Download THE EVOLUTION OF BIAS ABSTRACT Bias in the

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
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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
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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. This research was supported in part by NASA
under grant NAG5-2835.
REFERENCES
Bernardeau, F. 1994, A&A, 291, 697
Coles, P. 1993, MNRAS, 262, 1065
Davis, M., & Djorgovski, S. 1985, ApJ, 299, 15
Davis, M., & Geller, M. J. 1976, ApJ, 208, 13
Dressler, A. 1980, ApJ, 236, 351
Frieman, J., & Gaztañaga, E. 1994, ApJ, 437, L13
Fry, J. N. 1984, ApJ, 279, 499
———. 1994, Phys. Rev. Lett., 73, 215
Fry, J. N., & Gaztañaga, E. 1993, ApJ, 413, 447
Fry, J. N., & Seldner, M. 1982, ApJ, 259, 474
Goroff, M. H., Grinstein, B., Rey, S.-J., & Wise, M. 1986, ApJ, 311, 6
Grinstein, B., Politzer, H. D., Rey, S.-J., & Wise, M. B. 1987, ApJ, 314, 431
Hamilton, A. J. S. 1988, ApJ, 331, L59
Jain, B., & Bertschinger, E. 1994, ApJ, 431, 495
Juszkiewicz, R., Bouchet, F. R., & Colombi, S. 1993, ApJ, 412, L9
Kaiser, N. 1984, ApJ, 284, L9
Katz, N., Quinn, T., & Gelb, J. M. 1993, MNRAS, 265, 689
Peebles, P. J. E. 1980, The Large-Scale Structure of the Universe (Princeton:
Princeton Univ. Press)
Postman, M., & Geller, M. J. 1984, ApJ, 281, 95
Saunders, W., Rowan-Robinson, M., & Lawrence, A. 1992, MNRAS, 258, 134
Strauss, M. A., Davis, M., Yahil, A., & Huchra, J. P. 1992, ApJ, 385, 421