Accelerated Expansion of the Universe in the Model with Non-Uniform Pressure
Elena Kopteva, Irina Bormotova, Maria Churilova, Zdenek Stuchlik
aa r X i v : . [ g r- q c ] J a n Draft version January 22, 2020
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Accelerated Expansion of the Universe in the Model with Non-Uniform Pressure
Elena Kopteva, Irina Bormotova,
1, 2
Mariia Churilova, and Zdenek Stuchlik Institute of Physics and Research Centre of Theoretical Physics and Astrophysics,Faculty of Philosophy and Science in Opava, Silesian University in Opava,746 01 Opava, Czech Republic Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research141980 Dubna, Russia
ABSTRACTWe present the particular case of the Stephani solution for shear-free perfect fluid withuniform energy density and non-uniform pressure. Such models appeared as possiblealternative to the consideration of the exotic forms of matter like dark energy that wouldcause the acceleration of the universe expansion. These models are characterised bythe spatial curvature depending on time. We analyze the properties of the cosmologicalmodel obtained on the basis of exact solution of the Stephani class and adopt it tothe recent observational data. The spatial geometry of the model is investigated. Weshow that despite possible singularities, the model can describe the current stage of theuniverse evolution.
Keywords:
Stephani solution — inhomogeneous cosmology — cosmological acceleration INTRODUCTIONAlthough ΛCDM model, based on the Friedmann solution, is most popular for explanationof the observed cosmological acceleration, it faces some fundamental problems, like the prob-lem of “dark energy” and the coincidence problem (Weinberg 1989). Thus different attemptsto find a possible alternative in this regard arise. The consideration of the inhomogeneous cos-mological models is among them. The Stephani solution (Stephani 1967) has drawn atten-tion of cosmologists so long as it allows to build the model of the universe with acceleratedexpansion (Dabrowski & Hendry 1998; Stelmach & Szydlowski 2004; Stelmach & Jakacka 2008;Balcerzak, Dabrowski & Denkiewicz 2015; Ong et al. 2018). This is a non-static solution for theexpanding perfect fluid with zero shear and rotation, which contains the known Friedmann solu-tion as a particular case. The Stephani solution was discussed much in the literature (see e.g.(Krasinski 1983; Sussman 1987, 1988a,b, 2000; Dabrowski 1993; Korkina, Kopteva & Egurnov2016; Ong et al. 2018) and references therein). Originally it has no symmetries, but the special caseof spherical symmetry is of particular interest in cosmology. It is known that spatial sections of the
Corresponding author: Elena [email protected]
Kopteva E., Bormotova I., Churilova M., Stuchlik Z.
Stephani space-time in this case have the same geometry as if they were subspaces of the Friedmannsolution. Therefore these models may have an intuitively clear interpretation being in close connec-tion with the Friedmann ones. The spatial curvature in the Stephani cosmological models is arbitraryfunction of time. This very property allows to obtain the appropriate behaviour of the cosmologicalacceleration.According to our knowledge only a few of the cosmological models based on the Stephani solutionwere studied concerning their correspondence to the observational data.In present work, we consider a rather general case of this solution restricted by the choice ofthe energy density in the same form as for the Friedmann dust. We analyze the properties of theresulting cosmological model and its applicability to the description of the current stage of theuniverse evolution.The paper is organized as follows. In Sec. 2 we introduce the special case of the Stephani solutionfor our model and fit it to the current values of the cosmological parameters. The geometry of thespatial part of the obtained solution is explored in Sec. 3. In Sec. 4 we investigate the dynamics ofthe universe evolution in our model, build the R-T-regions for the resulting space-time and discusssingularities of the model. In Sec. 5 we consider the cosmological implications of our model. Theconclusions are presented in Sec. 6. SPECIAL CASE OF THE STEPHANI SOLUTIONIt is known that for the perfect fluid described by 4-velocity vector field u α there exist four mainkinematic characteristics (see e.g. (Stephani et al. 2003)), which are the acceleration ˙ u α , the volumeexpansion Θ, the rotation ω αβ and the shear σ αβ . These parameters are defined as follows:Θ = u α ; α , (1) ω αβ = u [ α ; β ] + ˙ u [ α u β ] , (2) σ αβ = u ( α ; β ) + ˙ u ( α u β ) − Θ h αβ / , (3) h αβ = g αβ + u α u β , (4)where h αβ is the projection tensor, greek indexes run from 0 to 3, square/round brackets standardlymean antisymmetryzation/symmetryzation by corresponded indexes. Here and further in the paperdot means the partial derivative with respect to time and the geometric units are used where c ≡ πG ≡ u = 1 / √ g , u i = 0 , i = 1 , , s = D d t − R ( t ) V ( t, χ ) (cid:0) d χ + χ d σ (cid:1) , (5)where d σ is the usual metric on the unit 2-sphere and V = 1 + 14 k ( t ) χ (6) D = F ( t ) (cid:18) VR (cid:19) . (cid:18) VR (cid:19) − . (7) ccelerated Expansion of the Universe with Non-Uniform Pressure ε ( t ) = 3 C ( t ) . (8)The function k ( t ) is defined by the expression k ( t ) = (cid:20) C ( t ) − F ( t ) (cid:21) R ( t ) , (9)and corresponds to the curvature parameter, which in the Friedmann solution is constant normalizableto 0 , ±
1. Here C ( t ), F ( t ), R ( t ) are arbitrary functions.By use of the following relations r ( t, χ ) = R ( t ) V ( t, χ ) , (10) ψ ( t ) = 1 F ( t ) , (11) a ( t ) = R ( t ) , (12) ζ ( t ) = k ( t ) R ( t ) . (13)the metric (5) may be rewritten in the formd s = ˙ r r ψ d t − r (cid:0) d χ + χ d σ (cid:1) , (14)which we shall use in further consideration as more convenient for our purposes. Here r ( t, χ ) = a ( t )1 + ζ ( t ) a ( t ) χ , (15) ζ ( t ) = ε ( t ) − ψ ( t ) , (16)where ζ ( t ), ψ ( t ) and ε ( t ) are arbitrary functions, a ( t ) is the function related to the scale factor ofthe Friedmann solution.According to the Einstein equations the pressure is defined by the expression p ( t, χ ) = − ε ( t ) − ˙ ε ( t ) r ( t, χ )3 ˙ r ( t, χ ) . (17)It is clear that the pressure is non-uniform in the Stephani solution.The function ζ ( t ) is the spatial curvature. It is easy to verify that the scalar curvature R of thespatial sections t = const of the metric (14) is R = 6 ζ ( t ) and the Kretschmann invariant in this3-dimensional case is K = R αβµν R αβµν = 12 ζ ( t ). As far as t = const, one obtains the subspaceof everywhere constant curvature. It is from here that the identity for the spatial geometries of theStephani and the Friedmann solutions follows. Kopteva E., Bormotova I., Churilova M., Stuchlik Z.
Further in this article we sometimes omit the variable of the function provided it will not cause theconfusion.As it can be deduced from the following consideration, the equation (16) is a generalization of theknown Friedmann equation ˙ a ( t ) a ( t ) + ka ( t ) = 13 ε ( t ) , (18)where k = 0 , ±
1, the factor 1 / ε ( t ).The function ψ ( t ) turns out to be connected with the Hubble parameter H . In the case of inho-mogeneous cosmological model, the definition of the Hubble parameter should be generalized as itdepends both on the time and spatial position. We shall use the generalization introduced by (Ellis2009): H = 1 l d l d τ = 13 Θ , (19)where l is some “representative” length that corresponds to the scale factor a ( t ) in the Friedmannmodels, and τ is the proper time given in the standard way byd τ = √ g d t. (20)Due to the definition of the comoving coordinates, we obtain for the metric (14) from the expression(1) Θ = 12 √ g g ∂∂t g = 32 rψ ˙ r rr ˙ r = 3 ψ. (21)From (19) and (21)it follows that H = ψ. (22)If ζ = 0, then (14) is a parabolic type of the Friedmann solution and it follows from the eq. (16) thatthe function ψ attains the sense of the critical energy density ε cr = 3 H which is also in accordancewith (22).For the parabolic Friedmann solution we have r = a ( t ) , g = ˙ r r ψ = 1 ⇒ g = ˙ a a ψ = 1 , (23)then for the function ψ ( t ) it follows that ψ ( t ) = ˙ aa . (24)Thus for the appropriate transition to the Friedmann limit in the metric (14) one should choose ψ ( t )in the form (24). 2.1. The model of the universe with the accelerated expansion
We now define our model of the universe with the accelerated expansion based on the mentionedparticular case of the Stephani solution.We suppose the universe to be filled everywhere with the expanding shear-free perfect fluid withuniform energy density ε = ε ( t ) and non-uniform pressure p = p ( χ, t ). ccelerated Expansion of the Universe with Non-Uniform Pressure s = ˙ r a r ˙ a d t − r (cid:0) d x + d σ (cid:1) , (25)where r ( t, x ) = 2 a ( t ) e x ζ ( t ) a ( t ) e x . (26)As discussed above, the main equation that governs the evolution of the model reads˙ a a + ζ ( t ) = ε ( t ) . (27)From here and (18) it evidently follows that the Stephani models with ζ = ± /a , l = a (cid:2) cos χ + ζ a sin χ (cid:3) (cid:0) d χ + sin χ d σ (cid:1) , e x = tan χ , (28)d l = a (cid:2) cosh χ + ζ a sinh χ (cid:3) (cid:0) d χ + sinh χ d σ (cid:1) , e x = tanh χ , (29)d l = a h ζ a χ i (cid:0) d χ + χ d σ (cid:1) , e x = χ . (30)For our description we shall choose the case (30) implying the space-time metricd s = ˙ r a r ˙ a d t − r (cid:0) d χ + χ d σ (cid:1) , (31) r = a ζ a (cid:0) χ (cid:1) . (32)The energy density is chosen the same as for the Friedmann dust: ε = a a , (33) a = const = a ( t ), where t corresponds to the current moment of time (our time).We take the spatial curvature in the form ζ = −| β | a k a k +2 , (34)where k = const , β = const <
0, that means that the spatial curvature is negative everywhere inthe universe. Such expression for the spatial curvature is induced by the form of the Friedmannequation (18) which may contain the sum of energy densities of several non-interacting sources. In
Kopteva E., Bormotova I., Churilova M., Stuchlik Z. the Friedmann models the energy density for all known components of matter (including those withnegative pressure) is expressed in terms of scale factor raised to the correspondent power.The models known in the literature are mostly the particular cases of (34) with fixed valueof k ((Dabrowski 1995), (Dabrowski & Hendry 1998) Model I: k = −
1, Model II: k =1, (Stelmach & Jakacka 2008; Stelmach & Szydlowski 2004): k = −
1, (Ong et al. 2018),(Gregoris et al. 2019): k = − k is presented in (Sussman 2000;Hashemi et al. 2014) in the frame of investigation of the Stephani universes with physically mean-ingful equations of state of matter.We carry out our consideration without fixing k but figuring out the range of its values thatcorrespond to the right behavior of the universe acceleration.The pressure in the model according to (17), (33) and (34) reads p = a a (cid:0) χ (cid:1) | β | k (cid:16) aa (cid:17) k − (cid:0) χ (cid:1) | β | ( k + 1) (35)We now express some cosmological parameters in terms of χ and a ( t ), which will somehow parametrizethe time coordinate. In this part we restore the dimensions as far as we are going to put the numericparameters of the model ( a , β , k ) in accordance with the observational data.1. The Hubble parameter From (27) one has˙ aa = (cid:20) a a + | β | a k a k +2 (cid:21) , (36)and hence H = c r a a + | β | a k a k +2 . (37)2. Matter density parameter Ω m = εε cr = a c a H (38)3. The radius of the universe at present time r r = Z χ √− g d χ = Z χ r ( t , χ )d χ (39) r = Z χ a ( t )1 + ζ ( t ) a ( t ) (cid:0) χ (cid:1) d χ. (40)Taking ζ ( t ) from (34) one obtains r = 4 a | β | arctanh( χ | β | ) . (41)Using this relation it is possible to find the value of the coordinate χ corresponding to thecurrent size of the universe: r ( χ , t ) = r . ccelerated Expansion of the Universe with Non-Uniform Pressure
74. Deceleration parameter q . We take the general definition of the deceleration parameter accord-ing to (Ellis 2009) q = − l d l d τ H . (42)From (19), (22) and (24) we have d l d τ = l ˙ aa (43)differentiating both parts of (43) with respect to the time we obtaindd τ d l d τ = ˙ aa d l d τ + l √ g dd t (cid:18) ˙ aa (cid:19) = H + r ˙ a ˙ ra (cid:18) ¨ aa − H (cid:19) (44)Finally for the deceleration parameter there is q = − (cid:20) r ˙ a ˙ ra (cid:18) ¨ aa ˙ a − (cid:19)(cid:21) , (45)or in the explicit form due to (37): q = k | β | (cid:0) χ (cid:1) (cid:16) aa (cid:17) k − ( k + 1) | β | (cid:0) χ (cid:1) (cid:16) aa (cid:17) k − + k | β | (cid:18)(cid:16) aa (cid:17) k − + | β | (cid:19) . (46)It is clear that the expression (45) in the Friedmann limit ( r = a ) turns to the right form for thedeceleration parameter in Friedman models: q = − ¨ aa ˙ a .2.2. Estimation of the model constants with respect to the observational data
In this subsection we introduce the comparison of some observable parameters from previous sub-section with their values obtained within standard cosmological model.The current values of cosmological parameters obtained within ΛCDM model (or FLRW modelwith nonzero curvature) may be found in (Hinshaw, G. et al. 2013). We shall assume the followingnumbers: H = 2 × − s − , Ω m = 0 . , r ≈ , × m . (47)According to this data due to (37) and (38), the constants of our model ( a , β, χ ) related to thecurrent moment of time can be defined as follows: H = ca p | β | , (48)Ω m = c a H , (49) a = 1 . × m , (50) β = − . , (51) χ = 2 . . (52) Kopteva E., Bormotova I., Churilova M., Stuchlik Z.
Singularities of the model
It was also widely discussed (Sussman 1988b; Krasinski 1983; Dabrowski & Hendry 1998) thatthe Stephani models contain some special singularities that should be taken into account if oneintends to build a cosmological model. In our case the model contains three true singularities.1. The initial singularity: a ( t ) = 0 ⇒ r = 0, ε → ∞ , p → ∞ .2. The singularity arising from g : χ = 2 (cid:16) aa (cid:17) k p | β | . (53)3. The singularity arising from the expression for pressure p : χ = 2 (cid:16) aa (cid:17) k p (1 + k ) | β | . (54)In the case of k = − k < − Mass function and horizons of the model
Let us first briefly introduce the notion of R- and T-regions of the spherically symmetric space-time(Novikov 2001).The spherically symmetric metric written in general formd s = e ν ( t,x ) d t − e λ ( t,x ) d x − r ( t, x )d σ (55)can locally be brought to the viewd s = A (˜ t, ˜ x )d˜ t − B (˜ t, ˜ x )d˜ x − ˜ x d σ (56)by coordinate transformation preserving the spherical symmetry:˜ t = ˜ t ( t, x ) , ˜ x = ˜ x ( t, x ) . (57)At the vicinity of a taken point two main situations are possible. First one is the case when the worldline ˜ x = const, θ = const, ϕ = const is time-like. In this case ˜ x is the spatial coordinate, and thefollowing inequality holds for the general metric (55) e ν − λ > (cid:18) d x d t (cid:19) . (58)Here d x/ d t is found from the equations ˜ x = r ( t, x ) = const, regarding the invariance of g and g under the transformation (57). The points for which the inequality (58) is satisfied are calledR-points. They form the R-region of the space-time with usual properties of the world and observers. ccelerated Expansion of the Universe with Non-Uniform Pressure x = const, θ = const, ϕ = const is space-like. In this case ˜ x cannot be the spatial coordinate, thus in the metric (56), coordinates ˜ x and ˜ t “change” their roles (itis implied, that the functions A (˜ t, ˜ x ) and B (˜ t, ˜ x ) have the needed signs). In this case the followinginequality holds for the general metric (55) e ν − λ < (cid:18) d x d t (cid:19) . (59)The points for which the inequality (59) is satisfied are called T-points. They form the T-region ofessential instability where static observer is impossible.The strict equality e ν − λ = (cid:18) d x d t (cid:19) (60)defines the boundary between R- and T-regions of the space-time, known as horizon.Regarding the condition ˜ x = r ( t, x ) = const we rewrite (65) as follows e − ν ˙ r = e − λ r ′ . (61)This will be referred to as the horizon equation. The prime here means the partial derivative withrespect to the spatial coordinate x .The coordinate condition (61) may also be expressed in terms of the so-called mass function(Korkina & Kopteva 2012), which for the metric (55) reads m = r (cid:0) e − ν ˙ r − e − λ r ′ (cid:1) . (62)The horizon equation then transforms to m = r. (63)For the metric (25), regarding (32) and (36), the mass function takes the form m = a χ (cid:18) − (cid:16) a a ( t ) (cid:17) k (cid:0) χ (cid:1) | β | (cid:19) . (64)The horizon equation (63) then gives the following expressions for two branches of the horizon χ , = 2 | β | vuuut(cid:18) a ( t ) a (cid:19) k (cid:18) a ( t ) a (cid:19) k − + | β | ± (cid:18) a ( t ) a (cid:19) k − s(cid:18) a ( t ) a (cid:19) k − + | β | . (65) GEOMETRYIn this section we investigate the spatial geometry of the obtained solution. To build the spatialsections of the space-time with metric (31) we fix the time at present moment t = t = constthat yields a = a = const in the formulae. To make it possible to visualize the 3-dimensional0 Kopteva E., Bormotova I., Churilova M., Stuchlik Z.
Figure 1.
The spatial section t = t , θ = π/ O is the centre of the pseudo-sphere. Points P and P ′ are the north and the south poles, respectively. The point N is the stereographicprojection of the point M . The observer is situated at the point P . The red line is the line r = r indicatingthe current size of the universe. The green line is one branch of the horizon, corresponding to the sign “ − ”in (65). The purple line is the so-called Hubble sphere, that expands with speed of light. hypersurface we also fix θ = π/
2. Applying these conditions to (31) we obtain the intrinsic metric ofthe hypersurface of our interest in the following formd l = a (cid:16) − | β | χ (cid:17) (cid:0) d χ + χ d ϕ (cid:1) . (66)By use of a new coordinate ρ = √ | β | χ the metric (66) can be rewritten in more familiar wayd l = a | β | − ρ ) (cid:0) d ρ + ρ d ϕ (cid:1) . (67)This is a metric of the pseudo-sphere in terms of the stereographic projection coordinates (see e.g.(Dubrovin 1992)) accurate within the similarity transformation with constant factor a / | β | . Thisstereographic projection maps the upper half of the pseudo-sphere represented by the hyperboloidof revolution onto the open disk ρ = x + y < z = 0 as shown at Figure 1. Sucha disk equipped with normalized metric (67) (so that a / | β | = 1) refers to the Poincare model ofLobachevsky geometry. It is seen that the spatial sections of the interval (31) are the Lobachevskyspaces.Figure 1 demonstrates the form of the spatial hypersurface of the universe within our model asan instantaneous snapshot at present moment of time, corresponding to the line T = 1 at Fig. 2,3.To restore the 3-dimensional picture from Fig. 1 one should imagine that the circles of the sections z = const are in fact 2-spheres. ccelerated Expansion of the Universe with Non-Uniform Pressure UNIVERSE EVOLUTION IN THE MODELTo investigate the evolution of the universe in the obtained model we consider an observer situatedclose to the symmetry center χ = 0, who observes the dynamics of the infinite number of theconcentric spheres marked by successive values of χ . The velocity of the expansion of some sphere χ may be found as follows v ( t, χ ) = dd t Z χ r ( t, χ )d χ. (68)Using the results obtained in Sec. 2, we now build the universe expansion velocity profile found from(68) and the deceleration parameter given by (46).Further in our discussion we shall use the dimensionless function T ≡ a ( t ) a , (69)which will be treated as time parameter. The differentiation with respect to the time will be carriedout taking into account that according to (36)d T d t = 1 a r T + | β | T k . (70)Figure 2 shows the universe expansion velocity profile in the model in terms of dimensionless units,where the time parameter T is given by (69). The concrete values of the index k are chosen only forillustrative purposes, with decreasing of k the picture qualitatively remains the same. The point ofintersection of the lines T = 1 and r = r defines the coordinate χ that indicates the sphere of radius r corresponding to the edge of the universe. It is seen that at present time the boundary of theuniverse belongs to the region of non-stationarity and expands with the velocity exceeding the speedof light as it is in standard Friedmann model. The central observer always belongs to the R-regionof permitted observers. It is also seen that the universe does not reach the singularity given by (53)up to its present age, and the singularity cannot be observed according to the causality principle.Figure 3 shows the behaviour of the deceleration parameter of the model. This profile is not affectedby the singularity. For k < − q = 0. Hence onecould expect that after some time the acceleration of the universe expansion changes into deceleration.However, even from the velocity profile (Fig. 2) it is clear that there is no deceleration in the future.It may be verified by direct calculations using (68) that the function d v ( t, χ ) / d t changes its sign onlyonce, from negative to positive. Thus, we conclude that in our model, unlike the Friedmann models,there is no correlation between the signs of the deceleration parameter and the acceleration of theuniverse expansion. REDSHIFT-MAGNITUDE RELATION IN THE MODELThe advantage of the classical redshift-magnitude test is the sensitivity of the relation between theapparent magnitude and the redshift of the source to the cosmological model. In this section wecompare the observational results concerning the redshift-magnitude relation for Supernovae type Iawith the theoretical predictions of our model. In this regard we shall use the Hubble diagram ofdistance moduli and redshifts for HST-discovered SNe Ia in the gold and silver sets represented at(Riess et al. 2004).2
Kopteva E., Bormotova I., Churilova M., Stuchlik Z.
Figure 2.
The universe expansion velocity profile for k = − . k = − . χ = 0. The line T = 1 corresponds to thecurrent moment of time t = t . The line r = r is the line along which the radius of the universe equals toits current size found from the observations. The line v = 1 defines the Hubble sphere that expands withspeed of light. The green line shows the “visible” branch of the horizon. The second branch lies behind thesingularity. T-region of essential non-stationarity is situated between two branches of the horizon. R-regioncorresponding to our world, lies under the “visible” branch of the horizon. The redshift-magnitude relation for the inhomogeneous Stephani model was derived first and stud-ied in (Dabrowski 1995),(Dabrowski & Hendry 1998) for two special cases near the observer posi-tion. Further, it was developed numerically by (Stelmach & Jakacka 2008) for one of these cases,but higher redshifts.For our model we derive the redshift-magnitude relation in terms of distance modulus analyticallywithout any supposition about values of the redshift.The distance modulus within cosmological scales (Mpc) is defined by (see e.g. (Ellis 2009)) µ ( z ) = m ( z ) − M = 5 log [ d L ( z ) / Mpc] + 25 , (71)where z is the redshift, m ( z ) is the apparent bolometric magnitude of a standard candle whoseabsolute bolometric magnitude is M ; d L ( z ) is the luminosity distance to the source d L ( z ) = r c ( z )(1 + z ) . (72)This relation holds rather general and does not depend on metric choice. Here r c ( z ) is comovingdistance to the source by apparent size, which in our case is usual comoving radial distance given by(Celerier 2000) r c ( t, χ ) = r ( t, χ ) χ = a T χ − T − k | β | (cid:0) χ (cid:1) (73) ccelerated Expansion of the Universe with Non-Uniform Pressure Figure 3.
The profile of the deceleration parameter for k = − . k = − . T = 1 corresponds to the current moment of time. The green lines are two branches of the horizon.T-region is situated between the branches of the horizon. R-region is the region outlined by the lower branchof the horizon. The line q = 0 is the line where the deceleration parameter equals to zero. The general definition for the redshift in any cosmological model reads (Ellis 2009)1 + z = ( κ α u α ) emitter ( κ α u α ) observer , (74)where u α = d x α / d s is usual 4-velocity of the cosmological medium and κ α = d x α / d λ is a vectortangent to the correspondent null-geodesic with affine parameter λ , i.e. the solution of the geodesicequations for the photon. Indexes ’emitter’ and ’observer’ mean that the quantity should be calculatedat the correspondent position.Applying these definitions to the interval (31) we shall act according to the following plan:1. Solving the geodesic equations for the photon radial motion we obtain κ α .2. Taking into account the fact that in comoving system the only nonzero component of the 4-velocity is u = 1 / √ g , we find the expression for the redshift in terms of the time and spatialcoordinate.3. Using previous results, we compose the distance modulus µ as a function of T and χ , accordingto (71)-(73). Thus we obtain the two-parametric area µ − z with T and χ being the parameters.4. Then we use the condition that for any time t there exists only one possible coordinate χ suchthat the light being emitted from the point ( t, χ ) will be received by the observer at the point( t = t , χ = 0). This is expressed in the following equation Z t t p g ( t, χ )d t = Z χ r ( t, χ )d χ, (75)4 Kopteva E., Bormotova I., Churilova M., Stuchlik Z. which in our notations (69),(70) gives Z T a p g ( T, χ ) p T − + T − k | β | d T = Z χ r ( T, χ )d χ. (76)5. Then we plot the numerical solution of the equation (76) as the line T ( χ ) at the two-parametricdiagram µ ( T, χ ) − z ( T, χ ). As a result we obtain the theoretical prediction for the redshift-magnitude relation in our model and compare it with observational data from (Riess et al.2004).The geodesic equations for the interval (31) in case of the photon radial motion readd κ d λ = 12 g (cid:0) − κ κ ˙ g − ( κ ) g ′ + ( κ ) g ′ (cid:1) , (77)d κ d λ = 12 g (cid:0) − κ κ g ′ + ( κ ) ˙ g − ( κ ) ˙ g (cid:1) , (78) κ = √− g √ g κ . (79)The metric coefficients in the interval (31) have the following explicit form: g = (cid:16) − T − k (1 + k ) | β | (cid:0) χ (cid:1) (cid:17) (cid:16) − T − k | β | (cid:0) χ (cid:1) (cid:17) , (80) g = − ( a T ) (cid:16) − T − k | β | (cid:0) χ (cid:1) (cid:17) . (81)Putting κ from (79) into (77) and taking into account that κ = d χ/ d λ , we obtain the differentialequation with separable variables and integrate it with a result κ = 4 a k T k (cid:16) − T − k | β | (cid:0) χ (cid:1) (cid:17) − T − k (1 + k ) | β | (cid:0) χ (cid:1) χ p | β | T − k − χ p | β | T − k ! Tk/ √ | β | √ T − + T − k | β | , (82)and hence due to (79) κ = g κ = − a k T k χ p | β | T − k − χ p | β | T − k ! Tk/ √ | β | √ T − + T − k | β | . (83)In our model the observer occupies the position χ = 0 and receives the signal at present time T = 1.Thus we obtain for the redshift1 + z = κ u κ u | χ =0 ,T =1 = T k − T − k | β | (cid:0) χ (cid:1) − T − k (1 + k ) | β | (cid:0) χ (cid:1) χ p | β | T − k − χ p | β | T − k ! Tk/ √ | β | √ T − + T − k | β | . (84) ccelerated Expansion of the Universe with Non-Uniform Pressure Figure 4.
The redshift-magnitude relation in terms of distance modulus with overplotted data for HST-discovered SNe Ia in the gold and silver sets taken from (Riess et al. 2004). The blue curve is the best-fitobtained by these authors in frame of the standard ΛCDM model with H = 66 , m = 0 . Λ = 0 .
71. The red line is the best-fit found within the Stephani model with respect to the current timeparameter T = 1 .
13, curvature parameters: β = − . k = − .
01, the Hubble constant current value H = 65 . Now we have everything to plot the Hubble diagram for our model. The Fig. 4 shows the redshift-magnitude dependence in terms of the distance modulus (71) with overplotted data taken from(Riess et al. 2004). It is seen that even in this particular case without concretizing the form of thefunction a ( t ) the Stephani model can in principle give an adequate interpretation of the observationaldata. CONCLUSIONSIn this work, a particular case of the Stephani solution was investigated as possible model of theuniverse with accelerated expansion. The R-T-structure of the obtained space-time was built, and itwas shown that the central observer belongs to the R-region of permitted observers. In this modelthe boundary of the observable universe belongs to the T-region and expands with velocity exceedingthe speed of light, as it is in standard Friedmann model. The correlation between the signs of thedeceleration parameter and the acceleration of the universe expansion is absent in this model.It is shown that the spatial sections of the universe are the Lobachevsky spaces. It turned out thatthe form of spatial section taken at present moment of time does not depend on the power of a ( t ) incurvature function ζ ( t ).It was established that the theoretical prediction for the redshift-magnitude relation in our modelis in good accordance with type Ia Supernovae observational data.The obtained results serve as an evidence in favour of the possibility that our world in principlemay be described by such model up to its recent stage without any harm from existing singularities.6 Kopteva E., Bormotova I., Churilova M., Stuchlik Z.
Another advantage of this approach is that it allows to stay within the general relativity with noneed for modifications and introducing any exotic types of matter.ACKNOWLEDGMENTSThis paper is supported by the Grant of the Plenipotentiary Representative of the Czech Republicin JINR under Contract No. 208 from 02/04/2019. The authors acknowledge the Research Centre ofTheoretical Physics and Astrophysics of the Faculty of Philosophy and Science, Silesian University inOpava for support. Z. S. acknowledges the Albert Einstein Centre for Gravitation and Astrophysicssupported by the Czech Science Foundation Grant No. 14-37086G. I. B. acknowledges the SilesianUniversity in Opava grant SGS 12/2019. Authors cordially thank Maria Korkina for suggesting theproblem and valuable discussion. REFERENCES
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