The Evolution of Bias - Generalized

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by  Lam Hui

The Evolution of Bias - Generalized
Lam Hui1,2 and Kyle P. Parfrey1,3
1Institute for Strings, Cosmology and Astroparticle Physics (ISCAP)
2Department of Physics, Columbia University, New York, NY 10027, U.S.A.
3Department of Astronomy, Columbia University, New York, NY 10027, U.S.A.
[email protected], [email protected]
(Dated: February 2, 2008)
Fry (1996) showed that galaxy bias has the tendency to evolve towards unity, i.e. in the long
run, the galaxy distribution tends to trace that of matter. Generalizing slightly Fry’s reasoning,
we show that his conclusion remains valid in theories of modified gravity (or equivalently,complex
clustered dark energy). This is not surprising: as long as both galaxies and matter are subject to
thesameforce, dynamicswould drivethemtowards tracingeach other. This holds,for instance, in
8
theorieswherebothgalaxiesandmattermoveongeodesics. Thisrelaxationofbiastowardsunityis
0
temperedbycosmicacceleration,however: thebiastendstowardsunitybutdoesnotquitemakeit,
0
unlesstheformationbiaswereclosetounity. Ourargumentisextendedinastraightforwardmanner
2
to the case of a stochastic or nonlinear bias. An important corollary is that dynamical evolution

n could imprint a scale dependence on the large scale galaxy bias. This is especially pronounced
a if non-standard gravity introduces new scales to the problem: the bias at different scales relaxes
J at different rates, the larger scales generally more slowly and retaining a longer memory of the

5 initial bias. A consistency test of the current (general relativity + uniform dark energy) paradigm
2 is therefore to look for departure from a scale independent bias on large scales. A simple way is
to measure the relative bias of different populations of galaxies which are at different stages of

] bias relaxation. Lastly, we comment on the possibility of directly testing the Poisson equation on
h
cosmological scales, as opposed toindirectly through the growth factor.
p
-
o PACSnumbers: 98.80.-k;98.80.Bp;98.80.Es;98.65.Dx;95.35.+d;95.36.+x;95.30.Sf
r
t
s
a I. INTRODUCTION bias evolution remain valid as long as both galaxies and
[ matter are subject to the same force.

3 In a classic paper, Fry [1] showed that galaxies, once Notethatthesameconclusionholdsif,insteadofmod-
v formed, would evolve to eventually trace mass. This re- ifyinggravity,oneintroducesadditionalsourcesoffluctu-
2 sult was subsequently extended to the case of a stochas- ations, for instance clustered dark energy. The presence
6
tic bias by Tegmark and Peebles [2]. Similar results of dark energy perturbations modifies the relation be-
1
were found in numerical simulations [3]. The linear bias tween matter perturbations and the metric, or the grav-
1
. paradigm has been applied to a variety of cosmological itational potential (just as modifying gravity does). But
2 models, including ΛCDM and quintessence [4]. Both [1] once this gravitational potential is given, both galaxies
1
and[2]assumedNewtonian/Einsteiniangravityfromthe andmatterparticlesrespondtoitinthesameway,there-
7
start. One of our principal aims is to show that relaxing foreguaranteeingeventualevolutiontowardsatrivialbi-
0
: this assumption does not alter the basic conclusions. In asing relation between them. This is consistent with the
v theprocess,wewilluncoveraninteresting,unanticipated fact that modifying gravity can be viewed as some com-
i
X corollary. plicatedformof darkenergy - no surprisegiventhat one
canalwaysshifttermsfromthelefthandsideoftheEin-
r Thebasicreasoningissimple: bothgalaxiesand(dark)
a matter canbe regardedas test particlesin some gravita- stein equation to the right [8]. Modifications of gravity
are indistinguishable from complex dark energy, at least
tional field; since their dynamics are identical, they are
expected to trace each other given sufficient time. This at the linear level, unless constraints are put on the na-
ture of dark energy, such as spatial homogeneity or zero
holds regardless of the origin of the gravitational field.
DeparturesfromNewtonian/Einsteiniangravityonlarge coupling to matter (e.g. [9, 10] and references therein).
scales have been suggested as a possible explanation for Why are we interested in proving these general but in
the observed cosmic acceleration. Most of these theories a sense rather obvious statements about bias evolution?
ofmodifiedgravity,forinstanceDGP[5]orscalar-tensor First, as we will see, there is an interesting and proba-
theories [6], are metric theories, and matter (or galaxy) bly not so obvious corollary: that bias evolution can im-
particles move on geodesics defined by the metric. This printascaledependenceeveniftheinitialbiaswerescale
is true aslongasmatter andgalaxiesareminimally cou- independent. The effect is especially large if one’s non-
pled to gravity in the relevant frame [7]. The statement standard theory of gravity/dark energy introduces new
thatmatterandgalaxiesshouldevolvetotraceeachother scales to the problem, making the growth factor scale
does not rely on how the metric comes about. All that dependent. A promisingwaytotest the current(general
matters is that they have the same dynamics under the relativity + uniform dark energy) paradigm is therefore
metric. Moregenerally,aswewillsee,ourconclusionson tolookforascaledependentbias. Oursecond(andorig-
2
inal)motivationistoaddressthe issueofdirectlytesting erage, v is the matter peculiar velocity, a is the scale
m
the Poisson equation. Most observations, such as grav- factor,φisthegravitationalpotentialormetricperturba-
′
itational lensing and the microwave background, tell us tion, G is Newton’s constant, denotes a derivative with
themetricfluctuationsorgravitationalpotential,theleft respect to conformal time η, and ∇ denotes a derivative
handsideofthePoissonequation. Thereareveryfewob- with respect to comoving position x.
servations that tell us directly what sources the gravita- Equations (1) and (2) express respectively mass and
tionalpotential, the righthand side of the Poissonequa- momentum conservation. Equation (3) is the Pois-
tion. The galaxy distribution provides one such handle, son equation, which tells us how matter fluctuations
but it is complicated by the uncertain galaxy bias. One source the gravitational potential according to Newto-
mighthope thatthe tendency ofthe galaxybias to relax nian/Einsteinian gravity.
towards unity could help bridge the gap between galax- Galaxies(or more precisely,somepopulation thereof),
ies and mass. However,as we will see, in an accelerating onceformed,obeythefollowingequationsonsub-Hubble
universe,the biasrelaxationprocessisquite slow. Modi- scales:
fications of gravitythat predict cosmic accelerationtend δ′ =−∇·[(1+δ )v ] (4)
to slow it further. The slow relaxation process makes g g g
′
it difficult to argue that galaxies that formed at high v′ +(v ·∇)v + a v =−∇φ (5)
redshifts tend to have a trivial bias today, unless they g g g a g
formed with a bias close to unity. On the other hand, as where δ ≡ (n −n¯ )/n¯ is the galaxy overdensity, with
g g g g
wewillsee,theslowbiasrelaxationisaboontothescale n being the galaxy number density and n¯ its spatial
g g
dependence test (motivation number one). average,andv isthegalaxypeculiarvelocity. Equation
g
Twoclarifyingremarksareinorder. First,throughout (4) crucially assumes number conservation, a subject to
this paper, the galaxybias is defined with respect to the which we will return in §VI. Equation (5) is momentum
matter fluctuations. It is importantto bear this in mind conservation,orthe equationofmotion, forthe galaxies.
since, in models that allow clustered dark energy, there It is worth noting that we do not assume v = v , as
m g
are different kinds of galaxy bias: with respect to mat- is often done. This is in part because we want to make
ter, dark energy or the sum. From the point of view of explicit the importance of subjecting the galaxies and
dynamical evolution, the bias with respect to matter is matter to the same force. If they are not subject to the
the natural one to consider. Secondly, an important as- same force, v and v would divergefrom eachother; if
m g
sumptionin[1]aswellashere isthatthe galaxynumber they are, the two velocities tend to converge, as we will
is conserved, just as mass is. In practice, galaxies are of see.
course created (they form) and destroyed (they merge). Equations (1) to (5) constitute the starting point for
A complete theory of galaxy bias needs to fold in these [1,2]ontheevolutionofbias. Ourkeyobservationisthat
processes. We will have more to say about this in §VI. Eq. (3) is unnecessary, at least on a qualitative level.
Dynamical evolution is in any case an important part of This is interesting in light of the recent interest in
the story. theories of modified gravity. By and large, these theo-
Thispaperisorganizedasfollows. Thebasicequations ries respect mass and momentum conservation; particles
arereviewedin§II. Theimplicationsforthelineargalaxy (matterorgalaxy)moveongeodesics. Theonlyequation
bias are worked out in §III. The extension to a (linear) thatismodifiedinthese theoriesisthereforethe Poisson
stochastic bias is presented in §IV, and the extension to equation (3). Similarly, in theories where dark energy
a nonlinear bias is givenin §V. The interesting corollary has non-trivial clustering behavior, dark energy can act
of a possible scale dependent imprint on the galaxy bias as an additional source for gravitational perturbations,
is derived and discussed at the end of both §III and IV. and therefore the simple form of the Poisson equation
In §VI, we conclude with some remarks about testing given in (3) is again modified.
theories of modified gravity or complex dark energy. Henceforth, our derivation relies on Eqs. (1), (2), (4)
and(5),butnoton(3). Inotherwords,ouronlyassump-
tionsare: thatmassandnumberareconserved,andthat
II. BASICS
matter and galaxies are subject to the same force.
According to the current structure formation
paradigm, the fundamental equations governing the III. LINEAR BIAS
dynamics of sub-Hubble matter fluctuations are [12]:
δ′ =−∇·[(1+δ )v ] (1) Keeping only terms linear in fluctuations, one can see
m m m
′ from Eqs. (2) and (5) that
a
vm′ +(vm·∇)vm+ avm =−∇φ (2) (v −v )′+ a′(v −v )=0 (6)
g m g m
∇2φ=4πGρ¯ a2δ (3) a
m m
which implies
whereδ ≡(ρ −ρ¯ )/ρ¯ isthematteroverdensity,with
m m m m
ρ being the matter mass density and ρ¯ its spatial av- (v −v )=u /a (7)
m m g m 0
3
where u is independent of time but dependent on posi- can define the bias b in Fourier space as well, in which
0
tion. Notethatinderivingthisresult,wedonotneedthe case Eq. (12) takes the form:
righthandsidesofEqs. (2)and(5)tobe the gradientof
δ (a ,k)
some potential; all we need is that they are identical i.e. b(a,k)=1+(b −1) m 0 . (13)
matterandgalaxiesaresubjecttothe sameforce. Equa- 0 δm(a,k)
tion (7) tells us that if v = v initially, it will remain
g m In the standard paradigm of general relativity plus uni-
so; any initial velocity bias, should it exist, decays away.
form dark energy, different sub-Hubble Fourier modes of
This holds in the context of linear perturbation theory
δ grow with the same (k independent) growth factor,
m
(i.e. large scales) [11]. In fact, [1, 2] set the galaxy and
hence aninitially scaleindependent bias(b )willremain
0
matter velocities to be equal from the outset.
scale independent (b). Note that according to the stan-
Similarly, linearizing Eqs. (1) and (4) and taking the
dard paradigm, the initial linear bias is expected to be
difference, we obtain
scale independent [13, 14, 15].
However, it is entirely possible that modifications to
(δ −δ )′ =−∇·(v −v )=−∇·u /a. (8)
g m g m 0 the Poisson equation (3) take a scale dependent form,
which would make the perturbation growth scale depen-
This can be integrated to give
dent. General relativity itself introduces such correc-
da tions, but they are generallysuppressedby (aH/k)2 and
δg−δm =−∇·u0Z a3H +∆0 (9) are therefore very small except close to the Hubble scale
where cosmic variance is large [16]. A less trivial exam-
where∆ isindependentoftimebutspatiallydependent, ple is a Yukawa-like modification of the Poisson equa-
0
andH istheHubbleparameter. Theintegraloveragen- tion: ∇2 → ∇2−µ2, where 1/µ defines some new scale
erallygivesadecayingtermincontrastwiththeconstant [17,18]. Scalar-tensortheoriesintroducemodificationsof
∆ term,andsotoleadingorder,the galaxybiasevolves this kind where µ2 is related to the second derivative of
0
as: the scalarpotential[19], in addition to the better known
effect of giving G a time dependence [20, 21, 22]. In
δg(a) ∆0 these cases, the linear growth factor δ (a,k)/δ (a ,k)
b(a)≡ =1+ (10) m m 0
δm(a) δm(a) will be k dependent. As a concrete example, we show
in the bottom two panels of Fig. 1 the linear bias as a
where we have inserted the argument a in each quantity
function of redshift and wavenumber k for two different
that is time dependent (while space dependence is kept
populations of galaxies, with the following modification
implicit). The quantity ∆ is independent of time, and
0 to the Poisson equation:
can be alternatively expressed as
(∇2−a2m2)φ=4πGρ¯a2δ (14)
m
∆ =(b −1)δ (a ) (11)
0 0 m 0 where m is a constant and a2m2 plays the role of µ2.
whereb istheinitialbiasatsomeearlytimecorrespond- We use µ ∝ a such that the Yukawa correction becomes
0
ing to scale factor a . From Eq. (10), we can see that as smallintheearlyuniverse[18],althoughsomeothertime
0
long as the matter fluctuation δ grows,the galaxy bias dependence is possible. There is also the question of
m
b evolves towards unity. Exactly how rapidly it evolves how such a scalar-tensor theory can be made consistent
dependsonthegrowthrateofδ ,whichcannotbedeter- with solar system constraints; this depends on the full
m
minedunlessthemodificationtoEq. (3),orlackthereof, Lagrangian of the theory - the kinetic term, the poten-
is specified. For small fluctuations, it is not unreason- tial and the coupling to curvature can always be cho-
abletoexpectthatδ growsroughlyontheHubbletime sen to satisfy such constraints [23]. In Fig. 1, we adopt
m
scale. m=0.05h/Mpcwhichappearstobeconsistentwithcur-
There is an interesting corollary that follows from Eq. rentobservations[17,18]. AflatcosmologyofΩm =0.27
(10). Suppose the galaxy bias is initially scale indepen- and ΩΛ =0.73 is assumed in computing the background
dent. This means δ (a ,x) = b δ (a ,x), where b is expansion rate.
g 0 0 m 0 0
the initial bias which is independent of position, and a From Eq. (14), one can see that the high k modes
0
denotes the scale factor at some early time. Equations (k/a ≫ m) are insensitive to the Yukawa correction.
(10) and (11) then tell us for a≥a : Hence, the high k end of Fig. 1 shows a bias evolution
0
thatisconsistentwithEinsteinian/Newtoniangravity. It
δ (a ,x) is worthstressing that while the bias wants to evolve to-
b(a,x)=1+(b −1) m 0 (12)
0 δ (a,x) wards unity, it does not quite make it to unity. This is
m
the result of cosmic acceleration: perturbation growth
which means b is independent of position, or is scale in- is much suppressed once the universe starts to acceler-
dependent, only if δ (a,x) preserves exactly the initial ate, and the bias relaxation process is stalled. In fact,
m
spatial dependence encoded in δ (a ,x). We know this even if we continue the evolution into the future, the
m 0
holds in general relativity (plus uniform dark energy) at bias will not become very close to unity, unless its ini-
the level of linear theory in the sub-Hubble limit. One tial value were close to unity to begin with. According
4
FIG. 1: The evolution of the linear bias and relative bias FIG.2: AnalogofFig. 1exceptthataDGPmotivatedmod-
as a function of wavenumber k. In the bottom panel is bA, ificationofgrowthrateisused: Eq. (15). Notethatthisdoes
the linear bias of galaxies that formed at redshift 5 with an not account for the scale dependence arising from the tran-
initialscaleindependentbiasofb =2. Inthemiddlepanelis sition from the scalar-tensor regime to the general relativity
0
bB, the linear bias of galaxies that formed at redshift 2 with regime, theso called r∗ effect.
an initial scale independent bias of b = 0.7. The top panel
0
shows the ratio of the two. The number labeling each curve
is the corresponding redshift. This makes use of the Yukawa
modification tothe Poisson equation: Eq. (14). We note in passing that the rather popular f(R) the-
ory is a scalar-tensor theory, and is therefore expected
to exhibit a scale dependent bias qualitatively similar to
to general relativity, δ (a = ∞)/δ (a = 1) ∼ 1.36 for that shown in Fig. 1. It appears, however,that the par-
m m
ouradoptedparameters,implyingbiasrelaxationviaEq. ticular f(R) theories usually considered in the literature
(13) is rather limited in the future. have an effective mass m (or µ) that is rather small [6],
which makes for a weak scale dependence. There is no
If general relativity were the whole story, the curves
fundamental reason, though, to focus only on these par-
in Fig. 1 would have been flat (aside from very small
(aH/k)2 corrections mentioned earlier). Instead, we see ticular realizations of a scalar-tensortheory.
thattheYukawacorrectionfurthersuppressesthegrowth Another example of a modified gravity theory is DGP
ofperturbationsatlowk’s(comparedtohighk’s),which [5], which is expected to introduce scale dependence to
significantlyslowstheirevolutiontowardsabiasofunity. the growthfactor as well. There are two sources of scale
This is qualitatively what one expects from theories of dependence. OneisthatDGPisexpectedtogivecorrec-
modified gravity which are constructed to yield cosmic tions to the growth factor of the order of aH /k, arising
0
acceleration: such theories tend to make gravity weaker fromitspeculiarnonlocalcharacterfromthe4Dperspec-
onlargescales. Inotherwords,the lower-kmodes retain tive (Scoccimarro,private communication). The other is
a longer memory of the initial bias, while the higher- that DGP introduces a special scale, often referred to
k modes catch up more efficiently (but not completely) as r∗, where gravity makes a transition from the gen-
with the matter perturbations. The result is a scale de- eral relativity regime to a scalar-tensorregime [24]. One
pendent bias, as shown in Fig. 1. Note that, in prin- expects the structure growth to have a feature around
ciple, this scale dependence is temporary – if the bias r∗. Its precise value is however uncertain, because it de-
does eventually reach unity in the far future, it becomes pendsontheexactnatureandcharacteristicsofthefluc-
scale independent once again. In practice, in an acceler- tuations. None of the existing calculations of structure
atinguniverse,thebiasrelaxationprocessissoslowthat growthin DGP gravityaddress growthin this transition
this does not actually occur, unless the initial bias were regime [25]. The general expectation is that r∗ is close
quite close to unity. This is a useful feature from the to the nonlinear scale, but its actual value could well be
perspective of preserving, and potentially observing, the larger. Asanillustration,inFig. 2,wesidestepther∗ is-
dynamically generated scale dependent bias. sue,andsimply modulate the DGP growthby anaH /k
0
5
correction (see also [16]): IV. (LINEAR) STOCHASTIC BIAS
δDGP(a,k)=δPoisson(a,k)(1−aH /k) (15) Having a stochastic bias means one introduces an ad-
m m 0 ditional parameter into the problem, following Dekel &
Lahav [28]. In addition to b, defined by,
where δPoisson denotes the matter fluctuation according
to the Pmoisson equation, with a background expansion b2 ≡hδg2i/hδm2 i (17)
chosen to match the expected DGP growth on small
one introduces r, defined by
scales (but larger than r∗) [26]. Here, the scale depen-
dence of the bias is much weaker than in the Yukawa
br≡hδ δ i/hδ2 i (18)
example, though one must remember that this ignores g m m
the r∗ effect. As mentioned before, general relativity it- Here, one can think of the ensemble averages in several
self introduces corrections of this sort, but they come in
different ways. For instance, hδ δ i can be thought of
at quadratic order (aH/k)2 and are even weaker. g m
as the ensemble average of the product of δ and δ ,
g m
Lastly, instead of modifying the Poisson equation (3) each smoothed on some given scale R. It can also be
on the left hand side, one could also modify it on the thought of as the two point correlation function of δ
g
righthandsidebyhavingdarkenergythatclusters. This andδ atsomegivencomovingseparation∆x. Another
m
would introduce new scales to the problem, such as the alternative is to think of it as the power spectrum at a
Jeansscale,andoneexpectsthegrowthofperturbations, wavenumber k: ∝[hδ (k)δ∗ (k)i+hδ∗(k)δ (k)i]/2. De-
g m g m
andthereforethebiasrelaxation,tobecomescaledepen- pendingthereforeonthedefinition,bandrareingeneral
dent. functions of R, ∆x or k respectively. It is worth noting
that cross-correlation measurements of galaxies suggest
We should add that in theories where gravity is mod-
r is consistent with unity on large scales [27].
ified, or clustered dark energy is introduced, it is quite
possiblethatthegalaxiesarebornwithascaledependent SquaringEq. (9), and keepingonly the leadingcontri-
bution to the ensemble average,we have
bias on large scales, contrary to expectations according
to the standard excursion set theory of halos [13, 14]. It
b2−2br+1=h∆2i/hδ2 i (19)
would be very surprising if any such initial scale depen- 0 m
dence conspires to cancel the scale dependence that de- where h∆2i is time independent, but hδ2 i is not. Alter-
velops dynamically. Checking for departure from a scale 0 m
natively, multiplying Eq. (9) by δ , and keeping only
m
independent bias on large scales therefore constitutes an
the leading contribution:
interesting test of(scale dependent) modifications to the
Poissonequation (3). One way to perform this test is to br−1=h∆ δ i/hδ2 i (20)
0 m m
comparethe galaxypowerspectrumandthe masspower
spectrum from lensing (the latter requires the Poisson Theevolutionofthelineargalaxybiasbandthecross-
equationtoconvertfrommetrictomassfluctuations). A correlationcoefficient r is therefore given by:
simplertestistocomparethelargescalebiasofdifferent
populations of galaxies, which are at different stages of h∆ δ i h∆2i 1/2
dynamicalbiasrelaxation. Forinstance,labelingthetwo b= 1+2 0 m + 0 (21)
(cid:20) hδ2 i hδ2 i(cid:21)
populations of galaxiesbyA andB, their relativebias is m m
1 h∆ δ i
0 m
r = 1+ . (22)
b (cid:20) hδ2 i (cid:21)
bA(a,k) 1+(bA−1)δ (aA,k)/δ (a,k) m
= 0 m 0 m (16)
bB(a,k) 1+(bB0 −1)δm(aB0,k)/δm(a,k) Recall that ∆0 is independent of time (Eq. [9]). As-
sumingthe matterfluctuationδ grows,wecansee that
m
at late times:
fora≥a . Aratioofthis kindcanbe obtainedfromthe
0
ratioofthe observedpowerspectraofAandB. Thetop h∆20i ≪ h∆0δmi ≪1 (23)
panels of Fig. 1 and 2 illustrate this ratio for two par- hδ2 i hδ2 i
m m
ticular populations of galaxies. It is worth noting that
the best chance for observing a scale dependent relative and therefore at late times:
biasistocompareoverbias(>1)galaxieswithunderbias
h∆ δ i
(<1) galaxies. This is because modified gravitytheories b∼1+ 0 m (24)
hδ2 i
tend to predict a higher bias for the overbiasedgalaxies, m
andalowerbiasfortheunderbiasedgalaxies. Thiseffect 1h∆2i 1 h∆ δ i 2
r∼1− 0 + 0 m (25)
gets stronger at lower k’s; the scale dependence is most 2hδ2 i 2(cid:20) hδ2 i (cid:21)
m m
visible in their relative bias. So far, there is no observa-
tionalevidencefordeparturefromscaleindependenceon and so, b → 1 and r → 1, as expected, with r tending
large (i.e. linear) scales [27]. to unity at a faster rate than b, although, as emphasized
6
before, cosmic acceleration slows the approach to unity that(b−1)decayslike1/δ(1),consistentwiththefinding
m
considerably. in §III. The second order terms go like
Our remarks in §III concerning the possible develop-
b
mentofascaledependence tothe largescalegalaxybias [(b−1)δ(2)+ 2(δ(1))2]′ = (30)
apply to b and r here as well. In other words, the quan- m 2 m
tities that controlthe evolutionof b and r, h∆0δmi/hδm2 i d3k˜ d3k eik˜·xk·k˜δ(1)′(k)∆(1)(k˜−k)
and h∆20i/hδm2 i, can develop a scale dependence (on R, Z (2π)3(2π)3 k2 m 0
∆x or k), due to possible scale dependent modifications
of the Poissonequation. where we have suppressed the x dependence on the left
handside,andthetime dependence onbothsides. Here,
∆(1) is the Fourier counterpart of ∆ in Eq. (10), and
0 0
V. NONLINEAR BIAS is therefore a first order quantity, and denoted as such.
Note also that it is time independent.
It is straightforwardto see that, to leading order:
Fry & Gaztan˜aga [29] introduced a systematic expan-
sion that relates the galaxy and matter fluctuations:
b =O[δ(2)/δ(1)3]+O[1/δ(1)] (31)
2 m m m
b
δg =bδm+ 22δm2 +... (26) One generally expects that δm(2) ∼δm(1)2, and therefore b2
tends towards zero at late times as long as δ(1) grows.
m
To find the evolution of b , we need to keep terms up to
2 The same caveat applies as before: cosmic acceleration
secondorder in fluctuations. Assuming gradientflow i.e.
significantly slows the relaxation of b .
v =∇φv and v =∇φv, Eqs. (2) and (5) tell us that 2
m m g g It is noteworthy that b develops a scale dependence
2
evenif it startedoff scaleindependent. This is true even
′
a
(φv−φv )′+ (φv −φv ) (27) if the Poisson equation (3) holds [1]. The same is not
g m a g m true for b: the Poisson equation (3) guarantees b stays
1
+ ∇(φv −φv )·∇(φv +φv )=0. scale independent if it started off so, as noted in §III.
2 g m g m
To linear order, this equation implies what we already
VI. DISCUSSION
know from Eq. (7), i.e. (φv −φv )(1) decays like 1/a,
g m
where the superscript (1) denotes the order we are con-
Ourresultsontheevolutionofbiasaresummarizedby
sidering. Whether (φv −φv )(2) decays depends on the
g m Eq. (10), (12) or (13) for the linear bias, by Eqs. (21)
behaviorof(φvg+φvm)(1) whichdependsonhowthePois- and(22) forthe (linear)stochasticbias,andby Eq. (31)
sonequation(3)mightbemodified. Itcanbeshownthat for the nonlinear bias. The Poisson equation (3) is not
(φvg−φvm)(2)decaysawayif(φvg+φvm)(1)growsslowerthan assumedin deriving these results,therefore leaving open
′
a [30]. Henceforth, we will assume so, following [1] who the possibility that gravity is modified or new sources of
essentially assumed vg =vm from the beginning. fluctuations (such as clustered dark energy) exist. The
Equation (1) can be rewritten, up to second order, as only assumptions made are mass and number conserva-
tion, and that matter and galaxies are subject to the
−∇·vm(x)=δm′ (x) (28) same force. For the linear bias and the stochastic bias,
d3k˜ d3k k·k˜ it is not even necessary that the force be the gradient of
−Z (2π)3(2π)3eik˜·x k2 δm′ (k)δm(k˜−k) some potential.
Followingtheseassumptions,ourmainconclusionsare:
where all time dependence is left implicit, and k and k˜ (1) the bias tends to relax towards unity, but this relax-
ationis significantly slowedby the diminishing structure
denote wavevectors. We have abused the notation a bit
growthoncecosmicaccelerationcommences;(2)the cor-
by using δ to stand for both the realspace andFourier
m relation coefficient r tends towards unity faster than the
space counterparts, distinguishing them purely by their
bias b does; (3) the large scale (linear) bias dynamically
arguments. A similar equation holds for the galaxies as
well. Assuming the difference v −v decays away to develops a scale dependence if modifications to the Pois-
g m
son equation are scale dependent.
secondorder(oriszerotobeginwith),wehavetherefore
Conclusion (1) implies, everything being equal, galax-
ies that formed at higher redshifts tend to have a more
d3k˜ d3k k·k˜
[δ (x)−δ (x)]′ = eik˜·x (29) relaxed bias today. The stalling of bias relaxation by
g m Z (2π)3(2π)3 k2
cosmic acceleration, however, implies we cannot auto-
[δ′(k)δ (k˜−k)−δ′ (k)δ (k˜−k)] maticallyconclude thatsuchgalaxieshavecloseto unity
g g m m
biastoday,unless they hadaninitialbias thatwasfairly
which holds to second order. Expanding δ as in Eq. close to unity. In other words,one cannot freely approx-
g
(26), the first order terms in the above expressiontell us imate δ ∼ δ by appealing to bias relaxation alone.
m g
7
This is unfortunate, because δ is about the only ob- scale bias as well. Distinguishing this possibility from
g
servational handle we have on δ without assuming the modifications to the Poisson equation should be feasi-
m
Poisson equation. Other observations, such as the mi- ble since there are other more direct ways to test for
crowave background and gravitational lensing, generally primordial non-Gaussianity, such as from the microwave
probe the metric perturbations, i.e. the left hand side of backgroundanisotropies [34].
the Poisson equation [31]. A direct test of the Poisson As emphasized in §I, our calculation of the bias evo-
equation requires independent knowledge of δm, i.e. the lution assumes galaxy number conservation, just as in
right hand side. It appears one has to be content with [1]. Ultimately, one would like to go beyond the treat-
indirect tests via the linear growth factor, such as those ment in this paper, and formulate a complete theory of
widely discussedin the literature[32]. Theytest forhow halo bias accounting for formation, mergers and dynam-
φ changes with time, which the system of equations (1), ics, extending the excursion set theory [13, 14] beyond
(2) and (3) predicts. The relation between φ and δm is the standard (general relativity + uniform dark energy)
tested only in so far as the prediction for linear growth paradigm. Such a theory would depend on the details of
from this whole system of equations is tested. To go be- how the Poisson equation is modified. It is interesting
yondthis, one would need some independent method for to note that the standard excursion set theory predicts
constraining the galaxy bias. Standard methods to do a bias evolution that is consistent with the one derived
so carry additional baggage. Methods using higher mo- here: b(a)−1∝1/D(a)whereDisthelineargrowthfac-
ments, for instance, assume Gaussian initial conditions tor (the halos are identified atsome scale factor a <a).
0
as well as the Poisson equation itself. This suggests our simple dynamical calculation already
Conclusion(3)is,inourview,themostinterestingone. captures much of the relevant physics. An additional
Itsuggestsasimplewaytotestforscaledependentmod- bonus of extending the excursion set theory beyond the
ifications to the Poissonequation, which are expected in standardparadigmis that one could address the issue of
many existing theories of modified gravity or clustered whethergalaxies/haloscouldbebornwithascaledepen-
dark energy (see §III). One can measure the relative dent bias in non-standard theories. We hope to pursue
large scale (linear) bias of two different populations of these issues in the future.
galaxies which are at different stages of bias relaxation
(Eq. [16]). The standard paradigm of general relativity
+ uniform dark energy predicts a scale independent lin-
Acknowledgments
ear bias (both at birth and in its subsequent evolution).
Checking for departures from scale independence there-
fore constitutes a test of the paradigm. As illustrated in We thank Enrique Gaztan˜aga, Roman Scoccimarro,
thetoppanelsofFig. 1and2,thistestisbestcarriedout Ravi Sheth, Sheng Wang and Pengjie Zhang for discus-
by comparing overbiased and underbiased galaxies. It is sions. LH thanks the Fermilab Theoretical Astrophysics
also important to focus on large scales, since the bias is Groupforareunionthatstimulatedsomeofthesediscus-
expectedtodevelopscaledependenceonnonlinearscales. sions. Research for this work is supported by the DOE,
It was recently noted by [33] that primordial non- grant DE-FG02-92-ER40699, and the Initiatives in Sci-
Gaussianity could give rise to a scale dependent large ence and Engineering Programat Columbia University.
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2
anycase,thestandard(generalrelativity+uniformdark [34] Modifications of gravity, which are generally a late time
energy)paradigmtypicallygivesagrowthto(φv+φv )(1) effect, does not significantly affect the microwave back-
g m
that is slower than a′, and modifications of gravity to ground except in a minor way through the integrated
Sachs-Wolfeeffect.Forcurrentconstraintsonprimordial
accommodatetheobservedcosmicaccelerationgenerally
non-Gaussianity from the microwave background, see P.
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JCAP 03, 05 (2007).
[31] These observations actually measure a combination of