Nonlinear approximation for perturbations in�CDM

Gerasimos Rigopoulos1 and Wessel Valkenburg2

1Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen, D-52056, Germany2Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

(Received 20 April 2012; published 17 August 2012)

We describe inhomogeneities in a �CDM universe with a gradient series expansion and show that it

describes the gravitational evolution far into the nonlinear regime and beyond the capacity of standard

perturbation theory at any order. We compare the gradient expansion with exact inhomogeneous �LTB

solutions (Lemaıtre-Tolman-Bondi metric with the inclusion of a cosmological constant) describing

growing structure in a �CDM universe and find that the expansion approximates the exact solution well,

following the collapse of an over-density all the way into a singularity.

DOI: 10.1103/PhysRevD.86.043523 PACS numbers: 98.80.Cq, 04.25.Nx

I. INTRODUCTION

The study of cosmological inhomogeneities throughperturbation theory has been one of the cornerstones ofmodern cosmology [1]. Indeed, the confrontation of theorywith increasingly precise observations of the CMB and thelarge scale clustering of galaxies through the extended useof linear perturbation theory has played an important rolein the emergence of the concordance�CDM cosmologicalmodel and has allowed meaningful speculation about thevery early universe [2]. At higher order, perturbation the-ory is also being used to probe subdominant nonlinearcorrections and the associated non-Gaussianity, promisingto shed additional light on the primordial universe.

Although crucially important for following the develop-ment of cosmic structures in the primordial universe, andeven more recent eras on large enough scales, standardperturbation theory is naturally limited. In the matter eraperturbations grow, the density contrast becomes nonlinearand the inhomogeneities separate themselves completelyfrom the background expansion; at this point standardperturbation theory breaks down completely at all orders.Some recent reformulations can somewhat extend its rangeof validity in the quasilinear regime [3] but the mostcomplete quantitative description of gravitational cluster-ing is necessarily achieved by Newtonian N-body simula-tions. However, analytic approximations are important forobtaining insights into the process of structure formation.In particular, the Zel’dovich approximation along with arange of improvements is a prominent analytic approxima-tion scheme and can be used to understand various featuresof the resulting cosmic web [4].

In this paper we use an expansion different from wellknown and developed cosmological perturbation theory.We approximate the metric perturbations of a �CDMuniverse by a series whose terms contain an increasingnumber of spatial gradients of the initial metric and notpowers of the metric functions (or the density), with eachterm multiplied by a specified function of time. At leading(‘‘zeroth’’) order this is just the ‘‘separate universe’’

approximation. However, the inclusion of higher orderterms allows for the subhorizon evolution to be traced aswell and already at the first nontrivial order the approxi-mation goes beyond standard perturbation theory and candescribe the nonlinear regime with gravitational collapse/void rarefaction. The inclusion of even higher orders ap-proximates the true evolution to an increasing degree, atleast if followed for a characteristic time scale. It is inter-esting to note that the gradient expansion as will be usedbelow can reproduce the Zel’dovich approximation for thedensity contrast but also allows for relativistic correctionsto be included [5,6]; the latter are indeed crucial forincreased accuracy during the later stages of gravitationalcollapse.The gradient expansion can be used to evolve any initial

configuration of perturbations, for example those describedby a post-inflationary scale invariant spectrum. In thiswork, after presenting the general formulation, we com-pare the first two nontrivial terms in the expansion withexact Lemaıtre-Tolman-Bondi (LTB) [7–9] solutions withthe inclusion of a cosmological constant (�LTB) fromRef. [10] in order to assess the accuracy of the gradientseries approximation. We find the approximation to pro-vide a fairly good description, certainly much beyond thecapacity of standard perturbation theory. We include termswith up to four spatial gradients and the inclusion of higherorder terms is expected to fare even better in accuratelydescribing the nonlinear gravitational evolution.

II. THE INHOMOGENEOUS �CDM UNIVERSE ASA SERIES IN SPATIAL GRADIENTS

The gradient expansion as a method for obtaining solu-tions to the Einstein equations dates back to [11] and hasbeen used as a means for approaching the initial singularityof an inhomogeneous universe [12]. It was discussed inmore general terms in Refs. [13,14] and more recentlyapplied to inflation in Refs. [15,16]. A leading order co-variant computation can be found in Ref. [17]. It as alsoproved useful in studying the backreaction of cosmic

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structure formation, see Ref. [18] and more recentlyRef. [19]. In most past works the gradient expansion hasbeen applied directly on the Einstein equations. However,in Refs. [5,6] a Hamilton-Jacobi (HJ) approach was usedwhich arguably offers a less laborious way to perform theexpansion. In this section we follow the HJ methods toobtain a gradient expansion for inhomogeneities in a�CDM universe.

The applicability of a gradient expansion does not relyon some quantity being a small perturbation around a givenbackground but would generally require the existence of a

hierarchy between spatial and temporal derivatives @tQ >

@iQ� �Q�t >

�QL where Q is either a matter or a metric

quantity with �t and L the characteristic time scale andlength scale of the system respectively. One would generi-cally expect the gradient expansion to be applicable onlength scales L � �t i.e., scales beyond causal contact.This has indeed been used in early universe cosmologywhere @tQ�HQ under the guise of the super-horizon( 1aH < x) ‘‘separate universe’’ approximation which is

nothing but the lowest order in a gradient expansion [20].However, the gradient expansion can also be used forlength scales within causal contact provided that the spatialvariation of the initial data is smooth enough and thespatial gradients small. One could then still use the gra-dient expansion but for a limited time span: bigger spatialvariations would correspond to shorter times for which thegradient series would apparently converge. We will makethese statements more precise below.

A. Hamilton-Jacobi for �CDM

Let us start by writing an action for gravity with acosmological constant � and cold dark matter (CDM)

S ¼Zd4x

ffiffiffiffiffiffiffi�gp �

1

2�ðð4ÞR�2�Þ�1

2�ðg��@��@��þ1Þ

�;

(1)

where � is a potential for the four-velocity of CDM,U� ¼ �g��@��, and � is the energy density. The latteracts as a Lagrange multiplier whose variation ensuresthat U�U� ¼ �1. Variation with respect to � gives the

continuity equation r�ð�@��Þ ¼ 0 while variation of the

CDM part of the action with respect to g�� gives the usualCDM energy momentum tensor T�� ¼ �U�U�. Gravity is

described by the standard Einstein-Hilbert term with the

addition of a cosmological constant with ð4ÞR the four-dimensional Ricci scalar, and � ¼ 8�G. The Arnowitt-Deser-Misner decomposition for the metric can now beused to develop a Hamiltonian formalism for this system.Writing the metric as

g00 ¼�N2þhijNiNj; g0i ¼ �ijN

j; gij ¼ �ij; (2)

and defining the canonical momenta

�ij � �S� _�ij

¼ffiffiffiffi�

p2�

Eij � �ijE

N; (3)

�� � �S� _�

¼ �ffiffiffiffi�

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ �ij@i�@j�

q; (4)

with

Eij ¼ 1

2_�ij �rðiNjÞ; E ¼ hijEij; (5)

we can write the action as

S ¼Z

d4x

��� @�

@tþ �ij

@�ij

@t� NU� NiUi

�; (6)

where

U ¼ 2�ffiffiffiffi�

p��ij�

ij � �2

2

��

ffiffiffiffi�

p2�

ðR� 2�Þ

þ ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ �ij@i�@j�

q; (7)

and

U i ¼ �2rk�ki þ ��@i�: (8)

Choosing the velocity potential � to define the time hyper-surfaces of the system implies that @i� ¼ 0 along with@�@t ¼ 1 and N ¼ 1. The metric then takes the form

ds2 ¼ �dt2 þ �ijðt;xÞdxidxj; (9)

and the spatial coordinate lines comove with the matter—there is no peculiar velocity. This choice then correspondsto the comoving synchronous gauge.1 It also means that��

now plays the role of the Hamiltonian density:� �� �H . By imposing the energy constraint U ¼ 0 we obtainin this gauge a Hamiltonian density for the gravitationalfield

H ¼ 2�ffiffiffiffi�

p �ij�kl

��ik�jl � 1

2�ij�kl

��

ffiffiffiffi�

p2�

ðR� 2�Þ;(10)

where R is the Ricci scalar of the three-dimensional timehypersurfaces. The action finally takes the form

S ¼Z

dtd3x

��ij

@�ij

@t�H þ 2Nirk�

ki

�; (11)

and defines a constrained Hamiltonian system where thecanonical momentum �ij is constrained to be covariantlyconserved

rk�ki ¼ 0: (12)

Let us now apply the HJ approach to the Hamiltoniansystem (11)—see, e.g., Ref. [22]. The action can be con-sidered as a function of the metric and time S½t; �ij�. Thenthe momentum is simply the variation of this action withrespect to the metric �ij ¼ �S=��ij, the Hamiltonian is

minus the rate of change of the action with time and the HJequation

1There is conceptual similarity of this choice to theLagrangian picture of cosmological perturbations [21]; seealso the discussion below on the relation to the Zel’dovichapproximation.

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@S@t

þZ

d3x

�2�ffiffiffiffi�

p �S��ij

�S��kl

��ik�jl � 1

2�ij�kl

�

�ffiffiffiffi�

p2�

ðR� 2�Þ�¼ 0;

(13)

contains all the dynamical information of the system. It is asingle partial differential equation for S as a functional of�ij and a function of t. Once S½t; �ij� is determined the

metric can be obtained from

@�ij

@t¼ 2ffiffiffiffi

�p �S

��kl

ð2�ik�jl � �ij�klÞ: (14)

Furthermore, the remaining constraint (12) will be auto-matically satisfied if

S ¼Z

d3xffiffiffiffi�

pF ðt; �ijÞ; (15)

where F ðt; �ijÞ is some scalar function of the metric mak-

ing S invariant under three-dimensional diffeomorphisms.Indeed, the variation of such a functional with respect tothe metric will yield a covariantly conserved tensor whichwill thus satisfy (12). Once a solution is obtained, the CDMdensity can be obtained from

�ðt;xÞ ¼ �0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ðt;xÞp ; (16)

where �0 is a constant. Equations (13) and (14) are thebasic equations that need to be solved in this approach.

B. The gradient expansion

The gradient expansion in this formulation begins bywriting an ansatz for F as [5,6]

F ¼ �2HðtÞ þ JðtÞRþ L1ðtÞR2 þ L2ðtÞRijRij þ � � � ;(17)

a series in powers of the three-dimensional Ricci curvature(involving an increasing number in gradients of �ij). The

HJ equation (13) can now be solved separately for termswith a different number of gradients. This gives for thetime dependent coefficients in (17)

dH

dtþ 3

2H2 ��

2¼ 0; (18)

dJ

dtþ JH � 1

2¼ 0; (19)

dL1

dt� L1H � 3

4J2 ¼ 0; (20)

dL2

dt� L2Hþ 2J2 ¼ 0; (21)

etc. The solution for H then determines all other functions.For � � 0 Eq. (18) gives

H ¼ffiffiffiffi�

3

scoth

� ffiffiffiffiffiffiffi3�

p2

t

�: (22)

As the notation anticipated, H is simply the backgroundHubble parameter. The other functions are then given by

JðtÞ ¼ e�R

t

tiHZ t

ti

1

2e

Rx

tiHdx

¼ 1ffiffiffiffiffiffiffi3�

p ½sinhffiffiffiffiffi3�

p2 t�2=3

Z ffiffiffiffi3�

p2 tffiffiffiffi

3�p2 ti

ðsinhuÞ2=3du; (23)

L1ðtÞ ¼ 3

4LðtÞ; L2ðtÞ ¼ �2LðtÞ; (24)

where

LðtÞ ¼ e

Rt

tiHZ t

ti

½e�R

x

tiHJ2�dx

¼ 2ffiffiffiffiffiffiffi3�

p�sinh

ffiffiffiffiffiffiffi3�

p2

t

�2=3 Z ffiffiffiffi

3�p2 tffiffiffiffi

3�p2 ti

ðsinhuÞ�2=3J2ðuÞdu;(25)

ti is the initial time and u ¼ffiffiffiffiffi3�

p2 t. Given the above, the

metric can now be obtained by solving Eq. (14). Up toterms with four gradients we obtain

@�ij

@t¼ 2H�ij þ JðR�ij � 4RijÞ þ L1ð3�ijR

2 � 8RRij

þ 8RjijÞ þ L2ð3�ijRklRkl � 8RikR

kj þ 3�ijR

jkjk

þ 4Rikjjk þ 4Rj

kjik � 4Rij

jkjk � 4�ijR

kljklÞ

þOð6Þ: (26)

So far, the resulting Eq. (26) is exact if one includesall orders. The gradient approximation now consists ofsolving this equation iteratively for �ij. At zeroth order

in gradients we have

@�ð0Þij

@t¼ 2H�ð0Þ

ij ; (27)

which gives

�ð0Þij ðtÞ ¼ A2ðtÞkij; (28)

where kij is the initial metric at time t ¼ ti and

AðtÞ ¼ e

Rt

tiH ¼

"sinh

ffiffiffiffiffi3�

p2 t

sinhffiffiffiffiffi3�

p2 ti

#2=3

: (29)

This is simply the separate universe approximationwhere each point evolves as a homogeneous Friedman-Robertson-Walker (FRW) universe. Up to second order ingradients we get

@�ð2Þij

@t¼ 2H�ð2Þ

ij þ JðRkij � 4RijÞ; (30)

where we have substituted the zeroth order result in

the terms that contain two gradients: Rij � RijðklmÞ and

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R � RðklmÞ are the Ricci tensor and Ricci scalar of theinitial metric kij. The solution of (30) gives

�ð2Þij ¼ A2ðtÞkij þ ðtÞðRkij � 4RijÞ; (31)

where

ðtÞ ¼ e2R

t

tiHZ t

ti

½e�2R

x

tiHJ�dx

¼ 2ffiffiffiffiffiffiffi3�

p�sinh

ffiffiffiffiffiffiffi3�

p2

t

�4=3 Z ffiffiffiffi

3�p2 tffiffiffiffi

3�p2 ti

ðsinhuÞ�4=3JðuÞdu:(32)

Proceeding to four gradients we have

@�ð4Þij

@t¼ 2H�ij þ JðRkij � 4RijÞ þ C1R

2kij

þ C2RklRklkij þ C3RRij þ C4RikR

kj

þD1Rjkjkkij þD2Rjij þD3Rij

jkjk; (33)

where a vertical bar denotes covariant derivation withrespect to the seed metric kij and the Bianchi identity has

been used to eliminate some terms. The C and D coeffi-cients then read

C1¼8J

A2�23

4

L

A2; C2¼�12

J

A2þ10

L

A2;

C3¼�28J

A2þ18

L

A2; C4¼48

J

A2�32

L

A2;

D�2J

A2�2

L

A2¼D1¼D2¼�1

4D3:

(34)

We obtain

�ð4Þij ¼ A2ðtÞkij þ ðtÞðRkij � 4RijÞ

þ�A2ðtÞ

Z t C1

A2du

�R2kij þ

�A2ðtÞ

Z t C2

A2du

�

� RklRklkij þ�A2ðtÞ

Z t C3

A2du

�RRij

þ�A2ðtÞ

Z t C4

A2du

�RikR

kj þ

�A2ðtÞ

Z t D

A2du

�

��Rjk

jkkij � 4Rijjkjk þ Rjij

�: (35)

One could proceed in this manner to obtain higher orderterms.2

C. Range of validity and the relationto the Zel’dovich approximation

Before proceeding further it is worth estimating therange over which the gradient expansion can be applied.Focusing at early enough times the cosmological constantdoes not play a role and the universe evolves as if it werematter dominated. The metric in the gradient expansionthen reads

�ij ’�t

ti

�4=3

kijþ 9

20

�t

ti

�2t2i ½Rkij� 4Rij�þ 81

350

��t

ti

�8=3

t4i

���4RlmRlmþ 5

8Rjk

jkþ 89

32R2

�kij

� 10RRijþ 17RliRlj� 5

2Rijjk

kþ 5

8Rjij

�; (36)

where we have ignored terms that are subdominant whent � ti. The two-gradient and four-gradient terms becomeof the same order of magnitude as the zeroth order one aftera time t� tcon where

tcon�OðfewÞ 1

t2i R3=2

or tcon�OðfewÞ 1

t2i ðr2RÞ3=4 : (37)

The exact coefficient is of course dependent on the form ofthe initial 3-metric kij. We can therefore estimate that the

expansion should be valid for t & tcon. After this time thecontributions to the metric of successive terms containingmore gradients are no longer perturbatively ordered and theiterative solution to (26) breaks down. However, note thatthe time scale over which the series breaks down is com-mensurate with the time of collapse for a region of positive

spatial curvature (where Rkij � 4Rij < 0). After that the

region either forms a singularity or a caustic is createdwhich would need extra physics for its resolution. Eitherway, one would not expect the current description to holdanymore and thus the limited time scale over which theexpansion holds does not appear to be a serious limitation.For negative curvature regions which keep on expandingthings are less clear-cut since there is no natural limit onthe evolution time scale. Nevertheless, it would seem thatthe gradient expansion can provide a fair description fortimes longer than tcon even in this case—see Ref. [19].Let us now apply the above to our universe. Assuming

the standard inflationary initial conditions, the initial seedmetric takes the form

kij ¼�tit0

�4=3

�ij

�1þ 10

3�ðxÞ

�; (38)

where �ðxÞ is the primordial Newtonian potential and wehave scaled the seed metric such that in the absence ofperturbations the scale factor would be unity today. Notethat the lowest order in the above expansion is simply theseparate universe approximation where each spatial pointscales as a Friedman-Robertson-Walker, CDM, universe

2A gradient expansion solution equivalent to (35) was alsoobtained in Ref. [23] using a method different from the HJapproach utilized here. It would be interesting to further explorethe relation with that approach. Our thanks to the referee forpointing it out.

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with scale factor a / t2=3. With kij as the initial metric,

we have

�ij ’�t

t0

�4=3

��ijþ 3

�t

t0

�2=3

t20�;ijþ�t

t0

�4=3

t40Bij

�þOð6Þ;

(39)

where

Bij ¼ 9

28½19�;il�

;l;j � 12�;ij�

;l;l

þ 3�ijðð�;l;lÞ2 ��;lm�

;lm�: (40)

The time scale for which the above approximation for themetric is accurate is estimated to be

tcont0

’ 3:4� H30

ðr2�Þ3=2 ; (41)

in accordance to (37). Equivalently, applying the Poissonequationr2� ¼ 3

2a2H2�m�m, we find an upper bound for

the matter over-density �m ¼ ��m=�m today

�m0 ’ 1:5=�m: (42)

We see that for describing an inhomogeneous universewhich resembles ours for the whole of t0, the metric caninclude perturbations down to about k ¼ 0:3h Mpc�1, or�m0 ’ 5. Of course, even shorter scales can be accuratelydescribed but for shorter times, comparable to the collapsetimes of the over-dense regions.

Given these approximations, the local density of mattercan be obtained from

�ðt;xÞ ¼ 1

6�Gt21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Det½�ij þ 3ð tt0Þ2=3t20�;ij þ ð tt0Þ4=3t40Bij�q :

(43)

Note that near order in � one recovers the result of linearperturbation theory for the density contrast in the synchro-

nous gauge. Dropping the Bij term we have

�ðt;xÞ ¼ 1

6�Gt21

Det½�ij þ 32 ð tt0Þ2=3t20�;ij�

; (44)

which reproduces the well known Zel’dovich approxima-tion. Thus, the gradient expansion in the comovingsynchronous gauge can be thought of as providing a rela-tivistic extension of the Zel’dovich approximation [5,6].3

In fact, as will also be demonstrated below, the higher orderterms are crucial for increased accuracy during the moreadvanced stages of the gravitational evolution.

At late timesH !ffiffiffi�3

qand the universe enters a de Sitter

phase in which case it is easy to see that the four-gradient

terms in the metric are exponentially suppressed, thetwo-gradient terms freeze while the zero-gradient termsare exponentially enhanced. Thus, the gradient expansionbecomes increasingly better with time, converging withtime to the separate universe picture as would be expected,apart from patches which suffer gravitational collapsebefore the de Sitter phase.

III. COMPARISON WITH �LTB

A. The LTB metric

Now let us assess the quality of the gradient expansionby comparing it to an exact metric that approximates theformation of structure in a �CDM universe. The LTBmetric is spherically symmetric, and can describe a centraldensity perturbation embedded in an exact Friedman-Lemaıtre-Robertson-Walker (FLRW) metric by choosingan appropriate density profile. With the inclusion of dust,curvature and a cosmological constant this hence describesthe rarefaction of a void as well as the initial expansion andlater collapse of an over-density in a�CDM universe.4 Themetric is given by

ds2 ¼ dt2 � X2ðr; tÞdr2 � Y2ðr; tÞd�2; (45)

with Xðr; tÞ � @rYðr; tÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2EðrÞp

. The metric is de-scribed by three free radial functions, being the curvatureEðrÞ, the matter distribution �ðrÞ and the big bang timetBBðrÞ. Due to the gauge freedom in the radial coordinate r,one of the three functions can be set arbitrarily, and wechoose a gauge in which

Rr0 Y

0ðr; tÞY2ðr; tÞ�ðr; tÞdr / r3.Therefore the choice of curvature profile EðrÞ and bang-time profile tBBðrÞ defines the metric completely.Moreover, now any deviation of tBBðrÞ from a constantcorresponds to a purely decaying mode. Following theinflationary paradigm we know that the decaying modemust be small such that is has decayed today, and we cansafely ignore it by setting tBBðrÞ ¼ 0 everywhere. All free-dom now hence lies in EðrÞ.The Einstein equations simplify to

�@tYðr; tÞYðr; tÞ

�2 ¼

�@taðr; tÞaðr; tÞ

�2 ¼ H2ðr; tÞ

¼ H20

�1

a3ðr; tÞ þ3EðrÞ

4�r2a2ðr; tÞ þ�

3H20

�; (46)

where Yðr; tÞ � raðr; tÞ, and we see that for EðrÞ / r2 weobtain exactly the FLRW solution. Moreover, if we chooseEðrÞ such that beyond a certain radius, say r ¼ L, EðrÞ /r2, while below that radius it has a different shape, that EðrÞdescribes an inhomogeneity embedded in FLRW. Wechoose,

3Planar solutions in the presence of � akin to Zel’dovichpancakes are described in Refs. [24,25] for general relativityand Newtonian Gravity respectively

4Generalizations to spheroidal configurations can be found inRef. [26].

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EðrÞ ¼ �r2kW7

�r

L; 0

�; (47)

where k ¼ � 45H20

8� and the function Wnðr=L; 0Þ is definedin Ref. [10], and interpolates from Wnð0; 0Þ ¼ 1 toWnð1; 0Þ ¼ 0 while remaining Cn everywhere, such thatthe metric is Cn�1. By choosing n ¼ 7 we are guaranteedto have all functions entering the problem to be at least C0

everywhere. Having Wnð0; 0Þ ¼ 1 and Wnð1; 0Þ ¼ 0, wehave defined a curvature profile with nonzero curvatureat the center of the metric at r ¼ 0, and zero curvature atfinite radius r ¼ L such that the metric there becomes theexact spatially flat FLRW metric with dust and a cosmo-logical constant for r > L. The magnitude of k is arbitrary,and we chose the value it has such that it describes a centralover-density that reaches the singularity today, and anunder-density that is nonlinear today but does not sufferform shell crossing yet.

We use the full solution to the Einstein equation as wellas Y0ðr; tÞ from Ref. [10] valid for any distribution of dustand curvature, until shell crossing occurs. In calculating thehigher order gradient terms we encounter second spatialderivatives of the spatial Ricci scalar, which contain bothY00ðr; tÞ and Y000ðr; tÞ. Since we have exact solutions toYðr; tÞ and Y0ðr; tÞ, we can safely evaluate the second andthird radial derivative of Yðr; tÞ numerically.

B. Comparison of the gradient expansionwith the exact solution

We calculate the initial conditions to Eq. (26) fromthe exact metric satisfying (46) at some early initial timeand numerically integrate the set of six coupled differ-ential equations that describe AðtÞ, JðtÞ, LðtÞ, ðtÞ,RðtÞJðtÞ=A2ðtÞdt and

RLðtÞ=A2ðtÞdt at any later time.

We then compare the resulting solution (35) to the exactsolution from (46).

In Fig. 1 we show the results for an over-density with

k ¼ 45H20

8� (left) and an under-density with k ¼ � 45H20

8�

(right). We plotffiffiffiffiffiffiffiffiffiffiffi�grr

p ¼ Xðr; tÞ as a function of radius rfor different times, where time is encoded by the coloringof the lines and dots. Solid lines correspond to the exactsolution, while dots present the expansion up to fourthorder in gradients. The expansion follows the features inthe metric very well, in both the compensating shells andthe central over- and under-densities. Obviously, for r > L,where the metric is exactly FLRW, the gradient expansioncorresponds to the exact solution with grr / a2ðtÞ.To clearly show howwell the gradient expansion follows

the gravitational collapse of an object, we show in Fig. 2the radial metric component

ffiffiffiffiffiffiffiffiffiffiffi�grrp ¼ Xðr; tÞ very close

to the centre. Exactly at the centre we have Xðr; tÞ � 08t,

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5

X(r

,t) /

X(L

,t)

r / L

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5

r / L

t = 10-4 Gyr

t = 6.7 Gyr

t0 = 13.5 Gyr

FIG. 1 (color online). The radial component of the metricffiffiffiffiffiffiffiffiffiffiffi�grr

p ¼ Xðr; tÞ as a function of radius r at a number of times, wheredifferent colors correspond to different times. The solid lines are the exact solutions, the colored dots are the gradient-expansionsolutions, up to fourth order in gradients. On the left we show the solution for an over-density, compensated by an under-dense shelland matching to FLRW at exactly r ¼ L. The center of this particular configuration reaches the singularity at t ¼ 13:5 G yr after theinitial singularity. On the right we show an under-density with the same curvature as the over-density, but opposite sign. By eye, bothscenarios are very well approximated by the gradient expansion. See Figs. 2 and 3 for quantification of this statement.

0 0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12 14

X(r

,t)

t [Gyr]

r/L = 10-5

Exact

2nd order GE

4th order GE

PT

FIG. 2 (color online). The radial component of the metricffiffiffiffiffiffiffiffiffiffiffi�grrp ¼ Xðr; tÞ at r ¼ 10�5 L (practically at the center) as a

function of time t, comparing the exact solution (solid red), linearperturbation theory (dotted black, uppermost line), second ordergradient approximation (dotted green, innermost curve) andfourth order gradient approximation (dotted blue, directly belowthe exact solution). While linear perturbation theory clearly doesnot showany formof collapse, both lowest (second) and next order(fourth) in the gradient approximation show the collapse. Theinclusion of fourth order gradient terms approximates the exactsolution better than the truncation to second order.

GERASIMOS RIGOPOULOS AND WESSEL VALKENBURG PHYSICAL REVIEW D 86, 043523 (2012)

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since it corresponds to raðtÞ in FLRW, which is why weplot Xðr; tÞ just off-centre, at r ¼ 10�5 L. For illustrationwe show the solution at first order in linear perturbationtheory, which shows no sign of collapse. The fourth ordergradient expansion follows the exact solution closer thanthe second order expansion, but both anyway show thecollapse to a central singularity.

In order to quantify the improvement of the expansionup to fourth order in gradients over the expansion upto second order in gradients, we calculate the square ofthe difference between the approximation and the exactsolution:

fðr; tÞ � ½Xgradientðr; tÞ � Xexactðr; tÞ�2;FðrÞ �

Z t0

0fðr; tÞdt:

(48)

One could also integrate fðr; tÞ over time, or both time andradius, but we focus on the time integrated goodness of fitat different radii. This way we can see in Fig. 3 that theexpansion up to fourth order in gradients everywhere out-performs the expansion up to second order reaching a time

integrated errorffiffiffiffiF

pof less than 10% for all radii. In both

configurations there is a point at r ’ 0:3 L where the first

gradient term Rkij � 4Rij ¼ 0 which signifies the transi-

tion between the central region and the compensatingshells. At this point the leading order correction is actuallythe four-gradient term and thus offers the smallest im-provement in approximation; the point behaves closest toa ‘‘separate universe’’. A better approximation at this pointwould presumably require the inclusion of higher orderterms. The overall approximation is better in the centralregion where the perturbation profile is relatively flat. In

the compensating shells, r * 0:3 R the value offfiffiffiffiF

pis

higher but the inclusion of the four-gradient term offerssignificant improvement. Eventually the approximationconverges the flat FLRW metric as r ! L. The reasonwhy FðrÞ does not go to zero for r > L, is simply thatwe are comparing a numerical integration to an exactsolution, and the difference between the solutions agrees

within the desired numerical accuracy in the integrationprocess, which is an arbitrary choice.

IV. DISCUSSION

The gradient expansion is an approximation scheme of adifferent nature from standard cosmological perturbationtheory. Instead of assuming perturbations of small ampli-tude around a background, it builds a solution from aninhomogeneous initial seed metric kij by including terms

with an increasing number of gradients of kij through

combinations of the 3-Ricci tensor Rij½kij� and its deriva-

tives: �ij �PnFnðtÞðRn

ij þP

rr2rRn�rÞ, where a hat indi-cates that quantities are computed from kij. The time

evolution is then encapsulated in the functions FnðtÞ. Theapproximation is expected to break down when F1ðtÞR� 1but this roughly coincides with the time of collapse of over-dense regions. The ability of describing collapse and con-

sequently the evolution in the regime when j��j� > 1 clearly

goes beyond the capability of conventional perturbationtheory at any order. The gradient series also contains theZel’dovich approximation in its lowest order while higherorders provide relativistic corrections which are in factrequired for greater accuracy.In this paper the gradient expansion was formulated

for the case of a �CDM universe and offers a way toapproximate structure formation in the nonlinear regime.Furthermore, in order to gauge the accuracy of theapproximation, we have used it on a spherically symmet-ric distribution of matter with a nontrivial radial profilefor which exact solutions are known in terms of the LTBmetric. We found the approximation to work rather well.The first nontrivial terms containing two gradients de-scribe the situation qualitatively well by showing col-lapse to a singularity in a manner comparable to theZel’dovich approximation. The inclusion of four gra-dients provides quantitative improvement by approach-ing the exact solution even better and it is expected thatthe inclusion of higher order terms would provide evenmore accuracy, an expectation that would be interesting

1.0⋅10-13

1.0⋅10-10

1.0⋅10-7

1.0⋅10-4

1.0⋅10-1

0 0.5 1 1.5 2 F

(r)

(arb

itrar

y un

its)

r/L

δ0 = 3.6⋅105

2nd order4th order

0 0.5 1 1.5 2

r/L

δ0 = -0.68

2nd order4th order

FIG. 3 (color online). The goodness of approximationFðrÞ, defined in Eq. (48), for both the over- and under-density. The expansion upto second order in gradients it plotted in solid red, up to fourth order in gradients in dotted blue. The units are irrelevant.Whatmatters is theimprovements of the fourth order over the second order solution. See the text for an explanation of the shape of these curves.

NONLINEAR APPROXIMATION FOR PERTURBATIONS IN . . . PHYSICAL REVIEW D 86, 043523 (2012)

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to confirm in future work. The expansion could also bechecked for less symmetric situations, perhaps by com-paring with other exact solutions or numerical simula-tions. Even if at the end of the day one has to resort tosimulations for the most accurate description of thenonlinear dynamics, it would pay to have analyticapproximations that capture the essential features ofnonlinear evolution in a relatively simple way and withsome level of realism as they would be applicable to avariety of problems. For this reason gradient expansion

techniques might be useful additions to the toolbox ofcosmologists as they strive towards a more accuratedescription of the universe.

ACKNOWLEDGMENTS

G.R. is supported by the Gottfried Wilhelm Leibnizprogramme of the Deutsche Forschungsgemeinschaft.W.V. acknowledges funding from DFG through the projectTRR33 The Dark Universe.

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