small-scale velocity dispersion and large-scale linear ffows. After transformation to redshift space, we find that the power spectrum is insensitive to the ...
THE ASTROPHYSICAL JOURNAL, 464 : L103–L106, 1996 June 20 q 1996. The American Astronomical Society. All rights reserved. Printed in U.S.A.
VELOCITY DISPERSION AND THE REDSHIFT-SPACE POWER SPECTRUM T. G. BRAINERD,1, 2 B. C. BROMLEY,1 M. S. WARREN,1
AND
W. H. ZUREK1
Received 1995 September 22; accepted 1996 April 1
ABSTRACT A large (N 1 17 3 10 ) high-resolution N-body simulation of a standard cold dark matter (CDM) universe is used to investigate the effects of peculiar velocities on the power spectrum of galaxies in redshift space. The unprecedented dynamical range of the simulation code allows galaxy halos to be resolved in the numerical data while maintaining good statistical sampling on large scales. We present evidence that the redshift-space power spectrum Ps (k) can be related to its real-space counterpart by means of a simple filter function which reflects both small-scale velocity dispersion and large-scale linear flows. After transformation to redshift space, we find that the power spectrum is insensitive to the normalization of the CDM model at scales below 20 h 21 Mpc. Hence Ps (k) does not provide an unambiguous cosmological constraint at small scales. Nonetheless, it is significant that the redshift-space power spectra from CDM models with two different normalizations both compare remarkably well with results from the galaxies in the IRAS 1.2 Jy survey (Fisher et al.) on scales between 1 and 50 h 21 Mpc. By excluding a fraction of the most tightly bound halos, we create a galaxy catalog with 80% of the original objects in the lower normalization model that matches both the IRAS power spectrum and the inferred pairwise velocity dispersion on megaparsec scales. Thus, in contrast to previous reports, we find that the CDM scenario does not produce excess power at small scales. Subject headings: cosmology: theory — galaxies: clusters: general 6
Bahcall 1993). These results stand in contrast to the conclusion of Fisher et al. (1993), who argue that COBE-normalized CDM models yield excessive power on small scales when compared to IRAS galaxy data in the 1.2 Jy survey. Unfortunately, the numerical component of these investigations suffered from limited mass and/or length resolution. Thus, either the statistical integrity of the large-scale wave-mode samples was low, or realistic dark halos of galaxies were beyond the resolving power of the simulation. The present work is an analysis of numerical data from code with sufficient dynamic range to resolve both dark halos of galaxies and extragalactic structure on all but the very largest scales observed in real galactic catalogs. The principal results reported here are (1) an analysis of Ps (k) in two CDM models differing in their normalization, (2) a filter function for quantifying the relationship between Ps (k) and Pr (k) in these models, and (3) a comparison of the CDM models with the observed redshift-space power spectrum of galaxies in the IRAS 1.2 Jy survey.
1. INTRODUCTION
The relative clustering of galaxies as a function of physical scale is a powerful constraint on theories of structure formation. One important measure of galaxy clustering is the power spectrum, P(k), which is determined observationally in redshift space rather than real space (da Costa, Vogeley, & Geller 1994; Feldman, Kaiser, & Peacock 1994; Fisher et al. 1993; Vogeley et al. 1992 and references therein). As compared to the real-space power spectrum Pr (k), the redshift-space analog Ps (k) contains distortions from the peculiar velocities of galaxies along the line of sight. These distortions have been modeled to yield velocity-related information such as the linear growth factor b 1 V 0.6 , where V is the cosmological density parameter (e.g., Kaiser 1987). Here we report an investigation of redshift-space power spectra from a high-resolution numerical simulation of the cold dark matter (CDM) scenario. We consider the clustering of dark galaxy halos on scales ranging from the strongly nonlinear regime (11 Mpc), in which redshift-space distortions appear as ‘‘fingers of God,’’ up to linear scales well above 10 Mpc, in which peculiar infall compresses the observed cosmic structure in redshift space. However, the principal focus of this Letter is the redshift-space power at small, nonlinear scales. It is at these scales, the most difficult for accurate numerical modeling, that CDM models have been reported to fail (e.g., Fisher et al. 1993). Previous determinations of redshift-space power in standard CDM cosmogonies, normalized by the Cosmic Background Explorer (COBE) observations of fluctuations in the cosmic microwave background on large scales, have shown good agreement between the CDM power spectrum and the observed power spectrum of galaxies (e.g., Brainerd & Villumsen 1993, 1994; Bahcall, Cen, & Gramann 1993; Gramann, Cen, &
2. NUMERICAL SIMULATION AND ESTIMATION OF THE POWER SPECTRUM
The high-resolution numerical data were generated from a simulation of 256 3 1 17 3 10 6 particles in a periodic cube of comoving length L 5 125 h 21 Mpc based on a standard CDM model (V 0 5 1, L 0 5 0, h 5 0.5 in units of 100 km s 21 Mpc 21 , and a power spectrum as parameterized by Efstathiou, Bond, & White 1992). The simulation was evolved from z 2 60 to z 5 0 using a tree code with force resolution of 10 h 21 kpc. Dark halos, candidates for galaxies, were identified using the method of Warren et al. (1992). From this simulation, two CDM models with normalizations of s 8 5 1.1 (approximately COBE-normalized) and s 8 5 0.74 were extracted by rescaling position and velocity output at different times. Here s 8 is the conventional rms density fluctuation within a spherical top-hat function of radius 8 h 21 Mpc normalized to the average mass within the top hat. In the high-normalization model there are
1 Theoretical Astrophysics, MS B288, Los Alamos National Laboratory, Los Alamos, NM 87505. 2 Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215.
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16,975 halos in the simulation, and for the low normalization model there are 16,379 halos. The COBE normalization for CDM refers to the standard model wherein the large-scale primordial power spectrum has the Harrison-Zeldovich form, P(k) 1 k n with n 5 1. However, the low-normalization model discussed here may be consistent with the COBE data when the generation of density perturbations in an inflationary era is properly taken into account. Recent work (e.g., Davis et al. 1992) suggests that the power-law index n for primordial fluctuations in inflationary scenarios should be less than unity as a result of the finite length of the inflationary epoch. In generic models, gravitational tensor modes also contribute to the fluctuation spectrum observed by COBE at a level which is dependent on n. Then the amplitude of mass-density fluctuations at scales below 11000 Mpc is lower than expected in a standard n 5 1 model because of the tilt in the primordial spectrum and the contribution to the COBE signal from tensor modes. While the input physics is speculative, quantitative estimates of a spectral index which is consistent with both the COBE data and the s8 value from our low-normalization model are in the range 0.85–0.95 (Davis et al. 1992; Zurek et al. 1994; Bunn, Scott, & White 1995). To obtain power spectra, point distributions of halos in real and redshift space were constructed. Redshift-space positions for a given observer are expressed in distance units by obtaining line-of-sight velocities and dividing by the Hubble parameter H 5 100 h. The observer peculiar velocities are either zero (corresponding to the cosmic background radiation rest frame) or the center-of-mass velocity of a halo on which the observer is located. Ultimately we will compare the model power spectra directly to Ps (k) obtained from real galaxies in a finite volume of the universe. With this in mind, all power spectra presented here are computed following Fisher et al. (1993) in their analysis of the IRAS 1.2 Jy galaxy survey. Their measure of power in either redshift or real space comes from random sampling in cylindrical window functions centered on the observer; in agreement with Fisher et al. (1993), we find that the effects of the finite windows in the resulting measure ˜ in their paper) are minimal for all but the largest scales (P considered, where power is systematically underestimated by no more than 10%. Nonetheless, we stress that hereafter all references to power spectra, in both real and redshift space, refer to this measure. 3. THE POWER SPECTRUM IN REAL SPACE AND IN REDSHIFT SPACE
Figure 1 shows Pr (k) versus Ps (k) for the halos in CDM models with normalizations of s 8 5 1.1 and s 8 5 0.74. In this section, power spectra were evaluated using 100 randomly placed observers at rest with respect to the cosmic microwave background frame, and the results shown are the mean power spectra for the 100 observers. In the figure it is clear that on small scales Ps (k) is strongly suppressed compared to Pr (k) for both normalizations. The relationship between Ps (k) and Pr (k) can be quantified by means of a filter function F(k) such that Ps~k! 5 F~k! Pr~k!
(1)
(e.g., Peacock & Dodds 1994; Gramann, Cen, & Bahcall 1993; Fisher et al. 1993; Bromley et al. 1996). The form of the filter can be motivated by a determination of its limiting behavior at small and large scales. Here we assume that the power is measured in a survey which is sufficiently distant that the
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FIG. 1.—Real-space and redshift-space halo power spectra in standard CDM models with normalizations of (a) s 8 5 0.74 and (b) s 8 5 1.1 for comoving observers. The arrows point from the real-space power spectrum to the predicted redshift-space profile (dashed lines) obtained with the filter in eq. (5) which lie close to the measured Ps (k). In (c) the measured redshift-space power spectra for the two CDM models are superimposed to illustrate the insensitivity to normalization at small scales. The dashed lines show the 1 s confidence intervals based on observer-to-observer scatter in the measured power of the low-normalization model. The width of the confidence intervals in the high-normalization case are similar.
observer’s line of sight lies in approximately the same direction for all objects in the survey. This is mostly a mathematical convenience; otherwise the Fourier transform of a field in redshift space has a somewhat nontrivial dependence on the observer’s location, which is not important after we average over all possible mode orientations with respect to the line of sight. In the small-scale regime, the finger-of-God effect blurs the redshift-space correlations along the line of sight. The degree of line-of-sight distortion is determined by a convolution of the real-space correlation function and the distribution function for radial pairwise velocities. In the Fourier domain where power measures correlations, the Fourier transform of this distribution function is the desired filter. Here we use the simulation to determine the radial pairwise velocity distribution function for galaxy halos on megaparsec scales and find that (Zurek et al. 1993, 1994) f ~ v! 1 exp ~2
Î2u v u/s v! ,
(2)
where s v is the radial pairwise velocity dispersion. This exponential form is ubiquitous in high-resolution simulations, and recent work by Marzke et al. (1995) with the CfA galaxy catalog indicates that this form is appropriate for the real universe. The resulting small-scale filter function, averaged over all angles of observation, is Fd~k! 2
kd k
tan 21
SD k
kd
kd [
,
Î2H sv
.
(3)
In this choice of Fd we diverge from previous work (e.g., Peacock & Dodds 1994) which is based on a Gaussian velocity distribution function, although the general behavior of the filter is the same in both cases. At large, linear scales where the density fluctuations relative to the mean are much less than unity, Kaiser’s (1987) analysis of line-of-sight mode amplification gives an angle-averaged filter of Ff ~k! 5 1 1
2b 3
1
b2 5
;
(4)
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here b 5 V 0.6 /b, where b is a linear bias parameter. For the CDM models considered here, this reduces to the value of 28/15. We now propose to connect the small-scale and large-scale filters with the following parameterization: F~k! 5 Fd~k!
Ff ~k! 1 k 2 /k 2x 1 1 k /k 2
2 x
,
kx 2
p 10 s 28
h Mpc 21 ,
(5)
where kx corresponds to a transition scale between the linear and nonlinear regimes where the linear theory begins to fail. Figure 1 illustrates the effectiveness of this parameterization for the present purposes. The virtually indistinguishable redshift-space power spectra of the two CDM models at small scales in Figure 1c indicates an insensitivity of Ps (k) to the normalization of the primordial power. We may gain some insight into this property through the following consideration. As a first approximation, we suppose that the mass in each CDM model obeys a similarity transformation as described by Davis & Peebles (1977). In this case, it is straightforward to show that at small, strongly nonlinear scales where the real-space correlation function goes as a power law, j (r) 1 r 2g , both the power and j (r) scale g as s 32 (see Peebles 1980, § 73). The pairwise velocity disper8 sion in a similarity model (or any hierarchical system) is required by the cosmic virial theorem (e.g., Peebles 1980, § 75) to scale as the square root of j (r) for fixed V and pair separation. With g 2 1.8, this scaling behavior and the filter function in equation (3) suggest that the redshift-space power spectrum at small scales has weak dependence on normalization, Ps~k! 1 s 0.6 8 .
(6)
While the similarity model suggests an explanation for the insensitivity of Ps (k) to changes in normalization, we find it does not strictly apply to the halos in the CDM simulations. To understand the behavior of Ps (k) more fully, we must also consider the bias of halos relative to mass with respect to spatial correlations and pairwise velocities. While a bias in the pairwise velocity dispersion of halos relative to mass is evident, it is at the same level in both CDM models, hence it does not play a strong role in the conspiracy to keep Ps (k) independent of s 8 . Spatial biasing, on the other hand, does decrease with increasing s 8 , and in a manner which further reduces the dependence of Ps (k) on normalization. 4. COMPARISON WITH THE IRAS 1.2 JANSKY RESULTS
We now compare Ps (k) for the CDM halos to that of the galaxies in the IRAS 1.2 Jy survey. Following the method of Fisher et al. (1993), a set of simulation observers is chosen whose local environment is similar to that of the Local Group. Specifically, observers are positioned at halos which satisfy the following criteria: (1) the peculiar velocity, v p , is 600 H 10 km s 21 ; (2) the halo is inside a region in which the fractional overdensity, d, averaged in a sphere of radius 450 km s 21 , is in the range 20.2 , d , 1.0; and (3) there is low shear such that uv p 2 ^ v&u, 0.3v p within a sphere of radius 450 km s 21 . In the high- and low-normalization data, 93 and 86 halos, respectively, satisfy these criteria. The mean Ps (k) for all the Local Group observers was computed, and results are shown in Figure 2; the points with error bars are Ps (k) measured by Fisher et al. (1993) for the IRAS galaxies. Clearly, over all the scales we can investigate in
FIG. 2.—Redshift-space power spectra measured by ‘‘Local Group’’ observers (see text) for CDM halos in the high-normalization model (solid curve; s 8 5 1.1 and s v 2 1400 km s 21), the low-normalization model (dotted curve; s 8 5 0.74 and s v 2 800 km s 21), and the cool subset of the low-normalization model (dashed line; s v 5 378 km s 21). The filled circles with 1 s error bars are the IRAS 1.2 Jy data from Fisher et al. (1993).
the simulation, there is excellent agreement between the value of Ps (k) for the halos and that for the IRAS galaxies, independent of s 8 . In particular, for the COBE-normalized CDM model, there is no excess small-scale power problem compared to P(k) for the IRAS galaxies. It is important to note that the discrepancy between the results shown here for the power spectrum of galaxies predicted by CDM (where we equate one halo to one galaxy) and the theoretical results of Fisher et al. (1993) has to do with a fundamentally different approach to the analysis of the N-body simulation. Here we have used resolved halos as galaxy tracers and evaluated Ps (k) using the actual velocity field produced by the simulation. Because of the large volume and small number of particles in their simulation, Fisher et al. (1993) were restricted to individual particles as galaxy tracers. In addition, prior to evaluating Ps (k), Fisher et al. cooled the velocity field of the particles to mimic the low small-scale velocity dispersion deduced from the IRAS galaxies. The effect of artificial cooling in the simulation is to reduce the small-scale smearing of the redshift-space power. However, this unphysical procedure destroys the balance between small-scale clustering and velocities that is evident in Figure 1c. Clearly the temperature (i.e., the small-scale velocity dispersion) of the halo velocity field has a strong influence on the measured Ps (k) on small scales. Given that s v at megaparsec 21 separation is measured to be 317 140 in the IRAS 249 km s catalog (Fisher et al. 1994), it is therefore a serious concern that any agreement between Ps (k) for the two CDM models with a hot velocity field and that of observed galaxies is pure serendipity. This is certainly an issue for the high-normalization CDM model for which s v 1 1400 km s 21 on a scale of 1 h 21 Mpc. In the low-normalization model, however, s v 1 800 km s 21 on a scale of 1 h 21 Mpc, in reasonable agreement with
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results from recent large optical redshift surveys (Marzke et al. 1995; Guzzo et al. 1995). Still, the s v value from IRAS is less than half this value. The reason for the low velocity dispersion in the IRAS catalog as compared to that of optically selected galaxies is presumably that infrared selection excludes high-velocity galaxies which are associated with high-density regions. We crudely mimic this selection by eliminating halos in the low-normalization catalog which (1) are within 1.5 Abell radii of the highest mass-density peaks and (2) have a peculiar velocity greater than 600 km s 21 . The resulting subset of 13,133 halos (80% of the original sample) has a velocity dispersion of 378 km s 21 , a value which agrees with IRAS to within 20% and is less than half of the s v value of the parent catalog. We emphasize that, unlike Fisher et al. (1993), we did not alter any of the halo velocities. Interestingly, the fraction of the objects left in our cool sample is close to the fraction of galaxies which are late type, and which presumably constitute the bulk of the IRAS sample. The redshift space power from this ‘‘cool’’ halo subset is seen in Figure 2. The figure illustrates that this cool catalog has nearly the same goodness of fit to the IRAS power as the other two data sets, yet it has a small-scale velocity dispersion which is consistent with the IRAS data at the 2 s level. Clearly, our assertions about the insensitivity of Ps (k) at small scales are reaffirmed. Indeed, in order to obtain a cool halo catalog by preferentially selecting low-velocity objects, one necessarily must omit halos in high-density regions which account for the small-scale power. Thus the balance between small-scale power and velocity dispersion is maintained and Ps (k) remains unchanged. 5. CONCLUSIONS
The power spectrum of resolved halos in a large N-body simulation of a standard CDM universe has been computed directly in redshift space for two normalizations of the model, s 8 5 1.1 (high normalization) and s 8 5 0.74 (low normalization). The low-normalization model may well be realistic in light of the recent recognition that inflation in the early universe tends to tilt the primordial power spectrum toward lower small-scale fluctuation amplitudes (Zurek et al. 1993, 1994, 1995; White et al. 1995). As a result of a trade-off in the growth of halo peculiar velocities and the growth of clustering in real space, the redshift-space power spectra of halos on scales =20 h 21 Mpc are identical for both normalizations. Therefore, it would appear to be futile to use Ps (k) observed
for galaxies on small to moderate scales to unambiguously constrain models of structure formation. With the limitations of redshift-space power in evidence, it is nonetheless noteworthy that on scales of 11– 60 h 21 Mpc the redshift-space power spectrum of the CDM halos as measured by ‘‘Local Group’’ observers is in good agreement with Ps (k) measured for the galaxies in the IRAS 1.2 Jy survey. In particular, in the COBE-normalized CDM universe there is no apparent excess of small-scale power relative to the power spectrum of the IRAS galaxies. However, this model predicts unrealistically high small-scale velocity dispersion, and its consistency with the IRAS power spectrum in redshift space serves only to exemplify the difficulties in model discrimination on the basis of Ps (k) alone. The low-normalization model predicts a somewhat lower small-scale dispersion, in agreement with reports from recent optical surveys but still well above the IRAS value. When we examine a cool subset of halos from the low-normalization model with s v 5 378 km s 21 , a value within the 2 s uncertainty bound from IRAS, we find that again the small-scale power fits the IRAS data remarkably well. Thus, to the extent to which our selection of the cool subset is realistic, the low-normalization CDM model simultaneously fits the redshift-space power and the velocity dispersion deduced from the IRAS data at small scales. Here we have focused only on Ps (k) and s v , two of the constraints which any viable model of structure formation must face. Clearly, a viable model of structure formation must pass a number of other important tests. For example, the angular correlation function of optical galaxies (Maddox et al. 1990) provides a serious challenge to the V 5 1 CDM models considered here. At present, these issues are not resolved, and it is a matter of speculation how theoretical models such as CDM will fare when the observational constraints are tightened. We are grateful to Peter Quinn for hospitality at the Mount Stromlo and Siding Spring Observatories Heron Island Workshop on Peculiar Velocities in the Universe, where part of this work was completed. We are indebted to the referee, Andrew Hamilton, for insightful criticism and suggestions which led to improvements in the content and presentation of the manuscript. The comments of John Peacock were also appreciated. This project was supported by a grant from the NASA HPCC program. The Cray Supercomputer used in this investigation was provided through funding from the NASA Offices of Space Sciences, Aeronautics, and Mission to Planet Earth.
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