Beyond relativistic Lagrangian perturbation theory. I. An exact-solution controlled model for structure formation Ismael Delgado Gaspar1 2Thomas Buchert2yand Jan J. Ostrowski1z

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Beyond relativistic Lagrangian perturbation theory.

I. An exact-solution controlled model for structure formation

Ismael Delgado Gaspar,1, 2, ∗Thomas Buchert,2, †and Jan J. Ostrowski1, ‡

1Department of Fundamental Research, National Centre for Nuclear Research, Pasteura 7, 02–093 Warsaw, Poland

2Univ Lyon, Ens de Lyon, Univ Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69007, Lyon, France

We develop a new nonlinear method to model structure formation in general relativity from a

generalization of the relativistic Lagrangian perturbation schemes, controlled by Szekeres (and LTB)

exact solutions. The overall approach can be interpreted as the evolution of a deformation ﬁeld on

an inhomogeneous reference model, obeying locally Friedmann-like equations. In the special case

of locally one-dimensional deformations, the new model contains the entire Szekeres family of exact

solutions. As thus formulated, this approach paraphrases the Newtonian and relativistic Zel’dovich

approximations, having a large potential for applications in contexts where relativistic degrees of

freedom are relevant. Numerical simulations are implemented to illustrate the capabilities and

accuracy of the model.

I. INTRODUCTION

The problem of cosmological structure formation has

been approached by a variety of methods. The exact,

inhomogeneous solutions of Einstein’s equations provide

valuable hints on relativistic eﬀects, absent in Newtonian

theory, however, their applications remain limited as the

actual cosmic web is much more complex than the scope

covered by the exact solutions. Alternatively, the stan-

dard cosmological perturbation theory can be used in an

attempt to trace the evolution of the density inhomo-

geneities in the linear regime, the obvious limit being the

overdensities or second derivatives of metric perturba-

tions reaching relative values of around unity. These limi-

tations could in principle be overcome with the use of rel-

ativistic numerical simulations (see [1] for a recent com-

parison of performance among the most popular codes

currently being developed). There are, however, several

drawbacks of these numerical methods with a hard to es-

timate inﬂuence on the outcomes, e.g., global weak ﬁeld

assumption, noncovariance of conformal decomposition,

all-time ﬁxed toroidal topology or current computational

capacity limitations, to name a few. In light of recently

revealed, as well as long known but persistent, tensions

haunting modern cosmology, it is very important to im-

prove on the analytical methods providing an essential

counterpart to the numerical eﬀorts. In this spirit, the

so-called silent universe model ([19], [45]) was proposed

and is believed to be able to describe structure formation

in the nonlinear regime, from a wide variety of initial

data, by an exact method. The silent universe model is

based on the speciﬁc restriction on the 1 + 3 decomposi-

tion of Einstein’s equations for dust, i.e. the gravitomag-

netic part of the projected Weyl tensor is put to zero.

Unfortunately, this model turned out to be insuﬃcient

to access the nonlinear regime (because of the absence of

∗Ismael.DelgadoGaspar@ncbj.gov.pl

†buchert@ens-lyon.fr

‡Jan.Jakub.Ostrowski@ncbj.gov.pl

rotation and the requirement for the shear to be diagonal-

izable and have two identical eigenvalues) and, although

as desired, it contains several known inhomogeneous ex-

act solutions as subcases, the span of admissible initial

data is very restricted.

In this context, the recent investigation [41] consoli-

dated and generalized an earlier insight by Kasai [60] on

the correspondence between Szekeres class II solutions

and the ﬁrst-order Lagrangian perturbation solutions in

relativistic cosmology. The relativistic Lagrangian per-

turbation schemes have been developed in the series of

papers [35], [34], [2], [3], [67] and [41]. One aspect of this

generalization is furnished by the result that an extrap-

olated version of the ﬁrst-order solution scheme, in the

spirit of the original proposal by Zel’dovich in Newtonian

cosmology [92], allows to extend this correspondence to

nonlinear functional expressions of the ﬁrst-order scheme,

not only for the density but also for the bilinear met-

ric form, the extrinsic and intrinsic curvatures including

their tracefree parts, and other variables. The exact body

of the functionally extrapolated perturbations is obtained

by setting the second and third principal scalar invariants

of the deformation ﬁeld to zero (for details the reader is

referred to [41]). In the relativistic case, this corresponds

to Szekeres class II solutions, while in the Newtonian

limit this corresponds to a class of 3D solutions without

symmetry obtained in [21] (with empty background), in

[22] for backgrounds with zero cosmological constant and

in [6] including a cosmological constant.

A further insight concerns the way we write the

Szekeres class II solutions: in the so-called Goode-

Wainwright parametrization [48,49] this class can

be written in the form of deviations oﬀ a global

FLRW (Friedmann-Lemaˆıtre-Robertson-Walker) back-

ground solution. Looking at the spatial average prop-

erties of the inhomogeneous deviations, we ﬁnd admis-

sible initial data for deviations that average out on this

background solution. This can be realized, as in Newto-

nian cosmological simulations, by setting periodic bound-

ary conditions on the deviation ﬁelds on some scale that

is commonly associated with a ‘scale of homogeneity’.

arXiv:2210.04004v2 [gr-qc] 18 Jan 2023

The resulting architecture of such a (relativistic) sim-

ulation has the topological structure of a ﬂat 3-torus

[41,73], very similar to Newtonian simulations, imply-

ing integrability of the deformation ﬁelds and an on av-

erage zero intrinsic scalar curvature (for the notion of in-

tegrability of deformations, see [3] and [26]). The results

on the correspondence with Szekeres class II solutions

carry over to average properties known for constructions

with 3-torus topology, in Newtonian theory and in gen-

eral relativity [27,31]. This can be summarized by the

property of zero cosmological backreaction on the scale

set by the size of the toroidal space (for Szekeres class

II solutions, see [41]). In general, cosmological back-

reaction can be nonzero [39], and the topology of spa-

tial sections enjoys rich possibilities in general relativity.

For constant-curvature models, there are relations to the

topology of spatial sections, but in general situations,

the scalar curvature function is not tightly constrained;

it has a nonzero average and it can also change its sign

during the evolution [20,29]. A diﬀerent evolution of

the scalar curvature compared with the FLRW evolution

of the constant curvature also furnishes an explanation

of the vividly discussed Hubble tension [11,51] in the

standard cosmological model.

In the present paper, we propose to generalize La-

grangian perturbation schemes. In the example of the

ﬁrst-order scheme we exploit the structure of Szekeres

class I solutions that can be written as deviations oﬀ a

‘local background’ obeying a Friedmann-type equation.

These deviations can still be modeled by the functional

expressions developed in relativistic Lagrangian pertur-

bation theory. A ‘global background’ can be deﬁned

through a spatial averaging operation of the full solution

that leads to nonzero cosmological backreaction, i.e. it al-

lows not only for the impact of the background evolution

on the evolution of inhomogeneities as in Newtonian cos-

mologies, relativistic Lagrangian schemes and standard

quasi-Newtonian perturbation theories, but also for the

impact of inhomogeneities on the evolution of this global

background that is conceived as the average model. A

numerical implementation of this new model will open

the door to answering many of the questions raised in

the context of inhomogeneous relativistic cosmology, and

it may provide a more general architectural setting for

relativistic numerical simulations.

We proceed as follows. In Section II we recall the

Einstein equations in terms of the Lagrangian coframe

ﬁelds and the deﬁnition of the Relativistic Lagrangian

Zel’dovich Approximation (RZA). Section III is devoted

to reviewing the Szekeres models in the Goode and Wain-

wright parametrization and their relation with RZA. In

Section IV, we develop the proposed generalization of

relativistic Lagrangian perturbation schemes to include

both classes of the Szekeres solutions. Then, its most

important subcases, namely the locally one-dimensional

solutions, LTB models and RZA, are discussed in Section

V. In Section VI, we present a family of simple models

and implement numerical simulations aimed at illustrat-

ing the capability of the approach to model realistic cos-

mological structures. Finally, in Section VII, we put all

the elements of our analysis together, discuss their physi-

cal motivation and conclude with a summary of the paper

and ﬁnal remarks.

The text is complemented by ﬁve appendices. Ap-

pendix Aprovides additional details about the Goode

and Wainwright parametrization. Then, in Appendix B,

we have a closer look at Szekeres class I solutions and

discuss the relationship between some of their common

parametrizations. The model equations are obtained in

Appendix C. Appendix Dcontains useful expressions to

compute the kinematical backreaction term and its evo-

lution. Finally, in Appendix E, we look at the family of

models examined in Section VI but from the perspective

of exact solutions.

II. THE RELATIVISTIC LAGRANGIAN

FORMULATION

In the 3 + 1 relativistic Lagrangian framework, the

Einstein equations’ dynamical freedom is completely en-

coded in the coframe functions ηa

i[35].1Restricting our

attention to an irrotational dust source and considering

a ﬂuid-ﬂow orthogonal foliation of the spacetime leads to

a set of nine purely spatial coframes, in terms of which

the line-element reads:

(4)g=−dt⊗dt+(3) gwith (3)g=Gabηa⊗ηb.(1)

Here, the initial metric coeﬃcients are encoded in the

Gram’s matrix:

Gij (X)≡gij (tini,X),(2)

where tini denotes some arbitrary initial time.

A. The Lagrange-Einstein system

The Einstein equations are rewritten as a system of 9

evolution equations and 4 constraint equations [3,35],

Gab ˙ηa

[iηbj]= 0 ; (3a)

2Jabcikl ˙ηa

jηbkηcl.=−Rij+ (4π% + Λ) δij; (3b)

2Jabcmjk ˙ηa

m˙ηbjηck=−R

2+ (8π% + Λ) ; (3c)

1

Jabcikl ˙ηa

jηbkηcl||i=1

Jabcikl ˙ηa

iηbkηcl|j.(3d)

1Indices i, j, k, ··· = 1,2,3··· denote coordinate indices, while

indices a, b, c ··· = 1,2,3··· are introduced as counters of com-

ponents, e.g., of vectors or diﬀerential forms. In this paper we

use units where the gravitational constant and the speed of light

are set to G=c= 1.

We call this system the Lagrange-Einstein system. In

the equations above, as throughout the text, the overdot

represents the partial time derivative (the covariant time-

derivative in the ﬂow-orthogonal foliation), while the sin-

gle and the double vertical slashes stand for the partial

and covariant spatial derivatives, respectively. Rij de-

notes the spatial Ricci tensor, R=Riithe spatial scalar

curvature, and the determinant of the 3 ×3 coframe ma-

trix is deﬁned as:

J≡det(ηa

i).(4)

The exact density ﬁeld follows from the integration of the

continuity equation,

%=%iniJ−1,with J=√g/√G , (5)

and %ini =%(tini).2The expansion tensor can be com-

puted considering its relation to the extrinsic curvature,

which in the assumed foliation of the spacetime takes the

following form:

Θij =−Kij =1

2˙gij ; (6)

Θij=eia˙ηa

j,with eia=1

2Jabciklηbkηcl.(7)

For irrotational dust, the expansion tensor consists of

only the expansion scalar (its trace part, Θ = ˙

J/J) and

the shear tensor (its tracefree symmetric part, σij ),

Θij=σij+1

3Θδij.(8)

The 3D intrinsic curvature and the gravitoelectric and

gravitomagnetic parts of the spatially projected Weyl

curvature tensor reduce to the following expressions in

terms of the coframes [3]:

−Rij=1

2Jabcikl ˙ηa

jηbkηcl.−(4π% + Λ) δij; (9a)

−R

2=1

2Jabcmjk ˙ηa

m˙ηbjηck−(8π% + Λ) ; (9b)

2Jabcik` ¨ηa

iηbkηc`= Λ −4π% ; (9c)

−Eij=1

2Jabcikl ¨ηa

jηbkηcl+1

34π% −Λδij; (9d)

−Hij=1

JGabikl˙ηa

jklηbk+ ˙ηa

jηbkkl.(9e)

B. Relativistic Zel’dovich Approximation (RZA)

The relativistic Lagrangian perturbation theory con-

sists of perturbing the trivial coframe set associated with

a homogeneous and isotropic spacetime ηa

i=a(t)δa

2In this paper we refer to the initial quantities (evaluated at tini)

with the subscript ‘ini’.

where a(t) is the solution of the Friedmann equations.

Then, RZA emerges as the ﬁrst-order perturbation of

the deformation ﬁeld, Pa

ηa=ηa

idXi=a(t) (δa

i+Pa

i)dXi.(10)

In this approximation, the line-element (1) takes the fol-

lowing quadratic bilinear form:

gij =a2(t)Gij +Gab δa

iPb

j+δbjPa

i+Pa

iPb

j .

(11)

Consequently, any relevant ﬁeld (the Ricci and Weyl cur-

vature tensors, rate of expansion, shear, etc.) is function-

ally evaluated in terms of the coframe ﬁelds (10).

III. SZEKERES EXACT SOLUTIONS

The Szekeres models are the most general exact cosmo-

logical solution of Einstein’s equations [13,64].3These

models lack symmetries, albeit their ‘quasisymmetries’

impose signiﬁcant restrictions on the spacetime’s inho-

mogeneity and anisotropy. The source of the original so-

lution consists of irrotational dust [86], which was later

generalized by Szafron [84] to include a homogeneous

perfect ﬂuid, furnishing the inclusion of the cosmological

constant (see [43] for a discussion of the Szekeres source).

For a detailed review of the properties of these solutions

see [14,64,78]. The general line-element is commonly

expressed as (see [4,85,90] for coordinate-independent

deﬁnitions of the Szekeres solution):

ds2=−dt2+e2αdz2+e2βdx2+ dy2,(12)

where α=α(t, r), β=β(t, r), and r= (x, y, z) are

comoving coordinates. The integration of the Einstein

equations is performed by splitting the model into two

classes, namely class I, when β,z 6= 0, and class II, when

β,z = 0. However, once the solution is obtained, the class

II can be formulated as a limit of class I [52].

In the Goode and Wainwright (GW) parametrization

[48] the line-element (12) is rewritten as:

ds2=−dt2+S2G2W2dz2+e2νdx2+ dy2 ,(13)

where the conformal scale factor, S(t, z), obeys a

Friedmann-like equation:

S2=−k0+2µ

S+Λ

3S2.(14)

3This statement requires some justiﬁcation. In the present anal-

ysis we follow [13] and consider as cosmological models those

exact solutions of Einstein’s equations with at least a nontrivial

subclass of FLRW solutions as a limit. There are other solutions

that can be regarded as more general than Szekeres, for exam-

ple, the Lemaˆıtre model [66] (although it is spherically symmetric

the source is an inhomogeneous perfect ﬂuid) and other models

containing heat-ﬂow [5,47,68,73], viscosity [72,87] or electro-

magnetic ﬁelds [69,88]. However, in cosmological applications,

the ﬁeld source is usually simpliﬁed to a dust ﬂuid and the cosmo-

logical constant; then, Szekeres models arise as the most general

(exact) cosmological class of solutions.

Here, k0is a constant taking the values 0,±1, and µ(z)>

0 is an arbitrary scalar function. Furthermore,

G=A(r)− F(t, z) = A(r)−β+f+−β−f−,(15)

where f±are the growing and decaying solutions of the

following equation:

F+ 2 ˙

S˙

F − 3µ

S3F= 0 .(16)

The energy-density is given by:

8π%(t, r) = 6µ

S31 + F

G,(17)

which can be rewritten as:

4π%(t, r) = 3µA

S3G≡M(r)

S3G,(18)

assigning to the term 3µAthe meaning of a conserved

rest mass, M. The functions f±can be regarded as the

growing and decaying deviation ‘modes’ at a ‘ﬁctitious

background’ (or reference model) with initial local den-

sity 3µ= 4πρ.

The form of the functions A,e2ν,W,β+(z), β−(z), f+

and f−vary depending on the class, and they satisfy:

Class I: S=S(t, z), µ=µ(z) and f±=f±(t, z).

Class II: S=S(t)≡a(t), µ= const. and f±=f±(t).

For better readability of the text, we display their full

expressions in Appendix A.

The quasispherical branch of the class I solutions has

been the most widely used subclass for modeling struc-

ture formation, describing nontrivial networks of cosmic

structures [7,9,12,16,37,57,58,61–63,65,75,81,

82,89] or even the formation of primordial black holes

[40,42,50]. Due to its importance, the physical and

mathematical properties of this subclass have also been

explored in depth [17,18,46,53–55,83,91]. The other

subclasses have found applications in cosmology and as-

trophysics as well [56,70,71,77], although they have

received much less attention.

A. Szekeres class I models with a normalized scale

factor

In contrast to class II, the class I solution’s confor-

mal scale factor is a function of the spatial coordinates,

S(tini, z) = Sini (z). To have a normalized initial function

in a given Szekeres model, we proceed as follows. First,

introduce the function χ=S−1(tini,r)>0; then, the

conformal scale factor and the spatial metric are rede-

ﬁned:

A(t, r)≡ S(t, z)χ; (19)

gij =A2Diag "(A−F)2W

χ2

,eν

χ2

,eν

χ2#,

(20)

where metric functions are displayed in Appendix A.

In terms of the rescaled scale factor, the Friedmann-

like equation reads:

A!2

=−ˆ

k(r)

A2+8π

ˆ%b(r)

A3+Λ

3,(21)

where:

k(r) = k0χ2,4π

3ˆ%b(r) = µ(z)χ3.(22)

Note that the growing and decaying functions still refer

to the local reference background (14), and obey the same

equation as in (16):

F+ 2 ˙

A˙

F − 4πˆ%b

A3F= 0 .(23)

B. Relationship between RZA and exact solutions

In [41] we exploited such a splitting into ‘background’

and the growing and decaying ‘deviation modes’ to con-

nect the Szekeres class II solution to RZA. The main re-

sults of interest for the present article can be summarized

as follows:

(i) RZA contains the class II of the Szekeres exact so-

lutions as a particular case with the identiﬁcations:

i= (−e

F/e

A)δa

3δ3i; (24a)

Gab = Diag he

A2, e2ν, e2νi.(24b)

In these equations, e

A ≡ A − Fini ,e

F ≡ F − Fini.

Also, a(t) is the scale factor of the global FLRW

background with 4π%b/3 = µ= const.

(ii) The class I can be interpreted as a (constrained) su-

perposition of nonintersecting world lines, satisfy-

ing RZA’s evolution equations. As for the previous

class, the associated deformation ﬁeld and Gram’s

matrix are given by the identiﬁcations:

i= (−e

F/e

A)δa

3δ3i; (25a)

Gab = Diag "e

A2W

χ2

,eν

χ2

,eν

χ2#,(25b)

with the proviso that these expressions need to be

evaluated for each world line independently. Given

an arbitrary world line with ﬁxed comoving coor-

dinates ri= (xi, yi, zi), A(t)≡A(t, zi) can be re-

garded as the scale function of an associated FLRW

‘local background’ or ‘reference model’ with ini-

tial constant density %b(tini) = ˆ%b(zi) and curva-

ture k0=ˆ

k(z). Thus, f±(t)≡f±(t, z)|z=ziare its

associated growing and decaying modes.

IV. GENERALIZED RELATIVISTIC

ZEL’DOVICH APPROXIMATION (GRZA)

Motivated by the mathematical connection between

the Szekeres class II solutions and RZA, we present a new

nonperturbative approach that generalizes RZA to con-

tain Szekeres class I. The approach retains the mathemat-

ical structure of RZA but without pre-assuming a global

background. Instead, we consider a space-dependent con-

formal scale factor obeying a Friedmann-like equation (as

in Eq. (22) for Szekeres class I).

Restricting the analysis to an irrotational dust source,

the coframe set (10) ﬁnds its generalization in the follow-

ing expressions:

ηa=ηa

idXi=Aδa

i+ˆ

idXi; (26a)

A=A(t, r),ˆ

i=ˆ

i(t, r),(26b)

where, as discussed above, Asatisﬁes a Friedmanian

equation for a reference model with curvature and mat-

ter density parameters ˆ

k=ˆ

k(r) and ˆ%b(r), respectively:

2¨

A/A +˙

A2/A2−Λ + ˆ

k/A2= 0 ; (27a)

A!2

=−ˆ

A2+8π

ˆ%b

A3+Λ

3.(27b)

The line-element keeps the bilinear quadratic mathemat-

ical structure for the deformation ﬁeld, as in Eq. (1):

gij =Gabηa

iηbj(28a)

=A2hGij +Gab δa

iˆ

j+δbjˆ

i+ˆ

iˆ

ji .(28b)

Since at the initial time Ais normalized and we assume

(without loss of generality) that the initial deformation

ﬁeld vanishes ( ˆ

i(tini) = 0), the Gram’s matrix is deﬁned

via the initial spatial metric:

Gij (r) := gij (ti,r).(29)

The subsequent approach consists of (i) obtaining

(Lagrange-)linear evolution equations for the deforma-

tion ﬁeld and, then, (ii) injecting the formal solution into

the exact nonlinear functional expressions. This scheme

retains the original Zel’dovich’s extrapolation idea and

generalizes RZA to include the whole family of Szek-

eres models within its locally one-dimensional deforma-

tion ﬁeld limit. In light of this, we call the resulting

approach Generalized Relativistic Zel’dovich Approxima-

tion (GRZA).

A. Functional evaluation

First, let us evaluate all relevant ﬁelds as exact func-

tionals of the local deformation and the conformal scale

factor. The exact determinant of the spatial coframe co-

eﬃcients is given by:

J=A3J=A3(J0+J1+J2+J3),(30)

where we have introduced the peculiar-determinant J,

and the quantities Jn≡(n)Jkkare deﬁned through:

(0)Jij=1

6abciklδa

jδbkδcl; (31a)

(1)Jij=1

6abcikl ˆ

jδbkδcl+1

3abciklδa

jˆ

kδcl; (31b)

(2)Jij=1

3abcikl ˆ

jˆ

kδcl+1

6abciklδa

jˆ

kˆ

l; (31c)

(3)Jij=1

6abcikl ˆ

jˆ

kˆ

l.(31d)

From its deﬁnition, we can see that J0= 1, as expected.

Henceforth, the exact inhomogeneous density ﬁeld fol-

lows from injecting (30) and (31) into (5).

Next, we express the expansion tensor as a functional

of the deformation ﬁeld. Writing out Eq. (7) yields:

Θij= 3 ˆ

HPn(n)Jij

J+1

J(1)Ii

j+(2)Ii

j+(3)Ii

j.(32)

To keep the above expression short, we have introduced

the shorthand notations:

H=˙

A/A ; (33a)

(1)Ii

j=1

2abcikl ˙

jδbkδcl; (33b)

(2)Ii

j=abcikl ˙

jˆ

kδcl; (33c)

(3)Ii

j=1

2abcikl ˙

jˆ

kˆ

l.(33d)

Taking the trace of the expansion tensor and using a

similar notation as before, In≡(n)Ik

k, we obtain the

functional for the expansion scalar:

Θ=3ˆ

H+1

JI1+I2+I3.(34)

Then, the exact functional for the shear tensor can be

computed from (8), (32) and (34).

The gravitoelectric part of the spatially projected Weyl

tensor reads:

Eij=−3¨

APn(n)Jij

J−2ˆ

HPn(n)Ii

J−Pn(n)b

Iij

−1

3(4π% −Λ) δij.(35)

Above, the quantities (n)b

Iijare deﬁned as (n)Ii

jin (33),

but replacing ˙

jby ¨

j. The gravitomagnetic part is

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BeyondrelativisticLagrangianperturbationtheory.I.Anexact-solutioncontrolledmodelforstructureformationIsmaelDelgadoGaspar,1,2,ThomasBuchert,2,yandJanJ.Ostrowski1,z1DepartmentofFundamentalResearch,NationalCentreforNuclearResearch,Pasteura7,02{093Warsaw,Poland2UnivLyon,EnsdeLyon,UnivLyon1,CNRS,Centred...

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Beyond relativistic Lagrangian perturbation theory. I. An exact-solution controlled model for structure formation Ismael Delgado Gaspar1 2Thomas Buchert2yand Jan J. Ostrowski1z

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