Non-Spherical Szekeres models in the language of Cosmological Perturbations

Gauge invariant perturbations on a Friedman–Lemaître–Robertson–Walker (FLRW) background, generically examined within the framework of Cosmological Perturbation Theory (CPT), constitute an important theoretical tool in cosmological research (see pioneering work in CPTh and more recent comprehensive reviews in CPT1 ; CPT2 ; CPT3 ). Linear CPT is specially adequate to study cosmic sources whenever near homogeneous conditions can be justified: the early Universe and scales comparable to the Hubble radius for late cosmic times. Since late time structure formation at deep sub–horizon scales becomes highly nonlinear and (at least locally) nonÑrelativistic, it is usually studied by means of non–linear and non–perturbative Newtonian gravity, either through analytic models (spherical newt1 ; newt2 ; newt3 ; barrow-silk and elliptic collapse barrow-silk ; newt4 ; newt5 ) or by sophisticated numerical N-body simulations numsim1 ; numsim2 . By considering higher order perturbations (Newtonian and relativistic) CPT has been so far extended to study the mildly nonlinear regime CPT21 ; CPT22 ; CPT23 ; CPT24 ; CPT25 ; CPT26 , leaving the description of fully nonlinear effects in large scale structure formation to Newtonian non–perturbative methods, though the need for considering relativistic corrections is still an open question (an extensive review is CPT2summary ).

A proper understanding of the evolution of the density and the peculiar velocities in cosmic inhomogeneities is essential to distinguish between specific cosmological models. While several dark energy or modified gravity background FLRW models reasonably fit late time observational data, it is the clustering properties of matter that allows us to distinguish between these models Lahav-Rees ; Bueno-Bellido ; Percival-Song . The growth of structure is characterised by a growth factor $f$ function computed in linear CPT, which is an observable derived from the anisotropy of the power spectrum in redshift space in the non–linear regime Scoccimarro . Yet, it is precisely at this non–linear regime that departures from Newtonian evolution arise, which suggests considering the evolution of inhomogeneities at non–linear level by introducing corrections from General Relativity (GR) theory.

Non–perturbative and fully non–linear GR theory is not a particularly favoured theoretical tool in cosmological research, since (given its high nonlinear complexity) any minimally realistic non–perturbative and relativistic modelling of structure formation necessarily requires numerical 3–dimensional codes to solve Einstein’s equations in a cosmological context, whether as a continuum model or as GR numerical simulations. This impressively difficult task is still in its early stages of development GRN1 ; GRN2 ; GRN3 ; GRN4 ; GRN5 ; GRN6 ; GRN7 . However, all numerical and perturbative work (whether Newtonian or relativistic) still requires simple analytic “toy models” that provide useful practical hints and comparative qualitative results. From a fully relativistic and non–perturbative approach, these toy models emerge from appropriate exact solutions of Einstein’s equations. Since cold dark matter (CDM) at cosmic scales can be adequately described by a dust source and a $\Lambda$ term can always be added to the dynamics as a phenomenological description of dark energy, the most useful exact GR solutions applicable to cosmology are the well known spherically symmetric dust Lemaître–Tolman–Bondi (LTB) models ltbold and their non–spherical generalisation furnished by Szekeres models Sz751 ; Sz752 (see comprehensive reviews in kras1 ; kras2 ; BKHC2009 ; BCK2011 ; EMM for both classes of solutions).

The usage of LTB and Szekeres models in cosmological applications can (and should) also be undertaken within the prevailing $\Lambda$CDM paradigm or Concordance Model. It is important to emphasise this point, since employing these exact solutions is often associated with the recent past attempt to challenge this paradigm by means of non–perturbative large scale (Gpc sized) density void configurations mostly constructed with LTB models LTBV1 ; LTBV2 and, to a lesser degree, with simplified Szekeres models (see reviews in kras1 ; kras2 ; BKHC2009 ; BCK2011 ; EMM ). Once LTB void models were ruled out LTBV3 ; LTBV4 ; LTBV5 , thus re–affirming the validity of the Concordance Model, interest on cosmological application of exact GR solutions decreased. However, the claim that non–perturbative GR is redundant for cosmological research (within or without the $\Lambda$CDM paradigm) is an open issue GRE1 ; GRE2 ; GRE3 ; GRE4 ; GRE5 . As long as the cosmological implementation of numerical GR remains under development, it should be appropriate to continue applying LTB and Szekeres models as theoretical tools to complement perturbative and numerical cosmological research.

Because of their spherical symmetry LTB models only allow us to describe the evolution of a single CDM structure (an overdensity or a density void) imbedded in a suitable FLRW background (see numerical examples in BKHC2009 ). Evidently, Szekeres models introduce more dynamical degrees of freedom, as can be appreciated in the above cited reviews and in theoretical studies GW ; barrow1 ; barrow2 ; theo1 ; theo2 ; theo3 ; theo4 ; bolsus ; sussbol ; buckley ; theo5 ; multi , as well as in the extensive literature on their application to structure formation and fitting of cosmological observations SFO1 ; SFO2 ; SFO3 ; SFO4 ; SFO5 ; SFO6 ; SFO7 ; SFO8 ; SFO9 ; SFO10 ; SFO11 ; SFO12 ; SFO13 ; SFO14 ; SFO15 ; SFO16 ; coarse . Practically all of these articles (see exceptions below) only consider the simplest type of structure formation scenario allowed by the models: a 2–structure dipolar configuration comprised by an overdensity evolving next to a density void (see numerical examples in BKHC2009 ). The possibility of modelling less restrictive structures was suggested already in earlier work bolsus ; buckley ; SFO1 ; SFO2 , but it only became recently implemented multi ; coarse by using the full extension of the dynamical freedom of the models for the description of elaborated networks consisting of an arbitrary number of CDM structures (overdensities and voids).

Since Szekeres models (specially those examined in multi ; coarse ) provide the less idealised exact GR solution in a cosmological context, they are specially suitable to examine the connection between non–perturbative GR and CPT based perturbations at different orders, as well as with non–perturbative Newtonian models. In the present article we explore this connection through a detailed rigorous comparison between the dynamical equations of Szekeres models and CPT for dust sources in the comoving isochronous gauge, thus extending and continuing a previously published perts similar comparison between LTB models and CPT. This represents the first stage in an effort to compare all these theoretical tools in terms of their structure modelling predictions and their usage in fitting observations and addressing open theoretical issues.

The contents of this article are summarised as follows. Section 2 introduces the metric of Szekeres models and their particular cases (spherical and axial symmetry) in spherical coordinates. In sections 3 and 4 we show how the dynamics of the models can be completely determined by suitable covariant variables defined in sussbol and used in multi ; coarse : the q–scalars associated with the density, the Hubble scalar and spatial curvature, their corresponding FLRW background variables and exact fluctuations that convey the inhomogeneity of the models. The evolution equations for these variables (as we show) can be adequately related to the CPT evolution equations as in the LTB case perts . Employing these variables is crucial, as the standard metric based description of Szekeres model (as used in all previous work on these models save for sussbol ; multi ; coarse ) is not useful for relating to CPT dynamics. We define in section 5 a linear regime in Szekeres models by suitable first order expansions of the exact fluctuations as done with LTB models in perts . In particular, we show that this linear regime is compatible with large deviations from spherical symmetry that allow for a description of networks of multiple evolving structures (overdensities and voids multi ; coarse ). The linearised Szekeres evolution equations and their solutions are examined in section 6.

In section 7 we examine the equivalence between the Szekeres and CPT evolution equations for dust sources in the isochronous CPT gauge. We follow the methodology of perts but with more depth and generality: while we focus primarily on first order linear equations, we show that the Szekeres evolution equations are equivalent to CPT equations at all approximation orders. We prove that the first order Szekeres spatial curvature fluctuation is conserved in time for all scales, just as its equivalent linear spatial curvature perturbation from CPT.

In section 8 we describe the “pancake” and spherical collapse morphologies allowed by Szekeres models, then we compute the Szekeres analogue of the growth suppression factor $f$ used in CPT to address the study of redshift space distortion Percival-Song ; Scoccimarro ; RSD1 ; RSD2 . We show that this analogue fully coincides with its linear equivalent from CPT and that is only sensitive to collapse morphologies in a non–linear regime (either second order CPT or exact). In section 9 we obtain the Szekeres equivalent CPT metric potentials (in the isochronous gauge) by linearising the Szekeres line element in terms of its deviation from the FLRW metric. This allows us to relate our dynamical equations with CPT quantities in any gauge.

In section 10 we present concluding remarks and a discussion of the main results listed above. We also present a conceptual comparison between the CPT density perturbation, given as a random field, and its Szekeres equivalent: the linearised Szekeres density fluctuation, given as a linearised expression that follows from a deterministic exact solution. We argue that the linear CPT perturbation is far more general, but is only valid in the mildly non–linear regime, whereas its equivalent linearised Szekeres fluctuation can be evolved into a fully non–perturbative exact expression valid throughout the models evolution.

Finally, we provide two appendices: A discusses the integration of the Friedman–like quadrature that determines the functional form of the metric variables (the scale factors), while B derives the linearised form of these scale factors.

We consider quasi–spherical Szekeres models of class I in terms of “stereographic” spherical coordinates kras2 ; multi ; coarse ¹¹1All further mention of “Szekeres models” will refer only to quasi–spherical models of class I (see kras2 for a broad discussion on their classification). The standard diagonal metric form these models and the transformation relating it to (1) is given in Appendix A of multi . For constant time slices with spherical or wormhole topology kras2 the metric (1) must be modified as explained in Appendix D of sussbol .

where $h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu},\,\,u^{\mu}=\delta^{\mu}_{t}$ and the nonzero components of $\gamma_{ij}$ are:

with the functions $\Gamma,\,U,\,{\cal{P}}$ and ${\rm{\bf W}}$ given by

where the free parameters $X,\,Y,\,Z,\,{\cal{K}}_{qi}$ depend only on $r$ (see the interpretation of ${\cal{K}}_{qi}$ in (21)), and ^′ denotes the radial derivative. The function ${\rm{\bf W}}$ has the mathematical structure of a dipole whose orientation is governed by the choice of the three dipole parameters $X,\,Y,\,Z$ (see comprehensive discussion in multi ). Different particular cases follow by specialising these parameters:

Spherical Symmetry: LTB models.

We can define for each Szekeres model the “LTB seed model” as the generic LTB model that follows as its spherical limit: $X=Y=Z=0\,\,\Rightarrow\,\,{\rm{\bf W}}=0$, whose metric is the following specialisation of (1)

$$\gamma_{rr}=\frac{\Gamma^{2}}{1-{\cal{K}}_{qi}r^{2}},\quad\gamma_{\theta\theta}=r^{2},\quad\gamma_{\phi\phi}=r^{2}\,\sin^{2}\theta.$$

(7)

Notice that any Szekeres model can be constructed from an arbitrary LTB seed model simply by introducing suitable nonzero dipole parameters $X,\,Y,\,Z$. Evidently, Szekeres models always “inherit” the properties of their LTB seed models, a feature that is very useful to study their properties.

Axial Symmetry.

Another important particular case is furnished by axially symmetric models: $X=Y={\cal{P}}=0,\,\,Z\neq 0$, leading to the specialised metric

	$\displaystyle\gamma_{rr}$	$\displaystyle=$	$\displaystyle\frac{(\Gamma-{\rm{\bf W}})^{2}}{1-{\cal{K}}_{qi}r^{2}}+{\rm{\bf W}}_{,\theta}^{2},\quad\gamma_{r\theta}=-r{\rm{\bf W}}_{,\theta},$
	$\displaystyle\gamma_{\theta\theta}$	$\displaystyle=$	$\displaystyle r^{2},\quad\gamma_{\phi\phi}=r^{2}\,\sin^{2}\theta,$		(8)

where ${\rm{\bf W}}=-Z\,\cos\theta$ is independent of $\phi$. The asymptotic spherical limit follows if $Z\to 0$ as $r\to\infty$.

Practically all work considering cosmological applications of Szekeres models kras1 ; kras2 ; BKHC2009 ; BCK2011 ; EMM examine the dynamics of the models in terms of the metric variables determined from a Friedman quadrature that follows directly from Einstein’s field equations (see A and B). This approach is not useful to relate the models to CPT. Instead, we determine the dynamical equations (including the metric functions) through the first order system of evolution equations derived in sussbol and used in multi ; coarse

with $\dot{}=\partial/\partial t$, and with quantities subject to the algebraic constraints:

Here the quasi–local (q–scalars) $A_{q}$ and their exact fluctuations ${\textrm{\bf{D}}}^{(A)}$ sussbol are given for each scalar $A=\rho,\,\Theta,\,{\cal{K}},$ (density, Hubble expansion and spatial curvature) by

and where $\Xi=\sqrt{1-{\cal{K}}_{qi}r^{2}}$. This last integral in (19) is evaluated in an arbitrary time slice (constant $t$) in a spherical comoving domain ${\cal{D}}$ bounded by an arbitrary fixed $r>0$. This leads to the scaling laws ²²2The lower bound of the integrals (19) is the locus $r=0$, analogous to the symmetry centre of spherical models bolsus . While Szekeres models are not spherically symmetric, the surfaces of constant $r$ are non–concentric 2–spheres kras2 ; multi . Notice that $A_{q}=A_{q}(t,r)$ even if the scalars $A$ depend on the four coordinates $(t,r,\theta,\phi)$ theo3 ; bolsus ; sussbol . Their relation with the average integrals is discussed in bolsus ; sussbol .

where ${\cal{G}}$ is defined in (14) and the subindex _i denotes evaluation at an arbitrary time slice $t=t_{i}$.

The first order evolution equations for the fluctuations $\Delta^{(\rho)}$ and ${\textrm{\bf{D}}}^{(\Theta)}$ can be combined into a single second order equation

which resembles an exact (non–linear) generalisation of the equation of linear dust perturbations in the synchronous gauge perts .

The initial conditions to integrate the system (9)–(16) are specified at $t=t_{i}$ and consist of the cosmological constant $\Lambda$ plus the following five free functions that depend only on $r$:

where the radial functions are common to the LTB seed model and the dipole functions govern the deviation from spherical symmetry. We obtain the initial values of $\Theta_{q},\,\Delta^{(\rho)},\,{\textrm{\bf{D}}}^{(\Theta)},\,{\textrm{\bf{D}}}^{({\cal{K}})}$ by solving the constraints (15)–(16) at $t=t_{i}$, the radial coordinate is chosen so that $a_{i}=\Gamma_{i}=1$, while the Big Bang time $t_{\textrm{\tiny{bb}}}$ and its gradient $t_{\textrm{\tiny{bb}}}^{\prime}$ can be obtained from $\rho_{qi},\,{\cal{K}}_{qi}$ and their gradients (see sussbol ; multi ; coarse and A, B).

The fluctuations ${\textrm{\bf{D}}}^{(A)}$ and $\Delta^{(A)}$ in (19)–(19) compare the scalars $A=\rho,\,{\cal{H}},\,{\cal{K}}$ with their associated q-scalars $A_{q}=\rho_{q},\,{\cal{H}}_{q},\,{\cal{K}}_{q}$ at the same values of $r$ for all $t$. As a consequence, the density fluctuation $\Delta^{(\rho)}$ is different from the notion of a “density contrast”. However, assuming the existence of an asymptotic FLRW background (see conditions for this in B), we can define exact fluctuations that yield the notion of a contrast by comparing the scalars $A$ at very point with their background values $\bar{A}=\bar{\rho},\,\bar{\Theta},\,{\cal{K}}$ ³³3Notice that the covariant background scalars $\bar{A}$ can be understood as asymptotic limits as $r\to\infty$ of the q–scalars $A_{q}$. In other words: the $\bar{A}$ are averages for an asymptotic averaging domain covering the whole time slice (see comprehensive discussion in perts ). . This leads to the following “asymptotic” fluctuations

with the $\bar{A}$ given by (FLRW quantities will be henceforth denoted by a overbar)

The corresponding evolution equations and constraints are

while the equivalent to the second order equation (23) is

which not only “resembles” but strictly provides the exact (non–linear) generalisation of the evolution equation of linear dust density perturbation in the synchronous gauge perts .

The system (28)–(33) is not self–contained, it thus needs to be supplemented by (9)–(10). However, the contrast fluctuations in (28)–(33) are related to the quasi–local fluctuations defined in the previous section by

Therefore, for all purposes it is more practical to solve first the background equations (28)–(29) and then use the solutions of (9)–(16) and (13)–(14) to compute the density contrast and Hubble scalar fluctuation from the relations (35)–(36).

The FLRW limit of Szekeres models can be defined rigorously and in a coordinate independent manner kras2 ; multi by the vanishing of the shear and electric Weyl tensors in the background: $\sigma^{a}_{b}=E^{a}_{b}=0$, while their fluctuations are both expressible as

where ${\hbox{\bf{e}}}^{\mu}_{\nu}=\hbox{diag}[0,-2,1,1]$ is a unique traceless tensor basis satisfying $\dot{\hbox{\bf{e}}}^{\mu}_{\nu}=0$ and the fluctuations
${\textrm{\bf{D}}}^{(\rho)}=\rho_{q}\Delta^{(\rho)}$ and ${\textrm{\bf{D}}}^{(\Theta)}$ are

where we used (16) and (21)–(22) and $\delta^{(\rho)}_{i},\,{\textrm{\bf{d}}}^{({\cal{K}})}_{i}$ are the initial fluctuations of the LT seed model given by

where we used the fact that $a_{i}=\Gamma_{i}=1$. Proceeding as in LTB models in perts , we define a linear regime for Szekeres models (understood as functional parameter “closeness” to a FLRW background, see B) by demanding that a positive dimensionless number $\epsilon\ll 1$ exists such that

holds for a given evolution range, with the term $O(\epsilon)\ll 1$ denoting quantities of the order of magnitude of $\epsilon$. Notice that the covariant objects $\Sigma,\,\Psi_{2}$ vanish at the FLRW background, hence the fluctuations in (42) also vanish at the background, and thus (from Stewart lemma stewart ; ellisbruni89 ) are gauge invariant quantities. We derive below the parameter restrictions that yield the necessary and sufficient conditions for a linear regime defined by (42).

The necessary (not sufficient: see (53)) condition for a Szekeres linear regime is the existence of a linear regime in the LTB seed model, which requires (see perts ) the radial initial conditions (24) satisfying for $A=\rho,\,\Theta,\,{\cal{K}}$

where we used the fact that the initial fluctuations of the LTB seed model are linked by the constraint (16) restricted to ${\rm{\bf W}}=0$ and evaluated at $t=t_{i}$. As a consequence of (43) and bearing in mind (19)–(22), (35)–(36) and the results of B, we have up to $O(\epsilon)$

which suggests introducing the following notation valid up to $O(\epsilon)$:

that will be used henceforth to denote both types of fluctuations we have used so far, as they are indistinguishably when linearised.

The linearised forms for the metric functions (see derivation in B⁴⁴4The derivation of all functions appearing in the analytic forms of $\Gamma-1$ and $a_{1}=a-\bar{a}$ and their linear expansions are given in A and B. Equations (49)–(50) simplify considerably if the decaying mode is suppressed (which imposes a link between $\delta^{(\rho)}_{i}$ and ${\textrm{\bf{d}}}^{(\kappa)}_{i}$).) are

where $\bar{\phi}^{m}(\bar{a}),\,\bar{\phi}^{k}(\bar{a})$ are dimensionless functions of $O(1)$ defined in (123)

where $\bar{\Omega}_{i}^{m}=8\pi\bar{\rho}_{i}/(3\bar{H}_{i}^{2}$ and $\bar{\Omega}_{i}^{k}=\bar{\cal{K}}_{i}/\bar{H}_{i}^{2}=\bar{\Omega}_{i}^{m}+\bar{\Omega}_{i}^{\Lambda}-1$ are the standard density fraction FLRW parameters. To obtain the sufficient condition we expand (39)–(40) in terms of $\delta^{(\rho)}_{i},\,{\textrm{\bf{d}}}^{({\cal{K}})}_{i}$ which (from (43)) are $\sim O(\epsilon)$ quantities. Considering that $\hat{\Omega}_{qi}^{m}-\bar{\Omega}_{i}^{m}$ and $\hat{\Omega}_{qi}^{k}-\bar{\Omega}_{i}^{k}$ are both $\sim O(\epsilon)$, together with (46)–(47), we obtain:

where (and this is important to notice) we did not restrict the dipole term ${\rm{\bf W}}$ to be small (i.e, we have in general $|{\rm{\bf W}}|\sim O(1)$).

For purely growing modes (see A.2), the suppression of the decaying mode yields a constraint between the initial fluctuations $\delta^{(\rho)}_{i}$ and ${\textrm{\bf{d}}}^{(\kappa)}_{i}$. Therefore, the expansions (49) and (50) take the simplified form

where the background quantities $\bar{\cal{F}}(\bar{a}),\,\bar{\Phi}_{i}^{m},\,\bar{\Phi}_{i}^{k}$ are defined in B. Their explicit forms for a $\Lambda$CDM background ($\bar{\Omega}_{i}^{k}=0$) are given by (125)–(126). The necessary and sufficient condition for a linear regime in generic Szekeres models are then the necessary conditions (43)–(47) for the linear regime of the seed LTB model plus the extra condition involving the dipole term:

It is important to emphasise that a linear regime in Szekeres models, as specified by (43) and (53), does not imply closeness to spherical symmetry (i.e. an “almost spherical model” complying with $|{\rm{\bf W}}|\ll 1$). As long as (53) holds, large local deviations from spherical symmetry associated with small $1-{\rm{\bf W}}$ (see multi ; coarse ) are perfectly compatible with a linear regime.

Applying the criterion for a linear regime given by (42) to the system (28)–(34) we obtain its linearised form consisting of:

• •

  FLRW background equations: are identical to (28), (29) and (32)

  	$\displaystyle\dot{\bar{\rho}}=$	$\displaystyle-\bar{\rho}\,\bar{\Theta},\,\,$
  	$\displaystyle\dot{\bar{\Theta}}=$	$\displaystyle-\frac{\bar{\Theta}^{2}}{3}-4\pi\bar{\rho}+8\pi\Lambda,$		(54)
  	$\displaystyle\left(\frac{\bar{\Theta}}{3}\right)^{2}=$	$\displaystyle\frac{8\pi}{3}\left[\,\bar{\rho}+\Lambda\,\right]-\bar{\cal{K}},$

• •

  Linearised evolution equations for the fluctuations $\Delta^{(\rho)}_{(\textrm{\tiny{as}})},\,{\textrm{\bf{D}}}^{(\Theta)}_{(\textrm{\tiny{as}})}$ (linearised forms of (30)–(31))

  	$\displaystyle\dot{\Delta}^{(\rho)}_{1}$	$\displaystyle=$	$\displaystyle-{\textrm{\bf{D}}}_{1}^{(\Theta)},$		(55)
  	$\displaystyle\dot{\textrm{\bf{D}}}^{(\Theta)}_{1}$	$\displaystyle=$	$\displaystyle-\frac{2}{3}\bar{\Theta}\,{\textrm{\bf{D}}}_{1}^{(\Theta)}-4\pi\bar{\rho}\,\Delta^{(\rho)}_{1},$		(56)

• •

  Constraint that defines the spatial curvature fluctuation (notice that in (33) we have $(\Theta_{q}-\bar{\Theta})^{2}\sim O(\epsilon^{2})$)

  $\displaystyle\frac{3}{2}{\textrm{\bf{D}}}_{1}^{({\cal{K}})}={4\pi}\bar{\rho}\Delta^{(\rho)}_{1}-\frac{\bar{\Theta}}{3}{\textrm{\bf{D}}}_{1}^{(\Theta)}.$

  (57)

• •

  Linearised form of the second order equation (34) for the density contrast fluctuation

  $\displaystyle\ddot{\Delta}^{(\rho)}_{1}+\frac{2}{3}\bar{\Theta}\,\dot{\Delta}^{(\rho)}_{1}-4\pi\bar{\rho}\,\Delta^{(\rho)}_{1}=0,$

  (58)

In what follows we examine the analytic solutions for this linearised system by assuming initial conditions at the last scattering surface ($t_{i}=t_{\textrm{\tiny{LS}}},\,\,z\sim 1100$), so that the $\Lambda$CDM background is very close to an Einstein–de Sitter background model (${\cal{K}}_{i}=\Lambda=0$, see B). The solutions follow by applying the approximations (127) to the exact forms (116)–(119) (see B) and by bearing in mind together the equivalences (44):

and where the subindex ${}_{1\textrm{\tiny{LS}}}$ denotes $O(\epsilon)$ quantities evaluated at $t=t_{i}=t_{\textrm{\tiny{LS}}}$ and $\bar{a}^{3/2}=(3/2)\bar{H}_{\textrm{\tiny{LS}}}(t-t_{\textrm{\tiny{LS}}})+1$ (so that $t=t_{\textrm{\tiny{LS}}}$ corresponds to $\bar{a}=\bar{a}_{i}=1$). The coefficients $C_{\pm}=C_{\pm}(r,\theta,\phi)$ identify the amplitudes of the growing ($+$) and decaying ($-$) modes. The remaining fluctuations follow from (55) and (57)

where we remark that (as expected) the linearised curvature fluctuation has no contribution from the decaying mode.

It is customary to eliminate the decaying mode by setting $C_{\tiny{\textrm{(--)}}}=0$ ⁵⁵5As argued in multi ; coarse , it is not strictly necessary to totally eliminate the decaying mode, which is equivalent to demanding a perfectly simultaeous Big Bang ($t_{\textrm{\tiny{bb}}}^{\prime}=0$). Linear initial conditions imply that a small amplitude decaying mode (of the order of initial fluctuations) produces a gradient $rt_{\textrm{\tiny{bb}}}^{\prime}$ that leads to age differences ($\sim 10^{3}-10^{4}$ years) among observers that are negligible in comparison with cosmic age. , which yields the following constraint linking the density and curvature fluctuations

and thus the pure growing mode solutions of (58) are (59)–(60) with $C_{\tiny{\textrm{(--)}}}=0$:

with $\Delta^{(\rho)}_{1\textrm{\tiny{LS}}}=C_{\tiny{\textrm{(+)}}}$. While these linear fluctuations are the Szekeres analogues of linear CPT fluctuations, there are important and subtle differences: they are deterministic while linear CPT perturbations are based on random field variables which contain the former. This is because the solutions in (61) are separable and thus the evolution is independent of the initial configuration. This universal evolution of the linearised fluctuations is known as the transfer function. We discuss this issue in section 10, in particular we compare $\Delta^{(\rho)}_{1}$ with the matter density CPT perturbation $\delta_{1}$.

The metric in the Cosmological Perturbation Theory formalism in the isochronous gauge can be written in a similar form as the Szekeres metric in (1) (for more details see e.g. EMM ):

where $\gamma_{ij}$ is the 3-metric or conformal spatial metric and $\tau$ is the conformal time related to physical cosmic time by $\tau=\int{{\rm{d}}t/\bar{a}(t)}$. The density contrast $\delta$ is defined by:

with $\bar{\rho}$ denoting the background density. The deformation tensor $\vartheta^{\mu}_{\nu}$ is given by:

where the isotropic background expansion is given by the conformal Hubble scalar $\bar{H}(\tau)$ and the projection tensor $h^{\mu}_{~\nu}=u^{\mu}u_{\nu}+\delta^{\mu}_{~\nu}$ must be computed with $u^{\mu}=\bar{a}(\tau)\delta^{\mu}_{\tau}$.

The evolution equations for the variables $\delta$ and $\vartheta^{\mu}_{\nu}$ are furnished by the continuity and Raychaudhuri equations:

with $\vartheta=\vartheta^{\mu}_{\;\mu}$. Since $\tau=\tau(t)$ and thus $\partial/\partial\tau=\bar{a}(t)\,\partial/\partial t$, and considering that $u^{\mu}_{;\nu}=\sigma^{\mu}_{\nu}+(\Theta/3)h^{\mu}_{\nu}$ holds for Szekeres models (as the 4–acceleration and vorticity associated with $u^{\mu}$ vanish), we can rewrite the CPT deformation tensor introduced in (65) and its trace $\vartheta$ in terms of variables we have used to examine Szekeres models: the shear tensor and the contrast Hubble scalar fluctuation as follows

where we used  (36) and (38). From here onwards we will compute all quantities in terms of cosmic time (hence we remove the “$(t)$” label).

If we rewrite the continuity and the Raychaudhuri equations (66)–(67) in terms of the variables we have used, we find that they match exactly with the corresponding equations for asymptotic fluctuations (30)-(31) when the following non–linear correspondences between the CPT and Szekeres fluctuations hold:

Since the evolution equations for these non–linear variables are mathematically identical, the perturbative equations to all orders should be also identified. We proceed to relate the spatial curvature perturbations with the curvature of asymptotic variables by means of the definition $6\mathcal{K}={}^{(3)}{\cal{R}}$. The 3–Ricci scalar of the spatial metric $h_{ik}$ can be expressed as

We have thus related our variables describing Szekeres exact solutions to the covariant and gague-invariant set of variables of Cosmological Perturbation Theory. Let us exploit our new relations and the solutions for the Szekeres variables to show that, through the Hamiltonian constraint, the spatial curvature scalar is time-independent. At first order in perturbations, the constraint in Eq. (33) drops the last term and it can be written as an expression for the 3–Ricci scalar at first order,