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Then the iuconüug wave function is o"seUU and the outgoing wave fuuction in the outside range (r02rg) Is The outgoiug wave function has a sngsuluidtv at r=rjj iu the inteerand. if we want to ect the outeoiug wave function iu the iuside range(r<ry). we have to do analytic continuing over this snguluitv. | Then the incoming wave function is $\phi^{in}=e^{-i\omega \nu}$, and the outgoing wave function in the outside range $(r>r_H)$ is The outgoing wave function has a singularity at $r=r_H$ in the integrand, if we want to get the outgoing wave function in the inside $(r<r_H)$, we have to do analytic continuing over this singularity. |
Namely. the caleulatious of imaginary part of the A inteeral across the classical horizon are as follows Note. the caleulatious here are somehow nonaigorous. | Namely, the calculations of imaginary part of the wave-number $k$ integral across the classical horizon are as follows Note, the calculations here are somehow non-rigorous. |
Two remarks ou this poiut are follows: 1) The integration f&(r)dr is actually divergent due to the siugularity at r=ry in the inteerand. aud hence In(2[k(r)dr) is actually mecanineless as without a proper reeularization: 2). The second step of caleulation (18)) just serves as such a reguluization which belougs to a artificial mathematical trick. and the final step is due to the Dirac foxiuula: 1/6e4e)=Ἑιτimdtr). | Two remarks on this point are follows: 1) The integration $\int k(r)dr$ is actually divergent due to the singularity at $r=r_H$ in the integrand, and hence $Im(2\int k(r)dr)$ is actually meaningless as without a proper regularization; 2), The second step of calculation \ref{im}) ) just serves as such a regularization which belongs to a artificial mathematical trick, and the final step is due to the Dirac formula: $1/(x\pm i\epsilon)=\mathcal{P} / x \mp
i\pi\delta(x)$. |
Now. the outgoing wave function inside the horizon is where for simplicity. we take the positive sigeu(if taking the negative sign. we can ect the same result[0])). | Now, the outgoing wave function inside the horizon is where for simplicity, we take the positive sign(if taking the negative sign, we can get the same ). |
The absolute value of ratio of the outgoing wavefuuctiou's amplitude outside the black hole to the oue inside is as follows | The absolute value of ratio of the outgoing wavefunction's amplitude outside the black hole to the one inside is as follows |
theEoR.. and as far back into the epoch of first star formation as possible (20z€ 7. 70MHzxv200 MHz). | the, and as far back into the epoch of first star formation as possible $20 \lesssim z \lesssim 7$ , $70\,\mathrm{MHz}
\lesssim \nu \lesssim 200\,\mathrm{MHz}$ ). |
The configuration of the low-frequency component of theSKA.. the so-called SKA-lo. remains a matter of active study. but the current specifications forsee approximately of the total collecting area within a diameter of | km and approximately of the total collecting area within a diameter of5 km (Schilizzietal..2007). | The configuration of the low-frequency component of the, the so-called SKA-lo, remains a matter of active study, but the current specifications forsee approximately of the total collecting area within a diameter of 1 km and approximately of the total collecting area within a diameter of5 km \citep{ska100}. |
. The resulting filling factors are f~0.5 (f~ 0.05) within the central 1 km (5 km). | The resulting filling factors are $f \sim
0.5$ $f \sim 0.05$ ) within the central 1 km (5 km). |
Scaling from the results in Table 4.. the central portion of the SKA-lo could therefore detect RRLs along the Galactic plane in integration times of order 5 hr. | Scaling from the results in Table \ref{tab:lwadetect}, the central portion of the SKA-lo could therefore detect RRLs along the Galactic plane in integration times of order 5 hr. |
Given the primary scientific driver for the SKA-Io. however. it is unlikely that it will be conducting deep observations along the Galactic plane. | Given the primary scientific driver for the SKA-lo, however, it is unlikely that it will be conducting deep observations along the Galactic plane. |
Rather. it will likely conduct its deep observations (=100 hr) at high Galactic latitudes. | Rather, it will likely conduct its deep observations $\gtrsim 100$ hr) at high Galactic latitudes. |
With reference to the three 21-cm cosmology epochs (?2)). the goal for the LRA is to probe at least the epoch of first star formation. and as deeply into the Dark Ages as possible (100.<zxIS. 10MHz<v€100 MHz). | With reference to the three 21-cm cosmology epochs \ref{sec:cosmo}) ), the goal for the LRA is to probe at least the epoch of first star formation, and as deeply into the Dark Ages as possible $100 \lesssim
z \lesssim 15$, $10\,\mathrm{MHz} \lesssim \nu \lesssim
100\,\mathrm{MHz}$ ). |
Some of theinitial designs for the LRA follow the configuration analysis of Lidzetal.(2008).. who advocate a highly compact "super core.” | Some of theinitial designs for the LRA follow the configuration analysis of \cite{lzmzh08}, who advocate a highly compact “super core.” |
For such a configuration. the filling factor is large. potentially as large as f~0.9. | For such a configuration, the filling factor is large, potentially as large as $f \sim 0.9$. |
The result would be to reduce the required integration times potentially by an order of magnitude. | The result would be to reduce the required integration times potentially by an order of magnitude. |
However. like theSKA-lo.. the LRA would most likely target fields well away from the Galactic plane. | However, like the, the LRA would most likely target fields well away from the Galactic plane. |
Similar comments apply to the LRA as to the regarding such observations. | Similar comments apply to the LRA as to the regarding such observations. |
There are a number of low radio frequency arrays either under construction or in design and development. including the Long Wavelength Array (LWA). the LOw Frequency ARray (LOFAR). the low-frequency component of the Square Kilometre Array (SKA-Io). and the Lunar Radio Array (LRA). | There are a number of low radio frequency arrays either under construction or in design and development, including the Long Wavelength Array (LWA), the LOw Frequency ARray (LOFAR), the low-frequency component of the Square Kilometre Array (SKA-lo), and the Lunar Radio Array (LRA). |
The SKA-lo and the LRA are still in the design phase. but might likely have higher filling factors. potentially larger than50%... leading to the detection of these lines in 10 hr or less. | The SKA-lo and the LRA are still in the design phase, but might likely have higher filling factors, potentially larger than, leading to the detection of these lines in 10 hr or less. |
RRLs have been studied extensively along the line of sight to the bright radio sourceA. | RRLs have been studied extensively along the line of sight to the bright radio source. |
. However. the weakness of the lines and the subsequent need for long integration times and careful data processing to remove even very low level radio frequency interference (RFI) have hampered efforts for the type of systematic. multi-frequency surveys required to probe the physics ofthe cool. low density ISM throughout the Galaxy. | However, the weakness of the lines and the subsequent need for long integration times and careful data processing to remove even very low level radio frequency interference (RFI) have hampered efforts for the type of systematic, multi-frequency surveys required to probe the physics ofthe cool, low density ISM throughout the Galaxy. |
At present. observations do not provide an explicit picture of how the chromospheric network magnetic field is structured. | At present, observations do not provide an explicit picture of how the chromospheric network magnetic field is structured. |
On one hand. we have increasing. observational evidence of something which car be interpreted loosely as a canopy structure: e.g.. fibrils in the Ca II infrared triplet lines: Vecchioetal. 2007.. large-scale canopy structures in combined Zeeman and Hanle studies: Biandaetal.(1998);Stenflo(2002). canopy-like expansior seen in magnetograms near the limb: Jones&Giovanelli (1983))). | On one hand, we have increasing observational evidence of something which can be interpreted loosely as a canopy structure: e.g., fibrils in the Ca II infrared triplet lines; \citealt{Vecchio+others2007}, large-scale canopy structures in combined Zeeman and Hanle studies: \cite{Bianda+others1998,
Stenflo+others2002}, canopy-like expansion seen in magnetograms near the limb; \cite{Jones+Giovanelli1983}) ). |
On the other hand. The appearance of network magnetic flux concentrations in circular polarization maps changes from unipolar at disk center to bipolar near the limb. | On the other hand, The appearance of network magnetic flux concentrations in circular polarization maps changes from unipolar at disk center to bipolar near the limb. |
This ts consistent with network magnetic fields expanding and fanning out with height as proposed by Gabriel(1976). | This is consistent with network magnetic fields expanding and fanning out with height as proposed by \cite{Gabriel1976}. |
. In the chromosphere. the fanning is clearly present: e.g.. Jones&Giovanelli(1983) found low-lying. 200-800 km. magnetic canopies in magnetograms taken near the solar limb. | In the chromosphere, the fanning is clearly present; e.g., \cite{Jones+Giovanelli1983} found low-lying, 200-800 km, magnetic canopies in magnetograms taken near the solar limb. |
More recently Kontaretal.(2008) has used hard X-ray observations from RHESSI to estimate the expansion and found that the magnetic. field. expandec noticeably at a height of = 900 km. | More recently \cite{Kontar+Hannah+MacKinnon2008} has used hard X-ray observations from RHESSI to estimate the expansion and found that the magnetic field expanded noticeably at a height of $\approx$ 900 km. |
Expansion of magnetic field with height has been studied in photospheric structures mostly using magnetic flux tube models. | Expansion of magnetic field with height has been studied in photospheric structures mostly using magnetic flux tube models. |
Indeed. flux tube models predict a rapid expansior of the field with height. | Indeed, flux tube models predict a rapid expansion of the field with height. |
Solankietal.(1999) shows that magnetic structures as different in size and flux as small flux tubes and sunspots have similar relative expansior rates. which agree with the thin flux tube approximation. | \cite{Solanki+others1999} shows that magnetic structures as different in size and flux as small flux tubes and sunspots have similar relative expansion rates, which agree with the thin flux tube approximation. |
A study of the characteristics of magnetic flux. structures i1 radiative magnetohydrodynamic (MHD) simulations revealec the expansion properties to be similar with the thin flux tube and sheet approximations (YellesChaoucheetal. 2009)). | A study of the characteristics of magnetic flux structures in radiative magnetohydrodynamic (MHD) simulations revealed the expansion properties to be similar with the thin flux tube and sheet approximations \citealt{Yelles+others2009}) ). |
The expansion Is seen in observations: e.g.. a thin flux tube model can simultaneously reproduce the observed Zeeman splittings of Mg 12.32 um. Fe 525.0 nm and Fe 1.56 jm lines. which span the upper to the lower photosphere in formation height (Bruls&Solanki1995). | The expansion is seen in observations: e.g., a thin flux tube model can simultaneously reproduce the observed Zeeman splittings of Mg 12.32 $\mu$ m, Fe 525.0 nm and Fe 1.56 $\mu$ m lines, which span the upper to the lower photosphere in formation height \citep{Bruls+Solanki1995}. |
. Bruls Solanki also showed that a flux tube model can explain the Mg 12.32 jim line profile shapes observed by Zirin Popp (1989). | Bruls Solanki also showed that a flux tube model can explain the Mg 12.32 $\mu$ m line profile shapes observed by Zirin Popp (1989). |
Additional evidence of expansion is that à canopy resulting from the expansion of a flux tube can best explain the observed photospheric asymmetric Stokes V profiles with weak zero-crossing shifts (Grossmann-Doerthetal.1985). | Additional evidence of expansion is that a canopy resulting from the expansion of a flux tube can best explain the observed photospheric asymmetric Stokes $V$ profiles with weak zero-crossing shifts \citep{Grossmann-Doerth+others1988}. |
. In this paper we use circular polarization maps from the Solar Optical Telescope (SOT. Tsunetaetal.2008)) on the Hinode satellite to study the expansion properties by characterizing how the appearance of network flux concentrations changes from the solar disk center to the limb. | \nocite{Zirin+Popp1989} In this paper we use circular polarization maps from the Solar Optical Telescope (SOT, \citealt{Tsuneta+others2008}) ) on the Hinode satellite to study the expansion properties by characterizing how the appearance of network flux concentrations changes from the solar disk center to the limb. |
The center-to-limb approach lets us examine the expansion at different viewing angles and at different heights due to the shift in the height spectral lines as a function of μμ Ξcos(8). where ϐ is the viewing angle). | The center-to-limb approach lets us examine the expansion at different viewing angles and at different heights due to the shift in the height spectral lines as a function of $\mu$ $\mu=cos(\theta)$, where $\theta$ is the viewing angle). |
To further expand the coverage we use SOT observations from the spectropolarimeter (SP) and the narrowband filter imager (NFI) NaD, channel. | To further expand the coverage we use SOT observations from the spectropolarimeter (SP) and the narrowband filter imager (NFI) $_{1}$ channel. |
We combine the observations with modeling the expansion of magnetic flux with height by using the thin flux tube and sheet approximations and more realistic 3-dimensional magneto-convection simulations, | We combine the observations with modeling the expansion of magnetic flux with height by using the thin flux tube and sheet approximations and more realistic 3-dimensional magneto-convection simulations. |
The SOT NFI NaD, (589.6 nm. effective ο factor 1.33) data used in this study consist of 19 circular polarization filtergrams of non-active region maps at different positions on the solar disk (see Table | for time. location. exposure time and field of view. where xcen and yee are orthogonal distances from Sun center and avcen is directed along the equator). | The SOT NFI $_{1}$ (589.6 nm, effective $g$ factor 1.33) data used in this study consist of 19 circular polarization filtergrams of non-active region maps at different positions on the solar disk (see Table \ref{tab:nfi-sp-dat} for time, location, exposure time and field of view, where $xcen$ and $ycen$ are orthogonal distances from Sun center and $xcen$ is directed along the equator). |
The data reduction was done using the Solar Software (SSW) package routineprep. | The data reduction was done using the Solar Software (SSW) package routine. |
pro. To increase the signal-to-noise ratio. frames taken within 34 minutes are summed together (number of frames is shown às a multiplication factor in from of the exposure time in Table 1)). | To increase the signal-to-noise ratio, frames taken within 3 minutes are summed together (number of frames is shown as a multiplication factor in from of the exposure time in Table \ref{tab:nfi-sp-dat}) ). |
Errors in pointing are corrected in the limb data sets by forcing 4:50 | Errors in pointing are corrected in the limb data sets by forcing $\mu$ =0 |
of a group of 189 stars with Ho excess emission in the field of 11987A that were studied in PaperllI with the same method applied to HST Wide Field Planetary 22 data. | of a group of 189 stars with $\alpha$ excess emission in the field of 1987A that were studied in I with the same method applied to HST Wide Field Planetary 2 data. |
The uncertainty on L(Ha) is typically ~12 uncertainty on the Ho photometry accounts for absolute sensitivity of the instrumental setup for another Owing to the rather wide passband of the F658N filter, ΔΗα includes a small contribution from the [NII] emission features at 6548 and 6584AA. | The uncertainty on $L(H\alpha)$ is typically $\sim 12$ uncertainty on the $\alpha$ photometry accounts for absolute sensitivity of the instrumental setup for another Owing to the rather wide passband of the F658N filter, $\Delta H\alpha$ includes a small contribution from the [NII] emission features at $6\,548$ and $6\,584$. |
. Even though such a contribution is small (see II, detail), it is a to correct for it as it is a explainedsystematic effect. | Even though such a contribution is small (see I, ), it is a good practice to correct for it as it is a systematic effect. |
The adoptedgood practiceaverage correction factor to the intensity for the ACS F658N filter, is 0.979. | The adopted average correction factor to the intensity for the ACS F658N filter, is $0.979$. |
Besides the luminosity of the Ho emission line, also its equivalent width can be derived from our photometry. | Besides the luminosity of the $\alpha$ emission line, also its equivalent width $W_{\rm eq}(H\alpha)$ can be derived from our photometry. |
To this aim, as W.g(Ha)shown in PaperIl, we used the model atmospheres of Bessell et al. ( | To this aim, as shown in I, we used the model atmospheres of Bessell et al. ( |
1998) to derive the magnitude Ho corresponding to the sole continuum in the Ha band from the V and J magnitudes. | 1998) to derive the magnitude $H\alpha^c$ corresponding to the sole continuum in the $\alpha$ band from the $V$ and $I$ magnitudes. |
The corresponding relationships, properly validated by comparison with spectro-measurements, are listed in the Appendix of II and allow us to derive W.g(Ha) from the Ha magnitude via the equation: where RW is the rectangular width of the filter (similar in definition to the equivalent width of a line), which depends on the characteristics of the filter and corresponds to 74.96 ffor the F658N filter used here. | The corresponding relationships, properly validated by comparison with spectro-photometricmeasurements, are listed in the Appendix of I and allow us to derive $W_{\rm
eq}(H\alpha)$ from the $\alpha$ magnitude via the equation: where $\mathrm{RW}$ is the rectangular width of the filter (similar in definition to the equivalent width of a line), which depends on the characteristics of the filter and corresponds to $74.96$ for the F658N filter used here. |
The equivalent widths obtained in this way are shown in reffig4 as a function of the V—J colour. | The equivalent widths obtained in this way are shown in \\ref{fig4} as a function of the $V-I$ colour. |
The 791 stars with a 4c Ha excess are indicated with thicker symbols (circles and squares). | The 791 stars with a $4\,\sigma$ $\alpha$ excess are indicated with thicker symbols (circles and squares). |
We take as bona-fide PMS stars all those with Weq(Ha)«-20AA,, indicated as thick dots (in red in the online version), corresponding to a total of 694 stars. | We take as bona-fide PMS stars all those with $W_{\rm eq}(H\alpha) < -20$, indicated as thick dots (in red in the online version), corresponding to a total of 694 stars. |
Note that, as customary, negative values of the equivalent width indicate emission lines. | Note that, as customary, negative values of the equivalent width indicate emission lines. |
Since at temperatures Te;>10000 KK or colours V—I<0 the sample could be contaminated by Be stars that are evolving off the MS, there we set a more stringent condition on the equivalent width, namely «—50AA. | Since at temperatures $T_{\rm eff} \ga
10\,000$ K or colours $V-I \la 0$ the sample could be contaminated by Be stars that are evolving off the MS, there we set a more stringent condition on the equivalent width, namely $W_{\rm eq}(H\alpha) <
-50$. |
. This limit is suggested by a(Πα) survey of the Ho equivalent width of about 100 Be stars in the Galaxy (Coté Waters 1987) in which only one star is found with W.g(Ha)«—50 aand the largest majority have values in the range from —4 tto —30AA. | This limit is suggested by a survey of the $\alpha$ equivalent width of about 100 Be stars in the Galaxy (Coté Waters 1987) in which only one star is found with $W_{\rm eq}(H\alpha) <
-50$ and the largest majority have values in the range from $-4$ to $-30$. |
. In fact, as we discuss in refphysi, all the objects with Teg>10000 KK and an Ha excess above 4c are most likely PMS objects with masses aroundMsolar. | In fact, as we discuss in \\ref{physi}, all the objects with $T_{\rm eff}> 10\,000$ K and an $\alpha$ excess above $4\,\sigma$ are most likely PMS objects with masses around. |
. White Ghez (2001) carried out a detailed study of 44 T Tauri stars in binary systems in Taurus-Auriga, using both photometry and spectroscopy with the HST. | White Ghez (2001) carried out a detailed study of 44 T Tauri stars in binary systems in Taurus–Auriga, using both photometry and spectroscopy with the HST. |
Their analysis shows that photometric and spectroscopic measurements of the equivalent width of the Ha emission line in these objects are in good agreement with one another. | Their analysis shows that photometric and spectroscopic measurements of the equivalent width of the $\alpha$ emission line in these objects are in good agreement with one another. |
More recently, Barentsen et al. ( | More recently, Barentsen et al. ( |
2011) carried out a Ha survey of T Tauri stars in 11396, a region with considerable nebular emission. | 2011) carried out a $\alpha$ survey of T Tauri stars in 1396, a region with considerable nebular emission. |
Their photometric determination of W.g(Ha) for 109 objects in common with the spectroscopic survey of Sicilia-Aguilar et al. ( | Their photometric determination of $W_{\rm eq}(H\alpha)$ for 109 objects in common with the spectroscopic survey of Sicilia–Aguilar et al. ( |
2005) also shows an excellent agreement, particularly for stars with Weq(Ha)«-10AA,, which are the objects of interest in our study. | 2005) also shows an excellent agreement, particularly for stars with $W_{\rm eq}(H\alpha)< -10$, which are the objects of interest in our study. |
When the bona-fide PMS objects identified in the previous section are placed on the CMD (see reffig5)), they clearly reveal two distinct groupsobjects. | When the bona-fide PMS objects identified in the previous section are placed on the CMD (see \\ref{fig5}) ), they clearly reveal two distinct groups. |
One is located above (i.e. brighter and redder than)the MS and thus is indicative of a population of young PMS objects. | One is located above (i.e. brighter and redder than)the MS and thus is indicative of a population of young PMS objects. |
This population was originally noticed by Nota et al. ( | This population was originally noticed by Nota et al. ( |
2006) and later studied by Gouliermis et al. ( | 2006) and later studied by Gouliermis et al. ( |
2007) and Sabbi et al. ( | 2007) and Sabbi et al. ( |
2007), although these earlier works could only identify candidate young («5 MMyr) PMS stars since no information was available to verify and characterise the PMS nature of each individual object. | 2007), although these earlier works could only identify candidate young $< 5$ Myr) PMS stars since no information was available to verify and characterise the PMS nature of each individual object. |
This is the reason why the works above could not identify the second group of bona-fide PMS objects, which overlap in the CMD with older MS stars in the field of the SMC. | This is the reason why the works above could not identify the second group of bona-fide PMS , which overlap in the CMD with older MS stars in the field of the SMC. |
Hennekemper et al. ( | Hennekemper et al. ( |
2008) As we show here below, these objects are older PMS | 2008) As we show here below, these objects are older PMS |
The future dark energy. experiments considered in this paper aim at measuring galaxy positions in three dimensions to studs barvon acoustic oscillations and other aspects of the matter power spectrum including its evolution through the erowth of structure. | The future dark energy experiments considered in this paper aim at measuring galaxy positions in three dimensions to study baryon acoustic oscillations and other aspects of the matter power spectrum including its evolution through the growth of structure. |
Phe matter power spectrum. contains important cosmological information through its evolving amplitude. its shape including the turnover reflecting the transition from radiation to matter domination. and the suppression due to massive neutrino free streaming. and the barvon acoustic oscillation features serving as a standard ruler. | The matter power spectrum contains important cosmological information through its evolving amplitude, its shape including the turnover reflecting the transition from radiation to matter domination and the suppression due to massive neutrino free streaming, and the baryon acoustic oscillation features serving as a standard ruler. |
One aspect of particular interest. is the distorted. anisotropic mapping between the real space density field and the measurements in redshift space. caused by. peculiar velocities (IxaiserLOST:Hamilton1998).. | One aspect of particular interest is the distorted, anisotropic mapping between the real space density field and the measurements in redshift space, caused by peculiar velocities \citep{kaiser,redist}. |
This redshift space distortion. has attracted. recent attention as a possible technique for detecting deviations from. general. relativity (see Peebles(2002):Linder(2008):Guzzoοἱal.(2008). for carly work) as it depencds on the relation between the density and. velocity fields. which can be altered by modifving the &ravitational theory. | This redshift space distortion has attracted recent attention as a possible technique for detecting deviations from general relativity (see \citet{peebgrav,linder-gamma,guzzo} for early work) as it depends on the relation between the density and velocity fields, which can be altered by modifying the gravitational theory. |
Thus the observed galaxy power spectrum contains several types of cosmological information. | Thus the observed galaxy power spectrum contains several types of cosmological information. |
The autocorrelation function £(r) is defined as the excess probability of finding masses at à separation r: = ο.avt) =p ; FL guaeeee(2) avhere p is.the mean mass density and ο is the density contrast. | The autocorrelation function $\xi(\mathbf{r})$ is defined as the excess probability of finding masses at a separation $\mathbf{r}$ : = _m), = ^2 (1 + )) dV_1, where $\bar{\rho}$ isthe mean mass density and $\delta_m\equiv(\rho-\bar\rho)/\bar\rho$ is the density contrast. |
The mass power spectrum is then the Fourier transform of the autocorrelation function: Pik) = (ΟΙ k the wave vector. | The mass power spectrum is then the Fourier transform of the autocorrelation function: P(k) =, with $\mathbf{k}$ the wave vector. |
Due to spatial isotropy. only the magnitude & will enter. | Due to spatial isotropy, only the magnitude $k$ will enter. |
We do not observe the power spectrum in real space. however. but. obtain the radial position through redshift measurements. convolving the real distance with additional redshifts due to peculiar velocities. | We do not observe the power spectrum in real space, however, but obtain the radial position through redshift measurements, convolving the real distance with additional redshifts due to peculiar velocities. |
This σας to the recshilt-space power spectrum 2 gaining an angular dependence through the linear Ixaiser factor (Ixaiser1987). multiplving the isotropic. real space mass power spectrum (4): (deg) = | fi(4) where ji ds the cosine of the angle that K makes with the line of sight. | This leads to the redshift-space power spectrum $\tilde P$ gaining an angular dependence through the linear Kaiser factor \citep{kaiser} multiplying the isotropic, real space mass power spectrum $P(k)$: ) = (b+f ^2)^2 P(k), where $\mu$ is the cosine of the angle that $\mathbf{k}$ makes with the line of sight. |
For notational simplicity. we suppress the tilde from now on. | For notational simplicity, we suppress the tilde from now on. |
We work in the linear regime. where the continuity equation between the galaxy peculiar velocity Ποια. and the galaxy. mass overcensity is linear (see [for example Hamilton (1998))). | We work in the linear regime, where the continuity equation between the galaxy peculiar velocity field and the galaxy mass overdensity is linear (see for example \citet{redist}) ). |
The dimensionless growth rate f is givenby : —)——————dua(5) where e is the scale factor. and D(a) is the growth factor. the amplitude 9,,(k.e)xDla) or PUR)xD7(a). | The dimensionless growth rate $f$ is givenby f =, where $a$ is the scale factor, and $D(a)$ is the growth factor, the amplitude $\delta_m(\mathbf{k},a) \propto D(a)$ or $P(k)\propto D^2(a)$. |
We also need to take into account. that galaxies. not directly mass density. are observed. | We also need to take into account that galaxies, not directly mass density, are observed. |
The bias 6 relatesthe galaxy overdensity 6, to the total mass overdensity through 6,=bs. | The bias $b$ relatesthe galaxy overdensity $\delta_g$ to the total mass overdensity through $\delta_g = b \delta_m$ . |
ὃν looking at the angular dependence of the power spectrum at each Á. Pik.) RID 2.2 where osx is the normalization of the power spectrum. we can in principle Iit for ak. bfax and fo. hence allowing us to measure 6 ancl f provided we have an appropriate measurement of m. | By looking at the angular dependence of the power spectrum at each $k$, ) _8^2 (b + f ^2)^2 = _8^2 b^2 + 2 _8^2 b f ^2 + _8^2 f^2, where $\sigma_8$ is the normalization of the power spectrum, we can in principle fit for $b^2 \sigma_8^2$, $b f \sigma_8^2$ and $f^2 \sigma_8^2$, hence allowing us to measure $b$ and $f$ provided we have an appropriate measurement of $\sigma_8$. |
‘This is challenging in practice due to noise. | This is challenging in practice due to noise. |
Another possible route to separating out the bias involves the use of higher order correlation functions (Scoccimarroetal.1999). | Another possible route to separating out the bias involves the use of higher order correlation functions \citep{scocc}. |
. Although we have three measurable quantities. (the hree coellicients of the fourth order polynomial in σα. 6)) | Although we have three measurable quantities (the three coefficients of the fourth order polynomial in Eq. \ref{eq:p4}) ) |
and three unknowns. we cannot determine all of them oeause the second is the geometric mean of the other wo. | and three unknowns, we cannot determine all of them because the second is the geometric mean of the other two. |
This is because we work in the linear regime and general relativity. where the galaxy density ancl peculiar velocity fields. are perfectly. correlated. | This is because we work in the linear regime and general relativity, where the galaxy density and peculiar velocity fields are perfectly correlated. |
But. should. one of hese hypothesis be relaxed (as in modified. gravity models with non-linearities FFinger-of-Cod elfects). we need o introduce the correlation coellicient between the [ields (Whiteetal.2009:Uzan2009) ες)(Kk) = where the subscript tT)denotes the galaxy density. field. and the divergence of the peculiar velocity field. | But should one of these hypothesis be relaxed (as in modified gravity models or with non-linearities Finger-of-God effects), we need to introduce the correlation coefficient between the fields \citep{white,uzan} r(k) =, where the subscript denotes the galaxy density field, and the divergence of the peculiar velocity field. |
Ideally this correlation would be predicted by the physical theory (Desjacquesetal.2009):: allowing r instead to be completely [ree significantly degrades the constraints on. f (Whiteetal. 2009). | Ideally this correlation would be predicted by the physical theory \citep{dessheth}; allowing $r$ instead to be completely free significantly degrades the constraints on $f$ \citep{white}. |
. We do not consider this situation further in this article. instead assuming the standard correlation ofunity. since we restrict our analysis to the linear regime and many classes of eravity theory maintain the correlation in this regime. | We do not consider this situation further in this article, instead assuming the standard correlation ofunity, since we restrict our analysis to the linear regime and many classes of gravity theory maintain the correlation in this regime. |
To incorporate a measure of the sensitivity. to the eravity theory we use the gravitational growth index formalism of Linder (2005).. which parameterizes the growth factor as Dia) =a [f WWE ) sO (0) where dui is the ratio of matter density to the total energy density at scale size. e=(1|2) . l | To incorporate a measure of the sensitivity to the gravity theory we use the gravitational growth index formalism of \citet{growth}, , which parameterizes the growth factor as D(a) = ( _0^a ), so , where _m(a) = is the ratio of matter density to the total energy density at scale size $a=(1+z)^{-1}$ . |
opThe summation. runs over all the | The summation runs over all the |
of time dependence. while in the past. fitting it well required enhancing the sodium abundance by a factor of about four (Dessart&Hillier2006a). | of time dependence, while in the past, fitting it well required enhancing the sodium abundance by a factor of about four \citep{DH_06a}. |
. Perhaps the most extreme case in our investigation is110830AÀ... which persists over the entire 6 weeks covered by the baseline model A. and for which. at the last ime in the sequence. we prediet a non-trivial absorption strongly blueshifted from line center (tat 1.038//m. equivalent to Kms)». and a weak and broad flat-toppedemission?. | Perhaps the most extreme case in our investigation is, which persists over the entire 6 weeks covered by the baseline model A, and for which, at the last time in the sequence, we predict a non-trivial absorption strongly blueshifted from line center (at $\mu$ m, equivalent to ), and a weak and broad flat-topped. |
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