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THE ASTROPHYSICAL JOURNAL, 461 : L65–L67, 1996 April 20 q 1996. The American Astronomical Society. All rights reserved. Printed in U.S.A. THE EVOLUTION OF BIAS J. N. FRY Department of Physics, 215 Williamson Hall, University of Florida, Gainesville, FL 32611-8440 Received 1996 January 8; accepted 1996 February 12 ABSTRACT Bias in the galaxy distribution is often conceived as something applied at the present or as independent of time. In fact, a bias arising physically in the process of galaxy formation will evolve afterward, as galaxies move under the influence of gravity. I calculate the evolution of bias in a model in which galaxies are formed at a fixed time by a process that may depend nonlinearly on density but follow motions determined by the gravitational potential thereafter. The equation of continuity then determines the evolution. An initial bias decays with time, and in the long term, the galaxy distribution relaxes to that of the mass, but with galaxy formation occurring at a modest redshift, an appreciable bias may remain to the present. The evolution of bias changes somewhat the dependence of the bispectrum amplitude on configuration shape, but a weak dependence on configuration shape still corresponds to a large bias. Subject heading: large-scale structure of universe second-order contributions to d g for this model in perturbation theory are given in § 2, the effects on moments of the galaxy distribution are presented in § 3, and § 4 contains a final discussion. 1. INTRODUCTION The evolution of the cosmological mass density by gravitational instability in the perturbation theory regime is well understood (Peebles 1980; Goroff et al. 1986; Jain & Bertschinger 1994). The galaxy distribution is less well understood, since it is clear that not all galaxies can trace mass: a stronger clustering of elliptical than of spiral galaxies (Davis & Geller 1976) can be traced to a dependence of morphology on environment (Dressler 1980; Postman & Geller 1984); clustering strength increases for galaxies with higher surface brightness (Davis & Djorgovski 1985) and for intrinsically brighter galaxies (Hamilton 1988); and optically selected galaxies are more strongly clustered than infrared-selected galaxies (Strauss et al. 1992; Saunders, Rowan-Robinson, & Lawrence 1992). Many analytic toy models of bias, in which galaxies represent preferred locations such as peaks in the mass distribution (Kaiser 1984), have been found to give a galaxy density that has an enhanced contrast relative to the mass density, d g 5 bd, where d g 5 [n(x) 2 n̄]/n̄ and d 5 [r(x) 2 r̄]/r̄. In fact, this is the leading behavior for an arbitrary local bias (Coles 1993), where the galaxy distribution is a function of the mass density n g (x) 5 f [r(x)], or equivalently dg 5 O k!1 b d . k k 2. THE EVOLUTION OF BIAS The cosmological mass density evolves by gravitational instability following the equation of continuity and the Euler equation in an expanding universe (Peebles 1980). It is useful to write these in terms of the velocity potential v 5 ȧ=Q and velocity divergence = z v 5 ȧu (Bernardeau 1994). For V 5 1, the evolution equations then become d 1 =Q z =d 1 ~1 1 d!u 5 0, a (2) u 1 3 1 u 1 =Q z =u 1 Q , ij Q , ji 1 d 5 0. a 2 2 (3) a a These equations can be solved systematically in perturbation theory, order by order in the initial density contrast d 0 5 d(x, t 0 ) at an early time t 0 (Peebles 1980; Fry 1984; Goroff et al. 1986). To linear order, the density contrast is simply the initial d 0 multiplied by an overall scale factor that depends on time; for V 5 1, this factor is the same as the cosmological scale factor, d (1) 5 a(t)d 0 , with u (1) 5 2d (1) . The full evolution is nonlinear, inducing higher order terms. To second order, the velocity divergence is (1) The usual linear bias parameter is b 5 b 1 . Galaxy correlations then depend on the bias parameters b k as well as on the underlying density correlations (Fry & Gaztañaga 1993). This ignores evolution. Physically, bias in the galaxy distribution is not applied today but arises at the epoch of galaxy formation. The process that determines where galaxies form cannot affect how they move; thus, after galaxy formation, the distribution evolves under the influence of gravity. In this Letter, I compute evolution of bias in a model in which galaxies form by an arbitrary local process as in equation (1) at the time of galaxy formation but then move with the streaming velocity given by the gravitational potential or ‘‘go with the flow’’ (Grinstein et al. 1987). Calculations of the first- and u ~2! 5 a 2 ~t!~ 73 d20 2 =D 0 z =d 0 1 74 D 0, ij D 0, ij !, (4) where D is the solution to = D 5 2d (Peebles 1980; Fry 1984), 2 D~x! 5 E d 3 x9 d~x9! . 4p u x 2 x9u (5) I calculate the evolution of d g in the model that galaxy formation takes place at an instant t* by a nonlinear, biased process, with subsequent motion determined by gravity. The L65 L66 FRY Vol. 461 galaxy number density contrast d g is then determined by its own equation of continuity, a d g 1 =Q z =d g 1 ~1 1 d g !u 5 0 a (6) with u known from above and with initial conditions at t* given by equation (1); in particular, b 1 (t*) 5 b*. At linear order, d g satisfies d~1! g 5 2u ~1! , a a (7) where u (1) 5 2a(t)d 0 , with solution d~1! g 5 ~a 1 b 2 1!d 0 . * (8) Normalizations are chosen so that a(t ) 5 1; this result applies * for a $ 1. The effective bias b eff 5 d g /d depends on time as b eff ~t! 5 a~t! 1 b* 2 1 . a~t! (9) With d g(1) as source, to second order in d 0 , the galaxy number contrast satisfies a d~2! g ~1! ~1! 5 2u ~2! 2 =Q ~1! z =d~1! g 2 u dg , a (10) 3 @~a 2 !d 2 a=D0 z =d0 1 D0, ij D0, ij # 1 b2 d . * 2 7 1 2 2 0 (11) Higher order contributions can be computed systematically. Q ij 5 ~ 10/ 7 ! a 2 1 2~b* 2 1!~a 2 2/ 7! a 1 ~a 1 b* 2 1! 2 ~a 1 b* 2 1! 3 k̂ i z k̂ j S D ki kj ~ 4/ 7!~a 2 1 b* 2 1! 1 1 kj ki ~a 1 b* 2 1! 2 3 ~k̂ i z k̂ j ! 2 1 3. MOMENTS After the two-point function and power spectrum, which are 2 , information is commonly each multiplied by the factor b eff extracted from third moments. Equations (8) and (11) have effects on these moments. The three-point amplitude S 3 5 ^d 3 &/^d 2 & 2 , averaged over a top-hat window (cf. Juszkiewicz, Bouchet, & Colombi 1993; Bernardeau 1994), is S 3, g 5 (15) where 2 2 5 2 d~2! g 5 a ~ 7 d 0 2 =D 0 z = d 0 1 7 D 0, ij D 0, ij ! 1 ~b 2 1! * 2 0 With the solutions (8) and (11), the bispectrum is B 123 5 Q 12 P 1 P 2 1 Q 13 P 1 P 3 1 Q 23 P 2 P 3 , where u (2) is given in equation (4), with solution 2 7 FIG. 1.—Evolution of the skewness amplitude S 3, g as a function of a(t) with initial b 5 4, b 2 /b 2 5 1. Curves show results for spectral index n 5 23, 22, * * * 21, 0, and 11 (top to bottom). The horizontal dashed lines show the result without bias, S 3 5 34/ 7 2 (3 1 n). b2 * . ~a 1 b 2 1! 2 (16) a 2 ~ 34/ 7! 1 ~6a 2 8/ 7!~b* 2 1! 1 3b 2 * ~a 1 b* 2 1! 2 2 ~3 1 n! a . ~a 1 b* 2 1! (12) For a 3 E, this reduces to S 3 5 34/ 7 2 (3 1 n), the result without bias; for a 5 1, S 3, g 5 S 3 /b* 1 3b 2 */b*2 , the nonlinear bias result for the initial conditions (Fry & Gaztañaga 1993). The behavior of S 3, g is shown in Figure 1 as a function of a(t), for n 5 23, 22, 21, 0, and 11 (top to bottom) for initial conditions b* 5 4, b 2 */b*2 5 1. This initial b*, somewhat larger than the usual value, gives b eff 5 2 at a 5 3, ‘‘today.’’ Further information is found in moments in Fourier space, the power spectrum and bispectrum, ^d̃ g ~k 1 !d̃ g ~k 2 !&5~2p! d D ~k 1 1 k 2 ! P~k! (13) ^d̃ g ~k 1 !d̃ g ~k 2 !d̃ g ~k 3 !&5~2p! 3 d D ~k 1 1 k 2 1 k 3 ! B 123 . (14) 3 FIG. 2.—Evolution of the bispectrum amplitude Q(u ) for configurations with sides k 1 5 1, k 2 5 21 , separated by angle u with spectral index n 5 21. Shown are initial conditions as in Fig. 1 (solid line) and the evolved result at a 5 2 (long-dashed line), a 5 5 (short-dashed line), a 5 10 (dot–long-dashed line), a 5 20 (dot–short-dashed line), and a 3 E, the result with no bias (dotted line). No. 2, 1996 EVOLUTION OF BIAS TABLE 1 BIAS FIT PARAMETERS a 2 3 5 10 20 .................. .................. .................. .................. .................. 1/b eff 1/b fit (b 2 /b 2 ) eff (b 2 /b 2 ) fit 2/5 3/6 5/8 10/13 20/23 0.349 0.430 0.546 0.702 0.824 16/25 16/36 16/64 16/169 16/529 0.731 0.571 0.392 0.216 0.112 The reduced bispectrum amplitude, Q123 5 B123 /(P1 P2 1 P1 P3 1 P2 P3 ), is independent of scale and depends only on the shape of the triangle defined by k1 , k2 , k3 . Its evolution is shown in Figure 2 for n 5 21 with the same initial conditions as in Figure 1. For nonlinear bias of the form in equation (1), the bispectrum amplitude is Q g 5 Q/b 1 b 2 /b 2 . Using this form and the known dependence of Q on configuration shape in perturbation theory has been proposed as a technique to extract the bias parameters from observations (Fry 1994). With evolution, the problem is not that the value of b changes with time, but that the factors appearing in Q do not all have the same time dependence, so that the shape dependence of Q changes with evolution. Fitting the curves in Figure 2 to the naive bias form with constant b’s gives the results in Table 1. In all cases, the rms difference between the fit and true curves is of order 0.001– 0.002. 4. DISCUSSION The results above show that when galaxy motions follow from the net cosmological gravitational potential, an initial bias in the galaxy distribution relative to the mass evolves with time, the galaxy distribution eventually relaxing to that of the mass. This confirms the qualitative expectation that if galaxy motions are determined by gravity, a large bias cannot be sustained. Expressed in terms of the redshift z of galaxy formation, the present bias is b 1z b0 5 * . 11z (17) The reverse of this is that any present bias must have been even larger in the past. Frieman & Gaztañaga (1994) have examined S 3, g in the Automatic Plate Measuring Facility catalog and find no need for bias to explain their results. However, even without evolution, the amplitude S 3 provides only one constraint on two bias parameters. Figure 1 shows that S 3, g for n 5 21 is L67 near its asymptotic value for a large span of its evolution, but at a closer look, the contributions from the different terms vary greatly. A stronger test is the dependence of Q 123 on configuration shape. The bispectrum of the Lick galaxy catalog (Fry & Seldner 1982) shows a very weak dependence on configuration shape, which at present can be explained only by bias, and the dependence on configuration shape has been proposed as a way to determine b (Fry 1994). Evolution would seem to make that more difficult, with three parameters, b , b 2 */b*2 , and * a(t)/a(t ) in equation (11). The epoch of galaxy formation can * presumably be determined by other observations, and a fit to the form in equation (16), although no longer a linear fit, can be performed. However, an evolving bias does not greatly change the qualitative or even quantitative behavior: a flat Q requires a large present value of b. Ignoring evolution and fitting to the form Q g 5 Q/b 1 b 2 /b 2 with Q as given by perturbation theory gives values of b fit that are within 5%–15% of the correct answer, smaller than present observational errors. The evolution of bias has been little addressed in numerical simulations. In one study that does, Katz, Quinn, & Gelb (1993) compare peaks in the initial density field with the halos that actually form in a cold dark matter simulation by tagging the particle closest to each peak and examining the distribution of these tagged particles at later times. They report that the degree of biasing decreases at larger expansion factors, but they do not quantify the effect. There has been no study to date of the evolution of the three-point correlation function or the bispectrum for such a tagged subset. It is important to separate a physical bias at the time of galaxy formation, which evolves, from a statistical bias in the definition of objects, such as clusters larger than some threshold. As discussed in Fry & Gaztañaga (1993), the composition of a sequence of biases generates a net effective bias. Additional nonlinear effects in the observation process (does the identification of a ‘‘galaxy’’ on a photographic plate or in a CCD image depend on whether there are other galaxies nearby?) and other subsequent processing can modify the results further. Although the skewness S 3 may be explainable without need for bias, there is at present no other way to explain a flat dependence of the bispectrum amplitude Q on configuration shape. I thank Adrian Melott for helpful suggestions on the presentation. 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