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Because of its proximity. Wdl provides the unique opportunity to directly determine the spatially resolved IMF of a SSC and. given that it is located well away from the GC in a supposedly “quiescent” region of the Galactic disc. potentially study physical properties imprinted by the formation process from the parental molecular cloud on the cluster. | Because of its proximity, Wd1 provides the unique opportunity to directly determine the spatially resolved IMF of a SSC and, given that it is located well away from the GC in a supposedly `quiescent' region of the Galactic disc, potentially study physical properties imprinted by the formation process from the parental molecular cloud on the cluster. |
With the aim to better determine basic cluster properties such as age. foreground extinction and distance. to directly measure the IMF. and to study the structural properties of 11. we have analysed deep near IR observations. | With the aim to better determine basic cluster properties such as age, foreground extinction and distance, to directly measure the IMF, and to study the structural properties of 1, we have analysed deep near IR observations. |
This paper deseribes the analysis of NTT/Sofl observations of Wdl covering a ~5’x field of view centred on the cluster core. which are sensitive to ~ solar mass stars at a distance of kkpe: a second paper details subsequent VLT/NAOS-CONICA adaptive optics observations of selected core regions sensitive to ~O.2 MM... | This paper describes the analysis of NTT/SofI observations of Wd1 covering a $\sim 5' \times 5'$ field of view centred on the cluster core, which are sensitive to $\sim$ solar mass stars at a distance of kpc; a second paper details subsequent VLT/NAOS-CONICA adaptive optics observations of selected core regions sensitive to $\sim$ $_{\odot}$. |
The structure of the paper is as follows: in section 2 we present the observations and the data analysis. | The structure of the paper is as follows: in section 2 we present the observations and the data analysis. |
The basic astrophysical quantities extinction. distance. and age of Wd | are derived in section 3. | The basic astrophysical quantities extinction, distance, and age of Wd 1 are derived in section 3. |
Section 4 discusses the mass function and the total stellar mass of Wd 1. | Section 4 discusses the mass function and the total stellar mass of Wd 1. |
Dynamical and structural parameters of Wd ] are addressed in section 5. followed by a comparison of Wd | to other well studied starburst clusters in section 6. | Dynamical and structural parameters of Wd 1 are addressed in section 5, followed by a comparison of Wd 1 to other well studied starburst clusters in section 6. |
Section 7 summarises the results. | Section 7 summarises the results. |
JHKs-broad band imaging observations of Wd | (centred on RA(2000) = 1647"03*. DEC(2000) = -45750/37") and a nearby comparison field (located =7’ to the east. and =13 to the south of Wd 1. and centred on RA(2000) = 16/747?43*, DEC(2000) = -46*03'47" ). each covering an area of 4.5x4.5' (corresponding to 4.8pex at the distance of Wd 1). obtained with NTT/SofI (PI: J. Alves) were retrieved from the ESO archive (see refimage,gb)). | JHKs-broad band imaging observations of Wd 1 (centred on RA(2000) = $^h$ $^m$ $^s$, DEC(2000) = $^\circ$ $'$ $''$ ) and a nearby comparison field (located $\approx 7'$ to the east, and $\approx 13'$ to the south of Wd 1, and centred on RA(2000) = $^h$ $^m$ $^s$, DEC(2000) = $^\circ$ $'$ $''$ ), each covering an area of $4.5' \times 4.5'$ (corresponding to $4.8\,{\rm pc} \times 4.8\,{\rm pc}$ at the distance of Wd 1), obtained with NTT/SofI (PI: J. Alves) were retrieved from the ESO archive (see \\ref{image_rgb}) ). |
S of Hsequippedwitha Hawaii HgCdTedetector. | SofI is equipped with a Hawaii HgCdTe detector. |
Theobsery res | The observations were obtained with a plate scale of $''$ /pixel. |
ultinginl2sof integrationtimeperditherposition. | Individual integration times were 1.2s (DIT) and 10 frames were co-added, resulting in 12s of integration time per dither position. |
Foreachfield | For each field and filter a total of 10 dither positions were used, resulting in a total integration time of 120s in each filter. |
Data reduction was carried out using the eclipse jitter routines (Devillard 2001)). | Data reduction was carried out using the eclipse jitter routines (Devillard \cite{devil01}) ). |
Because of the high degree of crowding in the cluster field. sky frames derived from the comparison field were used in the reduction of the 11 frames. | Because of the high degree of crowding in the cluster field, sky frames derived from the comparison field were used in the reduction of the 1 frames. |
The resolution on the final images ts z0.75” to 0.80" (see Table 1. for more details). | The resolution on the final images is $\approx 0.75''$ to $0.80''$ (see Table \ref{obslog} for more details). |
PSF fitting photometry in J. H. and Ks for 7000 stars in the Wdl area. and 5300 stars in the off-field was derived using the IRAF implementation of DAOPHOT (Stetson 1987)). | PSF fitting photometry in J, H, and Ks for 7000 stars in the Wd1 area, and 5300 stars in the off-field was derived using the IRAF implementation of DAOPHOT (Stetson \cite{stetson87}) ). |
Photometric zeropoints and colourterms were computed by comparisonof instrumental magnitudes of relatively isolated. brightsources with photometry from the 2MASS point source catalogue. | Photometric zeropoints and colourterms were computed by comparisonof instrumental magnitudes of relatively isolated, brightsources with photometry from the 2MASS point source catalogue. |
We first calculate the teusor of the velocity eracdicnt Vv in plivsical wits: and define the tensor A: A=[Vv|(Vv)2, where (Vv) is the trauspose of Vv. | We first calculate the tensor of the velocity gradient $\nabla {\bf v}$ in physical units: and define the tensor ${\bf A}$: ${\bf A}=[\nabla {\bf v}+(\nabla {\bf v})^{\rm tr}]/2$, where $(\nabla {\bf v})^{\rm tr}$ is the transpose of $\nabla {\bf v}$. |
Then the viscous stress tensor can be written as (in plivsical units): where Eis the ideutity matrix Gy=oj. aud the divergence of gas velocity is After calculating all non-zero componeits of IE for cach simulation zone. we finally calculate the viscous term V-II in thegas momentum equation 2 and IE:Vv in the eas energy equation 23: | Then the viscous stress tensor can be written as (in physical units): where ${\bf I}$ is the identity matrix $I_{\rm ij}=\delta_{\rm ij}$, and the divergence of gas velocity is After calculating all non-zero components of ${\bf \Pi}$ for each simulation zone, we finally calculate the viscous term $\nabla \cdot {\bf \Pi}$ in thegas momentum equation \ref{hydro2} and ${\bf \Pi}:\nabla {\bf v}$ in the gas energy equation \ref{hydro3}: : |
1.40+0.17 for the quiescent galaxy fit reported by Laueretal. (2007a). | $1.40 \pm 0.17$ for the quiescent galaxy fit reported by \citet{lauer07a}. |
. Both of these studies included lenticular and spiral galaxies with dynamical masses in their fitting samples. | Both of these studies included lenticular and spiral galaxies with dynamical masses in their fitting samples. |
Laueretal.(2007a) compiled decompositions from the literature. while FFOS assumed specific ratios of Lou/Li based on morphological The compendium of AGN black hole masses and bulge lummosities presented here offers several advantages over previous measurements compiled from the literature (e.. Wandel1999,2003:MeLure&Dunlop 2001)). | \citet{lauer07a} compiled decompositions from the literature, while FF05 assumed specific ratios of $L_{\rm
bulge}/L_{\rm total}$ based on morphological The compendium of AGN black hole masses and bulge luminosities presented here offers several advantages over previous measurements compiled from the literature (i.e., \citealt{wandel99a,wandel02,mclure01}) ). |
The masses result from a homogeneous analysis of reverberation-mapping data (Petersonetal.2004).. in contrast to many recent studies of AGN black hole — bulge relationships where black hole masses are inferred from single-epoch spectral measurements (e.g.. Greeneetal.2008:Kim 2008)). | The masses result from a homogeneous analysis of reverberation-mapping data \citep{peterson04}, in contrast to many recent studies of AGN black hole – bulge relationships where black hole masses are inferred from single-epoch spectral measurements (e.g., \citealt{greene08,kim08}) ). |
The bulge luminosities in this work are estimated from two-dimensional surface brightness decompositions of unsaturated high-resolution space-based images taken with the same instrument and the same filter. | The bulge luminosities in this work are estimated from two-dimensional surface brightness decompositions of unsaturated high-resolution space-based images taken with the same instrument and the same filter. |
The exceptions are the five objects that were Imaged through the F547M filter using WFPC2. but the bandpass is very similar to the ACS F550M filter employed for the other objects (A.(F530M)=5580 vversus \.(F547M)=5483AA.. and A\(F550M)=2547 vversus AA(F547M)=483 AA)). | The exceptions are the five objects that were imaged through the F547M filter using WFPC2, but the bandpass is very similar to the ACS F550M filter employed for the other objects $\lambda_{\rm c} {\rm
(F550M)}=5580$ versus $\lambda_{\rm c}{\rm (F547M)}=5483$, and $\Delta \lambda{\rm (F550M)} = 547$ versus $\Delta
\lambda{\rm (F547M)}=483$ ). |
Unfortunately. there is not a similarly consistent sample of high-quality images from which the bulge luminosities can be estimated for the quiescent galaxies with dynamical black hole masses. | Unfortunately, there is not a similarly consistent sample of high-quality images from which the bulge luminosities can be estimated for the quiescent galaxies with dynamical black hole masses. |
While high-quality observations do exist for these nearby and well-studied galaxies. they have not been obtained 1n a uniform fashion. | While high-quality observations do exist for these nearby and well-studied galaxies, they have not been obtained in a uniform fashion. |
The recent work by Graham(2007) attempts to compensate for the different methods and analysis techniques employed in several publications (MeLure&Dunlop2002 with an updated cosmology presented in MeLure&Dunlop2004;Marconi.Hunt2005:Erwinetal.2004)). all of which arrive at different values for the slope of the quiescent galaxy Mgy—Lou: relationship and/or different black hole masses predicted for à specific bulge luminosity. | The recent work by \citet{graham07} attempts to compensate for the different methods and analysis techniques employed in several publications \citealt{mclure02b} with an updated cosmology presented in \citealt{mclure04,marconi03,erwin04}) ), all of which arrive at different values for the slope of the quiescent galaxy $M_{\rm BH} -
L_{\rm bulge}$ relationship and/or different black hole masses predicted for a specific bulge luminosity. |
Graham carefully updated and revised the samples of objects included in these studies and found that the differences of the best-fit parameters found by each study are mitigated. the scatter in the measurements is significantly decreased. and that MgyL5. | \citeauthor{graham07}
carefully updated and revised the samples of objects included in these studies and found that the differences of the best-fit parameters found by each study are mitigated, the scatter in the measurements is significantly decreased, and that $M_{\rm BH} \propto L^{1.0}$ . |
Interestingly. he finds a somewhat shallower slope. e~0.75—0.88. when bulge luminosities are estimated by two-dimensional decompositions of B-band images and corrected for inclination-dependent dust extinction in the host galaxy disks. | Interestingly, he finds a somewhat shallower slope, $\alpha \approx
0.75 - 0.88$ , when bulge luminosities are estimated by two-dimensional decompositions of $B$ -band images and corrected for inclination-dependent dust extinction in the host galaxy disks. |
Only 13 objects are included in that particular analysis (an updated form of the study presented by Erwinetal.2004)) but it 15 intriguing nonetheless in its close agreement with the fit that we find for the AGNs. | Only 13 objects are included in that particular analysis (an updated form of the study presented by \citealt{erwin04}) ) but it is intriguing nonetheless in its close agreement with the fit that we find for the AGNs. |
While we expect that the AGN bulge luminosities 1n. this may be slightly underestimated based on the conservative galaxy fits we employed. there is also a small &-correction introduced. as the average redshift of the 26 AGNs is z0.1. | While we expect that the AGN bulge luminosities in this may be slightly underestimated based on the conservative galaxy fits we employed, there is also a small $k$ -correction introduced, as the average redshift of the 26 AGNs is $z \approx 0.1$. |
The portion of the SED observed through the medium-band V filters employed in the observations is fainter (~0.1 ddex in luminosity) than if the galaxies were at z=0. assuming their stellar populations resemble that of the bulge template of Kinneyetal.(1996). | The portion of the SED observed through the medium-band $V$ filters employed in the observations is fainter $\sim 0.1$ dex in luminosity) than if the galaxies were at $z=0$, assuming their stellar populations resemble that of the bulge template of \citet{kinney96}. |
.. Accounting for any such biases in the galaxy fitting or colors would intensify the apparent differences between the slopes measured for the AGNs and quiescent galaxies. | Accounting for any such biases in the galaxy fitting or colors would intensify the apparent differences between the slopes measured for the AGNs and quiescent galaxies. |
The black hole masses for the AGNs have already been sealed so that their Λήμη--σ. relationship is brought into agreement with the quiescent Mpy—o. relationship fit to the same sample of quiescent galaxies included in the FFOS study. | The black hole masses for the AGNs have already been scaled so that their $M_{\rm BH} - \sigma_{\star}$ relationship is brought into agreement with the quiescent $M_{\rm BH} -
\sigma_{\star}$ relationship fit to the same sample of quiescent galaxies included in the FF05 study. |
The differences may be mitigated if Mareontetal.(2008) are correct in their suggestion that neglecting radiation pressure leads to systematic under-estimation of black hole masses from reverberation mapping data (see. however. Netzer 2008)). | The differences may be mitigated if \citet{marconi08} are correct in their suggestion that neglecting radiation pressure leads to systematic under-estimation of black hole masses from reverberation mapping data (see, however, \citealt{netzer08}) ). |
A separate concern has been raised by Yu&Tremaine (2002)... Bernardietal. (2007).. Laueretal. (2007a).. and Tundoetal.(2007).. who present an apparent disagreement between the quiescent galaxy μη—0. and My—Lou: relationships and suggest that the quiescent galaxy sample is biased towards galaxies with overly large velocity dispersions for their luminosities (see. however. Graham2008 who examines the role of bars in this issue). | A separate concern has been raised by \citet{yu02}, \citet{bernardi07}, \citet{lauer07a}, and \citet{tundo07}, who present an apparent disagreement between the quiescent galaxy $M_{\rm BH} -
\sigma_{\star}$ and $M_{\rm BH} - L_{\rm bulge}$ relationships and suggest that the quiescent galaxy sample is biased towards galaxies with overly large velocity dispersions for their luminosities (see, however, \citealt{graham08} who examines the role of bars in this issue). |
There is no reason— to suspect the AGN sample of having the same bias. às the mass measurements are made using flux variability techniques and not dynamical techniques. | There is no reason to suspect the AGN sample of having the same bias, as the mass measurements are made using flux variability techniques and not dynamical techniques. |
Such a bias may help explain why some of the quiescent galaxies at the high luminosity end have black hole masses that are more than an order-of-magnitude larger than the active galaxies. although it may not completely resolve this disparity. | Such a bias may help explain why some of the quiescent galaxies at the high luminosity end have black hole masses that are more than an order-of-magnitude larger than the active galaxies, although it may not completely resolve this disparity. |
Finally. there may be no reason to expect that the Mgy—Logs; relationship is the same for the AGNs and quiescent galaxies in these samples. as there is only a modest number of objects in each sample and selection effects likely play an important role on both sides (Laueretal.2007b). | Finally, there may be no reason to expect that the $M_{\rm BH} -
L_{\rm bulge}$ relationship is the same for the AGNs and quiescent galaxies in these samples, as there is only a modest number of objects in each sample and selection effects likely play an important role on both sides \citep{lauer07b}. |
. Clearly. there remain several areas that are in need of Investigation. any of which may shed light on the apparently inconsistent fits to the Mgy—ως relationship for AGNs and. for quiescent galaxies. | Clearly, there remain several areas that are in need of investigation, any of which may shed light on the apparently inconsistent fits to the $M_{\rm BH} - L_{\rm bulge}$ relationship for AGNs and for quiescent galaxies. |
As the — relationship is an important and widely used means of MpyLouestimating black hole masses throughout cosmic history (e.g.. Marconietal.2004: 2004)). an accurate characterization of this | As the $M_{\rm BH}
- L_{\rm bulge}$ relationship is an important and widely used means of estimating black hole masses throughout cosmic history (e.g., \citealt{marconi04,shankar04}) ), an accurate characterization of this |
in increased ionization of the gas. | in increased ionization of the gas. |
The ionization can then trigger the magnetic instability. causing a much higher level of disk turbulence and accretion. an outburst. | The ionization can then trigger the magnetic instability, causing a much higher level of disk turbulence and accretion, an outburst. |
After an outburst. ihe remaining disk gas cools and is replenished by accreting gas from larger radii (e.g.Ar-Pringle2001:Zhuetal. 2009). | After an outburst, the remaining disk gas cools and is replenished by accreting gas from larger radii \citep[e.g.][]{armitage01,zhu09}. |
. The outburst then repeats at later times inan approximately periodic manner. | The outburst then repeats at later times in an approximately periodic manner. |
Such behavior provides a possible model for FU Ori outbursts in voung stars. | Such behavior provides a possible model for FU Ori outbursts in young stars. |
The gravitational instability. is thought to be a natural outcome of dead zones in a lavered disk. regions where the disk is nonturbulent near the disk mid-plane. but. turbulent due to the MBA near the disk surface (Gamunie1996). | The gravitational instability is thought to be a natural outcome of dead zones in a layered disk, regions where the disk is nonturbulent near the disk mid-plane, but turbulent due to the MRI near the disk surface \citep{gammie96}. |
. The surface disk turbulence occurs as a consequence of surface ionization by external sources of ionization. such as cosnic rays or X-rays. (hat then permit the MRI to operate (Glassgold.Najita&Igea2004). | The surface disk turbulence occurs as a consequence of surface ionization by external sources of ionization, such as cosmic rays or X-rays, that then permit the MRI to operate \citep{glassgold04}. |
. In a layered region containing a dead zone. steady state accretion is generally not possible and the dead zone gains mass [rom some of the aceretion flow near the disk surface. | In a layered region containing a dead zone, steady state accretion is generally not possible and the dead zone gains mass from some of the accretion flow near the disk surface. |
As mass builds up in the dead zone. it can become sell-gravitating. | As mass builds up in the dead zone, it can become self-gravitating. |
As discussed above. the sell-gravitating state is turbulent and can trigger a disk outburst. | As discussed above, the self-gravitating state is turbulent and can trigger a disk outburst. |
The purpose of (hisLetter is to describe the disk outburst evele in teris of transitions between steady state configurations of an accretion disk. | The purpose of this is to describe the disk outburst cycle in terms of transitions between steady state configurations of an accretion disk. |
We determine the limit evcle for the outbursts in a diagram that plots (he mass accretion rate M. versus disk surface density V. | We determine the limit cycle for the outbursts in a diagram that plots the mass accretion rate $\dot{M}$ versus disk surface density $\Sigma$. |
A similar approach was taken to explain the disk instability in dwarf novae (Bath&Pringle&Papaloizou 1933). | A similar approach was taken to explain the disk instability in dwarf novae \citep{bath82, faulkner83}. |
. The dwarl nova thermal-viscous instability can be understood by the S-shaped curve of steady state configurations in such a diagram. | The dwarf nova thermal-viscous instability can be understood by the S-shaped curve of steady state configurations in such a diagram. |
In this paper. we determine an analogous curve lor outbursts in lavered disks with gravitational and magnetic sources of turbulence. | In this paper, we determine an analogous curve for outbursts in layered disks with gravitational and magnetic sources of turbulence. |
Our results do not conform to an S-curve because the middle portion of the S is missing. | Our results do not conform to an S-curve because the middle portion of the S is missing. |
That is. unlike the S-curve case. there is range of accretion rates for which there are no steady state accretion configurations. | That is, unlike the S-curve case, there is range of accretion rates for which there are no steady state accretion configurations. |
In Section 2. we describe the equations lor (he disk model. | In Section \ref{model} we describe the equations for the disk model. |
In Section 3. we find steady state solutions to the disk equations and the resulting AZ versus X curves, | In Section \ref{steady} we find steady state solutions to the disk equations and the resulting $\dot{M}$ versus $\Sigma$ curves. |
In Section we describe the results of munerical simulations and show that the instability evele canbe understood in terms of transitions between the stable solutions in the XM. diagram. | In Section \ref{num} we describe the results of numerical simulations and show that the instability cycle canbe understood in terms of transitions between the stable solutions in the $\Sigma$ $\dot{M}$ diagram. |
Seclion 5 contains the discussion and conclusions. | Section \ref{discussion} contains the discussion and conclusions. |
The surface densitv evolution in an accretion disk is determined by mass ancl angular momentum conservation (Pringle 1981).. | The surface density evolution in an accretion disk is determined by mass and angular momentum conservation \citep{pringle81}. . |
The disk model we adopt is essentially the same | The disk model we adopt is essentially the same |
is defined. | is defined. |
Then the constants have to be set to some reasonable value. If surface density cata exists. in a file named. c.g.data.txt.. with columns A.XAR) and ANCA). the [inal fitting is carried out by use of theb. | Then the constants have to be set to some reasonable value, If surface density data exists in a file named, e.g., with columns $R$ ,$\Sigma(R)$ and $\Delta\Sigma(R)$, the final fitting is carried out by use of the. |
example. by the density [uctuation in spheres of radius Shot Alpe. ex). | example, by the density fluctuation in spheres of radius $8\hmpc$ , $\sigma_8$ ). |
For example. Jungman used. the quadrupole C? to fix the amplitude of the fluctuation spectrum. | For example, Jungman used the quadrupole ${\rm C}_2^{1/2}$ to fix the amplitude of the fluctuation spectrum. |
We use (6|D);/(2x)n an average over the total band D of multipoles that is accessible to the experiment. since this is most accurately determined. | We use $\avrg{\ell (\ell +1){\rm C}_\ell
/(2\pi)}_B^{1/2}$, an average over the total band $B$ of multipoles that is accessible to the experiment, since this is most accurately determined. |
Llowever the normalization parameters ex and P57"m(Ay,) are of sullicient interest that we also show the accuracies with which these can be determined. | However the normalization parameters $\sigma_8$ and ${\cal
P}_\Phi^{1/2}(k_n)$ are of sufficient interest that we also show the accuracies with which these can be determined. |
To characterize the tensor amplitude.. we use rj,—CSVος οinstead. of--. Pee). | To characterize the tensor amplitude, we use $r_{ts}\equiv {\rm C}^{(T)}_2/{\rm
C}^{(S)}_2$ instead of ${\cal P}_{GW}^{1/2}(k_n)$. |
:. ↓⊔↓⊔∐⋜∏↓∪⊔⊔↓⋯⇂∢⊾↓⊳∖⊳∩∷⊓↿∖∕⋅⋅⊔⊐↧↴≺↓⋅↿∖∕⋅⋅⊔∃↓⊳∖⊳∖↓⊔↓↓≻↓∙∖⇁↓⋅∢⊾↓⋜⋯⋅∠⇂∪ plipli. ⋠⋠ he tensor tilt. with small corrections dependent. upon the scalar and tensor tilts. so one of the four initial Ductuation xwameters is a function of the other three. here chosen to ος mis | In inflation models, ${\cal P}_{GW}^{1/2}(k_n) /{\cal P}_{\Phi}^{1/2}(k_n) $ is simply related to the tensor tilt, with small corrections dependent upon the scalar and tensor tilts, so one of the four initial fluctuation parameters is a function of the other three, here chosen to be $n_t$. |
Hrs also depends upon other cosmological parameters as well as the tilts 3ond. 1996. equation (6.38)) | $r_{ts}$ also depends upon other cosmological parameters as well as the tilts Bond 1996, equation (6.38)). |
Any parameter set which defines a coordinate system on he likelihood surface is a viable set. | Any parameter set which defines a coordinate system on the likelihood surface is a viable set. |
However. parameters for which the Fisher matrix analysis is particularly well suited ave those for which the first order expansion €;=C,(s«)|(ο(8.1098)(ss.) is more accurate than the sampling variance AC; for parameters s that lie within a few standard deviations [rom the target set s«. | However, parameters for which the Fisher matrix analysis is particularly well suited are those for which the first order expansion ${\rm C}_\ell = {\rm C}_\ell ({\bf s_\circ}) +
(\partial {\rm C}_\ell({\bf s_\circ})/\partial {\bf s})\cdot ({\bf s}
- {\bf s_\circ})$ is more accurate than the sampling variance $\Delta
{\rm C}_\ell$ for parameters ${\bf s}$ that lie within a few standard deviations from the target set ${\bf s_\circ}$. |
Phe set of variables that we rave adopted gives acceptably high accuracy for the CM experiments described in Section 3.. | The set of variables that we have adopted gives acceptably high accuracy for the CMB experiments described in Section \ref{sec:satparams}. |
We analvze three spatially flat (Qj= 0) target models and one with negative curvature. | We analyze three spatially flat $\Omega_k=0$ ) target models and one with negative curvature. |
For the canonical Lat universe we use a standard CDM. model (SCDM) with the ollowing parameters: n,=1. n;—0. O,,=1.0. Q,=0.05. te=0. On=095. Opin,= 0. h 0.5. τι=ὐ. lu.=0.28: it has ex=1.2. normalized to match CODI DAI. but with too many clusters to match the observations. | For the canonical flat universe we use a standard CDM model (SCDM) with the following parameters: $n_s = 1$, $n_t = 0$, $\Omega_{m} =1.0$, $\Omega_b = 0.05$, $r_{ts}=0$, $\Omega_{cdm} = 0.95$, $\Omega_{hdm} = 0$ , ${\rm h}=0.5$ , $\tau_C =
0$, $Y_{He} = 0.23$; it has $\sigma_8=1.2$, normalized to match COBE DMR, but with too many clusters to match the observations. |
The open model has h=0.6. O,,0.33. οι=0.035. Ὁμου=0.30 and oy= 0.4. the CODE-normalization. which ias too few clusters. | The open model has ${\rm h}=0.6$, $\Omega_m =0.33$, $\Omega_b=0.035$, $\Omega_{cdm} = 0.30$ and $\sigma_8=0.44$ , the COBE-normalization, which has too few clusters. |
We also cliscuss results for two other DAM-normatized spatially Hat. models that more closely match observations: an HICDM model with 2 species of massive neutrinos. ο Ξ0.2.m,=2.4eV. h=0.5 and a=O.T: a ACDAL model with O4= 0.67. bh=0.7 and ox=1.1. ( | We also discuss results for two other DMR-normalized spatially flat models that more closely match observations: an HCDM model with 2 species of massive neutrinos, $\Omega_{hdm}=0.2, m_\nu =2.4 \, \eV$, ${\rm h}=0.5$ and $\sigma_8=0.77$; a $\Lambda$ CDM model with $\Omega_\Lambda =0.67$ , ${\rm h}=0.7$ and $\sigma_8=1.1$. ( |
ALL models have a 19 Gye cosmological age.) | All models have a 13 Gyr cosmological age.) |
Computational errors in the derivatives of Cy can Lead o large errors in the covariance matrix. | Computational errors in the derivatives of ${\rm C}_\ell$ can lead to large errors in the covariance matrix. |
We cistinguish »etween (vo classes of error. one caused by inadequate semi-analvtic approximations to the C; and the second caused by numerical errors in C; and its derivatives computed from incar Boltzmann transport codes. | We distinguish between two classes of error, one caused by inadequate semi-analytic approximations to the ${\rm C}_\ell$ and the second caused by numerical errors in ${\rm C}_\ell$ and its derivatives computed from linear Boltzmann transport codes. |
The €; accuracy must x I'& or better. especially for high resolution experiments wobing multipoles {1000 where the expected. random errors on cach individual multipole become AA. | The ${\rm C}_\ell$ accuracy must be $1\%$ or better, especially for high resolution experiments probing multipoles $\ell \simgt 1000$ where the expected random errors on each individual multipole become $\simlt 3 \%$. |
Exvrors which are weakly correlated: with physical parameters. are particularly serious since these can artificially break real near-degeneracies between cosmological parameters. ancl lead to overly optimistic error estimates. sometimes by an order of magnitude or more. | Errors which are weakly correlated with physical parameters are particularly serious since these can artificially break real near-degeneracies between cosmological parameters and lead to overly optimistic error estimates, sometimes by an order of magnitude or more. |
Extreme care is therefore required. in computing the €; derivatives. | Extreme care is therefore required in computing the ${\rm C}_\ell$ derivatives. |
For this work we have used derivatives calculated with twoBoltzmann transport codes. an updated version of the multipole code described by Bond and Efstathiou (1987) generalized to low density universes and including tensor components (Bond 1996 and references therein). and the fast. path-history code developed. recently by Seljak ancl Zaldarriaga (1996). | For this work we have used derivatives calculated with twoBoltzmann transport codes, an updated version of the multipole code described by Bond and Efstathiou (1987) generalized to low density universes and including tensor components (Bond 1996 and references therein), and the fast path-history code developed recently by Seljak and Zaldarriaga (1996). |
Generally the Cys from these codes agree to better than lA. | Generally the ${\rm C}_\ell$ 's from these codes agree to better than $1 \%$ . |
We use intervals of typically 1.5% in the parameters s in computing numerical derivatives of Cy. small enough that the derivatives are insensitive to the size of the interval. but large enough that they are unallected by numerical errors in the €; coctlicients. | We use intervals of typically $1 - 5 \%$ in the parameters ${\bf s}$ in computing numerical derivatives of ${\rm C}_\ell$ , small enough that the derivatives are insensitive to the size of the interval, but large enough that they are unaffected by numerical errors in the ${\rm C}_\ell$ coefficients. |
The primary limitation on the error estimates for Table 20 should be the Cy linearization assumption made in deriving equation l.. although we believe that better than percent level accuracy in C; is needed: to achieve high precision in the nearly degenerate directions of parameter space. | The primary limitation on the error estimates for Table \ref{tab:satparams1} should be the ${\rm C}_\ell$ linearization assumption made in deriving equation \ref{eq:fishmat}, although we believe that better than percent level accuracy in ${\rm
C}_\ell$ is needed to achieve high precision in the nearly degenerate directions of parameter space. |
The cilferenees between our error estimates ancl analogous results of Jungmanal... which are large for some parameters. are caused. primarily w their use of semi-analvtie approximations to calculate C; and its derivatives. | The differences between our error estimates and analogous results of Jungman, which are large for some parameters, are caused primarily by their use of semi-analytic approximations to calculate ${\rm C}_\ell$ and its derivatives. |
Zaldarriaga LOOT have undertaken a similar analysis and come to similar conclusions as hose presented here. | Zaldarriaga 1997 have undertaken a similar analysis and come to similar conclusions as those presented here. |
They. also showed that polarization information can improve the accuracy of some variables. Te. if foregrounds are ignored. | They also showed that polarization information can improve the accuracy of some variables, $\tau_C$, if foregrounds are ignored. |
Little is known about row the polarization of foregrounds. will compromise the relatively weak polarizationsignal of primary anisotropies. especially at low (where much of the improvement. comes roni. | Little is known about how the polarization of foregrounds will compromise the relatively weak polarizationsignal of primary anisotropies, especially at low $\ell$ where much of the improvement comes from. |
See also related work on forecasting errors by Alageuijo and Llobson (1997). | See also related work on forecasting errors by Mageuijo and Hobson (1997). |
We have mentioned that a non-uniform Z"(s|prior) is particularly useful for parameters such as Yg, and o. where other experiments restrict their values to much higher accuracy than can be achieved from CMD. experiments alone. | We have mentioned that a non-uniform $P({\bf s}\vert {\rm prior})$ is particularly useful for parameters such as $Y_{He}$ and $T_0$, where other experiments restrict their values to much higher accuracy than can be achieved from CMB experiments alone. |
For other parameters. oy and war. it may be that the distribution derived. from a CAIB experiment is much narrower than anv reasonable prior.distribution. in which case we gain little by including prior information. | For other parameters, $\omega_b$ and $\omega_{cdm}$, it may be that the distribution derived from a CMB experiment is much narrower than any reasonable priordistribution, in which case we gain little by including prior information. |
There are also intermediate cases where prior information can help break degeneracies between parameters estimated from.the CMDalone. | There are also intermediate cases where prior information can help break degeneracies between parameters estimated fromthe CMBalone. |
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