On the equivalence between the effective cosmology and excursion set treatments of environment
aa r X i v : . [ a s t r o - ph . C O ] J a n Mon. Not. R. Astron. Soc. , 1–4 (0000) Printed 24 October 2018 (MN L A TEX style file v2.2)
On the equivalence between the effective cosmology andexcursion set treatments of environment
Matthew C. Martino & Ravi K. Sheth ⋆ Department of Physics & Astronomy, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19104, USA
24 October 2018
ABSTRACT
In studies of the environmental dependence of structure formation, the large scaleenvironment is often thought of as providing an effective background cosmology: e.g.the formation of structure in voids is expected to be just like that in a less denseuniverse with appropriately modified Hubble and cosmological constants. However,in the excursion set description of structure formation which is commonly used tomodel this effect, no explicit mention is made of the effective cosmology. Rather, thisapproach uses the spherical evolution model to compute an effective linear theorygrowth factor, which is then used to predict the growth and evolution of nonlinearstructures. We show that these approaches are, in fact, equivalent: a consequence ofBirkhoff’s theorem. We speculate that this equivalence will not survive in models wherethe gravitational force law is modified from an inverse square, potentially making theenvironmental dependence of clustering a good test of such models.
Key words: methods: analytical - dark matter - large scale structure of the universe
One of the standard predictions of nonlinear hierarchicalstructure formation models is the abundance of virializedstructures (Press & Schechter 1974; Sheth & Tormen 1999;Jenkins et al. 2001). Simulations show that this abundancedepends on the large scale environment: the ratio of mas-sive to low mass objects is larger in dense regions (e.g.,Frenk et al. 1988). Recent measurements in galaxy surveysappear to bear this out: the virial radii of objects in un-derdense regions are smaller, consistent with their havingsmaller masses (e.g., Abbas & Sheth 2007).This paper is motivated by the fact that there arecurrently in the literature three methods for estimatinghow the mass function of virialized halos depends on theenvironment which surrounds them. The first, and per-haps easist to implement, is based on the excursion setapproach (Mo & White 1996; Sheth & Tormen 2002). Thesecond argues that halos which form in, say, voids shouldbe thought of as forming in a less dense background cos-mology, so the mass function is that in a universe withΩ void = Ω (1 + ∆ void ) (e.g., Gottl¨ober et al. 2003). Thethird is similar, but notes that to correctly estimate thebackground cosmology, one must account not only for thelower density in a void, but for the fact the effective Hubbleconstant of the void cosmology is larger than in the back-ground (e.g., Goldberg & Vogeley 2004). One way of think- ⋆ E-mail: [email protected] ing about the effective Hubble constant is that it ensuresthat the effective cosmology has the same age as the back-ground cosmology. (The cosmological constant is, of course,constant, but when expressed in units of the critical den-sity in the effective model, it is modified because the crit-ical density depends on the effective Hubble constant.) InSection 2, we use the spherical evolution model to showthat the first and third methods are equivalent (althoughGoldberg & Vogeley 2004, state otherwise), and that bothare incompatible with the second method (which incorrectlyignored the change to the Hubble constant).There has been recent interest in the fact that the for-mation histories of halos of fixed mass depend on their envi-ronment (Sheth & Tormen 2004; Gao et al. 2005), an effectwhich is not predicted by the simplest excursion set meth-ods (e.g., White 1996). So one might have wondered if thisis where the difference between the excursion set approachand one based on the effective cosmology is manifest. In Sec-tion 3 we show that in this case also, the two approaches areequivalent.A final section summarizes our results, and speculatesthat the equivalence we have shown will not survive in mod-els models where the force law has been modified from aninverse square. c (cid:13) M. C. Martino & R. K. Sheth
The main point of the following calculation is to show ex-plicitly that, at least for cosmologies with no cosmologi-cal constant, the environmental dependence of halo abun-dances can be described using the excursion set approach(e.g., Mo & White 1996; Sheth & Tormen 2002). Namely,one need not worry about the details of the effective cosmol-ogy associated with the region surrounding the perturbation(as do Goldberg & Vogeley 2004); it is enough to computean effective growth factor using the spherical collapse model.Although we have phrased our discussion in terms of anΩ = 1 background cosmology, it is obviously applicable toarbitrary values of Ω . Our analysis suggests that this re-mains true when the background cosmology has Λ = 0.For what follows, it is useful to recall that the age-redshift relation in an Ω = 1 cosmology is given by H ( z ) t ( z ) = 2 /
3, where H is the Hubble constant. In anopen universe, this relation is H ( z ) t ( z ) = 11 − Ω( z ) − Ω( z ) / − Ω( z )) / cosh − „ z ) − « (1)where Ω( t ) = Ω /a ( t ) Ω /a ( t ) + (1 − Ω ) /a ( t ) , (2)with the convention that a ( t ) ≡
1, so a ( t ) ≡ (1 + z ) − , andΩ ≡ Ω( t ) (Peebles 1980).The linear theory growth factor is D ( t ) = a ( t ) if Ω = 1,and if Ω < D ( t ) = 5Ω / − Ω ) x + 3 r xx ln ˆ √ x − √ x ˜! (3)where x = a ( t ) (1 − Ω ) / Ω (Peebles 1980). The spherical evolution model describes the evolution of thesize R of a spherical region in an expanding universe:d R d t = − GM ( < R ) R . (4)It provides a parametric relation between the den-sity contrast predicted by linear theory δ ( t ) = D ( t ) /D ( t init ) δ ( t init ), the nonlinear overdensity ∆, andthe infall speeds v pec (Gunn & Gott 1972; Schechter 1980;Peebles 1980; Padmanabhan 1993; Bernardeau et al. 2002).Here D ( t ) is the linear theory growth factor at time t , andwe will often use the shorthand, δ = δ ( t ).If Ω = 1, then M πR ¯ ρ/ ≡ f ( θ ) , v pec HR ≡ g ( θ ) , and δ ≡ h ( θ ) , (5)where f ( θ ) = (9 /
2) ( θ − sin θ ) / (1 − cos θ ) (9 /
2) (sinh θ − θ ) / (cosh θ − ,g ( θ ) = (3 /
2) sin θ ( θ − sin θ ) / (1 − cos θ ) (3 /
2) sinh θ (sinh θ − θ ) / (cosh θ − ,h ( θ ) = (3 /
5) (3 / / ( θ − sin θ ) / − (3 /
5) (3 / / (sinh θ − θ ) / , where the first expression in each pair is for initially over-dense perturbations and the second is for underdense ones.Overdense perturbations eventually collapse, the final col-lapse being associated with the value θ = 2 π , at which timethe linear theory density is δ c1 ≡ (3 / π/ / = 1 . = 1.If Ω <
1, then only perturbations above some density δ min will collapse, and1 + ∆ = f ( θ ) f ( ω ) , v pec H ω R = g ( θ ) g ( ω ) − , and δ δ min = − h ( θ ) h ( ω ) + 1 , (6)where ω = arccos(2 / Ω −
1) if closedarccosh(2 / Ω −
1) if open , (7) H ω is the Hubble constant, and δ min = 92 sinh ω (sinh ω − ω )(cosh ω − − . (8)Complete collapse is again associated with θ = 2 π , and wewill write the critical linear density required for collapse as δ c ω = δ min [1 − δ c1 /h ( ω )] . (9)It happens that δ c ω depends only weakly on Ω . When Ω →
1, then δ c ω → (3 / π/ / = 1 . δ c ω → / → δ and ∆ is rather well ap-proximated by 1 + ∆ ≈ (1 − δ /δ c ω ) − δ c ω . (10)Similarly, it is also useful to have an approximation to theexact solution for the linear theory growth factor. WhenΩ ≤
1, then the linear theory growth factor is well approx-imated by D ( t ) ≈ (5 / a ( t ) Ω( t )Ω( t ) / + 1 + Ω( t ) / , (11)(Carroll et al. 1992), where a ( t ) denotes the expansion fac-tor at time t , and Ω( t ) is given by equation (2). This expres-sion is normalized so that D ( t ) = a ( t ) = 1 if Ω = 1. Suppose we consider the evolution of a spherical underdenseregion in an Ω = 1 universe. Let 1 + ∆ ω < ω , and secondbecause the region is expanding faster than the background,so it has an effective Hubble constant H ω which is larger.To see what equation (6) implies for the evolution, let1 + ∆ denote the density of a small patch respect to thebackground density (the subscript unity denotes the factthat this is the overdensity with respect to a backgroundwhich has critical density: Ω = 1). Now, suppose that thispatch is surrounded by a region U within which the averagedensity is 1 + ∆ ω with respect to the true background. Then c (cid:13) , 1–4 ffective cosmology and excursion set equivalence the smaller patch has overdensity (1 + ∆ ) / (1 + ∆ ω ) withrespect to its local background. If we wish to describe the lo-cal environment as has having its own effective cosmologicalparameters, then the local value of the Hubble constant H ω differs from the global one H : H ω /H = g ( ω ). Thus, theexpressions in equation (6) are really the statements that1 + ∆ = 1 + ∆ ω and v pec H ω R = v pec1 − u pec1 u pec1 , (12)where u pec1 is the peculiar velocity of the shell U withrespect to the background, had the mass within U beensmoothly distributed (we know it is not because the cen-tral region has density 1 + ∆ ). Now, the local value of Ω ω within U differs from the global value Ω = 1 both because∆ ω = 0 and because the different expansion rate means thatthe local value of the critical density is different:Ω ω ( t ) = Ω (1 + ∆ ω )( H ω /H ) = f ( ω ) g ( ω ) , (13)where we have used the fact that Ω = 1. Notice that thisrelation between Ω ω and ω is the same as equation (7). Inother words, we get the same description for the evolutionof the small scale patch if we treat it as having overdensity1 + ∆ with respect to the Ω = 1 background within whichthe Hubble constant is H , as if we describe it with respectto the local cosmological model Ω ω and H ω , and we rescaleour definitions of density and peculiar velocity accordingly.In addition, using the exact expression for the age of theuniverse given above, we can see that these definitions alsoguarantee that t is the same in the both the backgroundand the local cosmological model.If we write the linear theory overdensity associated with1 + ∆ ω as δ ω = h ( ω ) , (14)then δ = δ min − δ ω h h ( θ ) − δ ω i = δ c ω δ c1 − δ ω h h ( θ ) − δ ω i . The term in square brackets is simply the difference in lineartheory values for the background cosmology. If we think ofthis as an effective linear theory overdensity in the effectivecosmology, then the prefactor is the effective linear theorygrowth factor. It is straightforward to verify that, indeed, δ c ω /δ c1 − δ ω /δ c1 = D ω D or δ c ω D ω = δ c1 − δ ω D . (15)where D is the growth factor in the background cosmol-ogy, and D ω is the growth factor in the patch, at time t .This last point is important, as the expansion factor a ( t )in the patch cosmology is not equal to the expansion factorin the background cosmology. In particular, we know that a ω ( t ) /a ( t ) = (1 + ∆ ω ) − / . For completeness, we notethat δ cw = δ min „ π ) / (sinh ω − ω ) / « (16)(recall that we are in an underdense region).In the following, take a ( t ) = 1, so D = 1. The ap-proximate solution (10) of the spherical evolution model shows similar behaviour:1 + ∆ ≡ ω = „ − δ /δ c1 − δ ω /δ c1 « − δ c1 = „ δ c1 − δ δ c1 − δ ω « − δ c1 = „ − δ − δ ω δ c1 − δ ω « − δ c1 = „ − D ω δ − δ ω δ c ω « − δ c1 ≈ „ − δ − δ ω δ c ω /D ω « − δ c ω , (17)where δ denotes the linear theory value associated withthe nonlinear density ∆ for Ω = 1. The final approxi-mation follows from recalling that δ c ω depends only weaklyon cosmology. Comparison with equation (10) shows explic-itly that the relevant linear theory quantity is the differencebetween the Ω = 1 values for the perturbation and theenvironment, and this difference must be multiplied by thelinear growth factor D ω in the effective cosmology.Now, to estimate the mass function of virialized ob-jects, we are interested in the case when θ = 2 π . Theanalysis above shows that δ c ω /D ω = δ c1 − δ ω ; the ob-jects which form in a region of nonlinear density 1 + ∆ ω with respect to the background, with corresponding lin-ear overdensity δ ω , can either be thought of as forming inan effective Ω ω cosmology (e.g., Goldberg & Vogeley 2004),or as forming in the true Ω background cosmology butwith an effective linear theory overdensity which is off-set by δ ω to account for the surrounding overdensity (e.g.,Mo & White 1996; Sheth & Tormen 2002). The second de-scription is easier to implement, and follows naturally fromthe excursion set description. In particular, the analysisabove shows that approaches which do not correctly com-pute Ω ω (e.g., Gottl¨ober et al. 2003 ignore the fact that H ω = H ) are incompatible with the excursion set ap-proach. In any case, the analysis above suggests that suchapproaches are ill-motivated. The previous section showed that the excursion set approachresults in the same expressions for the environmental depen-dence of the present day linear theory growth factor as onederives from thinking of the environment as defining an ef-fective cosmology. So the question arises as to whether ornot the two approaches predict the same evolution. For ex-ample, one might have wondered if the formation historiesof objects are the same in these two approaches.To see that they are, it will be convenient to modify ournotation slightly. We showed that δ c (Ω ω ) D ω = δ c (Ω ) − δ L (∆ ) D (18)where the subscripts 0 mean the present time. The quantity δ L (∆ ) is what we previously called δ ω ; it is the value of theinitial overdensity extrapolated using linear theory (of thebackground cosmology) to the time at which the nonlineardensity is ∆ . Also, we previously had set the growth factorin the background universe at the present time to unity: D = 1. We have written it explicitly here to show that,had we chosen to perform the calculation for some earliertime, then we would have found δ c (Ω ω ) D ω = δ c (Ω ) − δ L (∆ ) D , (19) c (cid:13) , 1–4 M. C. Martino & R. K. Sheth where the subscript 1 denotes the earlier time. I.e., Ω ω isthe effective cosmology associated with the overdensity ∆ ,which itself is related to ∆ by the spherical evolution model(the region that is ∆ today was a different volume in thepast, but its mass was the same.) And, analogously to theprevious expression, δ L (∆ ) is the initial overdensity extrap-olated using linear theory to the (earlier) time at which thenonlinear density was ∆ . Since ∆ is closer to 0 than is ∆ , δ L (∆ ) is also closer to 0 than is δ L (∆ ).If one were to apply the excursion set approach to studyformation histories in the effective cosmology, one would beinterested in the difference between equations (19) and (18): δ c (Ω ω ) D ω − δ c (Ω ω ) D ω = δ c (Ω ) D − δ c (Ω ) D − » δ L (∆ ) D − δ L (∆ ) D – . (20)Now, the quantity in square brackets is δ L (∆ ) D − δ L (∆ ) D = » δ L (∆ ) D /D − δ L (∆ ) – D − = 0 , (21)because δ L (∆ ) and δ L (∆ ) are the same quantity (the ini-tial overdensity), evolved using linear theory to two differenttimes. In particular, δ L (∆ ) is closer to 0 than is δ L (∆ ) by δ L (∆ ) /δ L (∆ ) = D /D . Thus, δ c (Ω ω ) D ω − δ c (Ω ω ) D ω = » δ c (Ω ) D /D − δ c (Ω ) – D − . (22)Note that the expression on the right has no dependence onthe effective cosmology. Moreover, it is exactly the same asthe expression that one obtains when using the excursionset approach to study formation histories in the backgroundcosmology. It is in this sense that the formation historiesof objects are independent of the effective cosmology of theenvironment; the excursion set approach is a simple self-consistent way of exploiting this fact. The excursion set description provides a simple, self-consistent way of estimating the effect of environment onstructure formation and evolution. In particular, it is equiv-alent to using the fact that the large scale environment canbe thought of as providing an effective background cosmol-ogy of the same age (Section 2). Estimating the parametersof the effective cosmology is slightly more involved, but use-ful for running simulations which mimic the formation ofstructure in different environments.In essence, the equivalence between the excur-sion set and effective cosmology descriptions is aconsequence of Birkhoff’s theorem: the evolution ofa perturbation does not depend on its surroundings.There has been recent interest in models with modi-fied gravitational force laws (e.g., Shirata et al. 2005;Stabenau & Jain 2006; Shirata et al. 2007). SinceBirkhoff’s theorem does not apply in such mod-els (Martino et al. 2008; Sch¨afer & Koyama 2008;Dai et al. 2008; Clifton 2006; Capozziello et al. 2007),it will be interesting to see if this equivalence survives.If not, the enviromental dependence of clustering may beadded as another constraint on such models.
ACKNOWLEDGEMENTS
RKS thanks Bepi Tormen for asking about this equivalenceon more than one occasion, and the participants of the meet-ing on Cosmological Voids held in December 2006 at theRoyal Netherlands Academy of Arts and Sciences. We alsothank E. Neistein for insisting that the excursion set andeffective cosmology approaches could not be reconciled withone another.
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