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The S'T mass function produces the largest fraction of high redshift clusters while Press-Schechter produces the least (a factor of 4 fewer than ST for z1). | The ST mass function produces the largest fraction of high redshift clusters while Press-Schechter produces the least (a factor of 4 fewer than ST for $z > 1$ ). |
Changing the mass function also produces slight variations in the total number of clusters seen. | Changing the mass function also produces slight variations in the total number of clusters seen. |
As we will show re[sec:iconstraints)). an accurate redshift. clistribution will be crucial in. determiningὃν cosmologicale parameters and details of the gas physics. in particular for the experiment. | As we will show \\ref{sec:constraints}) ), an accurate redshift distribution will be crucial in determining cosmological parameters and details of the gas physics, in particular for the experiment. |
For that experiment. number counts alone do not contain sullicient information to be able to provide interesting constraints. | For that experiment, number counts alone do not contain sufficient information to be able to provide interesting constraints. |
The dillerences between models with these three forms [or P(y) as shown in Fig. | The differences between models with these three forms for $P(y)$ as shown in Fig. |
4. should be compared to the expected statistical errors from individual experiments. | \ref{fig:Pydep} should be compared to the expected statistical errors from individual experiments. |
In the case of the statistical errors (which should be close to Poissonianσου re[fsecuescosmo)) will be significantly larger. than the uncertainties due to the form of P(y). | In the case of the statistical errors (which should be close to Poissonian—see \\ref{sec:rescosmo}) ) will be significantly larger than the uncertainties due to the form of $P(y)$. |
For the experiment however. the statistical errors will be. much smaller. making the svstematic cilferences due to the form of PCy) the dominant. uncertainty. | For the experiment however, the statistical errors will be much smaller, making the systematic differences due to the form of $P(y)$ the dominant uncertainty. |
We next examine the οσο of varving the prescription or cluster. formation redshifts. | We next examine the effect of varying the prescription for cluster formation redshifts. |
“Phe formation redshift determines the clusters gas density. ancl pressure. which directly impact the detectability. of the cluster. | The formation redshift determines the cluster's gas density and pressure, which directly impact the detectability of the cluster. |
This is a potentially significant. οσο on the number counts of SZ clusters. | This is a potentially significant effect on the number counts of SZ clusters. |
At present it is not fully clear how a cluster’s ormation and merger histories influence the thermodynamic woperties of the eas it contains. | At present it is not fully clear how a cluster's formation and merger histories influence the thermodynamic properties of the gas it contains. |
While numerical simulations may soon clarifv this issue we have for now explored three possibilities for the formation redshifts of clusters (which we assume fix the thermodynamic properties of the gas as described in re[secumocdel)). | While numerical simulations may soon clarify this issue we have for now explored three possibilities for the formation redshifts of clusters (which we assume fix the thermodynamic properties of the gas as described in \\ref{sec:model}) ). |
Comparing the predictions. of the three formation redshift’ formulas in Fig. 5. | Comparing the predictions of the three formation redshift formulas in Fig. \ref{fig:form}, |
we find that the functions proposed hy Lacey&Cole(1993). and. Sasaki(1994) vieldl quite similar results. | we find that the functions proposed by \cite{lc93} and \cite{sasaki94} yield quite similar results. |
Llowever. the lower limit of τε=2, Vields substantially smaller number counts. | However, the lower limit of $z_{\rm f} = z_{\rm o}$ yields substantially smaller number counts. |
Interestingly. all three mioclels produce remarkably similar normalised redshift distributions. | Interestingly, all three models produce remarkably similar normalised redshift distributions. |
It is worth noting that the differences in number counts for our three formation-redshift distributions are greater than the dillerences resulting from the various forms for P(y) discussed in relsce:resAIP.. and at mJy become comparable to. the random errors expected in the experiment. | It is worth noting that the differences in number counts for our three formation-redshift distributions are greater than the differences resulting from the various forms for $P(y)$ discussed in \\ref{sec:resMF}, and at 1mJy become comparable to the random errors expected in the experiment. |
These dillerences highlight the need lor a better understanding of how a cluster's gas properties are determined in order to make sulliciently accurate calculations of the abundance of SZ clusters. and to allow cosmological ancl gas-cistribution parameters to be determined without systematic biases. | These differences highlight the need for a better understanding of how a cluster's gas properties are determined in order to make sufficiently accurate calculations of the abundance of SZ clusters, and to allow cosmological and gas-distribution parameters to be determined without systematic biases. |
The form of the assumed. gas density. ancl temperature profiles will allect the calculation of the SZ tux here we examine three temperature profiles. ( | The form of the assumed gas density and temperature profiles will affect the calculation of the SZ flux — here we examine three temperature profiles. ( |
Note that | Note that |
datai πα. the1 error values1 are now retative). | data (assuming the error values are now ). |
ΤΙlix restiltant slope αςis unchauged. although Taupe Is decreased frou 0.17 to 0.07. | The resultant slope is unchanged, although $\sigma_{slope}$ is decreased from 0.17 to 0.07. |
This znaller σον may be strouger evidence for an increasing abundance eradient. but only We arc OVCH “how on Ni mn Ma (as the “nating tlας awayout 10 nnbest-fit lac may nucimdiceate). | This smaller $\sigma_{slope}$ may be stronger evidence for an increasing abundance gradient, but only if we are overestimating the errors on the R23 oxygen abundances (as the small scatter about the best-fit line may indicate). |
ances We considersual it attequite uulikely.t based on arguments by? and others. that the errors iu the stroue line adances less than 0.2 - 3 dex. | We consider it quite unlikely, based on arguments by \cite{ercolano07} and others, that the errors in the strong line abundances are significantly less than 0.2 - 0.3 dex. |
| are viclel significantly baryonie lass for FigureNGC plots(Alled effective versus totalits oxvBen abundauce is 2915the weighted circle),mean of assunungonr five totaliicasured ΠΠ resious. and shows the cupirical relation of | Figure 4 plots effective yield versus total baryonic mass for NGC 2915 (filled circle), assuming its total oxygen abundance is the weighted mean of our five measured HII regions, and shows the empirical relation of \cite{tremonti04}. |
The total effective vield is computed from the observed metallicity. Z. aud the galaxw total eas fraction (nof inchiding dark matter. 0.70 for NCC 2915). ji such that very=Zflu(g iy | The total effective yield is computed from the observed metallicity, $Z$, and the galaxy total gas fraction (not including dark matter, 0.70 for NGC 2915), $\mu$, such that $_{eff}=Z/$ $(\mu^{-1})$. |
Tu this case. 12 | Los (O/ID. an oxveen abundance by umuber. is converted to a metallicity by mass using the conversion factor 11.728. assundue that helium accounts for of the total eas mass (?).. | In this case, 12 + Log (O/H), an oxygen abundance by number, is converted to a metallicity by mass using the conversion factor 11.728, assuming that helium accounts for of the total gas mass \citep{lee03}. |
The logarithm of the effective vield of NGC 2915 i 2.516. eiven its crroraweighted imeau oxvecu abundance of 12 | Log (O/T) = uted7.97. | The logarithm of the effective yield of NGC 2915 is $-$ 2.516, given its error-weighted mean oxygen abundance of 12 + Log (O/H) = 7.97. |
Alone with total stellar and gaseous lasses prese in Table 1. NGC 2915 falls exactly where it is expected to fall on this plot. | Along with total stellar and gaseous masses presented in Table 1, NGC 2915 falls exactly where it is expected to fall on this plot. |
The relation of ? seen in Figure | as a solid line is attributed to metal loss via ealactic Winds. | The relation of \cite{tremonti04} seen in Figure 4 as a solid line is attributed to metal loss via galactic winds. |
NGC 2915 lies near its turnover. where winds are thought to start plaviug au important role in blowing out metals. | NGC 2915 lies near its turnover, where winds are thought to start playing an important role in blowing out metals. |
Iu this coutext. NGC 2915 behaves just like other galaxies of its salue lass. gas fraction.n. aud metallicity. | In this context, NGC 2915 behaves just like other galaxies of its same mass, gas fraction, and metallicity. |
. Considering the effective vields of the immer aud outer coniponcuts.sate of NGC NCIC!290152915 «γηseparatΕκ:lv. however,vero Sivesοὔπτρς o:a very different result. | Considering the effective yields of the inner and outer components of NGC 2915 separately, however, gives a very different result. |
We present two simple cases in wlich we calculate the ONVeCu abundance meswe mugAnaat expect to measure in. the outer UUeascous disk. of NGC 2915 basedASC SOLOINsolely Ol its stellar aie1 nutra‘al Sasens concut. | We present two simple cases in which we calculate the oxygen abundance we might expect to measure in the outer gaseous disk of NGC 2915 based solely on its stellar and neutral gas content. |
IIu the first case. we estimateIts the effective vield in the outer-disk of NGC 2915. aud compare it to the ? relation between effective vield aud total mass. | In the first case, we estimate the effective yield in the outer-disk of NGC 2915, and compare it to the \cite{tremonti04} relation between effective yield and total mass. |
Wo assume the 1hand light: best traces the stellar mass. and colette a radius centered ou the galaxy that contaius of the stellar mass oof the LEbaud DIuinositv). rog. to be ~ LL. | We assume the I-band light best traces the stellar mass, and calculate a radius centered on the galaxy that contains of the stellar mass of the I-band luminosity), $_{90}$, to be $\sim44$ ". |
We then casure an approxiuate radius which contains all of the stellar light. including the outer ΠΠ regious aud faint exteuded UV enmüssion. ry. to be ~ 1507. | We then measure an approximate radius which contains all of the stellar light, including the outer HII regions and faint extended UV emission, $_{tot}$, to be $\sim150$ ". |
We note that ACS images presented in ? πο several elobular clusters at radii simular to the outer WL regions. | We note that ACS images presented in \cite{meurer03} show several globular clusters at radii similar to the outer HII regions. |
This population of older stars is included in our Thane luicasmvements, | This population of older stars is included in our I-band measurements. |
From the IIT dataati presented in ?..an we find thati roy contaius oof the total III mass. aud the aumilar region between rog aud ty,¢ contains oof the total TM amass. | From the HI data presented in \cite{meurer96}, we find that $_{90}$ contains of the total HI mass, and the annular region between $_{90}$ and $_{tot}$ contains of the total HI mass. |
Within this auuulu recion. we measure an effective vield (log τῇ) of -1.91 and log | Within this annular region, we measure an effective yield (log $_{eff}$ ) of -1.94 and log |
correlation function between any two points on the sky as αγ left(tiNg(k Me | correlation function between any two points on the sky as = ) ) k - k' ) ). |
As the fundamental domain has a particular orientation on the sky. the correlation is not simply a function of the angular separation between fi and f as it is in the infinite Case. | As the fundamental domain has a particular orientation on the sky, the correlation is not simply a function of the angular separation between $\hn$ and $\hnp$ as it is in the infinite case. |
From this expression. €; can be determined. using the orthogonality relations of the Legendre polynomials: i0 fam. AVL (o) where j/=AΠΠ. | From this expression, $C_\ell$ can be determined using the orthogonality relations of the Legendre polynomials: = d d, ) ), where $\mu = \hn \cdot \hnp$. |
Expandingqp. the exponential. and Legendre polynomials in terms of spherical harmonics. this becomes (2€ |td) κι | Expanding the exponential and Legendre polynomials in terms of spherical harmonics, this becomes (2 + ) ). |
As ¢byd.,) and the spectrum of eigenvalues are known for all six possible [lat topologies. we can use this expression to compute C; for each of the possible cases. | As $\left\langle \hat \Phi_{\vec k} \hat \Phi_{\vec k'}^* \right\rangle$ and the spectrum of eigenvalues are known for all six possible flat topologies, we can use this expression to compute $C_\ell$ for each of the possible cases. |
The simplest topology is the hypertorus. which is built out of a parallelepiped by identifving (rey.z)(Grphoy|bospe). | The simplest topology is the hypertorus, which is built out of a parallelepiped by identifying $(x,y,z)\rightarrow (x+h,y+b,z+c)$. |
The identification leads to a restriction of the cigenvaluc spectrum. &=2z(jfh.wíb.nfc) with the joes running over all integers. | The identification leads to a restriction of the eigenvalue spectrum, $\vec k=2\pi(j / h, w/b,n/c)$ with the $j,w,n$ running over all integers. |
With this restriction. the Cis become ANY where P(A)x1 for α flat power spectrum. | With this restriction, the $C_\ell$ s become k)^2, where ${\cal P}(k) \propto 1$ for a flat power spectrum. |
This is in agreement with (Stevens.Scott. | This is in agreement with \cite{sss}. |
&Silk1993).. In reffig:torus— we plot this expression for three cilferen topology scales for a flat. power spectrum. normalized. by what we would expect for a universe with no topology: that is €Vlg.(2x)ai32f((6|1) where V. is. the volume o the fundamental domain. | In \\ref{fig:torus} we plot this expression for three different topology scales for a flat power spectrum normalized by what we would expect for a universe with no topology; that is $C_\ell \times
{\rm V}^{-1} (2 \pi)^2 \ell (\ell+1)$, where $V$ is the volume of the fundamental domain. |
The normalization is aolute. such that an infinite universe would. be represented. by normalization ((f|LC)=1. Cosmic variance is estimatcc as C72nn(2411. as for an infinite universe. although the true variances for any given topology would x» slightly dilferent. | The normalization is absolute, such that an infinite universe would be represented by normalization $\ell (\ell+1) C_\ell = 1.$ Cosmic variance is estimated as $C_\ell \sqrt{2/({2 \ell +1})}$, as for an infinite universe, although the true variances for any given topology would be slightly different. |
There are a number things to note here. | There are a number things to note here. |
In the upper panel we see that the low £ modes are damped. with suppression becoming more severe as the topology scale ecreases. | In the upper panel we see that the low $\ell$ modes are damped, with the suppression becoming more severe as the topology scale decreases. |
From the cliseretization of the wave vector. A. it is clear that there is a minimum eigenvalue corresponding to the longest wavelength that can fit inside the fundamental domain. | From the discretization of the wave vector, $\vec k$, it is clear that there is a minimum eigenvalue corresponding to the longest wavelength that can fit inside the fundamental domain. |
This maximum wavelength can be associated with an angular scale above which we do not expect to find fluctuations. | This maximum wavelength can be associated with an angular scale above which we do not expect to find fluctuations. |
As the association between real space and ngular perturbations causes some averaging over & moces. 16 camping is smeared over a range of£ values. | As the association between real space and angular perturbations causes some averaging over $k$ modes, the damping is smeared over a range of $\ell$ values. |
In addition to the damping at low f£. a finite topology also causes jags at higher { values. extending to values above (=60 for the torus of size Οὐδη. | In addition to the damping at low $\ell$, a finite topology also causes jags at higher $\ell$ values, extending to values above $\ell = 60$ for the torus of size $.66 \Delta \eta$. |
The jagev features not only suppress many of he ügher Cis but actually cause at selected { values. | The jaggy features not only suppress many of the higher $C_\ell$ s but actually cause at selected $\ell$ values. |
This ringing in the C's can be understood as caused by the presence of a discrete set of harmonies of the fundamental domain in the matter power spectrum. | This ringing in the $C_\ell$ s can be understood as caused by the presence of a discrete set of harmonics of the fundamental domain in the matter power spectrum. |
The discretization not only draws. power away from values that are disallowed. but enhances power at certain typical angular scales. | The discretization not only draws power away from values that are disallowed, but enhances power at certain typical angular scales. |
Another way to understand his clleet is to consider the presence of multiple copies of he same point. | Another way to understand this effect is to consider the presence of multiple copies of the same point. |
One can imagine that given a topology scale. here are certain angles at which multiple images5 tend to all. while at other anglesDo such correlations are cisallowed | One can imagine that given a topology scale, there are certain angles at which multiple images tend to fall, while at other angles such correlations are disallowed |
the sauyple of 19 objects. | the sample of 49 objects. |
Again. we remove all known BAL QSOs from the sauuple. | Again, we remove all known BAL QSOs from the sample. |
With this sample. due to selection effects. it is not possible to achieve similar-sized subsamples with similar redshift distibutious by selecting au appropriate value at which to divide the sample. | With this sample, due to selection effects, it is not possible to achieve similar-sized subsamples with similar redshift distributions by selecting an appropriate value at which to divide the sample. |
Similar redshift distributions ave obtained for a,=1.3. but then the NB subsample contains oulv about a dozen QSOs. compared to 37 in the ΝΕ subsample. | Similar redshift distributions are obtained for $\aox=1.3$, but then the XB subsample contains only about a dozen QSOs, compared to 37 in the XF subsample. |
Similar subsample sizes ire obtained for a,,=1.L for which. however. the requirement of a detection in the NB sample results in a lower mean redshift. due mostly to extra QSOs between 0.1«ccoc0.2. | Similar subsample sizes are obtained for $\aox=1.4$, for which, however, the requirement of a detection in the XB sample results in a lower mean redshift, due mostly to extra QSOs between $0.1<z<0.2$. |
Since QSO spectral evolution between their mean redshitts (0.31 and 0.57. respectively) is negligible. we cluphasize the subsamples split at a,,=1.1. which is in any case closer to the dividing value for the LBQS subsamples. | Since QSO spectral evolution between their mean redshifts (0.34 and 0.57, respectively) is negligible, we emphasize the subsamples split at $\aox=1.4$, which is in any case closer to the dividing value for the LBQS subsamples. |
However. we also check the subsamples split at o,=1.3. to insure that evolution or Iuuinositv effects do not significantly bias our results. | However, we also check the subsamples split at $\aox=1.3$, to insure that evolution or luminosity effects do not significantly bias our results. |
We frst deredshift cach iudividual spectra by dividing he linear dispersion cocfiicicuts (imütial wavelength aud wavelength per pixel) bv (1|:). | We first deredshift each individual spectrum by dividing the linear dispersion coefficients (initial wavelength and wavelength per pixel) by $(1+z)$. |
An estimate of the restframe coutinmun flux f.; at a chosen uormalization »oiut is then derived. by fitting a linear continu between wo continmun bands. | An estimate of the restframe continuum flux $f_{c,l}$ at a chosen normalization point is then derived, by fitting a linear continuum between two continuum bands. |
The width and ceuter of these xuids. typically chosen to straddle an important enission ine. are listed for cach waveleneth region in Table 2. | The width and center of these bands, typically chosen to straddle an important emission line, are listed for each wavelength region in Table 2. |
We hen normalize the eutire spectrum via division by f... | We then normalize the entire spectrum via division by $f_{c,l}$. |
Each nonualized spectrum is first rebiuned to the dispersion of the composite spectrum. couserving flux via interpolation. | Each normalized spectrum is first rebinned to the dispersion of the composite spectrum, conserving flux via interpolation. |
We choose bius. similar to the majority of the individual LDOS spectra. | We choose bins, similar to the majority of the individual LBQS spectra. |
The spectra are binned toL185À.. as in the original short wavoleugth prime (SWP) spectra. | The spectra are binned to, as in the original short wavelength prime (SWP) spectra. |
Each normalized. rebinucd spectrum is then stored as a vector in a 2-D array. | Each normalized, rebinned spectrum is then stored as a vector in a 2-D array. |
Finally. for cach pixel in the completed array. the umber of spectra NV with fluxes in that restframe wavelength biu is tallied. | Finally, for each pixel in the completed array, the number of spectra $N$ with fluxes in that restframe wavelength bin is tallied. |
If N&3. the median aud the mean of all the fiux values in the bin are computed and stored in the final 1-D conrposite spectra. as is No in the final histogram array. | If $N\geq 3$, the median and the mean of all the flux values in the bin are computed and stored in the final 1-D composite spectra, as is $N$ in the final histogram array. |
A similar procedure is performed for each bin. uutil the spectra are completed. | A similar procedure is performed for each bin, until the spectra are completed. |
The strong sky liue at ADSTTA.. sometimes poorly subtracted. was omitted from the LBOQS composite. | The strong sky line at $\lambda
5577$, sometimes poorly subtracted, was omitted from the LBQS composite. |
Geocoronal lines were similarly excluded when building the coniposites. | Geocoronal lines were similarly excluded when building the composites. |
We chose four separate contimuni poiuts to generate LBOS composite spectra in four wavelength regions (Table 2). | We chose four separate continuum points to generate LBQS composite spectra in four wavelength regions (Table 2). |
Composites normalized at different coutimuu points may have different histograms. since they require the restframe contimmuu bands to be present in all contributing observed frame spectra. | Composites normalized at different continuum points may have different histograms, since they require the restframe continuum bands to be present in all contributing observed frame spectra. |
Histogrius of the munber of QSOs contributing to cach LDBOS composite are shown in Figure 1. | Histograms of the number of QSOs contributing to each LBQS composite are shown in Figure 1. |
The conrposites were all normalized at the waveleneth ofCIV.. and the histograms for those spectra are slow iu Figure 2. | The composites were all normalized at the wavelength of, and the histograms for those spectra are shown in Figure 2. |
The median composites appear to be smoother than the ameans. since they are less affected by spikes aud low S/N features in the individual spectra. | The median composites appear to be smoother than the means, since they are less affected by spikes and low S/N features in the individual spectra. |
As a result. the \? values of model fits are also lower for mediau composites. | As a result, the $\chi^2$ values of model fits are also lower for median composites. |
However. since the median omits QSOs with more extreme spectral properties. we prefer to analyze the average composites. | However, since the median omits QSOs with more extreme spectral properties, we prefer to analyze the average composites. |
We check the significance of every stroue difference between NB and NF average composites by imieasurement of the median composites. to determine whether outlicrs dominate the feature in question. | We check the significance of every strong difference between XB and XF average composites by measurement of the median composites, to determine whether outliers dominate the feature in question. |
No analysis of contiuuuni slope is preseuted for the LBOS spectra. since these observations are not fully spectrophotometric. | No analysis of continuum slope is presented for the LBQS spectra, since these observations are not fully spectrophotometric. |
Using the IRAF task SPECFIT (IXxiss 1991). we fit simple euiipirical models to the LBOS composite spectra in four wavelength regions. defined iu Table 2. | Using the IRAF task SPECFIT (Kriss 1994), we fit simple empirical models to the LBQS composite spectra in four wavelength regions, defined in Table 2. |
Iu cach region. we beein by fitting a powerlaw (hereafter. PL) continua using only comparatively line-free regions (listed for each region below). | In each region, we begin by fitting a powerlaw (hereafter, PL) continuum using only comparatively line-free regions (listed for each region below). |
The resulting continuum fit parameters are slope à and intercept fFiooo. such that flux fy=οἹππς | The resulting continuum fit parameters are slope $\alpha$ and intercept $f_{1000}$, such that flux $\flam = f_{1000}(\frac{\lambda}{1000})^{\alpha}$. |
Since the LBOQS spectra are not spectrophotometric. the continu fit results should be used oulv to derive exact composite equivalent widths frou (normalized) line fixes if desired. | Since the LBQS spectra are not spectrophotometric, the continuum fit results should be used only to derive exact composite equivalent widths from (normalized) line fluxes if desired. |
The intercept value reflects the arbitrary normalization of individual spectra to unitv at the chosen waveleneth À,, shown in Table 2. | The intercept value reflects the arbitrary normalization of individual spectra to unity at the chosen wavelength $\lambda_n$ shown in Table 2. |
The cussion line compoueuts are assed to be Gaussian and svinmetric (skew fixed at unity). so output from the fits includes fux. centroid. aud FWIAL for cach line. | The emission line components are assumed to be Gaussian and symmetric (skew fixed at unity), so output from the fits includes flux, centroid, and FWHM for each line. |
Results from these fits are shown in Table 3 and Figure 3. | Results from these fits are shown in Table 3 and Figure 3. |
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