# How Stochastic is the Relative Bias Between Galaxy Types?

###### Abstract

Examining the nature of the relative clustering of different galaxy types can help tell us how galaxies formed. To measure this relative clustering, I perform a joint counts-in-cells analysis of galaxies of different spectral types in the Las Campanas Redshift Survey (LCRS). I develop a maximum-likelihood technique to fit for the relationship between the density fields of early- and late-type galaxies. This technique can directly measure nonlinearity and stochasticity in the biasing relation. At high significance, a small amount of stochasticity is measured, corresponding to a correlation coefficient on scales corresponding to 15 Mpc spheres. A large proportion of this signal appears to derive from errors in the selection function, and a more realistic estimate finds . These selection function errors probably account for the large stochasticity measured by Tegmark & Bromley (1999), and may have affected measurements of very large-scale structure in the LCRS. Analysis of the data and of mock catalogs shows that the peculiar geometry, variable flux limits, and central surface-brightness selection effects of the LCRS do not seem to cause the effect.

^{†}

^{†}slugcomment: Submitted to ApJ

## 1 Motivation

Galaxies of different morphologies have different spatial distributions, as first noted by Hubble (1936). Early-type galaxies, such as ellipticals and S0s, are highly clustered and account for 90% of galaxies in the cores of rich clusters; late-type galaxies, such as spirals and irregulars, are less clustered and make up 70% of galaxies in the field (Dressler 1980; Postman & Geller 1984; Whitmore, Gilmore, & Jones 1993). A general way of expressing the relationship between the density fields of galaxies of different types on any scale is with the joint probability distribution ; that is, the probability at any location of finding an overdensity of early-type galaxies and an overdensity of late-type galaxies. This quantity is analogous to the joint probability distribution of galaxy and mass density introduced by Dekel & Lahav (1999).

The traditional method of measuring the properties of has been to compare the amplitude of the fluctuations in each density field, using the correlation functions or the power spectra. By these measures, the level of fluctuations in ellipticals is stronger than that of spirals by a factor of 1.3–1.5 (Davis & Geller 1976; Giovanelli, Haynes, & Chincarini 1986; Santiago & Strauss 1992; Loveday et al. 1996; Hermit et al. 1996; Guzzo et al. 1997). These relative clustering properties are successfully reproduced by current models of galaxy formation. For example, Blanton et al. (1999) examined hydrodynamical simulations and identified galaxies as dense, rapidly cooling clumps of gas. Older galaxies, which correspond to early-types, turned out to be clustered more strongly than younger galaxies, which correspond to late-types. The relative bias factor, , where , is approximately 1.5 between these populations. Semi-analytic models, which follow halos in collisionless -body simulations and use simple models for star-formation and feedback inside each halo and for the effect of halo mergers, find similar results (Somerville et al. 1999).

However, Blanton et al. (1999) also found that there was considerable scatter between the two density fields; that is, that there was not a one-to-one relationship between the number of old galaxies in a region to the number of young galaxies. A measure of this scatter is the correlation coefficient between the early-type overdensity field and the late-type overdensity field . In the simulations, –. On the other hand, the semi-analytic models of Somerville et al. (1999) find that the correlation coefficient ; that is, they find very little scatter. The essential difference between the predictions of this model and that of the hydrodynamic model is the effect of the temperature history of the gas in the hydrodynamic simulations and its relationship with large-scale structure. Thus, one can use the correlation coefficient between different galaxy types to distinguish between these models of galaxy formation.

Measuring this scatter requires a probe of which differs from the traditional statistics mentioned above. For example, two completely unrelated density fields () can have the same correlation function. To detect the scatter, one must compare the density fields point by point, not just compare the overall levels of the fluctuations. A direct approach to constraining the properties of is to measure the related joint probability distribution of finding early-type and late-type galaxies in a single cell of size . After all, this latter probability is simply convolved with Poisson distributions. If one notes that

(1) |

then one can write

(3) | |||||

where and are the average number of galaxies of each type expected in a cell of a given volume (and given selection criteria). Naturally, one can integrate Equation (3) over to obtain:

(4) |

the probability distribution of counts of early-type galaxies. As I show below, one can use Equation 3 to devise a maximum likelihood method to fit for , and Equation 4 to fit for .

Equation (3) provides a direct probe of the relationship between galaxy density fields , including its nonlinearity and scatter. Consider for contrast the work of Benoist et al. (1999), who infer nonlinearity in the relative bias of galaxies of different luminosities in the Southern Sky Redshift Survey from the scale dependence of the higher-order moments of the density fields. Using the same data, one could instead compare the observed to models and detect nonlinearity more directly. Furthermore, the joint distribution contains more information than a comparison of the moments of each density field. While moments of the density field yield averaged information about the fluctuations, yields a point-by-point comparison of two density fields, which can be much more powerful (Santiago & Strauss 1992). For instance, one can use this comparison to determine whether the effects detected by Benoist et al. (1999) are actually due to nonlinearity (as they propose), or perhaps due properties of the scatter in the relationship between low luminosity and high luminosity galaxies.

In this paper, I perform an maximum-likelihood analysis of this joint distribution for different spectral types of galaxies in the Las Campanas Redshift Survey (LCRS), using cells with volumes approximately equal to that of cubes 25 Mpc on a side. A similar analysis has been performed on the LCRS by Tegmark & Bromley (1999; hereafter TB99), to which I will compare my results throughout. Essentially, their method calculates the second moments of , namely , the variance of the density field of late-type galaxies, , the variance of early-type galaxies, and , the correlation coefficient between the two fields, which is unity if the fields are perfectly correlated and zero if the fields are completely uncorrelated. As I will show below, calculating second moments is probably not sufficient on the scales which TB99 probe (– Mpc), because it does not correctly account for the fact that density fields cannot be negative. For this reason, the resulting may overestimate the degree of scatter in the relationship between the two fields. On larger scales where , these differences would of course be much reduced. Furthermore, the moments method also yields no information on how nonlinear the relationship between the density fields of the two galaxy types is, which the maximum likelihood method described here will. Finally, I have detected important effects concerning the galaxy selection function which affect the results of this analysis and have consequences for the interpretation of TB99 and other measurement of large-scale structure in the LCRS.

This paper is organized as follows. In Section 2, I describe the details of the LCRS. In Section 3, I describe the method used to calculate the selection function for the survey. In Section 4, I describe in detail the maximum likelihood analysis of the counts-in-cells. In Section 5, I present the results of fitting for the relationship between the different galaxy types and demonstrate the presence of systematic errors in the selection of galaxies in this survey. In Section 6, I describe the results of an analysis of mock catalogs, in order to quantify a number of possible statistical and systematic effects as well as to evaluate the importance of cosmic variance. I conclude in Section 7.

## 2 Galaxies in the Las Campanas Redshift Survey

The LCRS (Shectman et al. 1996) consists of around 25,000 galaxy redshifts with a median of , covering an area of about 700 deg on the sky. Three long slices () were surveyed in the North Galactic Cap, and three in the South Galactic cap. Within each hemisphere, the slices had the same right-ascension limits but were separated by several degrees in the declination direction. -band photometry was obtained at the Las Campanas Swope 1m telescope using three different CCDs; spectra were taken at the Las Campanas Du Pont 2.5m telescope, first with a 50-fiber MOS and later with a 112-fiber MOS. The galaxies in the 50-fiber MOS fields were selected between ; the galaxies in the 112-fiber MOS fields were selected between . In addition, a magnitude-dependent central surface-brightness cut was applied in order to avoid putting fibers onto galaxies unlikely to yield useful spectra. This cut takes the form:

(5) |

where is the faint magnitude limit, is a central aperture magnitude consisting of the flux within a two pixel radius of the center of the image, and is 18.85 for the 112-fiber fields and 18.15 for the 50-fiber fields. Since each of the CCDs had a different pixel size, the size of the aperture with which is calculated varies within the survey between about 3 and 4 in diameter. The central magnitude cut excludes about 12% of the detected galaxies. As in all redshift surveys, this surface brightness cut can affect the relationship between the luminosity function and the selection function, since the selection is not purely based on apparent magnitude. Below, I test the dependence of my results on this cut.

Bromley et al. (1998) have used a spectral classification scheme to divide the LCRS galaxies into six “clans.” Their method performs a singular-value decomposition (SVD) on the set of galaxy spectra (converted to rest wavelengths) to obtain an orthogonal set of galaxy “eigenspectra.” They found that the galaxy spectra form a well-defined one-dimensional locus when projected onto the two-dimensional plane defined by the two most significant eigenspectra. The “clans” are defined by each galaxy’s position along this locus. The spectra of the “late-type” clans more closely resemble the spectra of emission-line galaxies with young stellar populations, while the spectra of “early-type” clans have more prominent absorption features. Clans are labeled from 1 to 6, in order of increasing “lateness” of the spectra. For my purposes, I will split the galaxies into just two groups: an early-type group consisting of clans 1 and 2 and a late-type group consisting of clans 3 through 6. I place absolute magnitude limits on the early-type group of and on the late-type group of . Outside of these limits there are only a handful of galaxies and it is risky to determine the luminosity function and to calculate the selection function there. This procedure yields about 10,000 galaxies in each group. I show the spatial distribution of each type of galaxy in Figures 1a and 1b.

The geometry of the LCRS complicates an attempt to perform a counts-in-cells analysis on it. To do so, I create 14 redshift shells, each with an equal volume; thus, the shells at higher redshift have a shorter radial extent. Figure 2 shows the boundaries of these shells. In the angular dimension, I divide the survey into cells which are 3 MOS fields on each side. In the right ascension direction, the fields are adjacent; in the declination direction, the fields from the three slices in each Galactic hemisphere are combined. The radial spokes in Figure 2 are representative of the division in the right ascension direction, although the actual cells are somewhat more complicated, because the MOS fields are not all perfectly aligned in right ascension. This procedure produces 518 cells total, each with a volume equivalent to a 15 Mpc radius sphere, of about cubical dimensions at (except for the gaps in the declination direction). The average number of galaxies in each cell of each type is about 20, meaning that the average contribution of Poisson noise to the variance is about , though the actual contribution varies considerably with radius due to the selection function. A reasonable variance of the underlying density field at these scales in standard cosmological models is about , meaning the Poisson contribution to the variance is only about 20%.

## 3 Selection Function

The selection function for a flux-limited survey is the fraction of galaxies in a given absolute magnitude range which are within the flux limits at a given redshift. If one considers galaxies with luminosities in the range between and , and take the luminosity function to be normalized in that range, one can write the selection function:

(6) |

where and are the minimum and maximum luminosities visible at redshift , given the flux limits of the field under consideration, and is the apparent magnitude corresponding to and . and contain information on the incompleteness of the survey. is the sampled fraction of galaxies in the current field; that is, the fraction of galaxies within the stated flux limits whose redshifts were obtained, for each field. Galaxies are missed for a number of reasons: the limited number of fibers, the central magnitude cut, the fact that fibers cannot be placed closer than 55”, and the failure to determine redshifts from spectra. One must account for the fact that these effects are not distributed evenly in apparent magnitude (in practice, due mostly to the magnitude-dependence of redshift failures). To do so, I adjust the probability of observing a galaxy by a factor which is the completeness fraction at each apparent magnitude (normalized to unity, since already accounts for the total number of missing galaxies).

Thus, one’s task is to calculate the luminosity function given the survey’s various selection effects. Although Bromley et al. (1998) and Lin et al. (1996) have already published luminosity functions for different galaxy types in the LCRS, for a number of reasons I was motivated to reexamine the determination of the luminosity and selection functions. In particular, some features of the selection functions seemed suspicious; in fact, this suspicious behavior remains in my analysis, and I will describe it in detail later. I calculated the luminosity function using the standard iterative, nonparametric maximum likelihood technique described in detail in Efstathiou, Ellis, & Peterson (1988). Their technique is based on maximizing the probability of having observed each galaxy of luminosity at its given redshift . If the galaxy luminosity function is , a galaxy at redshift and having luminosity is observed with a probability per unit redshift and luminosity of:

(7) |

Here, and are as defined above, is the density of galaxies at redshift , and and vary depending on what MOS field the galaxy is in, since each field has different flux limits. In order to relate apparent magnitudes to absolute magnitudes and luminosities , I assume a flat universe with , which yields a distance modulus:

(8) |

where the comoving distance is

(9) |

and the -correction is of the form:

(10) |

This -correction is approximately equivalent to that which is appropriate for Sbc galaxies (Frei & Gunn 1994). Since the entire analysis is independent of the value of , I will for simplicity assume the value of 100 km/s/Mpc.

Given the joint probability in Equation (7), the conditional probability of observing a galaxy of luminosity given its redshift is then:

(11) |

where the limits on the integral are determined by the apparent magnitude limits in each field, as implied by Equation (7). Note that this estimator is density-independent, as all reference to a density field or the angularly varying sampling fraction drops out. This approach differs somewhat from that of Lin et al. (1996), which weights the densest regions more heavily.

In the Appendix, I describe in detail the method for maximizing this probability and for calculating the normalization of the luminosity function . Using the derived luminosity function and its normalization, one can calculate the selection function and, from that, quantities such as the expected distribution of number with redshift or the expected counts in cells. I perform this fit separately for each combination of MOS field type and Galactic hemisphere, and for the early- and late-type galaxies separately. I will give results not in terms of itself, but in terms of the luminosity function expressed in logarithmic intervals of luminosity and normalized to the average density:

(12) |

## 4 Fitting a Counts-in-Cells Distribution

Here I describe a maximum likelihood method for constraining the relative clustering properties of early and late-type galaxies, assuming models for the density distribution and the relative bias . I will assume is distributed as a log-normal. For , I describe a number of deterministic models as well as a model which includes scatter.

### 4.1 Defining the Likelihood

Let us assume that one has divided a galaxy redshift survey into cells and counted the numbers of early- and late-type galaxies, denoted and , in each cell . (I describe in Section 2 how I do so for the LCRS). Given the selection function of the survey, which may depend on angle as well as redshift, I define the expected count in cell of early-type galaxies:

(13) |

where the integral is over the volume of cell , and define similarly. Then, given a probability distribution characterized by a set of parameters , I define the likelihood for that cell as

(14) |

where the quantitity on the right-hand side is determined by Equation (3). I have implicitly assumed that all sets of have equal Bayesian prior probability. I then minimize the quantity

(15) |

by varying the parameters in order to find the best fit.

In practice, it is time-consuming to fit simultaneously for the parameters of both and in Equation (1). Thus, I first fit for the parameters of the density distribution ; I do so with an equation analogous to Equation (14), using the probability given in Equation (4) in place of . Then I fix the parameters of and use Equation (14) to fit separately for those of . Experiments show that the difference between this approach and fitting for all the parameters simultaneously is negligible compared to the error bars.

Once one has found the maximum likelihood fit to the parameters, one would like to calculate the errors associated with them. I use three methods of calculating error bars. First, I perform Monte Carlo bootstrap estimates of the errors. Second, I again make a Monte Carlo estimate, only now creating realizations based on the model to which I have fit. (This procedure also serves to show that the method is unbiased). Third, I simply look at the likelihood contour in order to estimate the 1 error contour, which it is in the limit that is a paraboloid. All of these methods agree within 10–20%, and the listed error bars in this paper are those determined by bootstrap. (For a comparison of the likelihood method and the bootstrap method, look at Figure 6 in Section 5).

I would also like to compare different models using likelihood ratio tests. For instance, one may want to fit a model with parameters and ask whether this model is significantly better than fitting a model with the single parameter . In this case, one calculates the “likelihood ratio” between the models,

(16) |

To evaluate the significance of this likelihood ratio, I create a large number of Monte Carlo realizations based on the single-parameter model and ask how often one sees a likelihood ratio as large as the measured purely by accident. I will make extensive use of this technique below.

A general concern about using a counts-in-cells analysis in a flux-limited survey (e.g., TB99, Efstathiou et al. 1990) is that the measured density distribution in the cells may be affected by the variation of the selection function over their extent. As a simple example, consider a cluster sitting at the near edge of one cell, and another, identical cluster sitting at the far edge of an identical cell. In each case, the true is the same, as is , but will be systematically smaller in the second cell, where the cluster is more distant. Given the peculiar geometry of the cells one is forced to construct in the LCRS, variation of the selection function in the angular direction also can contribute to this effect. Thus, power on scales smaller than that of the cell can contribute to the variance in the counts-in-cells. Correcting for these effects properly evidently requires assumptions about the clustering on scales smaller than a cell. In Section 6, I will show that this problem is not significant for this analysis by considering mock catalogs of the LCRS.

### 4.2 Models for the Probability Distribution Function

In this subsection I describe models for the density distribution function ; in the next two subsections, I will describe the models for the conditional probability , which contains the information on the “bias relation” between the two groups of galaxies. For a review of the many different ways to model the density distribution function , see Strauss & Willick (1995). I have tried just three: a Gaussian model, a first-order Edgeworth expansition, and a log-normal model. My main results do not depend on which choice I pick, so here I will only present results for the log-normal model, which provides the best fit and which is the most mathematically convenient. It can be written:

(17) |

where . Blanton (1999) gives more details on various fits to the density distribution function using this data.

### 4.3 Deterministic Bias Models

Here I describe models for the relationship between early- and late-type density fields with no scatter. “Deterministic” is not meant to refer to underlying physical principles but is simply meant to express the fact that knowing the density of ellipticals tells you with certainty the density of spirals, modulo Poisson noise. In this case, the joint density distribution can be expressed as

(18) |

where is the density distribution function of early-type galaxies and is the Dirac delta function. Under this assumption, one can describe the models by the biasing function .

The simplest model for is linear bias:

(19) |

The obvious problem with this model is that if , can become less than . In this case, I simply set . Another way of handling the problem is to use power-law bias:

(20) |

which always remains greater than . It turns out, as I will show below, that the power-law bias is in fact a poorer fit to the data than linear bias in this analysis of the LCRS.

At the cost of an extra parameter, the linear bias case can be extended trivially to quadratic bias:

(21) |

Another possibility I try is “broken” bias, which is piecewise linear with one slope in overdense regions and another in underdense regions:

(22) |

For each model, I require that , because it is meant to represent the overdensity of late-type galaxies. In practice, this requirement sets , which is therefore not treated as a free parameter in any of the above expressions.

### 4.4 Stochastic Bias Models

If variables other than the local density field are important in determining where galaxies form, it may be that the different formation processes of early-type and late-type galaxies cause scatter in the relationship between their density fields. Thus, I also examine other models which do incorporate scatter, rewriting Equation (18) by replacing the Dirac delta-function with a Gaussian of finite width:

(23) |

where and are chosen as above. parameterizes the degree of scatter in the relationship. Note that because of the lower limit , in general, although the differences will not be important for my purposes. I do shift the peak of the Gaussian slightly to guarantee that . In the case that is Gaussian, and and are small, reduces to a bivariate Gaussian distribution, and the standard correlation coefficient is related to by

(24) |

Below, I will use Equation (23) to fit linear bias with Gaussian scatter to the relationship between galaxy types.

## 5 Results from the LCRS

In this section, I determine the selection function of the LCRS, fit for the density distribution function, and fit for the bias. I then investigate important systematic effects in the results which indicate that the stochasticity measured from the full set of cells described in Section 2 may overestimate the true stochasticity.

### 5.1 Selection Function and Expected Counts

I first fit for the luminosity function for the early- and late-type galaxies, in each MOS field type (112-fiber or 50-fiber) and Galactic hemisphere (north or south), denoting therefore each field as N112, S112, N50, or S50. I limit the redshifts of the galaxies I consider to the range where the counts-in-cells analysis will take place, from 5,000 km/s to 50,000 km/s; this limitation minimizes uncertainties due to possible systematic effects associated with identifying higher redshift galaxies ( km/s). For details on these fits, see Blanton (1999).

Figure 3 shows as the solid histogram the expected distribution of galaxies with redshift, assuming , for all four types of fields and for both early- and late-type galaxies. The actual distribution of galaxies is shown as the dotted line. Note that the two distributions follow each other reasonably well. The alert eye will be suspicious of the apparent underdensity of actual late-type galaxies relative to the expected late-type galaxies at low redshifts in N112 and S112, where an large apparent overdensity exists for the early-type galaxies. In addition, there are somewhat fewer than expected early-type galaxies at high redshift. These features turn out to be important, and I discuss below their ramifications.

### 5.2 Fitting the Bias Function

The first task is to fit for and , using the method of fitting for the probability distribution function described in Section 4. The results of fits to a log-normal distribution are shown in Table 1. It is already clear from the results in this table that the early-type galaxies are more clustered than the late-type galaxies by about 20–30%.

With the parameters of the distribution in hand, I now fit for the bias relation . I try all of the models described in Section 4, as listed in the top section of Table 1. For the linear fit I find ; that is, the late-type galaxies are underbiased with respect to the early-type galaxies, to a degree which agrees with the conventional wisdom. The power-law fit is worse than the linear fit, and I will not consider it further. Some nonlinearity is detected in the quadratic bias case, at rather high significance (6; although read below for words of caution in interpretation), in the sense that the slope of the bias steepens at large densities. The broken bias model also shows this effect, at somewhat lower significance. Stochasticity is detected at about 10 significance, with . Contours showing the model distributions of superposed on the data are shown in Figure 4 for the linear, quadratic, broken, and stochastic models. Notice that the quadratic and broken models curve upward relative to the linear model to fit the data better, while the stochastic model has fattened contours because of the scatter it adds to the relation.

One can investigate the difference between these contour plots a bit more quantitatively. I do so by finding a set of contours of the model probability which contain an increasing fraction of the model cells, from to . I can compare this fraction to the fraction of actual cells which are contained in each contour. The difference between the fraction of model cells to the fraction of actual cells is plotted for each model in Figure 5. It is clear that the model which follows the probability contours most closely is the stochastic model. A refinement of this analysis would be to compare each cell to the model convolved with the Poisson noise of that cell alone, and to calculate a two-dimensional statistic analogous to the Kolmogorov-Smirnov test (Lupton 1993).

The likelihood functions for each type of fit are shown in Figure 6. This plot shows that bootstrap gives errors comparable to those calculated from the likelihood function. The likelihoods relative to the linear fit are also shown in Table 1, and indicate that the stochastic linear bias is clearly the best model I have considered. I have compared each of the two-parameter fits to the linear fit by using a likelihood ratio test, whose results I show in the last column as the probability of getting the observed likelihood difference if the true bias relation were linear. Note that since I only ran 200 realizations for each estimate, there is a lower limit on of 0.005. From these results, it is clear that I detect nonlinearity at a statistically significant level, and stochasticity at an extremely significant level.

Are stochasticity and nonlinearity both present? To address this question, rather than fitting a nonlinear and stochastic model, which would be complicated and would introduce an extra parameter, I instead ask: is the detected nonlinearity consistent with what one would find if the stochastic, linear model was correct? For example, in a realization of the stochastic, linear model, it can happen that the few high-density cells happen to all scatter below the mean . In this case, a nonlinear fit will have a higher likelihood than a linear fit; it will not, on the other hand, be correct. Can the degree of improvement of the nonlinear fits over the linear fits be simply attributed to this effect? I test this by looking at the likelihood ratio between the quadratic fit and the linear fit in a set of Monte Carlo realizations of the stochastic bias model. This experiment shows that the likelihood ratio is on average . That is, if there is in reality some scatter around linear bias, the data always is better fit by the deterministic quadratic model than by the deterministic linear model, and the measured quadratic parameters are meaningless. In fact, since the measured likelihood ratios between the nonlinear and linear fits are in this range, I cannot truly claim to have measured nonlinearity here.

While these likelihood ratio tests reveal the relative quality of the fits, one can evaluate the absolute quality by looking at the likelihood distribution of fits to a set of Monte Carlo realizations of each model, and calculating the probability of getting a lower best-fit for the model; if this probability is near unity, then the data do not fit the model as well as they should according to the realizations, whereas if it is less than the model fits the data as well as could be expected. This method is not ideal, but it gives a rough calibration of whether the fit is decent. From the top section of Table 1, it appears that the only model that does a reasonable job of fitting the full set of cells is the stochastic model.

The level of stochasticity detected here corresponds to ; for comparison, the moments method of TB99 would estimate for this counts-in-cells distribution to be . Apparently a considerable amount of the “stochasticity” measured by the moments method is due to an inadequate characterization of the distribution of the densities, most likely because the method does not account for the non-Gaussianity of the Poisson distribution at low or for the lower limit on the overdensity fields of . On the other hand, as discovered in the next section, even this small level of measured stochasticity may be fictitious, due to systematic errors in the selection function.

### 5.3 Testing for Systematic Errors

In order to probe the susceptibility of these results to systematic errors, I have run a battery of tests. First, I briefly list a number of tests which made no significant difference in my main results. Second, I describe the effects of the central magnitude cut. Finally, in the next subsection, I show that there is a redshift-dependence of the results.

A number of tests have very little effect on the results concerning stochasticity. For instamce, whether or not or is used as the independent variable has no effect on the results. In addition, identical results are found in the Northern and Southern Galactic hemispheres. Furthermore, there is no dependence of the stochastic fit on changes in: the form of the density distribution function used, the -correction applied, the completeness correction applied, or the cell configuration used. (The nonlinear fits do depend somewhat on cell configuration, but I argued above that the nonlinear fits in this case were probably meaningless). A more complete discussion of these issues can be found in Blanton (1999).

A suspicious element of the selection of galaxies in the LCRS which I address here is the central magnitude cut. Because aperture magnitudes of fixed angular size exclude a varying fraction of galaxy light depending on redshift, the central magnitude defined by the LCRS is actually a redshift dependent quantity. The sense is that at low redshift, less of the total galaxy flux is contained within a central angular radius, and thus is likely to be higher. Thus, this cut will preferentially exclude low-redshift galaxies. An aperture magnitude with a diameter of 3–4, as used in the LCRS, depends much more strongly on redshift for an exponential profile (characteristic of late-type galaxies) than a de Vaucouleurs profile (characteristic of early-type galaxies) in the redshift range under consideration here. It is therefore conceivable that late-type galaxies are being preferentially excluded at low redshifts, causing an apparent low density relative to the densities of early-type galaxies in that regime.

However, the situation appears not to be that simple, as I show by performing the following experiment. Instead of using the central magnitude limit (as defined in Equation 5) used by the survey, which excludes 12% of the galaxies, I enforce a somewhat more stringent limit in (by about 0.2 magnitudes) which excludes about 24% of the galaxies. If the effect descibed in the previous paragraph is important, one would expect the estimated galaxy density field to change significantly. However, it does not. To understand why, consider Figure 7. The top panel in each column shows the ratio of the number of galaxies per unit redshift in the stringent sample to the number in the full sample, for the N112 fields, , for each galaxy type. Clearly, as described in the previous paragraph, the late-type galaxies at low redshift are preferentially excluded. However, as shown in the middle panel, the selection function for the stringent sample (shown as the dotted line) also changes relative to the full sample (shown as the solid line). This happens because the average number of faint galaxies relative to bright galaxies is underestimated for the stringent sample. Since faint galaxies are not observable at large distances, this change does not affect the selection function at large redshifts. Indeed, as the bottom panel shows, the ratio for the stringent sample is almost identical to that of the full sample. Correspondingly, the results of the density distribution and bias fits are unchanged as well. Thus, the effect of the central magnitude cut seems not to be crucial.

### 5.4 Redshift Dependent Selection Effects

In order to demonstrate the redshift-dependence of the results, I cut the two innermost rings of cells out of the sample, and fit to the rest, as shown in the bottom section of Table 1. This set of cells shows no nonlinearity, and a much reduced stochasticity. The contours showing the model superposed over the data for this set of cells are shown in Figure 8. Again, I compare the model to actual fraction of cells within each contour in Figure 9, finding this time no apparent difference between any of the models. I can again express the stochasticity in terms of the correlation coefficient ; the moments method of TB99 obtains for this set of cells. This result indicates that most of the signal for stochastic bias was coming from the two innermost rings of cells.

This dominance of the inner two rings cannot be ascribed to the higher signal-to-noise ratio of these cells. To demonstrate this fact, I perform Monte Carlo realizations using all the cells except the two inner rings, but using the parameters for linear stochastic bias determined using all of the cells (i.e., using the parameters in the top section of Table 1). I find that the maximum likelihood fit detects (the correct value for the tests) with a likelihood difference with respect to the linear fit between and , not as for the real data. Nor does the redshift dependence appear to be a result of luminosity-dependent bias (in the sense that fainter galaxies, observable only at low redshift, have a different relative bias between galaxy types). I show this by performing the analysis again, using only galaxies brighter than , the faintest galaxy luminosity observable at 24,000 km/s; I found little change in the results.

In order to understand the effect better, consider Figure 10. This figure shows the distribution of and among the cells. I have marked the cells in the inner two rings with square boxes and the other cells as crosses. It is clear that the nearby cells have a different distribution than the rest of the cells. What this indicates is that the selection function is either overestimated for late-type galaxies or underestimated for early-type galaxies at low redshifts, which is plausible on consideration of Figure 3. I have performed the same analysis using selection functions based on the luminosity functions of Bromley et al. (1998) and find the same effect.

In Figure 11, I show the luminosity function of the galaxies in the N112 sample again, this time fitting separately to the high-redshift portion ( km/s) and the low-redshift portion ( km/s). Clearly, there are significantly fewer early-type galaxies detected at high redshift than at low redshift. Given the relatively shallow depth of the survey, it is not likely that that this difference is due to evolution of the population of early-type galaxies. Thus, the large apparent overdensity of early-types in the low-redshift region (see Figure 3) might exist only because the normalization is underestimated due to early-type galaxies being missed at high redshifts. In this case, the observed stochasticity could be due simply to fluctuations in the apparent overdensity field caused by errors in the selection function. I test this by fitting for the bias in all cells, but determining the expected counts using the low-redshift luminosity function for the inner two rings of cells, and the high-redshift luminosity function in the rest of the cells. The results are listed in the second section of Table 1. The linear fit is basically unchanged. The nonlinear fits change dramatically, though as I showed above, such changes should not be too surprising. Finally, the stochasticity is reduced significantly, to . Remembering that adds quadratically, this result indicates that a large proportion of the scatter was indeed simply due to the redshift dependence. Figure 12 shows the joint counts-in-cells that these fits were based on, showing (as in Figure 10) the low redshift cells separately from the high redshift cells. One can see clearly that the estimated early-type densities of the low redshift cells are quite different from those in Figure 10, and that the distribution of low redshift cells now appears consistent with the distribution of the rests of the cells.

The question remains where this redshift dependence comes from. I have already shown that the central magnitude selection criterion does not affect the results; in any case, one would expect it to preferentially exclude late-type galaxies at low redshift, since late-types are more extended than early-types. Similarly, the use of isophotal magnitudes, which depend on redshift due to surface-brightness dimming, would also preferentially exclude late-type galaxies. In addition, I show below using mock catalogs that the use of isophotal magnitudes does not affect the results. Another possibility is that the spectral classification scheme is misclassifying galaxies. However, in the case of misclassification, one would expect to see comparable errors in the normalization of both galaxy types, not just the early-types. A final, speculative possibility is that early-type galaxies at high redshift tended not to be be identified as galaxies in the survey in the first place, since they are compact enough to look like stars. I am currently looking at the redshift distribution of galaxies and issues of galaxy selection using the superior imaging and spectroscopic data of the SDSS (Gunn & Weinberg 1995), and it is possible that this work will lend understanding to the problems faced in the LCRS.

Until the nature of the galaxy selection in this survey is more fully understood, I recommend taking most seriously the results in the bottom section of Table 1, which exclude the two innermost rings, and indicate a bias which is linear, with perhaps some mild scatter, and an amplitude of .

## 6 Results from Mock Catalogs

Because of the peculiar geometry and selection effects of the LCRS, it is necessary to test this method against mock catalogs where I have simulated all of the properties of the survey. I would also like to test whether some of the systematic trends with redshift found in the last section can be explained by selection effects in the survey.

### 6.1 Simulations

For current purposes I run particle-mesh simulations of a 300 Mpc box using particles and grid cells, using a code provided by Renyue Cen. I use the flat CDM model with and ; the angular diameter distances and the distance moduli for the mock catalogs are calculated using this model, although the analysis of the mock surveys are performed using the model for the redshift-distance relation, as they are for the real survey. To select the late-type galaxies, I simply pick dark matter particles at random. To select the early-type galaxies, I smooth the density field with a 3 Mpc Gaussian filter, and apply a threshhold of , at ; every dark matter particle above the threshold has an equal probability of becoming an early-type galaxy. Mock catalogs are drawn from three realizations of this model. Note that since the box size is smaller than the redshift limits of the survey, I must extend the box periodically in each direction in order to simulate the LCRS.

As a benchmark against which to compare the mock catalogs, I take the simulation and divide it into cubic cells 25 Mpc on a side. I subsample the galaxies such that there are about 20–30 galaxies of each type in each cell. I refer to this sample as the benchmark catalog. It is free of all of the selection effects associated with the real survey, as well as redshift-space distortions. The cells are equivalent in volume to the cells described in Section 2. I have listed in the top section of Table 2 the results of fitting the bias for all such cells for all three realizations simultaneously. I will evaluate the degree to which the selection effects affect my results by comparing realistic mock catalogs to the results for these benchmark cells.

To create realistic mock catalogs, I pick a random particle in the simulation to represent the observer. In order to evaluate the effects of the cell shapes independent of other selection effects, I create a catalog using the angular and redshift limits of all of the cells, without regard to flux limits. I refer to this catalog as a volume-limited catalog. For this catalog, I do implement redshift-space distortions, although these do not make a significant difference in the results. Again, I subsample the galaxies such that there are approximately 20–30 galaxies of each type in each cell.

To create flux-limited catalogs, I assign each galaxy an absolute magnitude randomly from the luminosity function determined in Section 5 for N112 galaxies, depending on which type of galaxy it is. I then “observe” the galaxies in the simulation box, using the angular and photometric limits of the LCRS (Shectman et al. 1996). For most of the catalogs, I assume the observers have the ability to measure total magnitudes, and apply only the apparent magnitude limits, not the central magnitude limits. I explore below the effects of using isophotal magnitudes and implementing the central magnitude limits. I create two types of flux-limited mock catalogs: first, fully-sampled catalogs, in which I take the redshift of every galaxy within the flux-limits of each field; second, undersampled catalogs, in which I select the targets in each field based on the number of fibers available for that field (allowing for about 5% of the fibers to be accidentally placed on stars). Furthermore, for the undersampled catalogs, there is a probability of failing to observe the galaxy which is a function of its magnitude, given by . In the real LCRS, fibers could not be placed more closely than ; I implement that restriction in the undersampled catalogs as well. Finally, in accordance with the stated photometric errors in Shectman et al. (1996), I included 1 magnitude errors of for and for , as well as 1 redshift errors of km/s.

In order to determine the effects of cosmic scatter and to test whether the survey constitutes a fair sample, I draw thirty undersampled mock catalogs from three realizations. After fitting for the bias in each catalog, I compared the standard deviation of the results to the estimated errors. They were almost identical, indicating that the errors due to cosmic scatter are no bigger than the other statistical errors.

### 6.2 Analysis of the Mock Catalogs

The results of these mock catalogs are listed in Table 2. First, I compare the benchmark catalog, which consists of all three realizations divided into cubical cells, to the volume-limited catalog, which uses cells of the same shape used in the actual survey. The benchmark catalog has much smaller error bars because it probes considerably more volume than the other catalogs. The fluctuation amplitudes and measured for the volume-limited catalog are significantly smaller than for the benchmark catalog, indicating that the cell shapes in the LCRS probe effectively larger scales than cubes of equivalent volume. However, the bias fits change very little between the two catalogs. No parameter changes more than about 1.5. Note that the bias implemented here is only slightly scale-dependent, and that if it were strongly scale-dependent, the difference between the volume-limited and benchmark catalogs might be larger, because the two catalogs probe somewhat different scales.

Second, I consider the fully sampled catalog, which implements the flux-limits of the survey, but not the finite number of fibers or the fiber collision effects. The differences in all the parameters is quite small. and do increase slightly by about 1.5; in addition, the stochasticity in the bias also increases by almost 1.5, but it remains much smaller than that measured in the data. It is possible that these increases are due to the variation of the selection function across the cells, as explained in Section 4. However, this effect is apparently too small compared to the noise to be important for the LCRS, though it may be of concern to larger surveys if they strive for more precision.

Third, the results of the undersampled catalog, which implements the effects of a finite number of fibers and fiber collisions, are again almost identical to the fully-sampled case. and are reduced by about apiece, which could just as easily be due to chance as it could be due to the sampling effects. The only large change is in , which does decrease rather substantially from the fully-sampled case.

Given the results in this section, I conclude that the selection function, variable flux limits, finite sampling, and fiber collisions do not affect the results significantly.

### 6.3 Isophotal and Central Magnitude Limits

The catalogs I examine above are purely flux-limited and assume observers can measure total magnitudes. However, I also want to probe the effects of using isophotal magnitudes, as well of implementing the central magnitude cut. In order to do so, I must also model the surface brightness profiles and characteristic radii of the galaxies. I adopt a very simple picture here, since I am interested not in a perfect model of the galaxy distribution but only in some estimate of how adding these realistic observational effects changes one’s estimate of the galaxy density field. I model the early-type galaxies as pure de Vaucouleurs profiles:

(25) |

where is the distance from the center of the galaxy and is a characteristic scale length. I model the late-type galaxies as bulge components with de Vaucouleurs profiles, which characteristic scale , plus disk components with exponential profiles:

(26) |

where again is a characteristic scale length. For these galaxies I fix and (see Binney & Merrifield 1998 for the definition of ), which are appropriate choices for Sbc galaxies (Kent 1985). In order to determine the scale lengths, I follow Sodré & Lahav (1993) and write:

(27) |

where represents Gaussian noise with a standard deviation . In accordance with semi-analytic models I set (Dalcanton, Spergel, & Summers 1997). Furthermore, I choose , , Mpc, and Mpc. These parameters are not unique; they simply produce a distribution of and which is reasonably like the data. I tried several variations on these parameters as well, with no significant change in the results. The profile of each galaxy is scaled to the appropriate redshift and convolved with the seeing, which for simplicity I assume to be Gaussian with a FWHM of . Using a Gaussian seeing profile is somewhat unrealistic; it will make very little difference to the central magnitude, but it might cause the difference between the isophotal and total magnitudes to be underestimated at large redshifts. As a further simplification, I assume all galaxies are face-on and axisymmetric. This assumption exaggerates the difference between isophotal and total magnitudes.

To determine the isophotal magnitude, , I use a limiting isophote of about 23 mag/arcsec, corresponding approximately to 15% of the sky brightness in , which are the stated isophotal limits of Shectman et al. (1996). The total flux within this isophote is used to calculate the apparent magnitude of the galaxy in the sample. To determine the central magnitude, I take a circular aperture of diameter . I add central magnitude errors with a dispersion of , similar to the stated errors in the photometry of faint galaxies in the survey. I then apply the same flux and central magnitude cuts on the mock survey as to the real data.

I produce three mock catalogs to test these observational effects, all taken from the same vantage point in the same realization, so I can compare their density fields directly. First, I produce a mock catalog in which the observers have been able to measure the total magnitudes of the galaxies. Second, I produce one which is flux-limited, but based on isophotal magnitudes as described above. Third, I produce one which uses isophotal magnitudes and is also -limited; that is, the central magnitude cuts have been applied. I can compare these different catalogs by looking at as well as for each, as determined by fitting for the luminosity function and calculating the selection function; I make this comparison in Figure 13, which is analogous to Figure 7 for the observations. Using isophotal magnitudes evidently causes the systematic elimination of galaxies at high redshifts, where dimming and the effects of seeing start to become important. Meanwhile, as I showed for the real observations in Figure 7, the cut elimates galaxies at low redshift. However, the changes to the luminosity function caused by these eliminations seems to cause the selection functions to be reasonable estimates of the probability of observing a galaxy at all redshifts. The bottom panels of Figure 13 show, correspondingly, that the density fields of galaxies are thus unaffected by these changes in galaxy selection. I perform the counts-in-cells analysis on these galaxies, and as shown in Table 3, these observational effects do not appear to be able to cause the sort of stochasticity observed in the real sample.

## 7 Summmary and Conclusions

I have presented a straightforward maximum likelihood method to determine the relationship between the density fields of different galaxies types on a point-by-point basis by looking at the joint counts-in-cells distribution . Using mock catalogs, I have demonstrated the reliability of the method. I have applied the method to the LCRS in an attempt to constrain the nature of the segregation of different galaxy spectral types (as classified by Bromley et al. 1998). At most a small amount of stochasticity affects the relationship between early- and late-type galaxies in the LCRS, corresponding to , a larger correlation coefficient than found using the simple moments method of TB99. In addition, it is likely that even this result is low because of poorly understood selection effects in the survey, and that the true value of is closer to .

In either case, the large scatter predicted by Blanton et al. (1999) from hydrodynamic simuations does not seem to exist, and the results are more consistent with the semi-analytic predictions of Somerville et al. (1999). It is not clear yet what the implications of this result are, but there are at least three possibilities. First, because the survey is selected in the band and is surface-brightness limited, there may not be a sufficient range of galaxy types represented to reveal the predicted stochasticity. The fact that the relative bias between early- and late-type galaxies is also smaller than predicted (1.2 instead of 1.5) is consistent with this explanation. Second, since the simulations of Blanton et al. (1999) are low resolution and cannot resolve galactic disks, it may be that the simulations are not modelling important effects on subgrid scales which would considerably reduce the stochasticity. Third and most interesting (though probably least likely), is the possibility that the fundamental principles behind the way galaxy formation is approximated in the simulations are flawed, and need to be revised. Improved simulations and the analysis of new, larger, and more complete redshift surveys such as the SDSS will help answer these questions.

In addition to the main result, I have found suspicious behavior of the selection function derived for the sample (both my own and that of Bromley et al. 1998). A thorough investigation of possible causes of these errors, using the data itself as well as mock catalogs, has turned up no likely cause of this effect, including surface-brightness selection effects, the use of isophotal magnitudes, and errors in the -correction. On the other hand, it is possible that some of the mock catalog experiments presented here are misleading because the model I used for galaxy profiles was inadequate (for instance, if I used an inaccurate distribution of galaxy sizes). Analysis of larger surveys with better quality images and spectra, such as the SDSS, may thus be more useful than the mock catalogs in understanding the effect. I must note that the distinct, though unlikely, possibility remains that the low redshift portion of the LCRS is indeed an unusual section of the universe, either due to a rapid evolution of galaxy properties between and today or because of some peculiar local phenomenon.

The inadequacy of the selection function may have consequences for other results based on the LCRS. First, I have shown here that the low correlation coefficients measured by TB99 may be in doubt. Second, the excess large-scale power in this survey claimed by Landy et al. (1996) may be due to this effect. In fact, the largest amplitude wave in the survey that those authors detect is in the “outward,” redshift direction, which might indicate that redshift dependent selection effects could be contaminating their results; on the other hand, they also detect large waves tangent to the redshift direction, which might not be so readily explained.

In any case, the method presented here is applicable to any comparison of counts-in-cells of different galaxy populations. It may be most useful in surveys which are volume-limited and have simpler geometries. Such surveys would also make it easier to explore the scale-dependence of the relative bias of galaxies; this task is difficult in the LCRS, since looking at larger scales forces one to change the geometry of one’s cells, which as I have shown affects the results. In particular, the Sloan Digital Sky Survey (SDSS; Gunn & Weinberg 1995) and the Two-Degree Field (2DF; Colless 1998) would allow one to make powerful tests of the nature of morphological segregation. It is possible, of course, to compare the galaxy densities in different surveys using this method (Seaborne et al. 1999). For example, one might use the future -selected redshift survey based on the Two-Micron All Sky Survey (2MASS; Beichman et al. 1998) to compare in the appropriate volume to the SDSS or 2dF.

In conclusion, the details of morphological segregation contain much information about how galaxies formed. This paper has attempted to extract some of this information by measuring the stochasticity in the relative clustering of galaxy types. Future redshift surveys and more sophisticated galaxy formation models will be able to make much more powerful and informative tests.

## Appendix A Maximum Likelihood Calculation of the Luminosity Function

The goal of this section is to describe how to maximize the condition probability:

(A1) |

in terms of a model for the luminosity function , using the method described by Efstathiou, Ellis, & Peterson (1988).

Any interpolation scheme used to approximate the factors in Equation (A1) can be expressed as:

(A2) | |||||

(A3) |

Here refers to the luminosity function determined in bin , bounded by and . In the case of piecewise constant interpolation, which is sufficient for my purposes,

(A4) | |||||

(A5) |

The formulae for piecewise linear interpolation are given by Koranyi & Strauss (1997). Using these approximations, multiplying the conditional probabilities given by Equation (A1) for each galaxy in the survey, and imposing the maximum likelihood condition that

(A6) |

for all of the parameters, yields an iterative equation for each :

(A7) |

which reduces to Equation (2.12) of Efstathiou, Ellis, & Peterson (1988) in the case of piecewise constant interpolation, which I will use here. Note that for the case of piecewise linear interpolation, this formula differs from the one given by Koranyi & Strauss (1997), which is missing terms in the denominator of the numerator. Experiments have shown that the difference between using this formula and theirs is fairly small.

As described in Efstathiou, Ellis, & Peterson (1988), the luminosity function is derived by starting with some initial guess for the and iterating until the improvement in the likelihood per iteration is small. Throughout, one maintains the normalization condition:

(A8) |

The errors in can be calculated by inverting the matrix:

(A9) |

The diagonal elements of are the errors in each parameter , while the off-diagonal elements represent the covariances.

This procedure has determined the shape of the luminosity function, but one has yet to determine its amplitude. To do so one must calculate the selection function, which with the interpolation scheme can be approximated by

(A10) |

The most straightforward estimate of the normalization of the luminosity function (though not the minimum variance estimator) is

(A11) |

where is the size of the volume probed, and the error can be estimated as

(A12) |

Given , one can express the number of galaxies per unit volume per unit luminosity by .

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