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Non-Spherical Szekeres models in the language of Cosmological Perturbations Roberto A. Sussman,1 Juan Carlos Hidalgo,2 Ismael Delgado Gaspar,3 and Gabriel Germán2 1Instituto de Ciencias Nucleares, Universidad Nacional Autońoma de México, A. P. 70–543, 04510 México D. F., México,∗ 2Instituto de Ciencias F´ısicas, Universidad Nacional Autońoma de México, 62210 Cuernavaca, Morelos, México,† 3Instituto de Investigación en Ciencias Ba´sicas y Aplicadas, Universidad Autońoma del Estado de Morelos, Av. Universidad 1002, 62210 Cuernavaca, Morelos, México. (Dated: March 30, 2017) We study the differences and equivalences between the non–perturbative description of the evolution of cosmic structure furnished by the Szekeres dust models (a non–spherical exact solution of Einstein’s equations) and the dynamics of Cosmological Perturbation Theory (CPT) for dust sources in a ΛCDM background. We show how the dynamics of Szekeres models can be described by evolution equations given in terms of “exact fluctuations” that identically reduce (at all orders) toevolutionequationsofCPTinthecomovingisochronousgauge. WeexplicitlyshowhowSzekeres 7 linearisedexactfluctuationsarespecific(deterministic)realisationsofstandardlinearperturbations 1 of CPT given as random fields but, as opposed to the latter perturbations, they can be evolved 0 exactly into the full non–linear regime. We prove two important results: (i) the conservation of 2 thecurvatureperturbation(atallscales)alsoholdsfortheappropriatelinearapproximationofthe exact Szekeres fluctuations in a ΛCDM background, and (ii) the different collapse morphologies r a of Szekeres models yields, at nonlinear order, different functional forms for the growth factor that M follows from the study of redshift space distortions. The metric based potentials used in linear CPTarecomputedintermsoftheparametersofthelinearisedSzekeresmodels,thusallowingusto 8 relateourresultstolinearCPTresultsinothergauges. Webelievethattheseresultsprovideasolid 2 starting stage to examine the role of non–perturbative General Relativity in current cosmological research. ] c PACSnumbers: 98.80.-k,04.20.-q,95.36.+x,95.35.+d q - r g I. INTRODUCTION A proper understanding of the evolution of the den- [ sityandthepeculiarvelocitiesincosmicinhomogeneities is essential to distinguish between specific cosmological 2 Gauge invariant perturbations on a Friedman– v Lemaˆıtre–Robertson–Walker (FLRW) background, models. While several dark energy or modified gravity 9 generically examined within the framework of Cos- background FLRW models reasonably fit late time ob- 1 servational data, it is the clustering properties of matter mological Perturbation Theory (CPT), constitute an 8 that allows us to distinguish between these models [20– important theoretical tool in cosmological research (see 0 22]. Thegrowthofstructureischaracterisedbyagrowth pioneering work in [1] and more recent comprehensive 0 factor f function computed in linear CPT, which is an . reviews in [2–4]). Linear CPT is specially adequate 1 observablederivedfromtheanisotropyofthepowerspec- to study cosmic sources whenever near homogeneous 0 truminredshiftspaceinthenon–linearregime[23]. Yet, conditions can be justified: the early Universe and 7 it is precisely at this non–linear regime that departures 1 scales comparable to the Hubble radius for late cosmic from Newtonian evolution arise, which suggests consid- : times. Since late time structure formation at deep v eringtheevolutionofinhomogeneitiesatnon–linearlevel sub–horizon scales becomes highly nonlinear and (at i X least locally) nonrelativistic, it is usually studied by byintroducingcorrectionsfromGeneralRelativity(GR) theory. r means of non–linear and non–perturbative Newtonian a gravity, either through analytic models (spherical Non–perturbative and fully non–linear GR theory is [5–8] and elliptic collapse [8–10]) or by sophisticated not a particularly favoured theoretical tool in cosmolog- numerical N-body simulations [11, 12]. By considering ical research, since (given its high nonlinear complex- higher order perturbations (Newtonian and relativistic) ity) any minimally realistic non–perturbative and rela- CPT has been so far extended to study the mildly tivistic modelling of structure formation necessarily re- nonlinear regime [13–18], leaving the description of fully quires numerical 3–dimensional codes to solve Einstein’s nonlinear effects in large scale structure formation to equations in a cosmological context, whether as a con- Newtonian non–perturbative methods, though the need tinuum model or as GR numerical simulations. This for considering relativistic corrections is still an open impressively difficult task is still in its early stages of question (an extensive review is [19]). development [24–30]. 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. ∗ [email protected] From a fully relativistic and non–perturbative approach, † hidalgo@fis.unam.mx these toy models emerge from appropriate exact solu- 2 tions of Einstein’s equations. Since cold dark matter article we explore this connection through a detailed rig- (CDM) at cosmic scales can be adequately described by orous comparison between the dynamical equations of a dust source and a Λ term can always be added to SzekeresmodelsandCPTfordustsourcesinthecomov- the dynamics as a phenomenological description of dark ing isochronous gauge, thus extending and continuing energy, the most useful exact GR solutions applicable a previously published [78] similar comparison between to cosmology are the well known spherically symmet- LTB models and CPT. This represents the first stage in ric dust Lemaˆıtre–Tolman–Bondi (LTB) models [31] and an effort to compare all these theoretical tools in terms their non–spherical generalisation furnished by Szekeres of their structure modelling predictions and their usage models [32, 33] (see comprehensive reviews in [34–38] for in fitting observations and addressing open theoretical both classes of solutions). issues. The usage of LTB and Szekeres models in cosmolog- The contents of this article are summarised as follows. ical applications can (and should) also be undertaken Section 2 introduces the metric of Szekeres models and within the prevailing ΛCDM paradigm or Concordance their particular cases (spherical and axial symmetry) in Model. It is important to emphasise this point, since spherical coordinates. In sections 3 and 4 we show how employing these exact solutions is often associated with thedynamicsofthemodelscanbecompletelydetermined the recent past attempt to challenge this paradigm by by suitable covariant variables defined in [57] and used means of non–perturbative large scale (Gpc sized) den- in [60, 77]: the q–scalars associated with the density, the sity void configurations mostly constructed with LTB Hubble scalar and spatial curvature, their corresponding models [39, 40] and, to a lesser degree, with simplified FLRW background variables and exact fluctuations that Szekeresmodels(seereviewsin[34–38]). OnceLTBvoid convey the inhomogeneity of the models. The evolution models were ruled out [41–43], thus re–affirming the va- equations for these variables (as we show) can be ade- lidity of the Concordance Model, interest on cosmolog- quatelyrelatedto theCPTevolutionequations asinthe ical application of exact GR solutions decreased. How- LTB case [78]. Employing these variables is crucial, as ever, the claim that non–perturbative GR is redundant the standard metric based description of Szekeres model for cosmological research (within or without the ΛCDM (as used in all previous work on these models save for paradigm) is an open issue [44–48]. As long as the cos- [57, 60, 77]) is not useful for relating to CPT dynam- mological implementation of numerical GR remains un- ics. We define in section 5 a linear regime in Szekeres der development, it should be appropriate to continue models by suitable first order expansions of the exact applying LTB and Szekeres models as theoretical tools fluctuations as done with LTB models in [78]. In partic- to complement perturbative and numerical cosmological ular, we show that this linear regime is compatible with research. large deviations from spherical symmetry that allow for Because of their spherical symmetry LTB models only a description of networks of multiple evolving structures allowustodescribetheevolutionofasingleCDMstruc- (overdensitiesandvoids[60,77]). ThelinearisedSzekeres ture (an overdensity or a density void) imbedded in a evolution equations and their solutions are examined in suitable FLRW background (see numerical examples in section 6. [36]). Evidently, Szekeres models introduce more dy- In section 7 we examine the equivalence between the namical degrees of freedom, as can be appreciated in SzekeresandCPTevolutionequationsfordustsourcesin the above cited reviews and in theoretical studies [49– the isochronous CPT gauge. We follow the methodology 60], as well as in the extensive literature on their appli- of [78] but with more depth and generality: while we cation to structure formation and fitting of cosmologi- focus primarily on first order linear equations, we show cal observations [61–77]. Practically all of these articles that the Szekeres evolution equations are equivalent to (seeexceptionsbelow)onlyconsiderthesimplesttypeof CPT equations at all approximation orders. We prove structure formation scenario allowed by the models: a thatthefirstorderSzekeresspatialcurvaturefluctuation 2–structure dipolar configuration comprised by an over- is conserved in time for all scales, just as its equivalent density evolving next to a density void (see numerical linear spatial curvature perturbation from CPT. examples in [36]). The possibility of modelling less re- In section 8 we describe the “pancake” and spherical strictivestructureswassuggestedalreadyinearlierwork collapse morphologies allowed by Szekeres models, then [56,58,61,62],butitonlybecamerecentlyimplemented wecomputetheSzekeresanalogueofthegrowthsuppres- [60,77]byusingthefullextensionofthedynamicalfree- sionfactorf usedinCPTtoaddressthestudyofredshift dom of the models for the description of elaborated net- space distortion [22, 23, 79, 80]. We show that this ana- works consisting of an arbitrary number of CDM struc- logue fully coincides with its linear equivalent from CPT tures (overdensities and voids). and that is only sensitive to collapse morphologies in a Since Szekeres models (specially those examined in non–linearregime(eithersecondorderCPTorexact). In [60, 77]) provide the less idealised exact GR solution in section 9 we obtain the Szekeres equivalent CPT metric a cosmological context, they are specially suitable to ex- potentials (in the isochronous gauge) by linearising the aminetheconnectionbetweennon–perturbativeGRand Szekeres line element in terms of its deviation from the CPT based perturbations at different orders, as well as FLRW metric. This allows us to relate our dynamical withnon–perturbativeNewtonianmodels. Inthepresent equations with CPT quantities in any gauge. 3 In section 10 we present concluding remarks and a Spherical Symmetry: LTB models.: We can define discussion of the main results listed above. We also for each Szekeres model the “LTB seed model” present a conceptual comparison between the CPT den- as the generic LTB model that follows as its spher- sity perturbation, given as a random field, and its Szek- ical limit: X = Y = Z = 0 ⇒ W = 0, whose eres equivalent: the linearised Szekeres density fluctua- metric is the following specialisation of (1) tion, given as a linearised expression that follows from Γ2 a deterministic exact solution. We argue that the linear γ = , γ =r2, γ =r2 sin2θ. (7) rr 1−K r2 θθ φφ CPT perturbation is far more general, but is only valid qi in the mildly non–linear regime, whereas its equivalent NoticethatanySzekeresmodelcanbeconstructed linearisedSzekeresfluctuationcanbeevolvedintoafully fromanarbitraryLTBseedmodelsimplybyintro- non–perturbative exact expression valid throughout the ducingsuitablenonzerodipoleparametersX, Y, Z. models evolution. Evidently, Szekeres models always “inherit” the Finally, we provide two appendices: A discusses the propertiesoftheirLTBseedmodels, afeaturethat integration of the Friedman–like quadrature that deter- is very useful to study their properties. mines the functional form of the metric variables (the scalefactors),whileBderivesthelinearisedformofthese Axial Symmetry.: Another important particular case scale factors. isfurnishedbyaxiallysymmetricmodels: X =Y = P =0, Z (cid:54)=0, leading to the specialised metric (Γ−W)2 II. SZEKERES MODELS AND THEIR γ = +W2, γ =−rW , SUB–CASES. rr 1−Kqir2 ,θ rθ ,θ γ =r2, γ =r2 sin2θ, (8) θθ φφ We consider quasi–spherical Szekeres models of class I where W = −Z cosθ is independent of φ. The intermsof“stereographic”sphericalcoordinates[35,60, 77] 1 asymptotic spherical limit follows if Z →0 as r → ∞. ds2 =−dt2+h dxidxj, i,j =r,θ,φ, ij h =h δµδν =g δµδν =a2γ , a=a(t,r), (1) ij µν i j µν i j ij III. DYNAMICS THROUGH QUASI–LOCAL where h =g +u u , uµ =δµ and the nonzero com- SCALARS µν µν µ ν t ponents of γ are: ij Practically all work considering cosmological applica- (Γ−W)2 tionsofSzekeresmodels[34–38]examinethedynamicsof γ = +(P +W )2+U2P2, (2) rr 1−K r2 ,θ ,φ the models in terms of the metric variables determined qi from a Friedman quadrature that follows directly from γ =−r(P +W ), γ =r sinθUP , (3) rθ ,θ rφ ,φ Einstein’s field equations (see A and B). This approach γ =r2, γ =r2 sin2θ, (4) θθ φφ is not useful to relate the models to CPT. Instead, we determine the dynamical equations (including the met- with the functions Γ, U, P and W given by ric functions) through the first order system of evolution ra(cid:48) equations derived in [57] and used in [60, 77] Γ=1+ , U =1−cosθ, (5) a ρ˙ =−3ρ H , (9) q q q P =Xcosφ+Y sinφ, W=−Psinθ−Zcosθ,(6) Θ2 Θ˙ =− q −4πρ +8πΛ, (10) wherethefreeparametersX, Y, Z, Kqi dependonlyonr q 3 q (see the interpretation of Kqi in (21)), and (cid:48) denotes the ∆˙(ρ) =−(1+∆(ρ))D(Θ) (11) radial derivative. The function W has the mathematical (cid:18) (cid:19) structure of a dipole whose orientation is governed by D˙ (Θ) = −2Θ +D(Θ) D(Θ)−4πρ ∆(ρ), (12) the choice of the three dipole parameters X, Y, Z (see 3 q q comprehensive discussion in [60]). Different particular Θ a˙ =a q, (13) cases follow by specialising these parameters: 3 Γ−W G˙ =GD(Θ), G = , (14) 1−W with˙=∂/∂t,andwithquantitiessubjecttothealgebraic 1 Allfurthermentionof“Szekeresmodels”willreferonlytoquasi– constraints: sphericalmodelsofclassI(see[35]forabroaddiscussionontheir classification). Thestandarddiagonalmetricformthesemodels (cid:18)Θ (cid:19)2 8π and the transformation relating it to (1) is given in Appendix q = [ρ +Λ]−K , (15) A of [60]. For constant time slices with spherical or wormhole 3 3 q q topology [35] the metric (1) must be modified as explained in 3 Θ AppendixDof[57]. 2D(K) =4πρq∆(ρ)− 3qD(Θ). (16) 4 Here the quasi–local (q–scalars) A and their exact fluc- of Θ , ∆(ρ), D(Θ), D(K) by solving the constraints (15)– q q tuations D(A) [57] are given for each scalar A=ρ, Θ, K, (16) at t = ti, the radial coordinate is chosen so that (density, Hubble expansion and spatial curvature) by ai =Γi =1,whiletheBigBangtimetbb anditsgradient t(cid:48) canbeobtainedfromρ , K andtheirgradients(see (cid:82) bb qi qi Aq = (cid:82)DAΞdVp, (17) [57, 60, 77] and A, B). FdV D p rA(cid:48) D(A) =A−A = q , (18) q 3(Γ−W) IV. DYNAMICS THROUGH CONTRAST PERTURBATIONS. D(ρ) ρ−ρ ∆(ρ) = = q, (19) ρ ρ q q The fluctuations D(A) and ∆(A) in (18)–(19) compare (cid:113) with dV = det(g )d3x the scalars A = ρ, H, K with their associated q-scalars p ij A = ρ , H , K at the same values of r for all t. As q q q q a3r2(Γ−W) sinθ = drdθdφ, (20) a consequence, the density fluctuation ∆(ρ) is different Ξ fromthenotionofa“densitycontrast”. However,assum- (cid:112) ing the existence of an asymptotic FLRW background and where Ξ = 1−K r2. This last integral in (17) qi (see conditions for this in B), we can define exact fluctu- is evaluated in an arbitrary time slice (constant t) in a ations that yield the notion of a contrast by comparing spherical comoving domain D bounded by an arbitrary the scalars A at very point with their background values fixed r >0. This leads to the scaling laws 2 A¯=ρ¯, Θ¯, K3 . Thisleadstothefollowing“asymptotic” ρ = ρqi, K = Kqi, Θq = a˙, (21) fluctuations q a3 q a2 3 a ρ−ρ¯ 1+∆(ρ) 2 2 +∆(K) D(A) =A−A¯, ∆(ρ) = , (26) 1+∆(ρ) = i , +∆(K) = 3 i , (22) (as) (as) ρ¯ G 3 G where G is defined in (14) and the subindex denotes withtheA¯givenby(FLRWquantitieswillbehenceforth i evaluation at an arbitrary time slice t=t . denoted by a overbar) i Thefirstorderevolutionequationsforthefluctuations ∆(ρ)andD(Θ)canbecombinedintoasinglesecondorder ρ¯= ρ¯i, Θ¯ = a¯˙ =H¯, K¯ = K¯i, a¯(t)= lim a, equation a¯3 3 a¯ a¯2 r→∞ (27) 2(cid:104)∆˙(ρ)(cid:105)2 The corresponding evolution equations and constraints ∆¨(ρ)− +2Θ ∆˙(ρ)−4πρ ∆(ρ) (cid:16)1+∆(ρ)(cid:17)=0, are 1+∆(ρ) 3 q q (23) ρ¯˙ =−ρ¯Θ¯, (28) which resembles an exact (non–linear) generalisation of the equation of linear dust perturbations in the syn- Θ¯˙ =−Θ¯2 −4πρ¯+8πΛ, (29) chronous gauge [78]. 3 The initial conditions to integrate the system (9)–(16) ∆(˙ρ) =−(cid:104)1+∆(ρ)(cid:105) D(Θ), (30) arespecifiedatt=t andconsistofthecosmologicalcon- (as) (as) (as) i (cid:20) (cid:21) stantΛplusthefollowingfivefreefunctionsthatdepend D˙ (Θ) =− 2Θ¯ − 4Θ +D(Θ) D(Θ)− 2(cid:0)Θ −Θ¯(cid:1)2−4πρ¯∆(ρ), only on r: (as) 3 q (as) (as) 3 q (as) (31) The “radial” functions: ρ , K . (24) qi qi The “dipole” functions: X, Y, Z. (25) where the radial functions are common to the LTB seed (cid:18)Θ¯(cid:19)2 8π = [ρ¯+Λ]−K¯, (32) model and the dipole functions govern the deviation 3 3 from spherical symmetry. We obtain the initial values 3D(K) =4πρ¯∆(ρ) − ΘqD(Θ)+ 1(cid:0)Θ −Θ¯(cid:1)2, (33) 2 (as) (as) 3 (as) 6 q 2 The lower bound of the integrals (17) is the locus r = 0, anal- ogous to the symmetry centre of spherical models [56]. While Szekeres models are not spherically symmetric, the surfaces of 3 Notice that the covariant background scalars A¯ can be under- constant r are non–concentric 2–spheres [35, 60]. Notice that stood as asymptotic limits as r → ∞ of the q–scalars Aq. In Aq = Aq(t,r) even if the scalars A depend on the four coor- otherwords: theA¯areaveragesforanasymptoticaveragingdo- dinates (t,r,θ,φ) [54, 56, 57]. Their relation with the average maincoveringthewholetimeslice(seecomprehensivediscussion integralsisdiscussedin[56,57]. in[78]). 5 while the equivalent to the second order equation (23) is where we used (16) and (21)–(22) and δ(ρ), d(K) are the i i initial fluctuations of the LT seed model given by (cid:20) (cid:21)2 ˙ ∆(ρ) ∆¨(ρ) − (as) +(cid:20)2Θ¯ − 4Θ (cid:21) ∆(˙ρ)− δ(ρ) =∆(ρ)| = rρ(cid:48)qi, d(K) =D(K)| = rKq(cid:48)i, (as) 1+∆(ρ) 3 q (as) i i W=0 3ρqi i i W=0 3 (as) (41) (cid:104) (cid:105) (cid:16) (cid:17) 4πρ¯∆(ρ) −2(Θ −Θ¯)2 1+∆(ρ) =0, (34) (as) q (as) whereweusedthefactthata =Γ =1. Proceedingasin i i which not only “resembles” but strictly provides the ex- LTBmodelsin[78],wedefinealinearregimeforSzekeres act (non–linear) generalisation of the evolution equation models (understood as functional parameter “closeness” of linear dust density perturbation in the synchronous to a FLRW background, see B) by demanding that a gauge [78]. positive dimensionless number (cid:15)(cid:28)1 exists such that The system (28)–(33) is not self–contained, it thus needstobesupplementedby(9)–(10). However,thecon- all of |∆(ρ)|, |D(ρ)|, |D(Θ)|, |D(K)| ∼O((cid:15)) trast fluctuations in (28)–(33) are related to the quasi– ⇒ Σ, Ψ ∼O((cid:15)), (42) local fluctuations defined in the previous section by 2 (cid:18)ρ (cid:19)(cid:16) (cid:17) holdsforagivenevolutionrange,withthetermO((cid:15))(cid:28)1 ∆((ρas))−∆(ρ) = ρ¯q −1 1+∆(ρ) denoting quantities of the order of magnitude of (cid:15). No- ticethatthecovariantobjectsΣ, Ψ vanishattheFLRW =(cid:20)ρqi a¯3 −1(cid:21) 1+∆(iρ), (35) background, hencethefluctuations2in(42)alsovanishat ρ¯ a3 G the background, and thus (from Stewart lemma [81, 82]) i 31(cid:16)D((aΘs))−D(Θ)(cid:17)=13(cid:0)Θq−Θ¯(cid:1)= aa˙ − aa¯¯˙. (36) araremgetaeurgereisntvraicrtiiaonntsqtuhaanttiytiieesld. Wtheedneercivesesabreylowantdhesupffia-- cient conditions for a linear regime defined by (42). Therefore, for all purposes it is more practical to solve The necessary (not sufficient: see (53)) condition for a first the background equations (28)–(29) and then use Szekeres linear regime is the existence of a linear regime the solutions of (9)–(16) and (13)–(14) to compute the in the LTB seed model, which requires (see [78]) the ra- density contrast and Hubble scalar fluctuation from the dial initial conditions (24) satisfying for A=ρ, Θ, K relations (35)–(36). (cid:12)(cid:12)Aqi−A¯i(cid:12)(cid:12)∼O((cid:15)), (cid:12)(cid:12)rA(cid:48)qi(cid:12)(cid:12)∼O((cid:15)) ⇒ δ(ρ), d(K), d(Θ) ∼O((cid:15)), (43) i i i V. THE SZEKERES LINEAR REGIME. where we used the fact that the initial fluctuations of The FLRW limit of Szekeres models can be defined the LTB seed model are linked by the constraint (16) rigorously and in a coordinate independent manner [35, restricted to W=0 and evaluated at t=t . As a conse- i 60]bythevanishingoftheshearandelectricWeyltensors quence of (43) and bearing in mind (18)–(22), (35)–(36) inthebackground: σa =Ea =0,whiletheirfluctuations b b and the results of B, we have up to O((cid:15)) are both expressible as A A A a Eνµ =Ψ2eµν, with Ψ2 =−43π D(ρ), (37) Aq, A¯q, A¯, a¯, Γ≈1 D(Θ) ⇒∆(ρ) ≈∆(ρ), D(Θ) ≈D(Θ), D(K) ≈D(K), (44) σµ =Σeµ, with Σ=− , (38) (as) (as) (as) ν ν 3 which suggests introducing the following notation valid where eµ = diag[0,−2,1,1] is a unique traceless tensor ν up to O((cid:15)): basis satisfying e˙µ =0 and the fluctuations ν D(ρ) =ρ ∆(ρ) and D(Θ) are q ∆(ρ) =∆(ρ) =∆(ρ), D(A) =D(A) =D(A), 1 (as) 1 (as) D(ρ) =ρqi 1−Γ+δi(ρ), (39) a1 ≡a−a¯, (45) a3 Γ−W 4πρ (1−Γ+δ(ρ))−a(cid:104)K (1−Γ)+ 3d(K)(cid:105) thatwillbeusedhenceforthtodenotebothtypesoffluc- D(Θ) = qi i qi 2 i , tuationswehaveusedsofar,astheyareindistinguishably a3 (Θq/3)(Γ−W) when linearised. (40) The linearised forms for the metric functions (see 6 derivation in B4) are where φ¯m(a¯), φ¯k(a¯) are dimensionless functions of O(1) defined in (B3) 8πρ ρ K K Ωˆm(r)= qi =Ω¯m qi, Ωˆk(r)= qi =Ω¯k qi, a =a−a¯≈φ¯m(Ωˆm−Ω¯m)+φ¯k(Ωˆk −Ω¯k)∼O((cid:15)), qi 3H¯2 i ρ¯ qi H¯2 i K¯ 1 qi i qi i i i i i (46) (48) whereΩ¯m =8πρ¯/(3H¯2 andΩ¯k =K¯ /H¯2 =Ω¯m+Ω¯Λ−1 Γ−1≈−φ¯mδ(ρ)−φ¯kd(κ) ∼O((cid:15)), (47) i i i i i i i i i i are the standard density fraction FLRW parameters. Toobtainthesufficientconditionweexpand(39)–(40)intermsofδ(ρ), d(K)which(from(43))are∼O((cid:15))quantities. i i Considering that Ωˆm−Ω¯m and Ωˆk −Ω¯k are both ∼O((cid:15)), together with (46)–(47), we obtain: qi i qi i ρ¯ (1+φ¯m)δ(ρ)+φ¯kd(κ) (cid:20) (cid:18) (cid:15) (cid:19)(cid:21) D(ρ) ≈ i i i 1+O , (49) 1 a¯3 1−W 1−W (cid:104) (cid:105) 3Ω¯m[(1+φ¯m)δ(ρ)+φ¯kd(κ)]−a¯ Ω¯k(φ¯mδ(ρ)+φ¯kd(κ))+ 3d(κ) (cid:20) (cid:18) (cid:19)(cid:21) D(Θ) ≈ 2 i i i i i i 2 i 1+O (cid:15) , (50) 1 (Θ¯/3)a¯3(1−W)H¯−1 1−W i where (and this is important to notice) we did not restrict the dipole term W to be small (i.e, we have in general |W|∼O(1)). For purely growing modes (see A2), the suppression of the decaying mode yields a constraint between the initial fluctuations δ(ρ) and d(κ). Therefore, the expansions (49) and (50) take the simplified form i i ρ¯ (1+F¯)δ(ρ) (cid:20) (cid:18) (cid:15) (cid:19)(cid:21) D(ρ) ≈ i i 1+O , (51) 1 a¯3 1−W 1−W (cid:2)3Ω¯m(1+F¯)−a¯(cid:0)Ω¯kF¯− 3Φ¯m/Φ¯k(cid:1)(cid:3)δ(ρ) (cid:20) (cid:18) (cid:15) (cid:19)(cid:21) D(Θ) ≈ 2 i i 2 i i i 1+O , (52) 1 a¯3(Θ¯/3)(1−W)H¯−1 1−W i wherethebackgroundquantitiesF¯(a¯), Φ¯m, Φ¯karedefinedinB.TheirexplicitformsforaΛCDMbackground(Ω¯k =0) i i i are given by (B5)–(B6). Thenecessaryandsufficientconditionforalinearregime VI. LINEARISED EVOLUTION EQUATIONS in generic Szekeres models are then the necessary con- AND THEIR SOLUTIONS. ditions (43)–(47) for the linear regime of the seed LTB modelplustheextraconditioninvolvingthedipoleterm: Applyingthecriterionforalinearregimegivenby(42) tothesystem(28)–(34)weobtainitslinearisedformcon- (cid:15) sisting of: (cid:28)1 ⇒ (cid:15)(cid:28)1−W. (53) 1−W • FLRWbackgroundequations: areidenticalto(28), (29) and (32) ItisimportanttoemphasisethatalinearregimeinSzek- eresmodels,asspecifiedby(43)and(53),doesnotimply ρ¯˙ =−ρ¯Θ¯, cicloaslemneosdsetl”ocsopmheprliycianlgsywmitmhe|tWry|((cid:28)i.e.1)a.nA“sallmonogstassp(h5e3r)- Θ¯˙ =− Θ¯2 −4πρ¯+8πΛ, (54) 3 holds,largelocaldeviationsfromsphericalsymmetryas- (cid:18)Θ¯(cid:19)2 8π sociated with small 1 − W (see [60, 77]) are perfectly = [ρ¯+Λ]−K¯, compatible with a linear regime. 3 3 • Linearised evolution equations for the fluctuations ∆(ρ), D(Θ) (linearised forms of (30)–(31)) (as) (as) 4 Thederivationofallfunctionsappearingintheanalyticformsof ˙ Γ−1anda1=a−a¯andtheirlinearexpansionsaregiveninA ∆1(ρ) =−D(1Θ), (55) andB.Equations(49)–(50)simplifyconsiderablyifthedecaying modeissuppressed(whichimposesalinkbetweenδ(ρ)andd(κ)). D˙ (Θ) =−2Θ¯ D(Θ)−4πρ¯∆(ρ), (56) i i 1 3 1 1 7 • Constraint that defines the spatial curvature fluc- with∆(ρ) =C . Whiletheselinearfluctuationsarethe tuation (notice that in (33) we have (Θ −Θ¯)2 ∼ Szekere1sLSanalog(+u)es of linear CPT fluctuations, there are q O((cid:15)2)) important and subtle differences: they are deterministic whilelinearCPTperturbationsarebasedonrandomfield 3D(K) =4πρ¯∆(ρ)− Θ¯D(Θ). (57) variables which contain the former. This is because the 2 1 1 3 1 solutions in (61) are separable and thus the evolution is independent of the initial configuration. This universal • Linearised form of the second order equation (34) evolution of the linearised fluctuations is known as the for the density contrast fluctuation transfer function. We discuss this issue in section 10, in ∆¨(ρ)+ 2Θ¯ ∆˙(ρ)−4πρ¯∆(ρ) =0, (58) ppaerrttuicrublaatriowneδco.mpare∆(1ρ) withthematterdensityCPT 1 3 1 1 1 In what follows we examine the analytic solutions for VII. SZEKERES MODELS & COSMOLOGICAL this linearised system by assuming initial conditions at PERTURBATION THEORY: EQUIVALENCE OF the last scattering surface (t = t , z ∼ 1100), so that i LS EQUATIONS. the ΛCDM background is very close to an Einstein–de Sitter background model (K =Λ=0, see B). The solu- i The metric in the Cosmological Perturbation Theory tions follow by applying the approximations (B7) to the formalism in the isochronous gauge can be written in a exact forms (A9)–(A12) (see B) and by bearing in mind similarformastheSzekeresmetricin(1)(formoredetails together the equivalences (44): see e.g. [38]): C ∆(1ρ) =C(+)a¯+ a¯3(/–2), (59) ds2 =a¯2(τ)[−dτ2+γij(x,τ)dxidxj], (63) with C =−3D(1KLS), C =∆(ρ) + 3D(1KLS), whereγij isthe3-metricorconformalspatialmetricand (+) 5 H¯2 (–) 1LS 5 H¯2 τ is the conformal time related to physical cosmic time LS LS (cid:82) by τ = dt/a¯(t). The density contrast δ is defined by: andwherethesubindex denotesO((cid:15))quantitieseval- 1LS uatedatt=ti =tLS anda¯3/2 =(3/2)H¯LS(t−tLS)+1(so ρ(x,τ)=ρ(τ)+δρ(x,η)=ρ¯(τ)(1+δ(x,τ)), (64) that t=t corresponds to a¯=a¯ =1). The coefficients LS i C = C (r,θ,φ) identify the amplitudes of the growing with ρ¯ denoting the background density. The deforma- ± ± (+)anddecaying(−)modes. Theremainingfluctuations tion tensor ϑµ is given by: ν follow from (55) and (57) Θ¯ Θ¯ 1da¯ D(1Θ) =−C(+) + 3C(–), D(1K) = 5C(+). (60) ϑµν =a¯uµ;ν − 3hµν, 3 =H¯ = a¯dτ, (65) H¯ a¯1/2 2 a¯3 H¯2 3 a¯2 LS LS wheretheisotropicbackgroundexpansionisgivenbythe where we remark that (as expected) the linearised cur- conformal Hubble scalar H¯(τ) and the projection tensor vaturefluctuationhasnocontributionfromthedecaying hµν =uµuν +δµν must be computed with uµ =a¯(τ)δτµ. mode. The evolution equations for the variables δ and ϑµ are ν Itiscustomarytoeliminatethedecayingmodebyset- furnishedbythecontinuityandRaychaudhuriequations: ting C = 0 5 , which yields the following constraint (–) ∂δ linking the density and curvature fluctuations +(1+δ)ϑ=0, (66) ∂τ D(1KLS) = 35H¯L2S∆(1ρL)S, (61) ∂∂ϑτ + Θ¯3ϑ+ϑµνϑνµ+4πGa¯2ρ¯δ =0, (67) and thus the pure growing mode solutions of (58) are with ϑ=ϑµ. Since τ =τ(t) and thus ∂/∂τ =a¯(t)∂/∂t, µ (59)–(60) with C(–) =0: and considering that uµ =σµ+(Θ/3)hµ holds for Szek- ;ν ν ν eres models (as the 4–acceleration and vorticity associ- ∆(ρ) =∆(ρ) a¯, D(Θ) =−∆(1ρL)S H¯ , D(K) = 5∆1(ρL)S H¯2 , ated with uµ vanish), we can rewrite the CPT deforma- 1 1LS 1 a¯1/2 LS 1LS 3 a¯2 LS tion tensor introduced in (65) and its trace ϑ in terms of (62) variables we have used to examine Szekeres models: the shear tensor and the contrast Hubble scalar fluctuation as follows 5 Asarguedin[60,77],itisnotstrictlynecessarytototallyelim- (cid:18)1 (cid:19) inate the decaying mode, which is equivalent to demanding a ϑµ =a¯(t) D(Θ)hµ +σµ , (68) perfectly simultaeous Big Bang (t(cid:48) = 0). Linear initial condi- ν 3 (as) ν ν bb tionsimplythatasmallamplitudedecayingmode(oftheorder ϑ(t)=ϑµ(t)=a¯(t)D(Θ), (69) ofinitialfluctuations)producesagradientrt(cid:48)bb thatleadstoage µ (as) differences (∼103−104 years) among observers that are negli- 1(cid:16) (cid:17) gibleincomparisonwithcosmicage. σµν = 3 −D((aΘs))+Θq−Θ¯ eµν, (70) 8 whereweused (36)and(38). Fromhereonwardswewill where a = a (t,r) is defined in (46). At the homoge- 1 1 computeallquantitiesintermsofcosmictime(hencewe neous level the form of K¯ in Eq. (27) immediately shows remove the “(t)” label). that(3)R isconstantintimeintheFLRWbackground. γ If we rewrite the continuity and the Raychaudhuri Using the evolution equations (28) and (29) for the ho- equations (66)–(67) in terms of the variables we have mogeneous quantities, as well as the linearized evolution used, we find that they match exactly with the corre- of the fluctuations (Eqs. (55)–(56)) we find at first order sponding equations for asymptotic fluctuations (30)-(31) the following important result when the following non–linear correspondences between the CPT and Szekeres fluctuations hold: d (cid:104)(3)R (cid:105)=12d (cid:0)a¯a K¯(cid:1)−4a¯2K¯D(Θ) =0. (77) ϑ↔a¯D(Θ), δ ↔∆(ρ). (71) dt 1γ dt 1 1 (as) (as) Since the evolution equations for these non–linear vari- whichisconsistentwiththeresultfromperturbationthe- ables are mathematically identical, the perturbative ory [83–85], because D(K) dictates the amplitude of the 1 equationstoallordersshouldbealsoidentified. Wepro- growingmodeofthelinearizeddensitycontrast,asshown ceed to relate the spatial curvature perturbations with explicitly in (59)–(60), and also expected from CPT. the curvature of asymptotic variables by means of the Having the time dependence of D(K) at hand, we can definition 6K = (3)R. The 3–Ricci scalar of the spatial (as) readtheamplitudefromthepowerspectrumatanytime metric h can be expressed as ik and evolve the fluctuations in time with the solutions of (3)R=6(cid:104)K¯ +D(K)(cid:105). (72) Eq. (60) and, subsequently, with the non–linear solution (as) in the non–perturbative regime. Substituting this definition and the correspondences of Eq.(71)inthenon–linearHamiltonianconstraintofCPT (see e.g. [82]), VIII. GROWTH FACTOR AND COLLAPSE 4 MORPHOLGIES ϑ2+ Θ¯ϑ−ϑiϑj +(3)Ra¯2 =16πGa¯2ρ¯δ, (73) 3 j i werecoverthehomogeneousconstraint,Eq. (32),aswell The observable that accounts for the growth of struc- asthefluctuationsconstraint,Eq. (33),atthenon–linear ture is the growth factor f, usually parametrised (in the level. linear regime) as a function of the matter density frac- WehavethusrelatedourvariablesdescribingSzekeres tion. In the language of CPT, exact solutions to the covariant and gague-invariant set ofvariablesofCosmologicalPerturbationTheory. Letus f ≡ dlnδ1 =Ωγ , (78) exploit our new relations and the solutions for the Szek- 1 dlna¯ m eresvariablestoshowthat,throughtheHamiltoniancon- straint, the spatial curvature scalar is time-independent. where γ is known as the growth index that distinguishes Atfirstorderinperturbations,theconstraintinEq.(33) between different gravity theories and background mod- dropsthelasttermanditcanbewrittenasanexpression els [80]. However, the proper interpretation of observa- for the 3–Ricci scalar at first order, tions is subject to considering non–linear effects in col- lapsing structures. Since Szekeres solutions are an exact 4 (3)R a¯2 =6a¯2D(K) =16πa¯2ρ¯∆(ρ)− ΘD(Θ). (74) non–linear extensionof CPT thatcomprises all orders of 1 1 1 3 1 approximation, we examine in this section how the ad- Wenotethatthethree-Riccicurvaturescalarforthecon- missible collapse morphologies associated with these so- formal metric γij relates to the Ricci scalar above as6: lutionscanmodifytheprescriptionsforstructuregrowth at non–linear order. (3)R =a2 × (3)R, (75) γ and thus, as a consequence, the time derivative of Eq. (74) can be employed to show that the (3)R is con- γ A. Szekeres collapse morphologies stant in time order by order. Up to O((cid:15)) we can write (cid:104) (cid:105) (3)Rγ =(a¯+a1)2 (3¯)R+(3)R1 +O((cid:15)2), The geometry of the collapse of a dust source is dic- =(cid:0)a¯2+2a¯a (cid:1)(cid:104)6K¯ +6D(K)(cid:105)+O((cid:15)2), tated by the time evolution along the principal space di- 1 1 rections associated with the eigenvalues (and their asso- ciatedscalefactors)ofthedeformationtensorϑµ defined =6a¯2K¯ +12a¯a K¯ +6a¯2D(K)+O((cid:15)2). (76) ν 1 1 in(65). From (68)–(70)thedeformationtensorforSzek- eres models can be easily written in terms of the exact asymptotic fluctuations as 6 Thetree-Ricciscalarforthefullspatialmetric(3)Risconformal tothethree-Ricciscalarforγij,denotedthroughoutthetextby a¯(cid:104)(cid:16) (cid:17) (cid:105) (3)Rγ SeeEqns.(63)and(1). ϑµν = 3 −D((Θas))+Θq−Θ¯ eµν +D((aΘs))hµν . (79) 9 We can identify the three the nonzero eigenvalues of ϑµ aminethegrowthfactorf forSzekeresmodelsatvarious ν and their associated scale factors: approximations and compare with its form in CPT. (cid:96)˙ (cid:20) 1 (cid:21) ϑ =ϑµeµ e(1) =a¯ (1) =a¯ D(Θ)+ (Θ −Θ¯) , (1) ν (1) ν (cid:96)(1) 3 q log f 10 (80) (cid:96)˙ (cid:96)˙ ϑ =ϑ =ϑµeµ e(2) =ϑµeµ e(3) =a¯ (2) =a¯ (3) (2) (3) ν (2) ν ν (3) ν (cid:96) (cid:96) (2) (3) a¯ = (Θ −Θ¯), (81) 3 q Γ−W a¯ a (cid:96) = , (cid:96) =(cid:96) = , (82) (1) 1−W a¯ (2) (3) a¯ where we used (14), and the triad vectors eµ satisfy (i) h eµ eν =δ . µν (i) (j) (i)(j) Szekeres models admit spherical and pancake types of collapse morphology [8, 60, 77, 86]. By looking at the eigenvalues of ϑµ in (80)–(81) we can see that a “pan- ν cake”collapseoccursinanexpandingmodel(Θ , Θ¯ >0) q along the principal direction marked by ϑ (see numer- (1) ical example in [77]). This is straightforward to verify: since in regions where Γ−W ≈ 0 the scale factor (cid:96) (1) decreases while (cid:96) = (cid:96) keep growing and the eigen- (2) (3) value ϑ also increases, diverging as a shell crossing is log Ωm (1) 10 approached Γ−W → 0, all this happening as the re- maining two eigenvalues ϑ =ϑ remain positive and (2) (3) bounded. As a contrast, spherical collapse occurs for FIG.1. Thelinearsuppressionfactor. Logarithmicgraphof regions around r =0 with Θq <0 where dust layers col- the Szekeres linear growth factor f1 obtained in (87) vs the lapse in an expanding background (thus Θ¯ > 0 remains ΛCDM background parameter Ω¯m, both expressed as func- finite). The three eigenvalues (80)–(81) and diverge at tions of (a¯,Ω¯m) for the values Ωm =0.2, 0.35. The line with 0 0 theBigCrunchcollapsesingularitywhenΘ →−∞and slope6/11isrepresentedbythecircles. Thefigureshowshow q D(Θ) → −∞ as the three scale factors (82) tend to zero f1 ∼ [Ω¯m]6/11 is a good approximation for f1 in the ranges Ω¯m ≈1. (see (21) and (40)). B. The Szekeres growth factor Linear regime: At first order the growth factor (85) takes the form The growth factor in linear CTP is defined by the following well known expression valid up to O((cid:15)) D(Θ) f =− 1 , (86) f =dlnδ1 = δ˙1 , (83) 1 ∆1(ρ)H¯ 1 dlna¯ δ H¯ 1 where D(Θ) and ∆(ρ) can be computed from the Θ¯ a¯˙ (cid:2)Ω¯m−Ω¯ka¯+Ω¯Λa¯3(cid:3)1/2 1 1 H¯ = = =H¯ i i i , (84) leading terms in (49)–(50) (or (51)–(52)) together 3 a¯ i a¯3/2 withtheformfor1−ΓderivedinB.Aslongasthe decaying mode is suppressed (see A2 and B), for where for a ΛCDM background we have Ωk = 0, hence i whatever FLRW background that may be chosen ΩΛ =1−Ωm. Consideringtheequivalencerelations(71) i i the linearised factor f exhibits the characteristic between the linear CPT δ , ϑ and the exact Szekeres 1 1 1 features of the linear CPT growth factor: it is nec- fluctuations ∆(ρ), D(Θ), the exact generalisation of the (as) (as) essarily a bounded quantity that is insensitive to linear growth factor (83) is collapse morphologies and depends only on back- f = ∆((˙aρs)) =−(cid:104)1+∆((ρas))(cid:105) D((aΘs)), (85) garnodunddi(κv)acriaanbcleels,osuintc(ebtehceauisneitioafl fl(Auc8t)uaatnidon(sBδ6i()ρ)) ∆(ρ) H¯ ∆(ρ) H¯ and the terms a2, a3 and 1−W in the denomi- (as) (as) nators of (49)–(50) also cancel out. In particular, where we used (30), (35)–(36) and (39)–(40). Since the for a ΛCDM background with suppressed decaying equivalence relations (71) are valid at all orders, we ex- mode with t = t (present cosmic time) we have i 0 10 (from (51)–(52)) Non–linear approximation and exact form: Up to second order (85) becomes f = 3 a¯−Ω¯m0 Φ¯k(a¯) , (87) (1+∆(ρ))D(Θ)+D(Θ) 1 2(cid:2)Ω¯m0 +(1−Ω¯m0 )a¯3(cid:3) Φ¯k f2 =− 1∆(ρ)1H¯ 2 , (89) 1 where 0 denotes evaluation at t0 and we used the wherethesecondordertermD(Θ)canbecomputed form for 1−Γ derived in (B6) with Φ¯k and Φ¯k0 de- from (50) or (52). It is evident2that the non–linear fined in (B5). As expected (see figure 1), the Szek- (but still perturbative) f is now sensitive to the 2 eres linear growth factor (87) also complies with collapsemorphologies,since(89)isnolongerasim- the well known linear CPT approximation ple quotient of fluctuations, and thus, even if sup- pressingthedecayingmode,theinitialfluctuations f1 ≈[Ω¯m]6/11, (88) and terms a2, a3 and 1−W in the denominators of (49)–(50)nolongercancelout(Themorphology- whereΩ¯m(a¯)=Ω¯m/[Ω¯m+(1−Ω¯m)a¯3]. Thisresult dependence in the growth function is also manifest 0 0 0 has been taken as a probe of gravity, arguing that in second–order CPT [87]). This sensitivity to col- deviations from the growth index γ = 6/11 imply lapsemorphologiesisevenmoremanifestlyevident a departure from the GR prescription [20, 21, 79]. if we compute the exact form of f in (85) f = (Γ−W()Ωâmq30/−Ω¯(m0Ωˆ)mq(01/+Ω¯m0δ0()ρ()1−+Wδ0(ρ))a¯−3 W)a¯3 (a/a¯)3/232(ΩΓˆmq−0(1W−)(cid:113)Γ+Ωˆmqδ0i(ρ−))Ωˆ−kq0aa(cid:104)+ΩˆkqΩ¯0Λ0(1a3−(cid:112)ΓΩ¯)m0+−32dΩ¯0(k0κ)a¯(cid:105)+Ω¯Λ0a¯3 −3(cid:18)ΘΘ¯q −1(cid:19), (90) where we used the exact forms (39)–(40) and their synchronous and comoving gauge [87]. relation with asymptotic fluctuations in (35)–(36). We examine the sensitivity to collapse morphologies by comparing (90) and the scale factors (82) associatedwiththem. Assuminganexpanding(but otherwise generic) FLRW background (so that a¯ is ever growing and H¯ >0) we have IX. CONNECTION TO METRIC BASED PERTURBATIONS. • Pancake collapse. It is evident that f can exhibit very large growth if Γ−W becomes sufficiently small for large a, hence the scale The conditions for a linear regime (43)–(47) and (53) factor (cid:96)(1) decreases for increasing (cid:96)(2) = (cid:96)(3) allow us to express the Szekeres line element (1) as the and f diverges (shell crossing) as (cid:96)(1) → 0. metricofaSzekeresmodelthatisclose(upto∼O((cid:15)))to Likewise, the eigenvalue ϑ(1) diverges for fi- anFLRWmodel. ConsideringthatthestrictFLRWlimit nite ϑ(2) =ϑ(3) as Γ−W→0. of the models follows from demanding that σµ =Eµ =0 ν ν • Sphericalcollapse. Thegrowthfactorcanalso holds everywhere, the conditions for this limit are (from increase as 0 ≈ a (cid:28) 1 for bounded Γ−W, (38)–(41)) given by so that the three scale factors (cid:96) ,(cid:96) = (cid:96) (1) (2) (3) decreasewhiletheeigenvaluesϑ ,ϑ =ϑ (1) (2) (3) diverge,acollapsesingularityoccursasa→0 K(cid:48) =ρ(cid:48) =0 ⇒ a(cid:48) =0, Γ=1 qi qi (all this happening with increasing and large ⇒ D(ρ) =D(K) =D(Θ) =0, a=a¯(t). (91) a¯ and H¯). Evidently, the behaviour of the exact growth factor (90) should be compared with the non–linear Applying these conditions to (1) necessarily transforms second order form (89). In particular, it is neces- this metric, for whatever choice of dipole parameters in sary to compare both forms (89) and (90) with the W (cid:54)= 0, into a FLRW metric in an unusual coordinate growth factor occurring in previous work using rel- representation. This FLRW metric is ativistic non–linear perturbations [14–16]) and its comparison with the second order Newtonian solution worked out for a ΛCDM background in the ds2 =−dt2+h¯ dxidxj, h¯ =a¯2γ¯ (92) ij ij ij