An inhomogeneous universe with thick shells and without cosmological constant
aa r X i v : . [ a s t r o - ph . C O ] J un An inhomogeneous universe with thick shells andwithout a cosmological constant
Stefano ViaggiuDipartimento di Matematica, Universit´a ”Tor Vergata”’,Via della Ricerca Scientifica, 1Rome, Italy 00133,[email protected] 26, 2018
Abstract
We build an exact inhomogeneous universe composed of a centralflat Friedmann zone up to a small redshift z , a thick shell made ofanisotropic matter, an hyperbolic Friedmann metric up to the scalewhere dimming galaxies are observed ( z ≃ .
7) that can be matched toa hyperbolic Lemaˆıtre-Tolman-Bondi spacetime to best fit the WMAPdata at early epochs. We construct a general framework which permitsus to consider a non-uniform clock rate for the universe. As a result,both for a uniform time and a uniform Hubble flow, the decelera-tion parameter extrapolated by the central observer is always positive.Nevertheless, by taking a non-uniform Hubble flow, it is possible toobtain a negative central deceleration parameter, that, with certainparameter choices, can be made the one observed currently. Finally, itis conjectured a possible physical mechanism to justify a non-uniformtime flow.
PACS numbers: 98.80.-k,98.80.Jk,95.36.+x,04.20.-q
Supernovae type Ia (SNIa) observations of the past decade seem to indi-cate an accelerating universe ([1, 2, 3]). In the standard approach with theFriedmann-Lemaˆıtre models (FRLW), an accelerating universe invokes thepresence of a large amount of the so called dark energy. In the FRLW pic-ture, this dark energy is given by the cosmological constant. The dark energy1epresents a puzzle and perhaps the biggest problem in modern cosmology.In fact, a direct detection of a cosmological constant is still lacking. Inthe last decade, many attempts have been made (see [4]-[24] and referencestherein) to obtain physically sensible inhomogeneous models. Some authors(see for example [16, 17, 20, 21, 22, 23, 24]) showed that inhomogeneities cangenerate an accelerating universe by using Lemaˆıtre-Tolman-Bondi (LTB)metrics (see [25, 26, 27]), but several conditions must be imposed (see [4])in order to build regular physically viable models. In particular, in [21, 22]it is shown that LTB metrics can mimic the distance-redshift relation of theFRLW models at least at the third order in a series expansion with respectto the redshift near the center where the observer is located. More gen-erally, in the LTB solutions, apparent acceleration in the redshift-distancerelation seen by a central observer can be shown to coexist with a volumeaverage deceleration on a spacelike hypersurface (see [28]). The assumptionof spherical symmetry is (obviously) not in agreement with the Copernicanprinciple. In any case, a spherical symmetry can be justified as the outcomeof a smoothing out with respect to the angles: the metric so obtained be-comes spherical.An accelerating universe can also be built by averaging inhomogeneities (see[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]) by means of the techniques depictedin [8, 10, 11]. The approaches dealing by averaging on spatial domains aredifferent with respect to the ones dealing with exact spherical solutions.First of all, averaged cosmologies retain the Copernican principle, althoughin a generalized statistical sense. Further, the exact spherically symmetricmodels try to describe apparent acceleration by means of a large amountof inhomogeneities. Conversely, the ”Copernican” cosmologist introduce in-homogeneities without special symmetries and then try to understand themodifications to the average evolution from backreaction. For a review ofinhomogeneous cosmological models see [29, 30]. In particular, we studythe idea developed in Wiltshire’s papers [13, 14, 15]. There, the dimmingof the distant galaxies is interpreted as a ”mirage” effect by means of a”Copernican” statistical model. This effect is due to the different rate ofclocks located in averaged not expanding galaxies, where the metric is spa-tially flat, with respect to clocks in voids where the spatial curvature isnegative. With a negative spatial curvature can be associated a positivequasilocal energy. This gravitational energy is non-local, according to thestrong equivalence principle. In this picture the universe is composed by acosmic web of regions evolving asymptotically like an Einstein-de Sitter uni-verse (our local universe) ”matched” with local voids evolving like a Milneuniverse. The matching conditions are imposed with a uniform Hubble flow2y equating the radial null sections of the cosmic web. Although this match-ing is reasonable, it cannot be enough to define a geometry. In this context,inspired by Wiltshire’s idea, we introduce clock effects, but with an exactspherically symmetric model and without introducing backreaction. In thisway we can build a geometry with a non-uniform time flow by imposing theusual matching conditions required by general relativity, which are missingin [13, 14, 15]. To mimic Wiltshire’s idea, our model is built with a centralzone up to a small redshift z ( < /h M pc ) made of a flat Friedmannmetric, where the observer is located. In practice, the inhomogeneities canbe taken into account by glueing together different homogeneous portionsof universe (see for example [24]). It should be noticed that cosmologicalmodels with only two metrics can be exhaustively found in [31]. By meansof a thick shell of anisotropic matter, the flat zone can be smoothly matchedwith an hyperbolic Friedmann metric up to the zone where dimming galax-ies are observed [32]. Finally, we stress that the hyperbolic metric can besmoothly matched to a hyperbolic LTB solution on a comoving boundarysurface, according to WMAP data [33]. Actually, we could choose the flatFriedmann one (without cosmological constant!), but WMAP predicts a flatmetric only at early times, while at later epochs a negatively curved metricis more appropriate. Our solution is exact and no approximations have beenmade, thus providing a general framework to analyze the effects of a possiblenon-uniform time flow. In this context, we show that, after a formal Taylorexpansion of the distance-redshift relation near the center at z = 0, a neg-ative deceleration parameter is only compatible with a non-uniform Hubbleflow within a non-uniform time flow. The physical content of the thick shellis studied together with the energy conditions. Finally, it is conjectured thata non-vanishing heat flow term in the energy-momentum tensor of the thickshell can explain the possibly non-uniform time flow.In section 2 we introduce the metrics of our model. In section 3 the junctionconditions are discussed. In section 4 the matter content of the thick shellis studied. In section 5 the case of a uniform time flow is analyzed, whilein section 6 the case of a uniform Hubble flow is presented. Section 7 isdevoted to the study of the general case. Finally, section 8 collects somefinal remarks and conclusions. All the astrophysical observations agree with the assumption that our ”near”local universe is rather inhomogeneous. Nevertheless, at sufficiently large3cales ( ∼ /h M pc ) the inhomogeneities can be averaged to obtain a sta-tistically homogeneous universe. In any case, from the perturbations of theprimordial inflation, a model with an overdense density surrounded by voidsseems to be the most probable. Hence, we can build an inhomogeneous uni-verse by glueing together different homogeneous volumes. As a consequence,the central region, with the observer located at the center, is provided by aflat Friedmann metric ds B = − dt B + a B ( t B )( dη B + η B d Ω ) ,a B ( t B ) = a Bi (cid:18) t B t Bi (cid:19) , (1)where in (1) we have assumed a dust model and a Bi , t Bi are initial values.With (1), the Hubble flow H B is given by H B = 1 a B da B dt B = 23 t B . (2)The zone where the dimming galaxies are observed ( z ≤ .
7) is modeledwith an hyperbolic Friedmann metric with negative spatial curvature and acommon centre with (1). In appropriate coordinates, the metric can be putin the form ds F = − dt + a ( t ) [ dη F + sinh η F d Ω ] , (3) H i t = Ω i − Ω i ) (sinh ξ − ξ ) ,a ( t ) = a i Ω i − Ω i ) (cosh ξ − ,a ( t ) H (1 − Ω) = 1 , where Ω i is an initial density parameter, H i an initial Hubble constant and a i an initial expansion factor to be specified. An observer in the portion ofuniverse given by (3) measures the observables by means of the comovingtime t . With respect to this time, an observer in (3) measures an Hubbleflow with a time dependent Hubble constant H given by H = 1 a dadt = 2 H i sinh ξ Ω i (cosh ξ − (cid:0) − Ω i (cid:1) . (4)The only way to smoothly match the metrics (1) and (3) is by means of athick shell living in the region z ∈ [ z , z ]. Without loss of generality, we can4hoose the metric of the thick shell mimic the expression of a LTB metric ds thick = − e G ( τ,η ) dτ + R ,η ( τ, η ) f ( η ) dη + R ( τ, η ) d Ω , (5)where e G ( τ,η ) denotes the lapse function. Furthermore, to achieve agreementwith WMAP data [33], the universe beyond the dimming zone could bemodeled with an hyperbolic LTB spacetime (see [25, 26, 27]) ds = − d ˜ τ + ˜ R , ˜ η (˜ τ , ˜ η )˜ f (˜ η ) d ˜ η + ˜ R (˜ τ , ˜ η ) d Ω , ˜ f (˜ η ) > . (6)In fact, WMAP forces us to conclude that at early epochs the LTB metricmust approach a flat one and, as a result, we must impose the condition thatthe density parameter Ω m approaches unity at early times (recombinationera), i.e. Ω m (˜ τ rec ) →
1. According to [34], at later epochs the metric couldas well be taken to have a negative spatial curvature.
First of all, it should be noticed that the matching of the Friedmann metricsof this paper can be obtained only by taking a thick shell. We perform thematching along comoving surfaces (the boundary of the thick shell) givenby η B = η B (1) , η = η (1) , η = η (2) , η F = η F (2) , (7)where the subscripts (1) − (2) denote the boundaries of the shell. For thethick shell, the continuity of the first and the second fundamental form[35, 36] leads to (cid:18) dt B dτ (cid:19) (1) = e G ( τ,η (1))2 , (cid:18) dtdτ (cid:19) (2) = e G ( τ,η (2))2 , (8) R ( τ, η (1) ) = η B (1) a B ( t B ) ,R ( τ, η (2) ) = a ( t ) sinh η F (2) , (9) f ( η (1) ) = 1 , f ( η (2) ) = cosh η F (2) , (10) G ,η ( τ, η (1) ) = G ,η ( τ, η (2) ) = 0 . (11)Together with equations (8)-(11), we have the ”gauge” condition dtdt B = J ( ξ ) , H ( ξ ) = α ( ξ ) H B ( t B ) . (12)5he function J ( α ( ξ )) depends upon the chosen function α . With condition(11), the heat flow vanishes at the boundaries of the thick shell (see thenext section). To integrate the system (8)-(11), we can fix an expression for G satisfying equation (11) and the relations t = t ( τ, η ) , t B = t B ( τ, η ) thatsatisfy equation (8). In this way, thanks to the equations (9), the behaviourof R ( τ, η ) is fixed at the boundaries (1) − (2), and, as a result, we have thefreedom to choose R ( τ, η ) inside the shell and so also the function f withconditions (10).For a smooth matching between (3) and (6) on a comoving boundary surface η F = η F (3) , ˜ η = ˜ η (3) we have˜ τ = t, (13)˜ R (˜ τ , ˜ η (3) ) = a ( t ) sinh η F (3) , (14)˜ f (˜ η (3) ) = cosh η F (3) , (15)It should be noticed that there exists a relationship between the metrics (5)and (6). First of all, at the recombination era (early epochs) we have: τ rec ≃ ˜ τ rec , G ( τ rec , η ) ≃ , R ( τ rec , η ) ≃ ˜ R (˜ τ rec , ˜ η ) . (16)Furthermore, thanks to the matching conditions (8) and (13), we have (re-member that η (2) is a constant) e G ( τ,η (2) ) dτ = g ( τ ) dτ = d ˜ τ , (17)Finally, without loss of generality, we could also set η = ˜ η in (5) and (6).Since the matching conditions have been discussed, we can give the formalexpressions for the angular distance d A and the distance luminosity d L where d L = d A (1 + z ) (see [37, 38, 39]). In fact, thanks to conditions (8)-(11) and(13)-(15), we obtain (see [24]) d A = a B ( t B ) η B , z ≤ z , (18) d A = R ( τ, η ) , z ∈ [ z , z ] ,d A = a ( t ) sinh η F , z ∈ [ z , z ] ,d A = ˜ R (˜ τ , ˜ η ) , z ≥ z , where, z ( > .
7) represents the ”starting point” of the LTB metric. Forour purposes, we are interested in the patch of universe where the dimminggalaxies are observed, i.e. the third of equations (18).6
Energy-momentum tensor for the thick shell
In this section we study the metric (5). The most general energy-momentumtensor T ab compatible with it is T ab = EV a V b + P ⊥ [ W a W b + L a L b ] ++ P η S a S b + K [ V a S b + S a V b ] , (19)with V a = h − e G , , , i , (20) W a = [0 , , R, ,L a = [0 , , , R sin θ ] ,S a = (cid:20) , R ,η f , , (cid:21) , (21)where E is the energy-density, P η the radial pressure, P ⊥ the tangentialpressure, K being the ”heat flow term” or radial energy flux. In particular,Einstein’s equations for K give K = − f R ,τ G ,η RR ,η e G . (22)The regularity conditions require that ( R ,η , R ,τ , R, f ) = 0. Hence, fromequation (22), it follows that K = 0 if and only if G ,η = 0. Therefore, a nontrivial lapse function is only compatible with a non-vanishing energy flux.If we take the most general expression for a spherically symmetric metricthat is ds = − e G ( τ,η ) dτ + A ,η ( τ, η ) f ( η ) dη + B ( τ, η ) d Ω , (23)the energy flux for (23) vanishes if and only if G ,η = 2(ln B ,τ ) ,η − A ,τ ) ,η B ,η B ,τ . (24)To the best of our knowledge, the only non-static metric with a non-barotropicequation of state satisfying equation (24) is the Stephani metric [40]. Re-member that a time dependent spherical matter cannot have a barotropicequation of state ( E = E ( P )) within a finite radius [41]. It is a simple mat-ter to see that the spherically symmetric Stephani space-time cannot satisfythe matching conditions (8)-(11). As a result, a link between the heat flow7nd a non-trivial lapse function can be conjectured, at least for sphericallysymmetric space-times. Should this argument be correct, we would have apossible physical mechanism to generate clock effects.In what follows, starting from conditions (8)-(11), we analyze the possibilityto build a physically reasonable anisotropic thick shell. For simplicity, westudy the case G = 0 ( t = t B ). Hence, the matching conditions (9)-(10) canbe fulfilled by taking, for example R = η t Y + a ( t ) η ˜ Y , (25) f ( η ) = Y + q η ˜ Y , (26) Y = 1 − ( η − η ) ( η − η ) , ˜ Y = Y = 1 − ( η − η ) ( η − η ) , where, without loss of generality, we have taken (only for this section!): a Bi ( t rec ) = 1 , t Bi = t rec = 1 , η = sinh η F , η B = η, (27)and η , η denote the location of the comoving shell. Furthermore, we haveset in (27) the scale of times to be the unity at the recombination era.Regularity conditions impose that R > , R ,η >
0. By performing, afterfixing the time, a Taylor expansion near the boundaries of the thick shell,we have E ( t, η ≃ η ) = 43 t + o (1) , (28) E ( t, η ≃ η ) = 3 a ,t − a + o (1) ≃ t + o (1) ,P η ( t, η ≃ η ) = s ( η − η ) + o (1) ,P η ( t, η ≃ η ) = 2 η ( η − η ) a + o (1) P ⊥ ( t, η ≃ η ) = sη ( η − η ) + o (1) ,P ⊥ ( t, η ≃ η ) = − a ,t,t a + a ,t a ≃ − t + o (1) s = 4 t − η ( η − η ) [ t (18 q η −
9) + 4 aη −− η t a ,t,t − η ta ,t ] , ≃ on the right hand side means that the expression isevaluated for t >>
1. From the equations (28), although the tangentialpressure can have a negative value near η = η , the energy conditions follownear the boundaries of the thick shell. This means that physically reason-able thick shells can be built with our matching conditions. It should bestressed again that the junction conditions (8)-(11) only fix the behaviouron the boundaries (1) − (2).In the next three sections we apply the technology developed above to phys-ically interesting situations. G = 0 First of all, we must integrate along the past null cone inward. Generally,we have there an equation given by dT = − A ( T, η ) f ( η ) dη, (29)where t B = τ = t = T .Following C´el´erier (see [21, 22]), we obtain for the redshift zdTdz = − A ( T ( η ) , η )(1 + z ) A ,T ( T ( η ) , η ) . (30)We are interested in the determination of the distance-redshift formula forthe dimming galaxies. Therefore, from the third of equations (18), we get d L = (1 + z ) a ( t ) sinh η F . (31)Obviously, we must solve equation (30) in the three regions (hyperbolicFriedmann, thick shell and flat Friedmann) up to an observer located at z = 0.For z ∈ [ z , z ], ( A ( T ( η ) , η ) = a ( T ) , f ( η ) = 1) we have1 + z z = a ( T ) a ( T ) . (32)For z ∈ [ z , z ], we have ( A ( T ( η ) , η ) = R ,η )ln (cid:18) z z (cid:19) = − Z T T R ,η,T R ,η dT = γ ( T , T ) . (33)9rom (33) we get 1 + z z = e γ , (34)where γ > e γ is of the order of unity( ≥ t = t B = T = Ω H (1 − Ω ) (sinh ξ − ξ ) . (35)Finally, thanks to (35), for z ≤ z we obtain ( A ( T ( η ) , η ) = a B ( T ) , f ( η ) = 1)1 + z = a B ( T ) a B ( T ) = (cid:18) sinh ξ − ξ sinh ξ − ξ (cid:19) , (36)with the subscript ”0” denoting the actual time at z = 0. By multiplyingequations (32)-(36), we get1 + z = e γ a ( T ) a ( T ) a B ( T ) a B ( T ) . (37)For η F along the past null cone we have η F = ξ − ξ , ξ ≤ ξ . (38)As a result, since Ω( ξ ) = ξ , equation (37) becomes1 + z = 2 F (1 + z ) (1 − Ω )Ω (cosh ξ − , (39)where ”1” refers to the time T (or ξ ) and F = e γ (cosh ξ − ξ − . (40)Physical plausibility requires that F be of order of unity. Thanks to (3) and(39), the formula (31) becomes d L = Ω (1 + z )(1 + z ) H (1 − Ω ) F (1 − Ω )Ω sinh( ξ − ξ ) . (41)Equations (39) and (41) hold for z ≥ z and obviously the junction condi-tions only imply the continuity of d L and not its analyticity when crossing10he boundaries of the model. This obviously applies to any inhomogeneousmodel built by matching two or more metrics (as for example in [13, 15, 24]).Nevertheless, to make contact with the astrophysical data at intermediateredshifts, a central observer can formally expand expression (41) in a Taylorseries in the range z << , z − z << ≃ Ω , F ≃ z we formally obtain d L = zH obs + o ( z ) , (42) H obs = ǫ H (1 − Ω ) Ω F (1 − Ω )(1 + z )Ω P ,P = q F ǫ (1 − Ω )(1 + z ) ,ǫ = − Ω [ F (1 + z ) −
1] + F (1 + z ) . Therefore, in our inhomogeneous universe, after writing the correct matchingconditions, a central observer extrapolates an effective Hubble flow given by(42). It is worth noticing that, in the limit F = 1 , Ω → Ω , z = 0 , ǫ = 1, inwhich the full space-time is composed only with the hyperbolic Friedmannmetric, we have H obs → H , a correct result. Furthermore, the inequality H obs = H in a general inhomogeneous universe is compatible with the factthat in such space-times we have not a unique definition of an observedHubble flow (see [42]): a direct way is to infer its value by a formal Taylorexpansion (if this is possible) near the observer. The extrapolated centraldeceleration parameter q is given by q = − H obs d dz ( d L ( z = 0)) + 1 . (43)Note that, from equation (36), we could express Ω as a function of z , ξ ,although this is not necessary for our purposes. Equation (43), thanks to(41), gives q = Ω ǫ . (44)From equation (42) we must have ǫ > H obs >
0) and therefore q ≥ T and no ”formal” acceleration is perceived by the central observerby considering the distance-redshift function. At early times ( ξ ≃
0) we have q → and q → + asymptotically (for ξ → ∞ ).11 Uniform Hubble flow
The case of a uniform Hubble flow has been studied in [13, 15] in the contextof the Buchert equations with backreaction ([8]). In [13, 15] the overdensityevolves asymptotically as an Einstein-de Sitter space-time, while the under-density as a Milne universe. As a result, our model can be considered asrepresenting the far future limit ( t → ∞ ) of [13, 15], where backreaction isasymptotically vanishing. With the uniform Hubble gauge we have H = H B , α = 1 , J ( ξ ) = dtdt B = 32 (1 + cosh ξ )(2 + cosh ξ ) . (45)The calculations are similar to the ones of the last section. However, thecentral observer measures the redshift with respect to its proper time t B .As a result, along the past null cone we have dt B = − aJ ( ξ ) dη F → η F = ξ − ξ z z = J ( ξ ) J ( ξ ) a ( ξ ) a ( ξ ) . (46)Instead of the equation (33) we read1 + z z = e γ ≃ J ( ξ ) J ( ξ ) (1 + z )(1 + z ) , (47)(1 + z )(1 + z ) = − Z τ τ e G R ,η (cid:18) R ,η e G (cid:19) ,τ dτ, (48) z , z being the redshifts measured by a comoving observer with time τ . Theapproximation ( ≃ ) in (47) has been given as an example and is valid when ξ ≃ ξ . It does not enter in the calculations of this section. The expression(31) becomes d L = (1 + z ) a ( t ) J ( ξ ) sinh( ξ − ξ ) . (49)Concerning the relation between t B and t we get t B = Ω H (1 − Ω ) (cosh ξ − (1 + cosh ξ ) . (50)Instead of equation (36) we have1 + z = (cid:18) t B t B (cid:19) . (51)12fter defining e γ = F (cosh ξ − ξ −
1) (2 + cosh ξ )(2 + cosh ξ ) (1 + cosh ξ )(1 + cosh ξ ) , (52)and with the same technique of the last section, we obtaincosh ξ = −
12 + Q z ) ++ q z ) + 2 Q (1 + z ) + Q z ) , (53) Q = F Ω (1 + z )(1 − Ω )(2 + Ω ) , (54) d L = (2 + cosh ξ )(cosh ξ −
1) sinh( ξ − ξ )3 H (1 − Ω ) (1 + cosh ξ ) (1 + z ) . (55)Performing a formal Taylor expansion of (55) at z = 0, we again get aneffective observed H obs , and by means of equation (43) we can obtain thecentral deceleration parameter. In any case, we always have q = at earlytimes. Furthermore, the extrapolated parameter q remains positive andapproaches zero as follows: q ( ξ → ∞ ) → Q . As a result, in presence of auniform Hubble flow q goes to zero more rapidly than in the case G = 0 (seeequation (44)), albeit always from positive values. In practice, we recoverthe results of [14], but imposing the correct matching conditions. In the general case H = α H B . Hence, the relation between t B and t becomes t B = α ( ξ ) Ω H (1 − Ω ) (cosh ξ − (1 + cosh ξ ) . (56)For the lapse factor J ( ξ ), we obtain J ( ξ ) = dtdt B = 32 β ( ξ ) , (57) α ,ξ (cosh ξ − ξ + α (2 + cosh ξ )(1 + cosh ξ ) = β − ( ξ ) . To explore the case with a non-uniform Hubble flow with G = 0 we can take β ( ξ ) = (1 + cosh ξ )( A cosh ξ + 3 − A ) , A > . (58)13he case with A = 1 has been studied in the section above. The calculationsare similar to the ones of the section 6 and after posing e γ = 1 + z z = F Ω (1 − Ω )Ω (1 − Ω ) (2 A − A Ω + 3Ω )(2 A − A Ω + 3Ω ) , (59)we get d L = B (1 + z )Ω H (1 − Ω ) sinh( ξ − ξ ) , (60)1 + z = B (1 + cosh ξ )( A cosh ξ + 3 − A )(cosh ξ − , (61) B = F (1 + z )(1 − Ω )(2 A − A Ω + 3Ω )Ω . (62)The expressions for H obs and q are rather cumbersome. However, somegeneral remarks can be done on the behaviour of the extrapolated centraldeceleration parameter q at different values of the parameter A . First ofall, ∀ A ∈ (0 , ∞ ), q ( ξ = 0) = . Furthermore, ∀ A ∈ [1 , ∞ ), the parameter q is always positive and asymptotically q ( ξ → ∞ ) → + . Conversely, ∀ A ∈ (0 , q becomes negative at some value of B , andafter an absolute minimum, reaches 0 − asymptotically but from negativevalues. As an example, for A = , q = 0 for B ≃ . B ≃
10 wehave q min ≃ − . A = , q = 0 for B ≃ . q min ≃ − . B ≃
8. For A = , q = 0 for B ≃ . q min ≃ − .
31 at B ≃ A = , q = 0 for B ≃ . q min ≃ − .
51 at B ≃
5. In anycase, ∀ A ∈ (0 ,
1) it is always possible to have, from the equation (62),reasonable values for F ( ≃
1, with z − z <<
1) and Ω compatible withan apparent acceleration at some later time calculated by means of theequation (36), provided that B > B . The inequality B > B imposes aconstraint on Ω ≃ Ω . As an example, for F = 1 . , z = 0 .
01 and A = we have an accelerated universe if and only if Ω ≃ Ω < .
24, while for A = and F (1 + z ) = 1 , z <<
1, we have Ω ≃ Ω < .
22. Further, for A = , F = 1 . , z = 0 .
01 we have formal acceleration when Ω ≃ Ω < . A ≤ a value for q compatible with theactual observations. Furthermore, note that the models with A ∈ (0 , α ( ξ ) < H B > H (this can be see by notingthat the equation (57) has, in the limit ξ >>
1, the tracker solution α = A ).As a result, the clock effects depicted in this paper can mimic a model witha large underdensity surrounded by an overdensity (see [24]). Concluding,14f the clock effects depicted in this paper (and in [13, 15]) there exist inthe real universe, the actual data at intermediate redshifts are in agreementwith a non-uniform Hubble flow. We built a model for the universe without dark energy, by means of anexact spherically symmetric solution taking into account the observed inho-mogeneous universe. The main purpose of this paper is to show how thenon-uniform time flow depicted in [13, 15] can be obtained within an exactsolution of Einstein’s equations by imposing the correct matching conditionsrequired by general relativity. In this sense, since the backreaction is absentin our model, our approach is different from Wiltshire’s, where the backre-action is analyzed in a statistical (”Copernican”) model within the Buchertformalism.The model is composed of three regions, a central flat Friedmann metric, anhyperbolic Friedmann zone and eventually a bulk LTB hyperbolic metric,according to WMAP. Within our exact solution , it is shown that, aftera ”formal” Taylor expansion of the distance-redshift relation near the ob-server and by imposing the correct matching conditions, a uniform Hubble”gauge” (present in [13, 15]) does not lead to an ”apparent” accelerationas extrapolated from the redshift-distance relation. In a purely sphericallysymmetric universe, such an acceleration can only be obtained with a non-uniform Hubble flow. Furthermore, in our model we have a parameter ( A in the paper) at our disposal that permits us to obtain a large amount of”apparent” acceleration which is consistent with the other parameters ofthe model (for example the thickness z − z << < θ > ) fi = 0 , ( < σ > ) fi = 0 , (63)( < θ > ) s = 3 H s , ( < σ > ) s = 0 , (64)( < θ > ) v > , ( < σ > ) v = 0 , (65)where, following the notation of [13], ” fi ” stands for ”finite-infinity” and” v ” for ”voids”, H s is the averaged Hubble parameter for the thick shell.Hence, as suggested by the matching conditions, a third scale between ” fi ”and ” v ” with a non-vanishing shear is introduced: this is the scale at whichvariations in the flux energy are appreciable.Finally, note that the dark energy appears in two ”phase transitions” for theuniverse: the formation of the big structures and the end of it. Hence, sincecosmic strings and superstrings in the context of the M-theory are supposedto have acted during the inflation epoch to give (in principle) observableeffects a later times, a link between them and the thick structures depictedin this paper can be suggested (see [43]). References [1] Perlmutter S et al
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