves adiabatically ( | q | glyph[lessmuch] | Ω 2 q | ) or when m → 0 but m Ω 2 = κ remains constant then the MOF model reduces to the Barton and Calogeracos (BC) model [2] of a partially transmitting moving mirror. The free parameter in the BC model γ is related to the mirosc parameters of the MOF model ( m, Ω , λ ) by γ = λ 2 / (2 m Ω 2 ). The 'auxiliary field' model of Golestanian and Kardar [45, 46] arises from the MOF model in the limit that m → 0. In this limit, there is no mirosc dynamics and q ( t ) becomes an auxiliary variable. In the quantum theory, q ( t ) may have any possible realization (see (2.29)), which manifests as a Dirichlet boundary condition on the field at the location of the mirror and thus perfectly reflects incident radiation (see (2.31)). We also showed that our MOF model reduces to the phenomenological model of a mirror interacting with a cavity field via the radiation pressure exerted on the mirror's surface when a number of assumptions are made (though these may not all be necessary to derive the Nx -coupling in other setups). This ' Nx -coupling' is often used to describe laboratory setups but may be extended by the MOF model to scenarios where the mirosc does not evolve adiabatically, which may exhibit interesting macroscopic (or perhaps mesoscopic) quantum phenomena. Additionally, Nx -type coupling provides the leading order corrections to the classical radiation pressure coupling when the cavity is occupied by low photon numbers. The model we present in this paper will remain useful even when the necessary conditions for it to match with models with Nx -type coupling are not met, for example, when the mirror motion is sufficient to excite field quant to higher modes. The bulk motion of the mirror in the MOF model, which may be relativistic depending on the application, can be derived from an action or a Hamiltonian. In either formulation, we find that the mirror moves with a time-dependent effective mass M eff that is composed of the mirror's rest mass M and the mirror's total internal energy, which comes from the energy of the mirosc itself and its interaction with the field. We also showed (in a purely classical setting) that the MOF model seems to admit physical solutions despite the use of a point particle description for the mirror's motion and despite the interaction between the mirror and field resembling that of a charged particle (which can be plagued by pathologies). We demonstrated that when the field is generated by its interaction with the mirosc alone so that there is no external source J ext and no initial field configuration present then the mirror will evolve on an inertial trajectory (i.e., constant velocity), which is the correct expected result, in contradistinction to the radiation reaction on a point charge in electrodynamics where the charge may exhibit run-away motions in the absence of any external forces acting on the charge. As an application of the MOF model, we studied the'cooling' of a mirror by its interactions with an external field in a purely classical context. We found that when the mirosc is weakly coupled to the field that the mirror, when perturbed, will oscillate around its equilibrium configuration while its displacement amplitude decays slowly in time. An interesting consequence of our MOF model for moving mirrors is that it relates to models of quantum Brownian motion (QBM) in a straightforward manner. The relation essentially follows because the field can be regarded as a continuum of harmonic oscillators. Hence, for N mirrors held at rest, the MOF model is equivalent to N harmonic oscillators in a bath of oscillators (from the field). For N = 1 , 2, the master equation for such a system in a general environment has been derived exactly [68, 69] and even for general N [70]. Consequently, the MOF model can be used to study the superposition of two mirrors, the decoherence by and the disentanglement of moving mirrors via a field, etc., so as to gain insight into these aspects of macroscopic quantum phenomena. We expect that the rich repository of technical tools and physical insights from the study of QBM can be carried over directly to our MOF model for a broad range of applications involving moving mirrors and quantum fields. For example, QBM results for systems at finite temperature may provide a simple way to incorporate thermal effects into the MOF model. We will begin to explore this theme in a follow-up paper [53] on the theory of OM from an open quantum system viewpoint. The generalization of the MOF model to 3 spatial dimensions can be made where the mirror is an extended body having some surface geometry. On this surface, we may place a layer of mirosc's that play the role of the electrons in a metal gas or dielectric medium providing the mirror's light degrees of freedom and responsible for reflection of incident radiation over some bandwidth of the electromagnetic spectrum (e.g., optical as in many metals). Incorporating the electromagnetic field in the MOF model should also be straightforward as its structure is similar to that of a minimallycoupled scalar field in the MOF model (see the appendix of [61]). In the second series on back-action effects we will study the full quantum mechanical evolution of the MOF system in the context of mirror cooling. Therein, we will derive the exact equations of motion describing the mirror's average position. In the most general case we will show that the mirror motion is described by an integro-differential equation exhibiting non-Markovian dynamics. The equations can be simplified through a series of approximations which directly relate to experimentally engineerable quantities, such as the cavity's quality factor, and the relevant timescales for the mirror's internal dynamics. Given the broad range of applicability, these results can be employed to guide theoretical and experimental investigations ranging from the cooling of the center of mass motion of moveable mirrors, having broad-band reflective properties, to the manipulation of trapped ions near surfaces, possessing narrow-band reflective properties [14]. In the third series we will address the moving mirror analog of the back-reaction of Hawking radiation [19] on the evolution of a black hole. There are controversies in some deep issues related to the end-state of black hole evaporation resulting from the Hawking effect, namely, whether complete evaporation of a black hole means the nonunitary evolution of quantum states (see, e.g., [71]) which violates the basic tenets of theoretical physics or if unitarity is preserved, and if so, how? One key ingredient, the back-reaction of the emitted radiation on the spacetime, has not been taken into account fully or correctly (for a recent update, see [72, 73] and papers cited therein.) There are analog studies on how information is shared in the black hole (harmonic) atom - quantum field system (see, e.g., [74] and references therein.) as well as moving mirror analog problem [17]. The connection was made between the s -wave component of Hawking evaporation and the emission of radiation from moving mirrors by the dynamical Casimir effect but, like the original calculation by Hawking, treated the effects of back reaction rather coarsely. Since the MOF model offers a large degree of flexibility and tractability, we were able to find exact equations of motion for the mirror incorporating the effects of back-reaction [60]. These exact solutions, as well as those from the atom-field analogs, can provide new insights into this basic issue in theoretical physics.", "pages": [ 23, 24 ] }, { "title": "IX. ACKNOWLEDGEMENTS", "content": "CG was supported in part by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory administered by Oak Ridge Associated Universities through a contract with NASA, and in part by a NIST Gaithersburg grant awarded to the University of Maryland when this work was started. RB gratefully acknowledges the support of the U.S. Department of Energy through the LANL LDRD program. BLH wishes to thank Professor Jason Twamley, Director of the the Centre for Quantum Computer Technology at Macquarie University for his warm hospitality in Feb-Mar 2011 during which this work was partly carried out. His research was partially supported by NSF grant PHY-0801368 to the University of Maryland. Copyright 2012. All rights reserved. does such as dimensional regularization [79, 80]. In QED, a high-frequency cut-off breaks the gauge symmetry and induces a mass for the photon [78].", "pages": [ 24, 25, 27 ] } ]

[ { "title": "Quench Dynamics in Bose condensates in the Presence of a Bath: Theory and Experiment", "content": "∗ Adam Ran¸con, Chen-Lung Hung, Cheng Chin, and K. Levin James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA In this paper we study the transient dynamics of a Bose superfluid subsequent to an interaction quench. Essential for equilibration is a source of dissipation which we include following the approach of Caldeira and Leggett. Here we solve the equations of motion exactly by integrating out an environmental bath. We thereby derive precisely the time dependent density correlation functions with the appropriate analytic and asymptotic properties. The resulting structure factor exhibits the expected damping and thereby differs from that of strict Bogoliubov theory. These damped sound modes, which reflect the physics beyond mean field approaches, are characterized and the structure factors are found to compare favorably with experiment. Understanding out-of-equilibrium dynamics and dissipation in superfluids has experienced a revival with recent experiments in cold atoms. These atomic systems afford access to new probes of non-equilibrium behavior, not available in condensed matter counterparts, among these a sudden change of the interaction strength [1, 2]. The equilibration process after this 'quench' necessarily involves a source of dissipation, the understanding of which can elucidate essential microscopic processes, not included in standard mean field approaches to superfluidity. The importance of understanding quantum coherence and dissipation was emphasized in the seminal work of Leggett and Caldeira [3] and subsequently explored by many others [4-8]. The goal of this paper is to apply these important ideas to the quench dynamics of a Bose superfluid and to show how to address related experiments [9]. Following earlier work [8, 10] we develop a formalism for calculations of the real time dynamics of a superfluid coupled to a rather general bath. This formalism is then applied to address experimental studies of two dimensional Bose gases [9]. In these experiments, one has direct access to the equal time density correlation functions, or structure factor S k ( t ). A key experimental observation was that the quench appears to excite acoustic waves, which interfere in both the spatial and temporal domains, leading to Sakharov oscillations. A simple Bogoliubov-level theory was applied to analyze these experiments (see also [11]), while leaving a few experimental features, such as damping in the Sakharov oscillations, unexplained. Here, in investigating the physics of dissipation, we re-enforce the observation of oscillatory behavior by incorporating the presence of damping in the data analysis. Our study supports the earlier observation of oscillatory sound modes for some range of k and t in S k ( t ). It also provides insights into why the simplest Bogoliubov-based scheme is more inadequate in situations in which the coupling constant g is suddenly increased. Moreover, our tractable formalism for treating such dissipation, should be a first step in developing tools for elucidating the microscopics of cold gases which go beyond the simplest mean field theories of the steady state. The subject and origin of quantum dissipation has a long history. In the Leggett Caldeira (LC) approach the bath is modeled by an infinite set of harmonic oscillators; the Hamiltonian is then quadratic and one can solve the equations of motion exactly by effectively integrating out the environment. One similarly introduces a source of 'noise' or dissipation into fermionic superfluids via time dependent Ginsburg-Landau theory [12]. The counterpart in bosonic superfluids has been addressed in the context of stochastic versions of the Gross-Pitaevski equation (SGPE) [13-15], where more microscopic approaches to the noise source have been highlighted. In all these superfluids, in the simplest terms the noise or dissipation corresponds to fluctuations or processes not included in mean field theory. Specifically, for Bose superfluids, the mechanism for irreversible loss of energy can be associated with the interaction between Bogoliubov quasi-particles. There are microscopic schemes to address these beyond-Bogoliubov damping effects due to Beliaev [16, 17]. Their inclusion into dynamics is usually via the Schwinger-Keldysh formalism which can be rather involved with the condensate wave function appearing non-linearly in functionals that are non-local in space and time. In general, extensive numerical simulations are necessary, making such schemes less physically transparent and often restricting their use to rather high temperatures [13, 14]. The alternative more macroscopic concept of the environmental bath, applied here, is to split the total system into two parts: the quantum system where dissipation occurs (say the Bose superfluid at the Bogoliubov level) and a so-called environment. Evidence for universality suggests [8] that the particular description of the bath will not affect the essential features of the dissipative process. The latter is often modeled presuming Ohmic dissipation. Throughout this paper we neglect trap effects; our focus is on reasonably short times where the trap geometry is not important. In the absence of the bath (as well as trap), the Hamiltonian of the Bogoliubov modes is given by ˆ H bog ( g ) = glyph[negationslash] where ˆ ψ ( † ) k annihilates (creates) an atom with momentum k = 0 (the dispersion glyph[epsilon1] k can be quite general in the presence of an optical lattice but we will focus on the free dispersion, i.e. glyph[epsilon1] k = k 2 / 2 m ). Here n 0 = µ/g is the condensate density, with µ the chemical potential and mg is the dimensionless interaction strength. We use the convention glyph[planckover2pi1] = k B = 1 through the paper. In the mean field approximation, interactions are only included between non-condensed bosons and the condensate; clearly, Bogoliubov level theory is inadequate as it does not include dissipation. One can view this dissipation as arising from interactions between thermal particles. Treating such interactions in a manner which leads to analytically tractable dynamics is not straightforward [13]. Thus, following the precedent of time dependent Ginsburg-Landau theories, one introduces a 'noise' term [12] via a bath [10]. The dynamics can be derived exactly and the calculations of the response functions such as the structure factors is then precise and fully consistent. We note that other sources of dissipation which enter into the actual experiments cannot be ruled out, as cold atoms systems are subject to lasers and other probes and are not truly isolated. Two obtain the appropriate analytic properties, one introduces two kinds of bosonic modes ˆ W ( † ) j, k and ˆ V ( † ) j, k , with Hamiltonian Here the index j represents the bath degrees of freedom. This bath then interacts with the system of interest via where η j, k and ζ j, k represent generalized coupling constants. The coupling is expected to take one particle from the bath and put it into the system of interest (or the opposite). Additional processes involve a particle of the system and one of the bath falling into the condensate, and the converse. The equations of motion of the fields i∂ t ˆ ψ k ( t ) = [ ψ k , ˆ H f ] and i∂ t ˆ ψ † -k ( t ) = [ ψ † -k , ˆ H f ] can be written where we have formally solved the equations of the bath operators. Here before the quench, the Hamiltonian with interaction strength g i , in contact with the bath is ˆ H i = ˆ H Bog ( g i ) + ˆ H bath + ˆ H c , while after an instantaneous quench the Hamiltonian consists of ˆ H f = ˆ H Bog ( g f ) + ˆ H bath + ˆ H c . While it is not essential, as in previous work [9, 11] we neglect the time variation of the condensate density n 0 (which fixes the chemical potential µ i/f = n 0 g i/f ). We define ω k = glyph[epsilon1] k -µ f + 2 g f n 0 , ˆ D k ( t ) = ∑ j η j, k e -iω j, k t ˆ W j, k (0) + ∑ j ζ j, k e iν j, k t ˆ V † j, k (0) and γ k ( t ) = ∫ ω Σ 2 ( k , ω ) e -iωt with ∫ ω = ∫ dω/ (2 π ). We define Σ 2 ( k , ω ) = 2 π ∑ j [ | η j, k | 2 δ ( ω -ω j, k ) -| ζ j, k | 2 δ ( ω + ν j, k ) ] . Here, ˆ D k ( t ) plays the role of a random force operator and γ k ( t ) reflects the damping. The relaxation to equilibrium will be insured by the satisfaction of the fluctuationdissipation relation In a more abstract form, our central result is a time dependent Bogoliubov equation which now includes a damping term. The equations of motion can be formally solved by introducing a matrix M k ( t ) which will depend, via γ k ( t ), on the precise form of Σ 2 ( k , ω ), such that The simplest and most conventional choice is to choose a so-called Ohmic bath (with spectral density of the bath, Σ 2 ( k , ω ), proportional to ω at small frequency) with the proper high-frequency regularization Σ 2 ( k , ω ) = 2Γ k ω/ (1+ ω 2 / Ω 2 ). Here Ω is a high-energy cut-off. Note that Γ k is dimensionless. On physical grounds, we expect this damping parameter to increase with increasing interaction. For this Ohmic bath, the time-evolution matrix M k ( t ) is given by (we dismiss a highly oscillating part with frequency of the order of Ω which does not play any role in the time evolution) where we define the 'damped parameters' ¯ ω k = ω k / (1+ Γ 2 k ), ¯ g ¯ n 0 = g f n 0 / (1 + Γ 2 k ), ¯ γ k = Γ k ¯ ω k , ¯ E 2 k = [ E 2 k ,f -Γ 2 k ( g f n 0 ) 2 ] / (1 + Γ 2 k ), with E 2 k ,α = glyph[epsilon1] k ( glyph[epsilon1] k +2 g α n 0 ) is the Bogoliubov energy. The matrix M k ( t ) can be recast as a sum of exponentials exp [ -(¯ γ k ± i ¯ E k ) t ] with complex frequencies. At large momenta, we find (¯ γ k ± i ¯ E k ) glyph[similarequal] (Γ k glyph[epsilon1] k ± iE k ) / (1 + Γ 2 k ) (assuming that Γ k is bounded at large | k | , which is physically reasonable). The oscillations will therefore be typically given by the Bogoliubov energy, with expected damping at high-| k | proportional to glyph[epsilon1] k t , even at short time. This is supported by experiment, as discussed below. One can observe that, depending on the bath parameter Γ k , there may exist a range of momenta such that ¯ E 2 k ≤ 0, implying that the dynamics is overdamped and M k ( t ) will not produce oscillations. This phenomenon, beyond Bogoliubov theory, may well have been observed in [9] for a quench to higher interaction strength at low-momenta. Out-of-equilibrium correlation functions We now specify the out-of-equilibrium observables we will study. Because our model Hamiltonian is quadratic, it is reasonably straightforward to derive the four correlation functions involving combinations of ψ and ψ † . Here we will concentrate on the density-density correlation functions, which are easily accessed in current cold atoms experiments [18]. The density operator is defined in momentum space as ˆ ρ q ( t ) = ∑ k ˆ a † k + q ( t )ˆ a k ( t ), where ˆ a k = √ n 0 δ k , 0 + ˆ ψ k . The different correlation functions will involve the usual time ordered normal ( G k ( t, s )) and anomalous ( F k ( t, s )) Green's functions. Because of the quench, these Green's functions depend separately on their two time arguments. We find as expected that in the long time limit, when the system reaches its new equilibrium state, they become functions of only t -s . This is a non-trivial test of the current theory which reflects the energy dissipation mechanism. The density-density correlation function χ q ( t, s ) = 〈 ˆ ρ q ( t )ˆ ρ -q ( s ) 〉 - 〈 ˆ ρ q ( t ) 〉〈 ˆ ρ -q ( s ) 〉 , (with t ≥ s ) is related to the structure factor S q ( t ) = χ q ( t, t ) /n 0 . It can similarly be written in terms of the Green's functions so that χ q ( t, s ) = -n 0 { G q ( t, s )+ G q ( s, t )+ F q ( t, s )+ F † q ( t, s ) } + ∑ k { G k + q ( s, t ) G k ( t, s ) + F † k + q ( t, s ) F k ( t, s ) } , where the sum is over all k different from 0 and -q . Because the condensate is macroscopically occupied, we will neglect the second term in brackets in our numerical calculation of the structure factor [9, 11]. It is convenient to define the 4-vector Υ( t, s ) = { F k ( t, s ) , G k ( s, t ) , G k ( t, s ) , F † k ( t, s ) } . These component Green's functions can be evaluated by solving equation (5) which yields ( t > s ) (7) where the matrix M k ( v ) = ∫ v 0 M k ( v -v ' ) .M -1 k ,i ( v ' ) dv ' and M -1 k ,i ( t ) is the inverse of the M k with g f → g i , with the definition ∫ v 0 M k ( v -v ' ) .M -1 k ( v ' ) dv ' = δ ( v ). We have introduced the vector Υ i ( t ) = { F k ,i ( t ) , G k ,i ( -t ) , G k ,i ( t ) , F † k ,i ( t ) } , corresponding to the equilibrium Green's functions of the initial Hamiltonian. These can be defined using the the equilibrium bosonic spectral functions [19]. Our equations correspond to Bogoliubov theory in the absence of a bath. While Eq. 7 may seem formally complex, the physics it contains needs to be emphasized. Importantly, this equation is consistent with the fluctuation-dissipation theorem. Stated alternatively, it is consistent with the proper asymptotic (long time) regime which requires that the final state of the system be time independent. It can be shown directly from Eq. 7 that the long time limit of Υ( t + τ, t ) → Υ f ( τ ), which means that at long times the proper equilibrium normal and anomalous Green's function of the quenched Hamiltonian ˆ H f are obtained. It should not be presumed that one can view this final state as a convolution of a damping term and simple Bogoliubov theory; the asymptote of the structure factor (for example) is finite, as a new equilibrium phase is reached, so that the oscillations are not simply damped out. Numerical results We discuss now the numerical solution of the previous equations. We introduce the characteristic momentum k 0 = √ n 0 mg i and time t -1 0 = k 2 0 / 2 m . Note that k 0 is the inverse healing length of the condensate at t = 0. The left panel of Figure 1 shows the evolution in time of the structure factor at a fixed | k | after the quench-up for different values of the bath parameter Γ k chosen to be independent of | k | for simplicity. We also plot the results from Bogoliubov theory which corresponds to the case Γ k = 0. Here we stress that because of damping there is no single frequency observed for each k . Nevertheless, an important effect of the bath (indicated by the arrow) is the shift toward earlier time of the first extrema of the oscillations, leading to an apparent frequency increase. This shift was observed in [9] for a quench-up (where the effect of the bath is expected to be more important in the dynamics) and cannot be explained by Bogoliubov theory. This effect was not seen for a quench-down, which might be expected, as in this case Γ k is smaller. The central and right panels of Figure 1 plot S k ( t ) at fixed time t = t 0 and t = 10 t 0 for small Γ = 0 . 01 in our theory as compared with Bogoliubov theory. For the latter, at long time one sees that S k ( t ) oscillates faster and faster while never reaching a new equilibrium. We find a moderately successful phenomenological fit to our calculations at low damping with S ph k ( t ) = S k ,f + f k ( t ) ( [ 1 + ( E 2 k ,i -E 2 k ,f ) / ( E k ,f ) sin( E k ,f t ) 2 ] S k ,i -S k ,f ) , where S k ,i ( S k ,f ) is the initial (final) equilibrium structure factor. For | k | glyph[greatermuch] g f n 0 , we observe from our theory that f k ( t ) = exp( -( | k | /K ( t )) 2 ), where K ( t ) = k 0 √ (1+Γ 2 k ) t 0 Γ k t . Fits to our theory are shown in the central and right panels of Figure 1. It should be stressed that for these small values of the damping, as illustrated here, there is not yet a signature of an apparent shift in frequency. Experimental results We have reported [9] the experimental observation of oscillatory behavior of the density structure factor associated with a sudden quench in a 2D Bose system in the almost-pure- superfluid phase of a cesium atomic gas. The emphasis of the present paper is on damping effects we observe in these oscillations. These are in contrast to an extensive literature focusing on shortcuts to adiabaticity after a fast change of the experimental parameters, in particular of 1D gases, (see for instance [20]). The data points in Figures 2 represent the measured equal-time structure factor as a function of time at fixed | k | = 0 . 9 µm -1 for the final interaction strength g f = 0 . 19 and for a quench up with g f /g i glyph[similarequal] 2 . 4 (Fig. 2b) and a quench down (Fig. 2a) with | k | = 1 . 2 µm -1 and g f /g i glyph[similarequal] 0 . 3. Importantly, in contrast to other experimental studies and deliberate quench protocols [21], dis- pation is evident and Sakharov oscillations will eventually be damped out in the steady state. We do not address this asymptotic long time regime because our focus is on reasonably short times where the trap geometry is not important. We plot as solid lines in Figure 2, theoretical curves with the same microscopic parameters. The color-coded curves represent the regime of moderate damping for different values of Γ = 0 . 1 (in a quench up) and Γ = 0 . 075 (in a quench down). Since our focus is on semiquantitative comparisons, we have allowed a global shift of the y-axis as well as the x-axis (to take care of time uncertainties on the order of glyph[lessorsimilar] 1 ms , due to time lapse between the quench and the detection). In overall comparison between theory and experiment we find that Γ k ≈ 0 . 1 describes the current data set. From this observation, one cannot yet characterize the microscopic nature and origin of dissipation, but our work should be viewed as a first step in the process. Additional experiments and more systematic comparisons can be anticipated in future. Nevertheless, with this approach, we can, however, address several features of the experiments which were difficult to understand within strict Bogoliubov theory such as the apparent frequency shifts and the damping both in time and k . A general theme of this paper has been to focus on damping effects. Nevertheless, our out of equilibrium studies necessarily have implications on the equilibrium behavior (of the structure factor, say) due to the important fluctuationdissipation relation. In summary, in this paper we have demonstrated that the introduction of a Leggett-Caldeira bath for a Bose superfluid implies that the dynamics can be derived exactly and the calculations of the response functions such as the structure factors is then precise and fully consistent. A.R. thanks J. Bonart for discussions. This work is supported by NSF-MRSEC Grant 0820054. C.L. and C.C. acknowledge support from NSF Grant No. PHY0747907 and under ARO Grant No. W911NF0710576 with funds from the DARPA OLE Program. where D k ( ω ) = G k ,i ( -ω ) G ∗ k ,i ( ω ) -( g i n 0 ) 2 . Note that the spectrum is gapless, as D k =0 (0) = 0 and that A k ,i ( ω ) has the sign of ω as required for bosons. In order to respect the commutation relation at equal time, we have ∫ ω A k ,i ( ω ) = 1 and ∫ ω L k ,i ( ω ) = 0.", "pages": [ 1, 2, 3, 4, 5 ] } ]