Linear perturbations of an anisotropic Bianchi I model with a uniform magnetic field
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Linear perturbations of an anisotropic Bianchi I model with a uniformmagnetic field
Federico Di Gioia a,1,2 , Giovanni Montani b,1,3 Physics Department, “La Sapienza” University of Rome P.le A. Moro 5, 00185 Roma, Italy INFN, Roma 1 Section P.le A. Moro 5, 00185 Roma, Italy ENEA, C.R. Frascati (Rome), Italy Via E. Fermi 45, 00044 Frascati (Roma), Italythe date of receipt and acceptance should be inserted later
Abstract
In this work, we study the effect of a magneticfield on the growth of cosmological perturbations. We de-velop a mathematical consistent treatment in which a per-fect fluid and a uniform magnetic field evolve together in aBianchi I universe. We then study the energy density per-turbations on this background with particular emphasis onthe effect of the background magnetic field. We develop afull relativistic solution which refines previous analysis inthe relativistic limit, recovers the known ones in the New-tonian treatment with adiabatic sound speed, and it addsanisotropic effects to the relativistic ones for perturbationswith wavelength within the Hubble horizon. This representsa refined approach on the perturbation theory of an isotropicuniverse in GR, since most of the present studies deal withfully isotropic systems.
The formation of large scale structures across the Universeis one of the most fascinating and puzzling questions, stillopened in theoretical cosmology. Among the long standingproblems of this investigation area is the determination ofthe basic nature and dynamics of the cold dark matter [1],responsible for the gravitational skeleton on which the bary-onic matter falls in, forming the radiative component of thepresent structures.However, also the peculiarity of the matter distributionacross the Universe, in particular the possibility for largescale filaments [2], as well as hypotheses for structure fractaldimension [3,4] call attention for a deeper comprehension.In this respect, we observe that the Universe plasma na-ture, both before the Hydrogen recombination and, for a part a e-mail: [email protected] b e-mail: [email protected] in 10 also in the later matter dominated era [5,6], has to betaken into account.At the recombination the Universe Debye length is of theorder of 10 cm and therefore the implementation of a fluidtheory, like General Relativistic Magneto-hydrodynamics isto be regarded as a valid and viable approach to treat theinfluence of the primordial magnetic field [7] on the evolu-tion of perturbations [6]. Nonetheless, the smallness of suchmagnetic field, as constrained by the Cosmic MicrowaveBackground Radiation (CMBR) up to 10 − G [8,9,10,11,12,13,14], significantly limits the impact of the plasma na-ture of the cosmological fluid on the evolution of pertur-bations. As shown in [6,15], the presence of the magneticfield is able to trigger anisotropy in the linear perturbationsgrowth and it can be inferred that in the full non-linearregime, such anisotropy grows up to account for the forma-tion of large scale filaments.Apparently, a weak point in the perspective traced aboveconsists of the small plasma component surviving when theHydrogen recombines and in the observation that the mostrelevant cosmological scales enter the non-linear regime insuch a neutral Universe. Instead, it can be surprisinglydemonstrated [5,6,15] that the coupling between the neu-tral and ionized matter is very strong at spatial scale of cos-mological interest (for overdensities of mass greater than10 solar masses, the Ambipolar Reynold number is muchgreater than unity for redshift 10 < z < z ∼ (alsoconstant during the Universe evolution), maintains active a a r X i v : . [ g r- q c ] N ov strong Thomson scattering process, even after the hydrogenis recombined into atoms [16,17,18,5,6].These considerations are to underline that a single fluidGeneral Relativistic Magneto-hydrodynamics formulation isan appropriate tool to investigate the impact of the Universeplasma features on structure formation, at least for a largerange of the cosmological thermal history.In this context many works have been developed, mainlyassuming as negligible the backreaction of the magnetic fieldon the isotropic Universe, see [19] and references therein.However, the presence of a magnetic field rigorously vio-lates the isotropy of the space and the (essentially) flatRobertson-Walker geometry must be replaced by a BianchiI model. This paper faces the general question of how thelinear perturbations evolve on a background Bianchi I cos-mology, thought as a weak perturbation of the isotropic case,but treated in its full generality for arbitrary large magneticfields.We discuss in detail the structure of the perturbationequations in the synchronous gauge and the specific formof the spectrum time dependence in specific important lim-its, like the large scale limit, when the dependence on thewavenumber can be suppressed, and the sub horizon limit,when the dependence on the wavenumber is dominant.Furthermore, the change of the Jeans scale, when pass-ing from the ionized to the (essentially) recombined Uni-verse, is determined for the small scales, shedding light onthe role of the magnetic field and on the real nature of thegauge perturbations.We recover the slowing-down of the growing mode insuper-horizon scales, long known in FRW models. This ef-fect is very small given the upper limits on the cosmologicalmagnetic fields, of order O (cid:0) v A (cid:1) (cid:28)
1. At sub-horizon scales,we generalise the solutions of [20] and [21], which in turngeneralise the results of [19]. While they consider random(i.e. isotropic) magnetic fields to preserve the FRW model,we work in the anisotropic case and also consider a nonvan-ishing sound speed.Finally, we stress that, along the whole analysis, we com-pare our results with previous achievements in literature,providing a significant contribution to the understanding ofthe different effects that the Universe anisotropy, due to themagnetic field, induces on the perturbation evolution andstability.We notice that there is another paper about this mat-ter [22], which was the first analytical study to address thisissue. There, the authors study the model in 3 different phys-ical limits with specific anisotropies, while we completelyrelate the background anisotropy to the magnetic field.The paper is structured as follows: in section 3 we sum-marize the exact GRMHD equations in the 3+1 covariantformalism; in section 4 we find the solution for the back-ground Bianchi I model, then we write the equations for the perturbations in synchronous gauge in section 5 and we findthe gauge modes in section 6; finally we solve our system insome specific cases in section 7 and we compare our resultswith present literature.
As we already said, it is impossible to accommodate a mag-netic field in a isotropic model. Moreover, although presentobservations show that the isotropic FRW model describesvery well the present universe, it is only a very special de-scription of the universe towards the initial singularity, whilethe general one should incorporate anisotropy [23,24].In the first stage of the universe evolution the mattercontribution is negligible, while it is necessary to have aisotropic matter field to achieve the isotropization of themodel [25,26]. The general solution is constructed throughthe Bianchi VIII and IX models [23, 24,26], but we will fo-cus for simplicity on a single Kasner era and so we will usea Bianchi I model.The Bianchi I model is similar to the FRW one, but withthree different scale factors. It is intrinsically anisotropicin vacuum, i.e. the three cosmic scale factors are never allequal; moreover, in vacuum one of the three scale factor al-ways decreases with time, meaning that one of the spatialdirection is contracting.Near enough to the cosmological singularity, any mat-ter source in the form of perfect fluid energy density, havingequation of state p = w ρ always behaves as a test fluid, i.e.it induces negligible backreaction, as far a 0 < w <
1. Sincethe background magnetic field energy density is a radiation-like term in the Universe and it is associated to an equationof state p = ρ /
3, near enough to the singularity, we can ex-pect a typical vacuum solution of the Kasner form [24,26].The more general Bianchi IX model can be describedas a succession of Kasner epochs, in which the different di-rections exchange time evolutions, alternating moments ofgrowing and decreasing [26]. For more detailed informa-tions regarding the Bianchi models we recommend [27].Clearly, as soon as the Universe expands enough, thematter source can no longer be negligible and, if thepressure term is isotropic, the solution must correspondinglyisotropize, i.e. the three scale factors tend to be equivalent.This process of isotropization is particularly efficient in thecase of an inflationary paradigm [28,26], when a vacuumenergy, having an equation of state p = − ρ is dominatingthe Universe dynamics.The relevance of our study for the structure formationtakes place when the isotropization process reduced theBianchi I cosmology to a flat Robertson-Walker Universe,except for the residual intrinsic anisotropy due to the pres-ence of a background magnetic field. There exist already a large number of studies regard-ing Bianchi I models, analysing cases with different val-ues for the barotropic index w of the matter source in ad-dition to the magnetic field. [29] was probably the first toaddress their stability. [25] studies the effect of a pure mag-netic matter component, [30] contains analytic solutions fordust w = w = /
3, [31] contains solutionsfor w = / ≤ w ≤ w = −
1. The natureof the solutions depends on the values of various constants,it can collapse isotropically or anisotropically, only in thelongitudinal or in the transverse direction towards the BigBang. In general the magnetic fields accelerates expansion(or decelerates collapse) in the transverse direction of themagnetic pressure and it decelerates expansion (or acceler-ates collapse) in the direction of the magnetic tension. Forgeneral properties of the solutions, see [33].Some interesting cases are analysed in [30]: if B / ρ → B / ρ does not approach 0, then it is constantand both fluids determine the dynamics, or the magneticfield causes a rapid expansion in the transverse direction andthis change of the dynamics causes B / ρ →
0. Moreover,[32] shows that in presence of a cosmological constant themagnetic field has a strong effect at early times, decelerat-ing the collapse in the transverse direction and acceleratingit in the longitudinal one, and is negligible at later times,when the vacuum energy causes accelerated expansion inboth directions; the authors also describe the shape of thesingularity.It should be noted that in general the presence of themagnetic field causes a slowing down in the process ofisotropization, making the shear more important; this waythe CMB gives a strong constraint on primordial homoge-neous magnetic fields [9,10].
We will now recap the fundamental equations we’ll needlater; their derivation can be found in [19]. Following [19]we define the magnetic field as the spatial part of the Fara-day tensor F µν in the frame comoving with the cosmologicalfluid; we will use the ideal MHD approximation to turn offthe electric field. These equations can be easily obtained inthe covariant 3+1 formalism [34,35,36,37], as done in [38,39,19,40]; we will solve them, however, in a fixedsynchronous gauge. We will assume geometric units for thespeed of light c and Newton’s gravitational constant G inwitch c = π G / c = g µν with positive spatial signature ( − , + , + , +) filled by a per-fect fluid with energy density ρ , isotropic pressure density p , 4-velocity u µ and energy momentum tensor T µν = ρ u µ u ν + ph µν , (1)where h µν is the comoving spatial projector h µν = g µν + u µ u ν , (2)and a uniform magnetic field with Faraday tensor F µν .The time derivative of a generic tensor T νµ is˙ T νµ = u ρ ∇ ρ T νµ , (3)its spatial projected derivativeD ρ T νµ = h σρ h αµ h νβ ∇ σ T βα , (4)the totally antisymmetric spatial tensor ε µνρ = η µνρσ u σ , (5)where η µνρσ is the totally antisymmetric tensor with η = / √− g , and the irreducible components of the ve-locity derivative are θ = ∇ µ u µ = D µ u µ (6a) σ µν = (cid:0) D µ u ν + D ν u µ (cid:1) − h µν h αβ D α u β (6b) ω µν = (cid:0) D µ u ν − D ν u µ (cid:1) , ω µ = ε µνρ ω µρ (6c) A µ = ˙ u µ = u ν ∇ ν u µ . (6d)It is now possible to describe the electromagnetic field inthe Lorentz-Heaviside units: the electric field is E µ = F µν u ν ;the magnetic field is B µ = ε µνρ F νρ /
2, with magnetic energy B = B µ B µ and energy momentum tensor T µν = B u µ u ν + B h µν + Π µν (7a) Π µν = B h µν − B µ B ν . (7b)The equations that describe our system are the Maxwell equa-tions˙ B (cid:104) µ (cid:105) = (cid:18) σ µν + ε µνρ ω ρ − θ h µν (cid:19) B ν (8a) ε µνρ D ν B ρ = h νµ J ν − ε µνρ A ν B ρ (8b) ω µ B µ = − J µ u µ (8c)D µ B µ = , (8d)where J µ is the electric 4-current, and the projected Einsteinequations R µν u µ u ν = ( ρ + p + B ) (9a) h νµ R νρ u ρ = h ρµ h σν R ρσ = (cid:18) ρ − p + B (cid:19) h µν + Π µν (9c) in which R µν is the Ricci tensor.The interaction between the fluid and the magnetic fieldis given by ∇ µ T EM µν = − F µν J µ . (10)It is possible to use the Maxwell equation (8a) to find theconservation law for the magnetic energy˙ ( B ) = − θ B − σ µν Π µν , (11)and to derive the fluid energy conservation law from the tem-poral part of the Bianchi identities u µ ∇ ν T µν = ρ = − ( ρ + p ) θ ; (12)from the spatial projected Bianchi identities h µρ ∇ ν T ρν = (cid:18) ρ + p + B (cid:19) A µ = − D µ p − ε µνρ B ν ε ραβ D α B β − Π µν A ν (13)which, using ε µνρ B ν ε ραβ D α B β =
12 D µ B − B ν D ν B µ , (14)gives (cid:18) ρ + p + B (cid:19) A µ = − D µ p −
12 D µ B + B ν D ν B µ − Π µν A ν . (15) We assume that our system is homogeneous and perturbedat first order by weak inhomogeneous perturbations. At thebackground level we have a homogeneous universe with anisotropic perfect fluid and a uniform magnetic field: suchfield cannot live with an isotropic metric, such as FRW, butit can be accommodated in an anisotropic model. We mustuse one of the Bianchi models because of the homogeneityand our model fits best in a Bianchi I universe, which is thesimplest anisotropic generalization of FRW, so our metric insynchronous gauge is g µν = diag (cid:0) − , a ( t ) , a ( t ) , a ( t ) (cid:1) . (16)These type of models were widely studied in literaturein different assumptions and physical limits (see for exam-ple [41,30,31,25,32]); we are here interested mainly in theirbehaviour after the matter-radiation equivalence, where themagnetic field can be reasonably small compared to the mat-ter component. This regime was already studied in differentworks, for example by [25] in radiation dominated universe; here we will recap [10], which accounts for different type ofanisotropic stresses in both radiation an matter dominateduniverse. We will, however, amend for their time behaviourin matter dominated universe and we will not neglect higherorder corrections in the isotropic components.We assume that the magnetic field is oriented along the3 axis, so the system is axisymmetric and a = a ; forsimplicity we call a = a = a and c = a . We have u µ = ( , , , ) .It is now straightforward to write the Einsteinequations (9)2 ¨ aa + ¨ cc = − (cid:0) ρ + p + B (cid:1) (17a)¨ aa + ˙ aa (cid:18) ˙ aa + ˙ cc (cid:19) = (cid:0) ρ − p + B (cid:1) (17b)¨ cc + aa ˙ cc = (cid:0) ρ − p − B (cid:1) (17c)and the energy conservation laws for the system (12) and (11)˙ ρ + (cid:18) aa + ˙ cc (cid:19) ( ρ + p ) = ( B ) + aa B = . (19)We define the Alfvén velocity, which is the energy ratiobetween magnetic field and fluid (note the factor 1 / v A = B / ρ , (20)witch is responsible for the intensity of the anisotropies, theisotropic expansion H and the anisotropy parameter S H = aa + ˙ cc , S = H (cid:18) ˙ aa − ˙ cc (cid:19) . (21)If we now assume a barotropic fluid with equation of state p = w ρ and w = const the Einstein equation (17a) becomes3 ˙ H + H (cid:18) + S (cid:19) = − (cid:20) ( + w ) + v A (cid:21) ρ , (22)subtracting equation (17c) from equation (17b) we get H ˙ S + ˙ HS + H S = v a ρ (23)and summing 2 times equation (17b) to equation (17c) weeventually have3 ˙ H + H = (cid:20) ( − w ) + v A (cid:21) ρ . (24)From the definition of v A (20) and from the energy conser-vations (18) and (19) we have˙ ( v A ) = ˙ ( B ) / − ˙ ρ v A ρ = v A H (cid:18) w − − S (cid:19) . (25) If we now assume that the magnetic field energy is smallcompared to the fluid energy we have v A (cid:28) H = H ( ) + H ( ) , ρ = ρ ( ) + ρ ( ) (26)with H ( ) , ρ ( ) = O (cid:0) v A (cid:1) it is easy to see from equations (22)and (24) that at 0-order in v A we recover FRW and we have H ( ) = ( + w ) t , ρ ( ) = H ( ) , S ( ) = . (27)The anisotropy is described by S and equation (23) be-comes at first order in v A ˙ S + − w + w St = + w v A t , (28)while equation (25) gives˙ ( v A ) = −
23 1 − w + w v A t . (29)The isotropic part is contained in equations (22) and (24),which form a system whose solution is ρ ( ) = + w H ( ) t − ( + w ) v A t (30)˙ H ( ) + H ( ) t = −
29 1 − w ( + w ) v A t . (31)We are interested only in anisotropies caused by themagnetic field so we will put to 0 the homogeneous solu-tion of each equation, with the exception of (29).4.1 Radiation dominated universeFor radiation dominated universe w = / v A = v A = const . (32)Equation (28) then gives S = v A = v A . (33)From equation (30) we get ρ .From the definitions (21) we can get the values of a and c . Finally we have v A = v A = const , t = const (34) a ∼ (cid:18) tt (cid:19) / (cid:18) + v A ln (cid:18) tt (cid:19)(cid:19) (35) c ∼ (cid:18) tt (cid:19) / (cid:18) − v A ln (cid:18) tt (cid:19)(cid:19) (36) H = t (37) ρ = t ( − v A ) . (38) 4.2 Matter dominated universeFor matter dominated universe w = v A = v A (cid:18) tt (cid:19) − / , v A , t = const . (39)From equation (28) we get S ( t ) = v A ( t ) . (40)For the isotropic part we proceed as before: eq. (31)gives H ( ) = − v A ( t ) t (41)From equation (30) we get ρ .From the definitions (21) we can get the values of a and c . Finally we have v A = v A (cid:18) tt (cid:19) − / (42) a ∼ (cid:18) tt (cid:19) / − v A (43) c ∼ (cid:18) tt (cid:19) / + v A (44) H = t ( − v A ( t )) (45) ρ = t (cid:18) − v A ( t ) (cid:19) . (46) We perturb all the quantities that govern our system whilekeeping synchronous gauge, thus the perturbed metric is g µν = g B µν + δ g µν (47a) δ g µ = , (47b)where B means the background value; we can define γ µν = δ g µν (48a) g µρ g ρν = δ νµ = ⇒ δ g µν = − γ µν , (48b)where the indices of γ µν are raised and lowered with theunperturbed metric g B µν . In the following we write the traceof γ µν as γ = γ kk . The fluid velocity perturbation is δ u µ ,with u µ u µ = − = ⇒ δ u = . (49)The fluid energy perturbation is δ ρ and the fluid pressureperturbation is δ p = v S δ ρ ; it holds˙ w = − H ( + w )( v S − w ) (50a) w = const = ⇒ v S = w , (50b) but we keep v S as an arbitrary function and possibly differentfrom w ; the reason of this choice will be clear in section 7.2.The perturbed magnetic field must remain pure spatial atall orders, as shown in appendix Appendix A, so the condi-tion B µ u µ = δ ( B µ B µ ) = δ ( B ) = γ B B + c δ B B (51a) B µ u µ = = ⇒ δ B = c B δ u . (51b)Accordingly to [24,16] the perturbed Christoffel sym-bols are δΓ ρµν = g ρσ B (cid:0) ∇ B µ γ νσ + ∇ B ν γ µσ − ∇ B σ γ µν (cid:1) (52)and the perturbed Ricci tensor is δ R µν = ∇ B ρ δΓ ρµν − ∇ B ν δΓ ρµρ . (53)We are now ready to perturb the exact equations of sec-tion 3. We notice that, because of the homogeneity of thebackground model, when applied to the perturbation of ascalar quantity the comoving time derivative ˙ s is the same asthe synchronous time derivative ∂ s , so we make no differ-ence between them in the following. The fluid energy con-servation (12) becomes˙ δ ρ + (cid:18) aa + ˙ cc (cid:19) ( δ ρ + δ p )+ ( ρ B + p B ) (cid:18) ∂ i δ u i +
12 ˙ γ (cid:19) = ( δ ( B )) + aa δ ( B ) + B ( ) · (cid:18) ∂ i δ u i − ∂ δ u +
12 ˙ γ −
12 ˙ γ (cid:19) = . (55)The Einstein 00 equation is (we will always use Einsteinequations with a lower and an upper index)12 ¨ γ + ˙ aa ˙ γ − (cid:18) ˙ aa − ˙ cc (cid:19) ˙ γ + ( δ ρ + δ p ) + δ ( B ) = , (56)while the 33 equation reads ∂ k ∂ γ k − (cid:16) ∂ k ∂ k γ + ∂ ∂ γ (cid:17) +
12 ¨ γ + (cid:18) aa + ˙ cc (cid:19) ˙ γ +
12 ˙ cc ˙ γ − ( δ ρ − δ p ) + δ ( B ) =
0; (57) to remove ∂ ∂ k γ k from the last equation we need to use thederivative of the 03 equation with respect to the 3 index ∂ (cid:16) ∂ ∂ k γ k (cid:17) − ∂ ∂ ˙ γ + aa ∂ ∂ k γ k − (cid:18) ˙ aa − ˙ cc (cid:19) ∂ ∂ γ − (cid:18) ˙ aa − ˙ cc (cid:19) ∂ ∂ γ = − ( ρ B + p B ) ∂ δ u . (58)If we had used equations (9) we would have found the sameresults.By imposing the null divergence of the magnetic field (8d)we get ∂ i δ B i + B ∂ γ = . (59)The last equation we need is the conservation of the mo-mentum (15) (note that A µ has only the first order compo-nent): we define an index P ∈ { , } that lies on the plane or-thogonal to the background magnetic field and we write thedivergence of the momentum conservation on the 12-plane( ∂ () + ∂ () ) ( ρ B + p B ) (cid:18) ∂ ∂ P δ u P + aa ∂ P δ u P (cid:19) + B (cid:20) ∂ ∂ P δ u P + (cid:18) aa + ˙ cc (cid:19) ∂ P δ u P (cid:21) + ∂ P δ u P ∂ (cid:18) p B + B (cid:19) + ∂ P ∂ P (cid:18) δ p + δ ( B ) (cid:19) − B ∂ ∂ P δ B P + B (cid:18) ∂ P ∂ P γ − ∂ P ∂ γ P (cid:19) = ( ρ B + p B ) (cid:18) ∂ ∂ δ u + cc ∂ δ u (cid:19) + ∂ δ u ∂ (cid:18) p B + B (cid:19) + ∂ ∂ (cid:18) δ p + δ ( B ) (cid:19) + aa B ∂ δ u − B ∂ ∂ δ B − B ∂ ∂ γ = . (61)The system (54)-(61) fully characterizes the evolution ofthe perturbed quantities and it is the ground of the followinganalysis. Compared to [22] we fully related the backgroundanisotropy to the magnetic field, without the need of addi-tional hypothesis. Fixing the synchronous gauge does not end the freedom ofcoordinate choice: we can still make a gauge transformationpreserving the synchronous gauge.We follow the same scheme as of [26]: we make a genericcoordinate transformation of the form x µ → x µ + ε µ (62)with small ε µ and we keep terms up to O ( ε ) .The metric tensor becomes g (cid:48) µν ( x (cid:48) ) = g µν ( x ) − g µσ ( x ) ∂ ν ε σ − g ρν ( x ) ∂ µ ε ρ . (63)If we define ∆ g µν = g (cid:48) µν ( x ) − g µν ( x ) = − g µλ ( x ) ∂ ν ε λ − g λν ( x ) ∂ µ ε λ − ε λ ∂ λ g µν ( x )= − ∇ µ ε ν − ∇ ν ε µ (64)to preserve the synchronous gauge we need ∆ g µ = ε = ε ( x j ) and ε P = ˜ ε P ( x j ) + ∂ P ε ( x j ) a (cid:90) d ta , (65a) ε = ˜ ε ( x j ) + ∂ ε ( x j ) c (cid:90) d tc , (65b)where ε ( x j ) and ˜ ε i ( x j ) are arbitrary functions of the spa-tial coordinates: we still have 4 unused degrees of freedomrepresented by the functions ε and ˜ ε i .If we take the functions ε and ˜ ε i of the same order of theperturbations then the transformation given by equation (64)can be seen both as a gauge transformation and as a transfor-mation of the functions γ µν within fixed synchronous gauge:in the latter case equation (64) gives the value of ∆ γ µν . Inthe same way the stress-energy tensor transforms as ∆ T µν = − T µλ ∂ ν ε λ − T λν ∂ µ ε λ − ε λ ∂ λ T µν = − T µλ ∇ ν ε λ − T λν ∇ µ ε λ − ε λ ∇ λ T µν (66)and if we see these as transformations on the physical vari-ables instead of the coordinates we obtain the gauge modesfor δ T µν . Substituting the explicit expression of T µν as thesum of the fluid and the magnetic field components we seethat the transformation acts separately on the two compo-nents and we get for the fluid density perturbation ∆ δ ρ = − ε ˙ ρ B = H ( ρ B + p B ) ε = H ( + w ) ρ B ε . (67)In section 5 we linearised the equations and so the gaugetransformations solve our equations and we call them gaugeperturbations or gauge modes: these solutions are not phys-ical because they correspond to a simple change in the ref-erence frame. We are looking for a physical solution for the time dependence of δ ρ so the most interesting gauge trans-formation is given by equation (67).Having the knowledge of gauge modes it is possible toconstruct gauge invariant variables, in a similar way as donein [42]. We have ∆ δ u i = ∂ i ε (68)so our main scalar variable should be δ ρ GI = ∂ i ∂ i δ ρ − H ( + w ) ρ B ∂ i δ u i (69a) ∆ δ ρ GI = . (69b)It is easy to check that it is exactly the variable used in [19],expressed in synchronous gauge. We will, however, not needit because the vorticity part H ∂ i δ u i decays in time with re-spect to ∂ i ∂ i δ ρ / ρ B and we are interested in late time dy-namics. We will also not need the laplacian, because we’lluse Fourier expansions so it will reduce to a multiplicativeterm: for late times we can assume δ ρ to be gauge invariant.It is possible to watch this approximation from anotherperspective, shown in Appendix Appendix B. If we write the perturbations as Fourier transforms we seethat the system imposes different evolution to the perturba-tions that propagates along the background magnetic field,with ∂ P ( . . . ) =
0, and the perturbations that propagates or-thogonally to the background magnetic field, with ∂ ( . . . ) =
0. These different modes are however coupled bythe magnetic stress-energy tensor tensorial nature.To simplify the equations we use the barotropic stateequation for the fluid, so p B = w ρ B with w = const and δ p = v S δ ρ , and the Fourier expansion for the spatial partof the perturbations, so the spatial dependence is of the forme i k j x j . We define the new variables ∆ = δ ρ ( + w ) ρ B (70) G = γ (71) T = γ (72) M = δ ( B ) B . (73)Our differential equation system is not simple but we cansolve it for small magnetic fields by keeping only terms up tofirst order in v A : we shall remember that S is already at firstorder while ∆ , G , T and δ u i have also a 0-order (FRW) part; M has only the 0-order part because it is always multipliedby v A because δ ( B ) = B M = ρ B v A M . In the same way,looking at our system also T is always multiplied by v A : this is because it does not affect density perturbations unlesssome anisotropy is present.We also use eq. (50b) to discard terms proportionalto w − v S or to ˙ ( v S ) , unless multiplied by k i k i or k k . Thisis because, while they are equal to 0 for w = const, we willneed them in sec. 7.2.The fluid energy conservation equation (54) in the newvariables reads˙ G = − ˙ ∆ − ∂ i δ u i . (74)Similarly we rewrite the magnetic energy conservation(55)˙ M = − (cid:0) ∂ P δ u P + ˙ G − ˙ T (cid:1) = (cid:0) ˙ ∆ + ˙ T + ∂ δ u (cid:1) , (75)where we found the last equality by using the fluid energyconservation.Combining Einstein 33 equation (57) with its derivativewith respect to time and using the derivative of Einstein03 equation (58) with respect to the 3-index in order to takecare of ∂ i ∂ γ i terms we get an equation for T . Because T only appears in the system in terms that are multiplied by v A ,we will only need this equation at 0-order:3 ( + w ) ... T +
10 ¨ Tt + − w + w ˙ Tt − ∂ δ u t + Gt +
23 1 − w + w ˙ Gt − (cid:0) − v S (cid:1) ˙ ∆ t + ( − v S ) + w + w ∆ t + ( + w )( k i k i ˙ T − k k ˙ G ) = G from the other equations. This way the Ein-stein 00-equation (56) reads¨ ∆ + H (cid:18) + S (cid:19) ˙ ∆ − ( + v S )( + w ) ρ∆ + ∂ ∂ i δ u i + H (cid:18) + S (cid:19) ∂ i δ u i + ( + w ) S ˙ Tt − ( + w ) v A Mt = . (77)We obtain the evolution equation for the divergence ofthe 4-velocity by summing eqs. (60) an (61); we then useequation (57) to remove the ∂ i ∂ γ i term and equation (59)to remove the divergence of the magnetic field. Doing so we find (cid:18) + + w v A (cid:19) ∂ ∂ i δ u i ++ (cid:20) ( − w ) H + (cid:18) v A + w + S (cid:19) ( + w ) t (cid:21) ∂ i δ u i == − v S ∂ i ∂ i ∆ − v A + w ∂ i ∂ i M + + w v A ∂ ∂ δ u + (cid:18) v A + w + S (cid:19) ( + w ) ∂ δ u t − + w v A (cid:20) ¨ T + + w ˙ Tt + ( + w ) ˙ Gt (cid:21) + ( + w ) ( − v S ) v A ∆ t . (78)We will need also equation (61) which reads, using equa-tion (51a) to remove ∂ δ B , ∂ ∂ δ u + (cid:18) − w − S (cid:19) H ∂ δ u + ∂ ∂ ( v S ∆ ) = . (79)Thus we restated the dynamical system (54)-(61) in amore suitable form which is more appropriate for the fol-lowing analysis.7.1 Radiation dominated universe at large scalesIn radiation dominated universe we have w = v S = / k ≈ k k ≈
0. It is easy to checkthat, once we get rid of the scale dependent terms, eq. (76),(77) and (78) reduces respectively to2 ... T + Tt − ∂ δ u t − ∂ ∂ i δ u i t − ¨ ∆ t −
23 ˙ ∆ t + ∆ t = ∆ + (cid:0) + v A (cid:1) ˙ ∆ t − ( − v A ) ∆ t + v A ˙ Tt − v A Mt + ∂ ∂ i δ u i + (cid:0) + v A (cid:1) ∂ i δ u i t = (cid:18) + v A (cid:19) ∂ ∂ i δ u i + + v A ∂ i δ u i t == v A ∂ ∂ δ u + v A ∂ δ u t − v A ¨ T − v A ˙ Tt + v A ˙ ∆ t + v A ∆ t . (82)This system, together with (75) and (79), is satisfied by apower law solution and could be reduced to a pure algebraicproblem, but we found simpler to solve it for v A = v A . We found ∆ = ∆ gauge t + ∆ grow t − v A + ∆ t / − v A + ∆ t / + v A . (83) It can be shown that the t / modes are related to a non-vanishing divergence of the background velocity ∂ i δ u i = ik i δ u i : strictly speaking, we should have imposedthe k i ≈ t and 1 / t modes: ∆ = ∆ gauge t + ∆ grow t − v A (84)and recovering the usual FRW solution in the limit v A → / t is a gauge mode,while t − v A is the physical growing mode, with the correc-tion due to the magnetic field.We find our solution simpler than the one of [19], andwith a clearer physical interpretation of the solutions, but ourphysical growing mode follows a slightly different temporallaw, although this correction is small given v A (cid:28)
1. We alsofind simpler the comparison of our solution with the nonmagnetic one of [16].We see in (84) that the magnetic field reduces the grow-ing rate of density perturbations, but by an amount of or-der O (cid:0) v A (cid:1) (cid:28)
1. This effect has long been known, and it isdue to the extra magnetic pressure. A similar behavoiur wasfound in [19] and [22], although with the differences statedabove.We finally note a difference between our solution (83)and the one of [19]: the non dominant mode is t / in ourformalism, while t − / in their. At a more careful analysis,our equations tend correctly to the ones of [16] for v A → t / mode. Such discrepancy is therefore be-tween the synchronous and covariant formalisms, and it isbesides the purposes of our paper.7.2 Matter dominated universe at small scalesIn this section we analyse the perturbations in a matter domi-nated universe ( w = i k j x j ,with k j = const, and we define k = k i k i .Being at small scales means k (cid:29) H and assuming v S , v A (cid:28) v S or v A that are multiplied by k and drop-ping terms of order v S and v A . This means that the effectof the sound speed and the Alfvén speed is relevant only atvery small scales, as we will see from the solutions of ourequations. This approximation, although still relativistic andso comparable to other result in literature, for example [19],will give the nonrelativistic limit, as shown in section 7.2.2 First we need some considerations regarding the soundspeed. From a formal point of view, the sound speed is re- lated to the barotropic index w by (50a) and w = const im-plies v S = w , so it should vanish. From a physical point ofview we need a nonvanishing sound speed and we can alsoestimate its value. While formally the best solution to thisproblem would be using a two fluid model, with a differ-ent equation of state for perturbations, here we will simplydrop the relation between v S and w and assume that the per-turbed fluid follows a different equation of state with respectto the background fluid. This is correct in the Newtonian ap-proximation and it’s in fact the standard way of handlingthings [16,6], while putting v S = v S = δ p δ ρ ∼ γ p ρ ∼ ρ γ − ∼ t ( − γ ) (85)where γ is the heat ratio. We write ν = γ − / ≥ v S = v S (cid:18) tt (cid:19) − ( ν + ) . (86)We can estimate more precisely the sound speed value,and it’s possible to show that the adiabatic sound speed is[16,6] v S (cid:12)(cid:12) z < z rec = k B T b σ m p + k B T b σ , v S (cid:12)(cid:12) z > z rec = k B T b m p , (87)where z rec is the redshift value at recombination, k b is theBoltzmann constant, T b is the baryons temperature, m p is the proton mass and σ is the specific entropy, whosevalue is σ = a SB T / n b k B ≈ . · , being a SB the Ste-fan–Boltzmann constant and T the gas temperature. We ne-glected any anisotropic effects in temperature, because theywould be related to the next order corrections. The baryonstemperature is the same of the photons until z ≈ T b | z > = T γ = T γ (cid:12)(cid:12) z = ( + z ) , T γ (cid:12)(cid:12) z = ≈ . T b | z < ∝ ( + z ) . (88b)Comparing the two expressions we see that right afterrecombination and until complete decoupling, so for z rec = > z > = z dec , we have ν = γ = / > z > ν (cid:39) / γ (cid:39) /
3. The plot of the sound speed andof the Alfvén speed is in fig. 1. - - - - - - - - - - - - - - - - t / t Rec v / c v S v A Fig. 1: Plot of the sound speed and the Alfvén speed. Wesee that sound speed dominates until recombination, wheresuddenly the Alfvén velocity becomes important.We define two constants addressing the effect of soundspeed and Alfvén speed after recombination. Taking the timedependence of k depending only on the 0-order part of thebackground metric because it always appears multiplied by v S or v A , we have respectively Λ S = v S k t γ − / , Λ A = v A k t . (89)For a more detailed discussion about the sound speedsee [43]. Using the assumptions of section 7.2 we can greatly simplifyour equations. The energy conservation (74) and the mag-netic field energy conservation (75) retain the same form.The Einstein 00-equation (77) now reads¨ ∆ + t ˙ ∆ − t ∆ + ∂ ∂ i δ u i + t ∂ i δ u i = . (90)The momentum conservation 78 becomes ∂ ∂ i δ u i + t ∂ i δ u i = − v S ∂ i ∂ i ∆ − v A ∂ i ∂ i M (91)and its counterpart along the z -axis remains (79): ∂ ∂ δ u + t ∂ δ u + v S ∂ ∂ ∆ = . (92)We need the Einstein 33-equation only at 0-order in the mag-netic field, after being multiplied by v A , so equation (76) inour limit reads v A ∂ i ∂ i ˙ T + v A ∂ ∂ ( ∂ i δ u i + ˙ ∆ ) = . (93)With some algebra it is possible to reduce this systemto a single equation. Expanding the spatial part in Fourier,defining the anisotropy parameter µ of the solution as k k = µ k (94) and using the constants (89) we find, after some algebra,9 t ∆ ( ) + t ∆ ( ) + (cid:2) + Λ S t − ν + Λ A (cid:3) t ∆ ( ) + (cid:2) + Λ S ( − ν ) t − ν + Λ A (cid:3) t ∆ ( ) + (cid:2) Λ S (cid:0) − ν + ν + µ Λ A (cid:1) t − ν − µ Λ A (cid:3) ∆ = , (95)where ∆ ( i ) is the i -th derivative of ∆ . This corresponds ex-actly to equation (29) of [6], except for a difference in thedefinition of v A and so in Λ A .We believe interesting to analyse separately the two casesof ν = ν = /
3, instead of studying them together asin [6].
For 1100 > z >
100 we have ν =
0. The solution of (95) is ∆ = ∆ i t x i , (96)where ∆ i are arbitrary constants and x = (cid:16) − + (cid:112) δ − (cid:17) / x = (cid:16) − − (cid:112) δ − (cid:17) / x = (cid:16) − + (cid:112) δ + (cid:17) / x = (cid:16) − − (cid:112) δ + (cid:17) / δ ± = δ ± (cid:112) δ (97c) δ = − Λ S − Λ A (97d) δ = (cid:0) − + Λ A + Λ S (cid:1) − µ Λ A (cid:0) − + Λ S (cid:1) . (97e)The only possible growing solution is x , and the require-ment is that it holds one of the conditions µ > Λ S <
23 (98a) µ = Λ S + Λ A <
23 ; (98b)using (89) and (26), making explicit the presence of New-ton’s constant we get ρ = / π Gt , conditions (98) become µ > k < k J = (cid:115) π G ρ v S (99a) µ = k < (cid:115) π G ρ v S + v A < k J . (99b)While the first one is the standard Jeans condition, the sec-ond one means that, orthogonally to the background mag-netic field, there is a heavier requirement dependent on thestrength of the magnetic field: some modes could grow inevery direction but the one of the field. The presence of the magnetic field also imposes a slowing down of the growingmode: x ≤ x | Λ A = = (cid:18) − + (cid:113) − Λ S (cid:19) , (100)where the equal sign holds only in absence of a magneticfield, that is only if Λ A = This is exactly the case analysed in [6]. For z <
100 wehave ν > ∆ = ∆ i t x i F (cid:20) a i , a i b i , b i , b i ; − Λ S t − ν ν (cid:21) , (101)where ∆ i are arbitrary constants, F is a generalized hy-pergeometric function with constant coefficients a i j , b i j de-pending only on the constants ν , Λ S , Λ A (see app. AppendixC for the explicit value of the coefficients) and x = (cid:16) − + (cid:112) δ − (cid:17) / x = (cid:16) − − (cid:112) δ − (cid:17) / x = (cid:16) − + (cid:112) δ + (cid:17) / x = (cid:16) − − (cid:112) δ + (cid:17) / δ ± = − Λ A ± (cid:113)(cid:0) − Λ A (cid:1) + µ Λ A . (102c)The solutions can grow only if the argument of the hy-pergeometric functions is small, i.e. if Λ S / ν t ν (cid:28) ∆ = ∆ i t x i (cid:18) + O (cid:18) Λ S t − ν ν (cid:19)(cid:19) . (104)Condition (103) is the standard Jeans condition [16]: us-ing (89) and (26), eq. (103) translates in [6] k (cid:28) k J = (cid:115) ν π G ρ v S . (105)The only solution in (104) that can grow is 3: x > < µ ≤ µ = Λ A < . (106b)The first one means that, in any direction but orthogonalto the background magnetic field, the only necessary con-dition is the standard one. The second one is an additionalcondition that must hold for perturbations propagating or-thogonally to the background magnetic field, and using (89)and (26) it reads [6] k < k A = (cid:115) π G ρ v A . (107) The presence of this new condition makes possible the ex-istence of Jeans unstable modes, that orthogonally to thebackground magnetic field are stabilized by the magneticpressure if k A < k J and k A < k < k J [6].Studying the growing rate of this solution with morecare, we see that x satisfies µ = = ⇒ x = x | Λ A = =
23 (108a) µ (cid:54) = = ⇒ x < x | Λ A = : (108b)orthogonally to the background magnetic field the grow-ing rate is unchanged, while in other directions it is sloweddown, depending on the field strength.7.3 Full relativistic caseIf we put v S = µ > µ = k < k A = (cid:115) π G ρ v A . (109b)Moreover, the solution is ∆ = ∆ i t x i (110)with x i given by (97) with Λ S =
0, or equivalently by (102).If we compare our result with [19], we identify theanisotropic behaviour and we obtain the correct Newtonianlimit of [6]. However, our solutions are different and we areunable to explain such discrepancy: we can argue they mayhave found some sort of average effect, however this is notclear, given the strong anisotropy of the model: the magneticJeans wavenumber is present only in one direction, the onewith µ = Λ A (cid:28) x ( ) = (cid:18) − ± (cid:113) − µ Λ A (cid:19) (111a) x ( ) = (cid:18) − ± (cid:113) − ( − µ ) Λ A (cid:19) (111b)and setting µ = / x and x recover eq. (31)of [20] and eq. (31) of [21], so our small scales solution ofsec. 7.2 is a generalization of their work, while including anonvanishing sound speed and pressure. To better show our results, we numerically integrated thesystem (54)–(61), using estimates from [44] to set the nu-merical values for the background functions. We followedthe same procedure of [6] to determine the initial conditions:we started the integration from a very early time and weverified that the initial perturbations were outside the Hub-ble horizon and we used the large scale solution to matchthe initial conditions to the growing mode; in our case suchconditions come from eq. (84).We assumed to perturb only the baryon componentof the universe, while leaving the CDM componentunperturbed; a rigorous treatment should rely on a multi–fluid model, but we ague that we can still extract meaning-ful information within our approximation. Practically speak-ing, this assumption means that every quantity present inour equations at perturbative level must be replaced by itsbaryonic component, while the background model still de-pends on CDM. Our equations are still correct, because thebackground interaction is only due to energy density, whileat perturbative level every dependence on CDM disappears,except from background quantities.We choosed to study the same scales of [6],i.e. k ≈ ( , . , . ) Mpc − normalized at presenttime, corresponding to baryonic masses of M ≈ ( . × , . × , . × ) M (cid:12) and roughlyequivalent respectively to a dwarf galaxy, a galaxy and agalaxy cluster. The results of the numerical integration areshown in figure 2.Our results must be compared to the ones of [6]. Untilequivalence ( z ≈ ∆ ( z ≈ ) / ∆ ( z ≈ ) has the same value in both the analysed cases, sothe main anisotropic contribution comes from theregion 3400 (cid:46) z (cid:46) μ = μ = - - - - - - - - t / t Rec δ / δ I n2 k =
17 Mpc - (a) Perturbations at dwarf galaxy scale: k (cid:39) − , M ≈ . × M (cid:12) . μ = μ = - - - - - - - - t / t Rec δ / δ I n2 k = - (b) Perturbations at galactic scale: k (cid:39) . − , M ≈ . × M (cid:12) . μ = μ = - - - - - - - - t / t Rec δ / δ I n2 k = - (c) Perturbations at galaxy cluster scale: k (cid:39) . − , M ≈ . × M (cid:12) . Fig. 2: Density perturbations evolution in time, relative totheir initial value. While some anisotropy is present in (a)because of the magnetic Jeans length (see sec. 8 and [6]),most of the anisotropic effects of [6] here are suppressedbecause of thermal pressure in the radiation dominated era. numbers; however, the qualitative evolution is the same, withthe µ = We developed above a self-consistent scheme for the analy-sis of cosmological perturbations in the presence of a mag-netic field. We set up in the synchronous gauge a dynamicalscheme which accounts for the effects induced by the mag-netic field both on the background and the first order formu-lation. To this end, we considered a Bianchi I model, whoseanisotropy with respect to the flat Robertson-Walker geome-try is due to the privileged direction defined by the magneticfield.We first solve in detail the equations describing theanisotropic background and then we analyse the perturba-tion dynamics, having awareness of the gauge contributionanalytical form.We amended for the previous analysis in [19] in the caseof a super-horizon wavelength of the perturbation. In par-ticular, our solution has a clearer comparison with the nonmagnetic one. We recovered the slowing-down of the grow-ing mode caused by the magnetic pressure, and so of or-der O (cid:0) v A (cid:1) (cid:28)
1. This effect has long been known in FRWmodels and has been analysed in Bianchi I models with par-ticular anisotropies by [22], while we worked always relat-ing the background anisotropy to the magnetic field withoutadditional assumptions.We refined the results of [19] for the sub-horizon wave-length of the perturbations, showing that an anisotropic treat-ment is required. We also generalised the results of [20]and [21], while including a nonvanishing sound speed andconsidering the anisotropic case.We finally enforced the Newtonian limit obtained in [6],completing it with the relativistic analysis, also facing a nu-merical treatment. We showed that the relativistic regimelimits the anisotropy induced by the magnetic field.Overall, despite the assumption of a Bianchi Ibackground, most of our solutions reproduce those obtainedon an FRW background. At a closer look, the Bianchi Ianisotropy enters the system via the S function defined in (21).At small scales the relevant terms are the ones with k , andnone of those are related to such anisotropy. However, whenthe condition H (cid:28) k does not hold, such terms becomeimportant; unfortunately, in this case the system would bemuch more complicated that the one of sec. 7.2. On the other hand, at large scales the background anisotropy sur-vives, and we argue that it is mainly related to the perturbedfluid velocity. In particular, it can be shown that the so-lutions proportional to t / in (83) are related to δ u i , andmore precisely in ∆ t / + v A we have both k i δ u i (cid:54) = k δ u (cid:54) =
0, while in ∆ t / − v A it holds k δ u =
0; the so-lutions ∆ grow t − v A and ∆ gauge / t , on the other hand, bothhave k i δ u i = k δ u = v A (cid:28)
1, and they become relevant only atsmall scales, due to the large wavenumber k (cid:29) H and tothe also small sound speed v S . This is clear by looking atfig. 1. Appendix A: Magnetic field at perturbative level
In literature there are different definitions of the magneticfield at a perturbative level, but it is easy to recognize thatnot all of them satisfy the required properties. After a care-ful analysis we concluded that the correct one, at least withrespect to the physical phenomenon we study here, it theone of [19] made through the 3+1 formalism. This way, themagnetic field is defined as the spatial projected part of theFaraday tensor F µν , while the electric field as the temporalone E µ = F µν u ν (A.1a) B µ = ε µνρ F νρ = η µνρσ F νρ u σ (A.1b)and we have B µ u µ = E µ = Appendix B: Gauge behaviour in late times
We will analyse here the FRW case, to clarify the mean-ing of δ ρ becoming gauge invariant for late times. Follow-ing [16] and using the Newtonian approximation we see thatthe solutions after recombination are δ ± ∝ t − / J ∓ ν (cid:18) Λ t − ν ν (cid:19) , (B.3)where γ = ν + / > / γ (cid:39) / δ = δ ρ / ρ , Λ = t γ − / v S k (B.4)is a constant, v S is the squared sound speed and k thewavenumber. The functions J a ( z ) are the Bessel functions:when their argument is large they oscillate, but when the ar-gument is small they behave like δ ± ∝ t ( − ± ) / . (B.5)The growing mode is the physical solution we are lookingfor, while the other one decays to zero.We cannot speak of gauge modes in Newtonian theory,but the decaying mode corresponds exactly to the relativisticgauge mode, and as expected it decays in time with respectto the growing one. This means that, for large times, gaugemodes naturally decay to zero and we can neglect them aslong as we are looking only for the growing ones.It should be noted that in our calculations we are in thesame situation: we cannot have a relativistic sound speeddifferent from w in a single fluid model, but we make thisapproximation in section 7 because from a physical pointof view we need a nonvanishing sound speed. This way we“break” the gauge invariance, but the gauge modes manifestthemselves in one of the decaying solutions. We are onlylooking for growing modes, so we can safely neglect them. Appendix C: Late times solution coefficients
We report here the values of the coefficients of the hyper-geometric function appearing in (101), using δ ± defined in eq. (102c): a ( ) = ∓ (cid:112) δ − / ν − (cid:113) − µ Λ A / ν (C.6a) a ( ) = ∓ (cid:112) δ − / ν + (cid:113) − µ Λ A / ν (C.6b) a ( ) = ∓ (cid:112) δ + / ν − (cid:113) − µ Λ A / ν (C.6c) a ( ) = ∓ (cid:112) δ + / ν + (cid:113) − µ Λ A / ν (C.6d) b ( ) = ∓ (cid:112) δ − / ν (C.6e) b ( ) = ∓ (cid:112) δ − / ν − (cid:112) δ + / ν (C.6f) b ( ) = ∓ (cid:112) δ − / ν + (cid:112) δ + / ν (C.6g) b ( ) = ∓ (cid:112) δ + / ν (C.6h) b ( ) = ∓ (cid:112) δ + / ν − (cid:112) δ − / ν (C.6i) b ( ) = ∓ (cid:112) δ + / ν + (cid:112) δ − / ν . (C.6j) References
1. C. Armendariz-Picon, J.T. Neelakanta, Journal of Cosmology andAstroparticle Physics , 049 (2014). DOI 10.1088/1475-7516/2014/03/0492. C. Gheller, F. Vazza, M. Brüggen, M. Alpaslan, B.W. Holwerda,A. Hopkins, J. Liske, Mon. Not. Roy. Astron. Soc. (1), 448(2016). DOI 10.1093/mnras/stw15953. J.J. Dickau, Chaos, Solitons & Fractals (4), 2103 (2009). DOI10.1016/j.chaos.2008.07.0564. P. Grujic, V. Pankovic, ArXiv e-prints physics.gen-ph , 0907.2127(2009)5. R. Banerjee, K. Jedamzik, Phys. Rev. D , 123003 (2004). DOI10.1103/PhysRevD.70.1230036. M. Lattanzi, N. Carlevaro, G. Montani, Phys. Lett. B (2), 255(2012). DOI 10.1016/j.physletb.2012.10.0677. M. Giovannini, International Journal of Modern Physics D (03),391 (2004). DOI 10.1142/S02182718040045308. A. Kosowsky, A. Loeb, Astrophys. J. , 1 (1996). DOI 10.1086/1777519. J.D. Barrow, P.G. Ferreira, J. Silk, Phys. Rev. Lett. , 3610(1997). DOI 10.1103/PhysRevLett.78.361010. J.D. Barrow, Phys. Rev. D , 7451 (1997). DOI 10.1103/PhysRevD.55.745111. E. Komatsu, K.M. Smith, J. Dunkley, C.L. Bennett, B. Gold,G. Hinshaw, N. Jarosik, D. Larson, M.R. Nolta, L. Page, D.N.Spergel, M. Halpern, R.S. Hill, A. Kogut, M. Limon, S.S. Meyer,N. Odegard, G.S. Tucker, J.L. Weiland, E. Wollack, E.L. Wright,The Astrophysical Journal Supplement Series (2), 18 (2011).DOI 10.1088/0067-0049/192/2/1812. D. Paoletti, F. Finelli, Phys. Rev. D (12), 123533 (2011). DOI10.1103/PhysRevD.83.12353313. L. Pogosian, Journal of Physics: Conference Series (1),012025 (2014). DOI 10.1088/1742-6596/496/1/01202514. Planck Collaboration, et al., Astronomy & Astrophysics , A19(2016). DOI 10.1051/0004-6361/20152582115. G. Montani, G. Palermo, N. Carlevaro, ArXiv e-prints (2017)16. S. Weinberg, Gravitation and Cosmology: Principles and Appli-cations of the General Theory of Relativity (John Wiley & Sons,1972)517. E. Kolb, M. Turner,
The Early Universe . Frontiers in Physics(Westview Press, 1994)18. S. Weinberg,
Cosmology (Oxford University Press, 2008)19. J.D. Barrow, R. Maartens, C.G. Tsagas, Physics Reports (6),131 (2007). DOI 10.1016/j.physrep.2007.04.00620. H. Vasileiou, C.G. Tsagas, Monthly Notices of the Royal As-tronomical Society (3), 2500 (2015). DOI 10.1093/mnras/stv241821. D. Tseneklidou, C.G. Tsagas, J.D. Barrow, Classical and QuantumGravity (12), 124001 (2018). DOI 10.1088/1361-6382/aac07f22. C.G. Tsagas, R. Maartens, Classical and Quantum Gravity (11),2215 (2000). DOI 10.1088/0264-9381/17/11/30523. V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz, Advances inPhysics (6), 639 (1982). DOI 10.1080/0001873820010142824. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields , Course of Theoretical Physics , vol. 2, 4th edn. (Elsevier Science,2013)25. I.B. Ze’ldovich, I.D. Novikov,
Relativistic Astrophysics, 2: TheStructure and Evolution of the Universe , vol. 2, revised and en-larged edition edn. (The University of Chicago Press, 1983)26. G. Montani, M.V. Battisti, R. Benini, G. Imponente,
PrimordialCosmology (World Scientific, 2011). DOI 10.1142/723527. M.P. Ryan, L.C. Shepley,
Homogeneous relativistic cosmologies (Princeton University Press, 1975)28. A.A. Kirillov, G. Montani, Phys. Rev. D , 064010 (2002). DOI10.1103/PhysRevD.66.06401029. V.G. LeBlanc, Classical and Quantum Gravity (8), 2281 (1997).DOI 10.1088/0264-9381/14/8/02530. K.S. Thorne, Astrophysical Journal , 51 (1967). DOI 10.1086/14912731. K.C. Jacobs, Astrophysical Journal , 379 (1969). DOI10.1086/14987532. E.J. King, P. Coles, Classical and Quantum Gravity (8), 2061(2007). DOI 10.1088/0264-9381/24/8/00833. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact solutions of Einstein’s field equations , 2nd edn. (CambridgeUniversity Press, 2003)34. J. Ehlers, General Relativity and Gravitation (12), 1225 (1993).DOI 10.1007/BF0075903135. G.F. Ellis, General Relativity and Gravitation (3), 581 (2009).DOI 10.1007/s10714-009-0760-736. G.F. Ellis, in Cargèse lectures in Physics , vol. VI, ed. by E. Schatz-mann (Gordon and Breach, 1973), vol. VI, pp. 1–6037. G.F.R. Ellis, H. van Elst, in
Theoretical and Observational Cos-mology , NATO Advanced Science Institutes (ASI) Series C , vol.541, ed. by M. Lachièze-Rey (1999),
NATO Advanced Science In-stitutes (ASI) Series C , vol. 541, pp. 1–11638. C.G. Tsagas, J.D. Barrow, Classical and Quantum Gravity (9),2539 (1997). DOI 10.1088/0264-9381/14/9/01139. C.G. Tsagas, Classical and Quantum Gravity (2), 393 (2005).DOI 10.1088/0264-9381/22/2/01140. C.G. Tsagas, A. Challinor, R. Maartens, Physics Reports (2–3), 61 (2008). DOI 10.1016/j.physrep.2008.03.00341. A.G. Doroshkevich, Astrophysics (3), 138 (1965). DOI 10.1007/BF0104193742. H. Noh, J. Hwang, Phys. Rev. D , 1970 (1995). DOI 10.1103/PhysRevD.52.197043. S.A. Bonometto, L. Danese, F. Lucchin, Astronomy and Astro-physics35