Inhomogeneous and anisotropic Universe and apparent acceleration
IInhomogeneous and anisotropic Universe and apparent acceleration
G. Fanizza
1, 2, ∗ and L. Tedesco
1, 2, † Dipartimento di Fisica, Universit`a di Bari, Via G. Amendola 173, 70126, Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy (Dated: May 6, 2019)In this paper we introduce a LTB-Bianchi I (plane symmetric) model of Universe. We study andsolve Einstein field equations. We investigate the effects of such model of Universe in particular theseresults are important in understanding the effect of the combined presence of an inhomogeneous andanisotropic Universe. The observational magnitude-redshift data deviated from UNION 2 cataloghas been analyzed in the framework of this LTB-anisotropic Universe and the fit has been achievedwithout the inclusion of any dark energy.
Keywords:
I. INTRODUCTION
A very important assumption of the standard modelof cosmology (ΛCDM) is based on the homogeneousand isotropic Friedmann-Lemaitre-Robertson-Walker so-lutions of Einstein’s equations. The homogeneity and theisotropy are considered on large scale in the Universe.The Universe is not isotropic or spatially homogeneus onlocal scales.The question of whether the Universe is homogeneousand isotropic is of fundamental importance to cosmol-ogy, but we have not decisive answers. On the otherhand neither observations of luminosity distance combi-nated with galaxy number counts nor isotropic CosmicMicrowave Background Radiation are able to say if theUniverse is spatially homogeneous and isotropic.The fundamental question consists in a simple obser-vation: this geometry is the only that is able to explainand to be compatible with experimental data? Are wesure that the assumption of homogeneity and isotropy isa logical and comforting way of thinking, or better is itan a-priori assumption?This is a pertinent question because we need that more96% of the content of our Universe must be dark (energyand matter) in order to have a compatible model withobservations. The solution of the dark energy puzzle isthe keystone of the modern cosmology.Are there observables that can prove the Universe ishomogeneous and isotropic on large scales? Very inter-esting studies has been done in this direction [1–3].The ΛCDM model of the Universe is remarkablysuccessfull, but we have important tensions between themodel and the experimental data [4, 5]. On the otherhand dark energy is the biggest puzzle in cosmology.There are many papers with more detailed discussionsabout dark energy, that are outside the scope of thispaper, see for example [6, 7] and references therein.There are many reasons that consider the ΛCDM modelfull of theoretical problems [8], one is that Λ has avalue absurdly small in quantum physics. Moreoverwe cannot expect that dark energy will have in future locally observable effects.The Cosmic Microwave Background (CMB) hashigh isotropy and this is considered as a strong evidenceof the homogeneity and isotropy of the Universe, thatis to say the Universe is well described by meansof a Friedmann-Lemaitre-Robertson-Walker (FLRW)model. The main indication for this model is due tothe Ehlers, Geren and Sachs theorem (EGS) [1] in 1968.This theorem is due to an earlier paper of Tauber andWeinberg [9] in 1961. In EGS theorem we considerthe observers in an expanding Universe, dust Universemeasures isotropic CMB and this implies that FLRWmetric is valid and the cosmological principle is alsovalid. This theorem is important because it permitsto have the homogeneity and isotropy not from exper-imental measurements of the isotropy of the Universebut from the CMB. But as we will discuss later, CMBradiation have small anisotropies with 10 − of amplitude!As regards the homogeneity of the Universe it isimportant to note that the mass density of the Universeis not inhomogeneous on scales much smaller than theHubble radius, in other terms the homogeneity is nottrue at all orders but we can assume to be valid ondistance greater that 100 Mpc. Many papers indicatethis feature, see for example [10] (and references therein),where the author indicates evidences that galaxy distri-bution is spatially inhomogenous for r <
100 Mpc/h.The strong interest in inhomogenous cosmologicalmodels, in particular the so called Lemaitre-Tolman-Bondi (LTB) model [11–13] (for more details see [14]and references therein), that represents a sphericallysymmetric exact solution to the Einstein’s equationswith pressureless ideal fluid, is due to the its simplicityand it is very useful. In fact it allows for studies ofinhomogeneities that cannot be analyzed as perturbativedeviations from FLRW and it permits to evaluatethe effect of inhomogeneities In particular it has beenstudied that LTB models without dark energy can fitobserved data. a r X i v : . [ g r- q c ] D ec The high precision cosmology is able to understandby more details our study about Universe. When weconsider the isotropy of CMB we must not forget thatthis is not sufficient to say that our region of space isisotropic [15].We have two very important observational evidencesshowing that we don’t have exact isotropy [16]. Both ev-idences may be caused by an anisotropic phase during theevolution of our Universe in other terms the existence ofanomalies in CMB suggests the presence of an anomalousplane-mirroring symmetry on large scales [17, 18]. Thesame anomalous features in seven-years WMAp data andPlanck data seems to suggest that our Universe could benon-isotropic .The first is the presence of small anisotropy deviationsas regards the isotropy of the CMB. In fact we have smallanisotropies with 10 − amplitude.The second is connected with the presence of large an-gle anomalies [19]. These anomalies can be considered in4 families:1) the alignment of quadrupole and octupole moments[20–23];2) the large scale asymmetry [24, 25];3) the very strange cold spot [26];4) the low quadrupole moment of the CMB, that isvery important because it may indicate an ellipsoidal- Bianchi type I anisotropic evolution of the Universe[27–30]. This is due to the fact that the low quadrupolemoment is suppressed at large scale and this suppressioncannot be explained by the common cosmological model.Some years ago it has been shown [31] that if we startwith a FLRW Universe, it is possible to have small de-viations from homogeneity and isotropy taking into ac-count small deviations in the CMB. In particular if weconsider an homogeneous and anisotropic Universe, thesmall quadrupole anisotropy in CMB implies a very smallanisotropy in the Universe. Next, general results havebeen established [32, 33], in which the authors does notassume a priori homogeneity and they found that smallanisotropies in CMB imply that the Cosmo is not exactlyFLRW but it is almost FLRW. Limits on anisotropy andinhomogeneity can be found starting from CMB.The cosmological model that takes into account all theseand stimulates many interest is the ”anisotropic Bianchitype I model” that can be an intriguing alternative to thestandard model FLRW, in which small deviations fromthe isotropy is able to explain the anisotropies and theanomalies in the CMB.The anisotropy considered in this work might be inter-preted as an imprinting, a primordial relic of an earlyanisotropy that appears in the context of a multi- di-mensional cosmological model of unified string theories.In this paper our goals are to study an anisotropic andinhomogeneous model of the Universe. In particular weintroduce a new approach to a Universe in which inhomo- geneities and anisotropies coexist, therefore we study inorder to obtain the relative Einstein’s equations. Thesemodels of Universe inhomogeneous and anisotropic hasbeen studied in different physical situations, as the roleof the diffusion forces in governing the large-scale dynam-ics of inhomogeneous and anisotropic Universe [34].The Supernovae observations are good tests about thestructure of the space-time on different scales. This is avery important point, in fact some years ago Zel’dovich[35] studied the importance of the effects of the inhomo-geneities on light propagation and also in the next years[36–42]. To check this model we calculate the luminositydistance in order to compare theoretical approach withexperimental data. We explain the acceleration of theUniverse without invoking the presence of a cosmologicalconstant or dark energy.The structure of this paper is the following. Inthe next Section we calculate the metric for this LTB-Bianchi I model of the Universe. In Section III afterproviding the calculation of various symbols we write theEinstein’s equations taking into account this geometry.Section IV is dedicated to calculate the luminositydistance and in Section V we compare the theoreticaldata with experimental data. Finally the discussion andconclusion are summarized in Section VI. II. LTB-BIANCHI 1 METRIC
In order to find the anisotropic-LTB metric, let us startwith a Bianchi type I space-time metric, spatially homo-geneous, descripted by the metric ds = dt − a ( t ) ( dx + dy ) − b ( t ) dz (1)with two expansion parameters a and b that are thescale factors normalized in order that a ( t ) = b ( t ) = 1and t present cosmic time. The metric (1) consid-ers the xy-plane as a symmetry plane. To our aimwe write the Bianchi type I metric in polar coordinate( x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ ): ds = dt − [ a ( t )sin θ + b ( t )cos θ ] dr − r [ a ( t )cos θ + b ( t ) sin θ ] dθ − r [ a ( t ) − b ( t )]sin θ cos θ dr dθ − r a ( t ) sin θ dφ . (2)In order to have a LTB-Bianchi I metric, we make thefollowing substitutions r a ( t ) → A (cid:107) ( r, t ) ≡ A (cid:107) (3) r b ( t ) → A ⊥ ( r, t ) ≡ A ⊥ . (4)In this way it is possible to obtain the general LTB-Bianchi I metric in polar coordinate, observing that a ( t ) = A (cid:48) and 2 r (cid:48) a ( t ) = ( A (cid:107) ) (cid:48) where (cid:48) ≡ ∂/∂r , wehave: ds = dt − ( A (cid:48) (cid:107) sin θ + A (cid:48) ⊥ cos θ ) dr − ( A (cid:107) cos θ ++ A ⊥ sin ) dθ − ( A (cid:107)(cid:48) − A ⊥(cid:48) ) sin θ cos θ dr dθ + − A (cid:107) sin θ dφ . (5)It is important to observe that the eq. (5) brings back toknown cases: A (cid:107) ( r, t ) = r a ( t ) and A ⊥ ( r, t ) = r b ( t ) , Bianchi I A (cid:107) ( r, t ) = A ⊥ ( r, t ) , LTB A (cid:107) ( r, t ) = A ⊥ ( r, t ) = r a ( t ) FRW . (6)Therefore the metric (5) is a non-homogeneous metricwith axial symmetry, that is simple referable to pure ho-mogeneous or pure isotropic case.Let us define the following quantity (cid:15) ( r, t ) = A ⊥ − A (cid:107) (7)that represents the degree of anisotropy of the Universe.From the definition of (cid:15) we obtain: A (cid:48)⊥ = A (cid:48)(cid:107) + (cid:15) (cid:48) (8a) A (cid:48)⊥ = A (cid:48)(cid:107) + (cid:15) (cid:48) + 2 A (cid:48)(cid:107) (cid:15) (cid:48) (8b) A ⊥ = A (cid:107) + (cid:15) + 2 A (cid:107) (cid:15) (8c) (cid:0) A ⊥ (cid:1) (cid:48) = (cid:16) A (cid:107) (cid:17) (cid:48) + (cid:0) (cid:15) (cid:1) (cid:48) + 2 A (cid:48)(cid:107) (cid:15) + 2 A (cid:107) (cid:15) (cid:48) . (8d)Let us introduce these relations in the metric (5) in orderto show it as a function of (cid:15) and A (cid:107) (or (cid:15) and A ⊥ ).Putting all togheter we have ds = dt − (cid:104) A (cid:48)(cid:107) + ( (cid:15) (cid:48) + 2 A (cid:48)(cid:107) (cid:15) (cid:48) ) cos θ (cid:105) dr + − (cid:2) A (cid:107) + ( (cid:15) + 2 A (cid:107) (cid:15) ) sin θ (cid:3) dθ ++ (cid:104)(cid:0) (cid:15) (cid:1) (cid:48) + 2 A (cid:48)(cid:107) (cid:15) + 2 A (cid:107) (cid:15) (cid:48) (cid:105) sin θ cos θ drdθ + − A (cid:107) sin θdφ ≡≡ (cid:16) g (LTB) (cid:107) µν + ∆ g ( AN ) (cid:107) µν (cid:17) dx µ dx ν (9)with our metric given by g µν ≡ g (LTB) (cid:107) µν + ∆ g ( AN ) (cid:107) µν (10)where: g (LTB) (cid:107) µν = − A (cid:48)(cid:107) − A (cid:107)
00 0 0 − A (cid:107) sin θ (11a)∆ g ( AN ) (cid:107) = − cos θ (cid:16) (cid:15) (cid:48) + 2 A (cid:48)(cid:107) (cid:15) (cid:48) (cid:17) (11b) ∆ g ( AN ) (cid:107) = sin 2 θ (cid:104)(cid:0) (cid:15) (cid:1) (cid:48) + 2 A (cid:48)(cid:107) (cid:15) + 2 A (cid:107) (cid:15) (cid:48) (cid:105) (11c)∆ g ( AN ) (cid:107) = − sin θ (cid:0) (cid:15) + 2 A (cid:107) (cid:15) (cid:1) . (11d)The script ”(LTB)” (up or down is the same) means thatthe quantity refers to Lemaitre-Tolman-Bondi Universe,while ”(AN)” refers to anisotropic Universe. In otherwords the metric (9) is be able to describe the inho-mogeneity and axial anisotropy of the Universe, on theother hand a very interesting thing is that it has beendecomposed in the sum of a LTB metric with null curveand a a term that contains whole information about theanisotropy, (cid:15) ( r, t ).For completeness reasons, it is possible to rewrite in thesymmetric way the metric as g µν ≡ g (LTB) ⊥ µν + ∆ g ( AN ) ⊥ µν (12)where g ( LT B ) ⊥ µν is obtained by eq. (11a) with the substitu-tion of A ⊥ instead of A (cid:107) and∆ g ( AN ) ⊥ = sin θ (cid:0) (cid:15) (cid:48) − A (cid:48)⊥ (cid:15) (cid:48) (cid:1) (13a)∆ g ( AN ) ⊥ = sin 2 θ (cid:104)(cid:0) (cid:15) (cid:1) (cid:48) − A (cid:48)⊥ (cid:15) − A ⊥ (cid:15) (cid:48) (cid:105) (13b)∆ g ( AN ) ⊥ = cos θ (cid:0) (cid:15) − A ⊥ (cid:15) (cid:1) (13c)∆ g ( AN ) ⊥ = sin θ (cid:0) (cid:15) − A ⊥ (cid:15) (cid:1) . (13d) III. EINSTEIN’S EQUATIONS IN LTB-BIANCHI IUNIVERSE
In this Section we want to write the Einstein’s equa-tions taking into account the LTB-Bianchi I metric. Tothis end we suppose a very small anisotropy of the Uni-verse, in order to have (cid:15) ( r, t ) (cid:28) A (cid:107) ( r, t ) (14a) (cid:15) (cid:48) ( r, t ) (cid:28) A (cid:48)(cid:107) ( r, t ) . (14b)These positions permit to expand our results to the firstorder in (cid:15) , in other terms:∆ g ( AN ) µν → δg ( AN ) µν (15)with δg ( AN ) (cid:107) = − A (cid:48)(cid:107) (cid:15) (cid:48) cos θ (16) δg ( AN ) (cid:107) = − A (cid:107) (cid:15) sin θ (17) δg ( AN ) (cid:107) = 2 A (cid:107) (cid:15) (cid:48) sin θ cos θ. (18)At this point we can calculate the Christoffel connectionto the first order in (cid:15) (we repeats that the script ( LT B )and ( AN ) are indifferently written up or down):Γ αµν = 12 g αρ ( ∂ µ g νρ + ∂ ν g ρµ − ∂ ρ g µν ) (cid:39)(cid:39) (cid:16) g αρ ( LT B ) + δg αρ ( AN ) (cid:17) (cid:104) ∂ µ ( g ( LT B ) νρ + δg ( AN ) νρ )++ ∂ ν ( g ( LT B ) ρµ + δg ( AN ) ρµ ) − ∂ ρ ( g ( LT B ) µν + δg ( AN ) µν ) (cid:105) , (19)that, neglecting the second order terms in (cid:15) , becomesΓ αµν (cid:39) g αρ ( LT B ) (cid:16) ∂ µ g ( LT B ) νρ + ∂ ν g ( LT B ) ρµ − ∂ ρ g ( LT B ) µν (cid:17) ++ 12 g αρ ( LT B ) (cid:16) ∂ µ δg ( AN ) νρ + ∂ ν δg ( AN ) ρµ − ∂ ρ δg ( AN ) µν (cid:17) ++ 12 δg αρ ( AN ) (cid:16) ∂ µ g ( LT B ) νρ + ∂ ν g ( LT B ) ρµ − ∂ ρ g ( LT B ) µν (cid:17) . (20)The first term in eq. (20) is just Γ α ( LT B ) µν , the Christoffelconnection with the metric tensor g ( LT B ) µν , and puttingΣ αµν ≡ g αρ ( LT B ) ( ∂ µ δg ( AN ) νρ + ∂ ν δg ( AN ) ρµ − ∂ ρ δg ( AN ) µν )(21)andΘ αµν ≡ δg αρ ( AN ) ( ∂ µ g ( LT B ) νρ + ∂ ν g ( LT B ) ρµ − ∂ ρ g ( LT B ) µν ) , (22)it is possibile to write the Christoffel connection at thefirst order as Γ αµν = Γ α ( LT B ) µν + Σ αµν + Θ αµν . (23)Let us calculate the Ricci tensor R να = ∂ µ Γ µνα − ∂ ν Γ µµα + Γ µµρ Γ ρνα − Γ µνρ Γ ρµα (24)with Γ µνα given by eq.(23). Therefore we have R να = ∂ µ (Γ µ ( LT B ) να + Σ µνα + Θ µνα ) + − ∂ ν (Γ µ ( LT B ) µα + Σ µµα + Θ µµα ) ++ (Γ µ ( LT B ) µρ + Σ µµρ + Θ µµρ )(Γ ρ ( LT B ) να + Σ ρνα + Θ ρνα ) + − (Γ µ ( LT B ) νρ + Σ µνρ + Θ µνρ )(Γ ρ ( LT B ) µα + Σ ρµα + Θ ρµα ) . (25)When we multiply in eq. (25), neglecting the second orderterms ΣΣ, ΘΘ, ΣΘ and ΘΣ, and putting R ( LT B ) να = ∂ µ Γ µ ( LT B ) να − ∂ ν Γ µ ( LT B ) µα ++ Γ µ ( LT B ) µρ Γ ρ ( LT B ) να − Γ µ ( LT B ) νρ Γ ρ ( LT B ) µα , (26) R (Σ) να ≡ ∂ µ Σ µνα − ∂ ν Σ µµα + Σ µµρ Γ ρ ( LT B ) να + Γ µ ( LT B ) µρ Σ ρνα + − Γ µ ( LT B ) νρ Σ ρµα − Σ µνρ Γ ρ ( LT B ) µα (27) R (Θ) να ≡ ∂ µ Θ µνα − ∂ ν Θ µµα + Θ µµρ Γ ρ ( LT B ) να + Γ µ ( LT B ) µρ Θ ρνα + − Γ µ ( LT B ) νρ Θ ρµα − Θ µνρ Γ ρ ( LT B ) µα , (28)the Ricci tensor to the first order in δ can be written as R να = R ( LT B ) να + R (Σ) να + R (Θ) να . (29)In order to consider the perturbations of the energy mo-mentum tensor, we consider a general anisotropic densityenergy given by: ρ mat ( r, t, θ ) ≡ ρ (cid:107) mat ( r, t ) sin θ + ρ ⊥ mat ( r, t ) cos θ == ρ (cid:107) mat ( r, t ) + δ mat ( r, t )cos θ (30)where δ mat ≡ ρ ⊥ mat − ρ (cid:107) mat . The density that we choosehas a planar symmetry, because of the consistency withthe metric that we are working with. This choice allowsus to rewrite the energy momentum tensor and its traceas: T νµ ≡ T ν ( LT B ) µ + ∆ T νµ (31) T ≡ T ( LT B ) + ∆ T (32)where, in general T ν ( LT B ) µ = diag[ ρ ( r, t ) , − p ( r, t ) , − p ( r, t ) , − p ( r, t )] . (33)However, all these definitions must be view as a first or-der correction to the usual energy momentum tensor inthe LTB case. This point of view becomes clear if welook at the usual perturbation theory provided by [47].In fact, our particular definition is fully consistent withthe Mukhanov’s one whether we fix δp = V = σ = 0 inEq. (5.1) of [47]: this means that we are only consideringthe perturbation in the energy density and we neglect theeffect of a different pressure along two directions (in par-ticular, we continue in using a pressureless matter fluideverywhere). For sure, what we did is a strong constraint.By the way, the particular choice of the perturbation isnot relevant for the purposes of this paper.In order to be explicit we write: T νµ = T ν ( LT B ) µ + δT ν ( AN ) µ (34)and T να = ( g ( LT B ) νµ + δg ( AN ) νµ )( T µ ( LT B ) α + δT µ ( AN ) α ) = (cid:39) T ( LT B ) να + δg ( AN ) νµ T µ ( LT B ) α + g ( LT B ) νµ δT µ ( AN ) α . (35)As regards the energy conditions, we consider the generalenergy-momentum tensor: T µν = ρ u µ u ν + p ( − g µν + u µ u ν ) (36)with u µ u µ = 1. Hence, energy conditions state: • weak energy condition: T µν u µ u ν ≥ • dominant energy condition: by defining W µ = T µν u ν , W µ W µ ≥ • strong energy condition: T µν u µ u ν ≥ T u µ u µ • null energy condition: T µν k µ k ν ≥
0, where k µ is alight-like vector.A well-known, the hierarchy among these conditions isthe following: strong implies null, dominant implies weakand weak implies null. In this way, by providing that thedominant condition holds, also weak and null are satisfiedas well. In particular, for energy-momentum (36), with p = 0, they become: • weak: ρ mat ≥ • dominant: ρ mat ≥ • strong: ρ mat ≥ • null: ρ mat ≥ R ν ( LT B ) µ + R ( LT B ) µν δg αν + ( R (Σ) µα + R Θ µα ) g να ( LT B ) =8 πG [ T ν ( LT B ) µ + δT ν ( AN ) µ + 12 δ νµ ( T ( LT B ) + δT ( AN ) )] . (37) IV. LUMINOSITY DISTANCE
The concept of distance depends on the assumed modelof the Universe and on the matter distribution in it. Themeasured distance are influenced by inhomogeneities andanisotropy of the Universe, see for example [43] and [44].The luminosity distance is one of the most importantquantity to understand the presence of dark energy inthe Universe, considering the photon coming from Su-pernovae Ia. In this section we want to calculate the lu-minosity distance for our metric eq.(9). The reciprocitytheorem by Etherington (1993) [45] and popolarized byEllis [46] connects the angular diameter distance d A andthe luminosity distance d L by d L = (1 + z ) d A (38)where d (ln d A ) = 12 ∇ α p α dτ (39)with τ temporal affine parameter and p α = dx α /dτ quadri-momentum of a generic signal that is started from the Supernova and reaches us. To our end it is necessaryto calculate ∇ α p α ≡ ∂ α p α + Γ αµα p µ : ∇ α p α = ∂ α p α + Γ αµα p µ = ∂ α p α + ∂ µ √− g √− g p µ == ∂ p + ∂ p + ∂ √− g √− g p + ∂ √− g √− g p (40)with − g = A (cid:107) ( g g − g ) sin θ ≡ A (cid:107) B ( t, r, θ ) sin θ (41)where we define B ( t, r, θ ) ≡ (cid:113) g ( t, r, θ ) g ( t, r, θ ) − g ( t, r, θ ) . (42)In this way we obtain: ∂ √− g √− g = (cid:0) ∂ A (cid:107) B + A (cid:107) ∂ B (cid:1) sin θA (cid:107) B sin θ = ∂ A (cid:107) A (cid:107) + ∂ BB (43a) ∂ √− g √− g = (cid:0) ∂ A (cid:107) B + A (cid:107) ∂ B (cid:1) sin θA (cid:107) B sin θ = ∂ A (cid:107) A (cid:107) + ∂ BB . (43b)This permits to write eq.(40) as ∂ p + ∂ p + (cid:18) ∂ A (cid:107) A (cid:107) + ∂ BB (cid:19) p + (cid:18) ∂ A (cid:107) A (cid:107) + ∂ BB (cid:19) p == ∂ p + ∂ p + 1 A (cid:107) dA (cid:107) dτ + (cid:18) ∂ BB p + ∂ BB p (cid:19) . (44)In the last equation we have considered the general rela-tion between partial derivates in the coordinates x α andtotal derivates in the affine time τ . In fact if Φ is a genericfunction that depends from the coordinates it is possibleto write: d Φ( x α ( τ )) dτ = ∂ Φ( x α ) ∂x β dx β dτ ≡ ∂ β Φ p β . (45)On the other hand as regards B , we must write dBdτ = ∂ B p + ∂ B p + ∂ B p , but we are considering radialsignal, therefore p ≡ dθ/dτ , that is to say θ ( τ ) = cost. In this way we can consider θ as a parameter that is ableto locate the trajectory of propagation of light. Thisemployment permits to write: B ( r, t, θ ) ≈ B ( r ( τ ) , t ( τ ) , θ ) ⇒ dBdτ ≈ ∂ Bp + ∂ Bp . (46)Therefore, eq. (40) becomes: ∇ α p α = ∂ p + ∂ p + 1 A (cid:107) dA (cid:107) dτ + 1 B dBdτ . (47)As regards the partial derivative of p α , remembering thatwe are considering the radial propagation of signals, therelevant components are: dp + Γ dx p + Γ (cid:0) dx p + dx p (cid:1) + Γ dx p = 0(48a) dp + Γ dx p + Γ (cid:0) dx p + dx p (cid:1) + Γ dx p = 0(48b)from which we have: ∂ p = − (cid:0) Γ p + Γ p (cid:1) (49a) ∂ p = − (cid:0) Γ p + Γ p (cid:1) . (49b)In order to complete the analysis observe that Γ =Γ = 0 andΓ = 12 g ( ∂ g + ∂ g − ∂ g ) ++ 12 g ( ∂ g + ∂ g − ∂ g ) == − (cid:0) g X∂ X + g F ∂ F (cid:1) (50)Γ = 12 g ( ∂ g + ∂ g − ∂ g ) ++ 12 g ( ∂ g + ∂ g − ∂ g ) == − (cid:0) g X∂ X + 2 g F ∂ F − X∂ X (cid:1) (51)where we have put g ≡ X g ≡ Y g ≡ F . (52)Now we work in small approximation of anisotropy, inorder to use eq.(23) to the lower order, in this way it ispossibile to write:Γ → Γ = ∂ ∂ A (cid:107) ∂ A (cid:107) (53a)Γ → Γ = ∂ A (cid:107) ∂ A (cid:107) (53b)where ∂ ≡ ∂ ∂r . Therefore eq.(47) is ∇ α p α ≈ − (cid:18) ∂ ∂ A (cid:107) ∂ A (cid:107) p + ∂ A (cid:107) ∂ A (cid:107) p (cid:19) + 1 A (cid:107) dA (cid:107) dτ + 1 B dBdτ . (54)Taking into account eq.(45) it is possible to write the firsttwo terms in eq.(54) as ∂ A (cid:107) dA (cid:107) dτ .Inserting eq.(54) in eq. (39) it is possible to obtain d A ,in fact we have: dd A d A ≈ (cid:18) A (cid:107) dA (cid:107) dτ + 1 B dBdτ − ∂ A (cid:107) d∂ A (cid:107) dτ (cid:19) dτ (55) and integrating in τ we obtain: d A ( r, t, θ ) = (cid:115) A (cid:107) ( r, t ) B ( r, t, θ ) ∂ A (cid:107) ( r, t ) . (56)This expression of the luminosity distance reduces to theisotropic limit of the LTB metric, in fact we have X ( r, t, θ ) → ∂ A (cid:107) ( r, t ) A ( r, t, θ ) → A (cid:107) ( r, t ) F ( r, t, θ ) → , (57)therefore we obtain the limit B ( r, t, θ ) → A (cid:107) ( r, t ) ∂ A (cid:107) ( r, t ) ⇒⇒ d A ( r, t, θ ) → d (LTB) A ( r, t ) = A (cid:107) ( r, t ) . (58) V. RELATION BETWEEN COORDINATES ANDREDSHIFT
In this section we want to calculate the luminositydistance in order to obtain an operative expression andtherefore to apply it to experimental data. The eq. (56)give us the luminosity distance: d L = (1 + z ) (cid:115) A (cid:107) ( r, t ) B ( r, t, θ ) ∂ A (cid:107) ( r, t ) , (59)but this expression is not directly applicable, becauseof it depends on ( r, t ) coordinates and on the redshift.Theferore it is necessary to find the relations r ( z ) e t ( z ).To this end let us consider the definition of redshift andlet us use the static observators classes that are geodeticalso ( Γ µ = 0). We consider light signal, that is to say p ∝ /δt , in this way we write:1 + z ≡ ( g µν u µ p ν ) em ( g µν u µ p ν ) oss = p p = δt oss δt em ⇒ z ( τ ) = δt oss δt ( τ )(60)where u µ = (1 , , ,
0) e g = 1. Now we derive withrespect to τ and we have dz ( τ ) dτ = − δt oss δt ( τ ) dδt ( τ ) dτ ≡ − z ( τ ) δt ( τ ) dδt ( τ ) dτ ⇒⇒ dδtdτ = − δt z dzdτ . (61)On the other hand for geodetic radial signals we have ds = 0 e dθ = dφ = 0 that gives dt − X ( r, t, θ ) dr = 0 ⇒ dt = ± X ( r, t, θ ) dr. (62)As regards the ambiguity of the sign we must consider theminus sign because of increasing the distance ( dr >
0) wehave a more ancient signal ( dt < t and t + δt ,the eq. (62) must be valid, therefore we have: dtdτ = − X ( r, t, θ ) drdτ (63a) d ( t + δt ) dτ = − X ( r, t + δt, θ ) drdτ . (63b)Eq.(63b) can be written as dtdτ + dδtdτ ≈ − [ X ( r, t, θ ) + δt ∂ X ( r, t, θ )] drdτ (64)that, taking into account eq.(63a), can be written as dδtdτ ≈ − δt ∂ X ( r, t, θ ) drdτ = − δt ∂ X ( r, t, θ ) drdz dzdτ . (65)Now eqs. (61) and (65) are equal, therefore we have drdz = 11 + z ∂ X ( r, t, θ ) . (66)As regards t ( z ) it is important to remember that dtdτ = dtdz dzdτ e drdτ = drdz dzdτ (67)in this way taking into account eq. (63b) we obtain therelation: dtdz = −
11 + z X ( r, t, θ ) ∂ X ( r, t, θ ) . (68)Putting all togheter, we are be able to write the luminos-ity distance as a function of the redshift z and the angle θ d L ( z, θ ) = (1 + z ) (cid:20) A (cid:107) ( r θ ( z ) , t θ ( z )) ∂ A (cid:107) ( r θ ( z ) , t θ ( z )) B ( r θ ( z ) , t θ ( z ) , θ ) (cid:21) (69a) dr θ ( z ) dz = 11 + z ∂ X ( r θ ( z ) , t θ ( z ) , θ ) (69b) dt θ ( z ) dz = −
11 + z X ( r θ ( z ) , t θ ( z ) , θ ) ∂ X ( r θ ( z ) , t θ ( z ) , θ ) . (69c)It is important to observe that subscript θ rememberus that the functions r and t are determined by theresolution of the system given by eqs (69b) and (69c),where the angle θ is fixed and considered as a constantparameter during the propagation of light. VI. COMPARISON WITH EXPERIMENTALDATA
The accelerating expansion of the universe is driven bymysterious energy with negative pressure known as Dark Energy. In spite of all the observational evidences, thenature of Dark Energy is still a challenging problem intheoretical physics, therefore there has been a new inter-est in studying alternative cosmological models [14].In the context of FLRW models the acceleration of theUniverse requires the presence of a cosmological constant.But it does not appear to be natural to introduce thepresence of a cosmological constant and does not appearto be natural to introduce the dark energy.In this Section we consider the comparison between ex-perimental data, in particular with Union 2 data set ofSupernovae Ia and our inhomogeneous and anisotropicUniverse. Let us suppose a small anisotropy in orderto write the functions A (cid:107) e A ⊥ as solutions of a LTBUniverse with null curvature and matter dominated. Wehave A (cid:107) ( r, t ) = r (cid:18) H (cid:107) ( r ) t (cid:19) (70a) A ⊥ ( r, t ) = r (cid:18) H ⊥ ( r ) t (cid:19) (70b)where we have considered the following parametrization H (cid:107) / ⊥ ( r ) = H (cid:107) / ⊥ + ∆ H (cid:107) / ⊥ exp (cid:18) − rr (cid:107) / ⊥ (cid:19) . (71)In this way we have the possibility to obtain again thesimple model in which H (cid:107) = H ⊥ , ∆ H (cid:107) = ∆ H ⊥ and r (cid:107) = r ⊥ . Let us consider that today and in our positionin the Universe ( t = 0 e r = 0) the Hubble constant is67 . ± . km s / Mpc [48]. We have the following conditions H (cid:107) + ∆ H (cid:107) = H ⊥ + ∆ H ⊥ = 67 .
3, therefore we have: H (cid:107) = 67 . − ∆ H (cid:107) (72a) H ⊥ = 67 . − ∆ H ⊥ . (72b)In this way we have not the six parameters of the model,now they are four: ∆ H (cid:107) , ∆ H ⊥ , r (cid:107) e r ⊥ . At this pointwe remember the limits given by (14), therefore for (cid:15) ∼ A ⊥ ( r, t ) ≈ A (cid:107) ( r, t ) ⇒ H (cid:107) ( r ) ≈ H ⊥ ( r ) . (73)This condition must be transfered to the four parame-ters.The condition eq. (73) is obtained when α ∼ ω ∼
1. On the other hand it also must be (cid:15) (cid:48) ∼ A (cid:48)(cid:107) / ⊥ = (cid:18) H (cid:107) / ⊥ t (cid:19) + r H (cid:48)(cid:107) / ⊥ t (cid:0) H (cid:107) / ⊥ t (cid:1) == A (cid:107) / ⊥ r + r H (cid:48)(cid:107) / ⊥ tA (cid:107) / ⊥ , (74)from which we obtain α ≡ r ⊥ r (cid:107) (75a) ω ≡ ∆ H ⊥ ∆ H (cid:107) . (75b) A (cid:48)(cid:107) − A (cid:48)⊥ = A (cid:107) − A ⊥ r + r t (cid:32) H (cid:48)⊥ A ⊥ − H (cid:48)(cid:107) A (cid:107) (cid:33) == (cid:15)r + r t (cid:18) − ∆ H ⊥ A ⊥ r ⊥ + ∆ H (cid:107) A (cid:107) r (cid:107) (cid:19) . (76)In this way, for (cid:15) ∼ (cid:15) (cid:48) ∼
0, we must write:∆ H ⊥ ∆ H (cid:107) = A ⊥ A (cid:107) r ⊥ r (cid:107) ⇒ ω = A ⊥ A (cid:107) α. (77)Therefore A ⊥ /A (cid:107) ≈
1, from which we obtain ω ≈ α . Μ Figure 1: Hubble diagram for type Ia supernovae by UNION2 catalog. The curve is the best fit.
In conclusion we have also three parameters: ∆ H (cid:107) , r (cid:107) and α . The advantage of this parametrization is that wecan change ∆ H (cid:107) and r (cid:107) as we want, taking into accountthat α (cid:39)
1. In our work we have changed α in the range[0 . , H (cid:107) (cid:104) km(s Mpc) − (cid:105) r (cid:107) [Gpc] α χ = 0 .
95. In fig. 1 we have the Hubble diagram for the557 Supernovae Ia of the UNION 2 catalog. The best fitcurve is in the same diagram. The fit of the cosmologicalobservational data is in very good agreement, withoutusing any dark energy!According to our ansatz, it is important to stress that ρ mat = ρ (cid:107) mat + δ mat cos θ , so dominant energy conditionup to first order gives: δ mat cos θ ≥ − ρ (cid:107) mat δ mat cos θ ≥ − ρ (cid:107) mat . (79)Furthermore, δ mat = ρ ⊥ mat − ρ (cid:107) mat so, from the Ein-stein’s equation, we have that: ρ (cid:107) mat = 18 π G (cid:32) ˙ A (cid:107) A (cid:107) (cid:33) + 2 ˙ A (cid:107) A (cid:107) ˙ A (cid:48)(cid:107) A (cid:48)(cid:107) δ mat = 18 π G (cid:32) ˙ A ⊥ A ⊥ (cid:33) + 2 ˙ A ⊥ A ⊥ ˙ A (cid:48)⊥ A (cid:48)⊥ − (cid:32) ˙ A (cid:107) A (cid:107) (cid:33) − A (cid:107) A (cid:107) ˙ A (cid:48)(cid:107) A (cid:48)(cid:107) . (80)Hence we have from solutions (70a) and (70b), withconditions (75a), (75b) and (77), that eq. (78) gives aconstraint on parameters which must be satisfied. Inparticular, by using the best-fit values for r (cid:107) and ∆ H (cid:107) ,dominant energy condition requires α < .
5, which is infully agreement with our analysis.
VII. CONCLUSION
In the present paper we have studied the possibleeffects of an anisotropy and inhomogeneity in theexpansion of the Universe.The motivation behind this choise is that singly, inhomo-geneous cosmological models and Bianchi I cosmologicalmodel of the Universe have motivations of thruth thatmust not be left out for one of the the two models. Bothmodels may be unified in a anisotropic expansion ofthe inhomogeneous Universe. The LTB-Bianchi I modelposseses important specific properties and at the sametime this is not too complicated from a physical andmathematical point of view.In particular we have connected this model on present-day observations as luminosity distance of the Super-novae Ia. We fit observational data from UNION 2catalog of the Supernovae Ia with a LTB-Bianchi Imodel of the Universe. The agreement is good. We havenot any dark energy in this model.We are sure that the voids in the Universe domi-nate, while matter is distributed in a filamentarystructure. Therefore photons must travel through thevoids and the presence of inhomogeneities can alter theobservable with respect to the corresponding FLRWmodel of Universe, homogeneous and isotropic.The key point is that in this model we have twocontributions to the Hubble diagram of the SupernovaeIa: inhomogeneity to the large scale geometry andanisotropy can generate dynamically effects that mayremove the need for the postulate of dark energy.This model must be intended as a first step towardsa most general case. The model is oversimplifying fordifferent reasons. First, we have considered only thefirst order in (cid:15) . Second, it is necessary to generalizethis paper, a very interesting open question that wewill study in future, is to obtain how to treat light-coneaverage in more realistic cosmological calculation. Thirdwe have considered he simple LTB model of the Universe,but may be very interesting to study more completedinhomogeneous model as for example Swiss-cheesemodel. 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