Cosmography of Cardassian model
ppreprint
Cosmography of Cardassian model
Yu.L. Bolotin, M.I.Konchatnyi, O.A. Lemets and L.G. Zazunov
A.I.Akhiezer Institute for Theoretical Physics, National Science Center ”Kharkov Instituteof Physics and Technology”,Akademicheskaya Str. 1, 61108 Kharkov, UkraineE-mail: [email protected]
Abstract.
The parameters of any model that satisfies the cosmological principle (the universeis homogeneous and isotropic on large scale), can be expressed through cosmographic param-eters. In this paper, we perform this procedure for the Cardassian model. We demonstrate anumber of advantages of the approach used before traditional methods.
Keywords: cosmographic parameters, Cardassian model Corresponding author. a r X i v : . [ g r- q c ] J u l ontents According to the current cosmological paradigm, we live in a flat, rapidly expanding Universe[1–4]. However, the physical origin of cosmic acceleration is still the greatest mystery toscience [5]. The Universe filled with ”ordinary” components (matter and radiation) shouldeventually slow down its expansion. The search for a solution to this problem is conductedin two directions, based on the following assumptions:1. Up to 75% of the energy density of the Universe exists in the form of an unknownsubstance (commonly called dark energy) with a large negative pressure which ensuresthe accelerated expansion.2. The general relativity theory (GR) needs to be revised on cosmological scales.The standard cosmological model (SCM) [6] represents an extremely successful imple-mentation of the first direction. Effectiveness of the SCM was so high that for its adequatedescription even the special term ”cosmic concordance” was introduced [7]. A fairly sim-ple model with a small number of parameters makes it possible to describe giant arrays ofobservational data at the level of 5% [8–10].However, unlike fundamental theories, physical models reflect only our current under-standing of a given process or phenomenon for which they were created. Gradually growinginternal problems and inevitable contradictions with observations [11–14] stimulate the searchfor more and more new models. In particular, it is tempting to explain the accelerated ex-pansion of the universe without attracting exotic dark components. Of course, this can bedone only by modifying the equations of general relativity. In other words, any ”innovation”activity in cosmology can be interpreted in terms of modifying either the left or the right sideof Einstein’s equations. SCM achieves agreement with cosmological observations by includingdark energy and dark matter in the energy-momentum tensor. The focus will be on the so-called Cardassian model (CM) [15, 16], which allows us to describe the accelerated expansionof a spatially flat universe without involving dark energy. Accelerated expansion is achievedby modifying the Friedmann equation H = ρ → H = g ( ρ ) , where the density ρ includesonly ”ordinary” components: matter and radiation. The function g ( ρ ) is chosen in such a waythat in the early Universe (for z (cid:29) ) it reproduces the standard Friedmann equation, whilefor z (cid:29) it generates the accelerated expansion of the Universe. In the simplest version, theCM is characterized by only two parameters. Our goal is to construct a cosmography of theCM. – 1 –he cosmography [17–22] represents an approach entirely based on the cosmologicalprinciple, stating that the Universe is homogeneous and isotropic on scales larger than ahundred megaparsecs. It allows us to select from whole possible variety of models describingthe Universe a narrow set of homogeneous and isotropic models. The cosmological principleenables us to build the metrics and make first steps towards interpretation of the cosmo-logical observations. The cosmography is just the kinematics of cosmological expansion. Inorder to build the key characteristic – time dependence of the scale factor a ( t ) – one needsto take equations of motion (the Einstein’s equations) and make an assumption about ma-terial content of the Universe, which allows constructing the energy-momentum tensor. Thecosmography is efficient because it allows testing any cosmological model which does not con-tradict the cosmological principle. Modifications of General Relativity or introduction of newcomponents (such as DM and DE) evidently change the dependence a ( t ) but do not affectrelations between the kinematic characteristics.To accomplish this goal, one must:1. Express the model parameters through cosmographic parameters.2. Find admissible (consistent with observational data) intervals of variation of these pa-rameters.3. Analyze the relationship to the results obtained within other cosmological models.We emphasize that all our results presented below are exact, being derived from identicaltransformations. As an alternative explanation for the observed accelerated expansion of the universe Freeseand Lewis [15] proposed a modified version of the first Friedmann equation H = g ( ρ m ) , (2.1)where the energy density ρ m of a flat universe includes only nonrelativistic matter (bothbaryon and dark) and radiation, but does not contain dark energy. In what follows, werestrict to the simplest version of the CM, which uses the additional power law on the right-hand side of the Friedmann equation H = Aρ m + Bρ nm . (2.2)In the standard FLRW cosmology coefficient B = 0 . Therefore, we must choose A = πG .Suppose that the universe is filled only with non-relativistic matter. In this case, withthe dominance of the second term (high densities, late Universe): H ∝ ρ n/ m ∝ a − n/ , ˙ a ∝ a − n/ , a ∝ t / n . Consequently, accelerated expansion can be realized for n < / . Atthe upper boundary for n = 2 / we have a ∝ t and ¨ a = 0 ; for n = 1 / we have a ∝ t (the acceleration is constant). For n > / the acceleration is diminishing in time, whilefor n < / the acceleration is increasing. It is interesting to note that if n = 2 / we have H ∝ a − : in a flat Universe term similar to a curvature term is generated by matter.Let’s represent the energy density of the CM in the form of a sum of densities of ordinarymatter ρ m and components with a density ρ x = ρ nm such that H ∝ ρ m + ρ x . As we saw– 2 –bove, in the case of dominance of the additional term in the Friedmann equation one has a ∝ t / n . Since a ∝ t w +1) we find that the parameter of the equation of state p x = w x ρ x is w x = n − . (2.3)This relation holds for an arbitrary one-component fluid with w x = const . In this case dρ x dz = 3 ρ x w x ( z )1 + z . (2.4)Taking into account that ρ x = ρ nm = (1+ z ) w x +1) n and substituting this into (2.4), we obtain w x = n − . (2.5)For < n < / − < w x < − / . (2.6)As you would expect, this values range of parameter w x generates a negative pressure, re-sponsible for the late time accelerated expansion of the universe. As one would expect, thisinterval of parameter values generates a negative pressure, responsible for the acceleratedexpansion of the universe at late times. In order to make more detailed description of kinematics of cosmological expansion it is usefulto consider the extended set of the parameters which includes the Hubble parameter H ( t ) ≡ ˙ aa ,and higher order time derivatives of the scale factor [17, 18] deceleration parameter q ≡ − C ,jerk parameter j ( t ) ≡ C snap parameter s ( t ) ≡ C etc, where C n ≡ a d n adt n H − n . (3.1)Note that all cosmological parameters, except for the deceleration parameter, are dimension-less. We give a number of useful relations for the deceleration parameter [22] q ( t ) = ddt (cid:0) H (cid:1) − q ( z ) = zH dHdz − q ( z ) = (1 + z ) H dH dz − q ( z ) = d ln H d ln(1+ z ) − q ( z ) = d ln Hdz (1 + z ) − . (3.2)Derivatives d n Hdz n can be expressed through the deceleration parameter and other cosmographicparameters as follows: dHdz = q z H ; d Hdz = j − q (1+ z ) H ; d Hdz = H (1+ z ) (cid:0) q + 3 q − qj − j − s (cid:1) ; d Hdz = H (1+ z ) (cid:0) − q − q − q + 32 qj + 25 q j + 7 qs + 12 j − j + 8 s + l (cid:1) . (3.3)– 3 –erivatives d ( i ) H dz ( i ) , i = 1 , , , in terms of cosmographic parameters have the form d ( H ) dz = H z (1 + q ); d ( H ) dz = H (1+ z ) (1 + 2 q + j ); d ( H ) dz = H (1+ z ) ( − qj − s ); d ( H ) dz = H (1+ z ) (4 qj + 3 qs + 3 q j − j + 4 s + l ) . (3.4)The current values of deceleration and jerk parameters in terms of N = − ln(1 + z ) are q = − H (cid:26) d ( H ) dN + H (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) N =0 ,j = H d ( H ) dN + H d ( H ) dN + 1 (cid:12)(cid:12)(cid:12)(cid:12) N =0 . (3.5)The derivatives of the Hubble parameter with respect to time can also be expressed in termsof cosmographic parameters ˙ H = − H (1 + q );¨ H = H ( j + 3 q + 2) ; ... H = H [ s − j − q ( q + 4) −
6] ; .... H = H [ l − s + 10 ( q + 2) j + 30( q + 2) q + 24] . (3.6)Let C n ≡ γ n a ( n ) aH n where a ( n ) is the -n-th derivative of the scale factor with respect to time, n ≥ and γ = − , γ n = 1 for n > . Then for the derivatives of the parameters C n withrespect to the redshift, we have relations (1 + z ) dC n dz = − γ n γ n +1 C n +1 + C n − nC n (1 + q ) (3.7)Using this relationship, one can express the higher cosmographic parameters through thelower ones and their derivatives: j = − q + (1 + z ) dqdz + 2 q (1 + q ) ,s = j − j (1 + q ) − (1 + z ) djdz ,l = s − s (1 + q ) − (1 + z ) dsdz ,m = l − l (1 + q ) − (1 + z ) dldz . (3.8)We then proceed with computing the deceleration parameter in the CM [23, 24]. For this, onenaturally has to go beyond the limits of cosmography and turn to the dynamics described bythe Friedmann equations. There is a complete analogy with classical mechanics. Kinematicsdescribes motion ignoring the forces that generate this motion. To calculate the acceleration,Newton’s laws are necessary, i.e. dynamics. Using q ( z ) = (1 + z ) H dH dz −
12 (1+ z ) E ( z ) dE ( z ) dz − ,E ≡ H H = Ω m (1 + z ) + (1 − Ω m ) (1 + z ) n (3.9)one finds q ( z ) = − (1 − n ) κ ( z )(1+ κ ( z )) ,κ ( z ) ≡ (cid:16) − Ω m Ω m (cid:17) (1 + z ) − − n ) (3.10)– 4 –or the current value of the deceleration parameter q = q ( z = 0) , we get q = 12 −
32 (1 − n ) (1 − Ω m ) (3.11)For n = 0 and Ω m = 0 the relation (3.11) correctly reproduces the value of the decelerationparameter q = − for the cosmological constant. Dunajski and Gibbons [25] proposed an original approach for testing cosmological modelswhich satisfy the cosmological principle. Implementation of the method relies on the followingsequence of steps:1. The first Friedmann equation is transformed to the ODE for the scale factor. To achievethis, the conservation equation for each component included in the model is used to findthe dependence of the energy density on the scale factor.2. The resulting equation is differentiated (with respect to cosmological time) as manytimes as the number of free parameters of the model.3. Temporal derivatives of the scale factor are expressed through cosmographic parameters.4. All free parameters of the model are expressed through cosmological parameters bysolving the resulting system of linear algebraic equationsThe above procedure can be made more universal and effective when choosing the systemof Friedmann equations for the Hubble parameter H and its time derivative ˙ H as a startingpoint. By differentiating the equation for ˙ H the required number of times (this number isdetermined by the number of free parameters of the model), we obtain a system of equationsthat includes higher time derivatives of the Hubble parameter ¨ H, ... H, .... H ...
These derivativesare directly related to the cosmological parameters by the relations (3.3). We implement thisprocedure for the CM.The evolution of CM is described by the system of equations H = Aρ m + Bρ nm (4.1) ˙ ρ m + 3 Hρ m = 0 (4.2)Differentiating Eq. (4.1) with respect to the cosmological time and using (4.2) we transformthe system (4.1)-(4.2) to the form H = Aρ m + Bρ nm (4.3) −
23 ˙ H = Aρ m + Bnρ nm (4.4)The solutions of this system are given by ρ m = − nH + ˙ HA (1 − n ) (4.5) B = H + ˙ Hρ n (1 − n ) (4.6)– 5 –o calculate the parameter n , we need an expression for ¨ H ,
29 ¨ HH = Aρ m + n Bρ nm (4.7)Substituting the above solutions (4.5) and (4.6) for ρ and B into this expression, we obtain
29 ¨ HH = − n + 23 ˙ HH (1 + n ) (4.8)Hence for the parameter n we find n = − (cid:16)
13 ¨ HH + ˙ HH (cid:17)
23 ˙ HH (4.9)Expressions (4.6) and (4.9) allow us to express the parameters of CM through cosmographicparameters. Using known expressions for time derivatives of the Hubble parameter in termsof cosmological parameters (3.6), we obtain Bρ nm H = 13 (1 − q )1 − n (4.10) n = 23 j − q − (4.11)Note that the parameters B and n are constants, which was explicitly used in deriving theabove relations. We verify that the solutions (4.10) and (4.11) agree with this condition.The requirement ˙ B = 0 is transformed into (4.8) and, consequently, it is consistent with theabove expression for n . The constancy of the parameters allows us to compute them for thevalues of the cosmological parameters at any time. Since the main body of information aboutcosmological parameters refers to the current time t , the relations (4.10), (4.11) can be putin the form Bρ n H = 13 (1 − q )1 − n (4.12) n = 23 j − q − (4.13)Otherwise, we must treat the time-dependent solution (4.5) for the density ρ m . It can berepresented in the form ρρ c = − n + (1 + q )1 − n , ρ c ≡ H πG (4.14)The current density in CM can be found by substitution q → q , H → H .It is interesting to note that the expression (4.11) for the parameter n coincides exactlywith the parameter s , one of the so-called statefinder parameters { r, s } [26, 27], r ≡ ˙ aaH , s = 23 r − q − (4.15)The coincidence is obvious, since r ≡ j . The reason for the coincidence can be explainedas follows. In any model with the scale factor a ∝ t α , there are the simple relations for thecosmographic parameters q and j q − − αα , j − − αα (4.16)– 6 –n CM a ∝ t n , from which it follows that s = n .Using the expression for ... H (3.6) and the solutions found (4.5), (4.6), (4.11), we obtainan equation relating cosmological parameters s + qj + (3 n + 2) j − q (3 n −
1) = 0 ,n = j − q − . (4.17)This fourth order ODE for the scale factor is equivalent to the Friedmann equation. For n = 0 Eq. (4.17) reproduces the known relation [25] between the cosmographic parameters in theLCDM, s + 2( q + j ) + qj = 0 (4.18)We now turn to the dimensionless form of the evolution equation for the scale factor (4.1).Coefficients of this equation must be expressed in terms of cosmological parameters. For thisusing ρ m = ρ a we represent Friedmann equation in the form ˙ a = Aρ a − + Bρ n a − n +2 . (4.19)Passing to the dimensionless time τ = H t and substituting in (4.19), ρ = − nH + ˙ H A (1 − n ) , B = H + ˙ H ρ n (1 − n ) (4.20)transform Friedmann equation (4.19) to the form dadτ = (cid:2) F ( q , j ) a − + Φ ( q , j ) a − n +2 (cid:3) / ,F = − n − (1+ q )1 − n , Φ = − (1+ q )1 − n ,n = j − q − . (4.21)This equation should be solved with constraints n < / (the condition ensuring the acceler-ated expansion of the universe) and n < (1 + q ) (the condition ensuring the positivity ofthe energy density). It is easy to see that the two conditions are consistent, since in the caseof the accelerated expansion one has q < .Fig. 1 illustrates the transition from delayed expansion to accelerated for late-timeUniverse in CM. The growth of the parameter j leads to a decrease of the parameter n : j = 0 . , n ≈ . ; j = 0 . , n ≈ , j = 0 . , n ≈ . . If j → (LDCM), as expected, n → . The considered here method of finding the parameters of cosmological models has some ad-vantages. Let us briefly dwell on them.1. Universality: the method is applicable to any braiding model that satisfies the cosmolog-ical principle. This procedure can be generalized to the case of models with interactionsbetween their components [28].2. Reliability: all the derived expressions are exact, as they follow from identical transfor-mations. – 7 – . 0 0 . 5 1 . 0 1 . 5 2 . 00 . 81 . 01 . 21 . 41 . 61 . 8 (cid:1) j = 0 . 3 5 j = 0 . 5 j = 0 . 7 d ad (cid:1) t j = 0 . 3 5 j = 0 . 5j = 0 . 7 q Figure 1 . Demonstration of the transition from delayed to accelerated expansion for different valuesof the parameter j , with q = − . for all three variants. Left figure: derivative da/dτ as a functionof dimensionless time. Right figure: the deceleration parameter q as a function of dimensionless time.
3. The simplicity of the procedure.4. Parameters of various models are expressed through a universal set of cosmologicalparameters. There is no need to introduce additional parameters. Let us illustrate thisstatement on the following example.The authors of the CM suggested the following procedure for estimation of the parameter B [15]. The original CM is described by a set of parameters { B, n } . We pass to a new setof parameters { B, n } → { z eq , n } , where z eq there is a redshift, in which the contributionsfrom the members Aρ m and Bρ nm are compared, Aρ ( z eq ) = Bρ n ( z eq ) . (5.1)Since ρ = ρ /a = ρ (1 + z ) , then BA = ρ − n (1 + z eq ) − n ) . (5.2)Using for the parameter A , A = H ρ − Bρ n − , (5.3)obtain Bρ n H = (1 + z eq ) − n ) z eq ) − n ) . (5.4)The CMB and supernovae data allow to limit the interval of change z eq , . < z eq < .Comparing (4.12) and (5.4), we are convinced of the obvious advantage of the cosmo-graphic approach: to find the parameter B , we did not have to introduce additionalparameters. The dimensionless parameter Bρ n H is determined by the current value ofthe fundamental cosmological parameter, the deceleration parameter q . From equating(4.12) to (5.4), we find a function z eq ( n, q ) that allows us to estimate the interval ofvariation of the parameter n corresponding to the interval . < z eq < . We see (seeFig. 2) that this interval includes parameters n ≤ . .– 8 – . 0 5 0 . 1 0 0 . 1 50 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0 n z e q Figure 2 . Function z eq ( n, q ) at the value of q = − . for the current deceleration parameter .
5. The method provides an interesting possibility of calculating the highest cosmologicalparameters from the values of lower parameters known with a better accuracy. Forexample, Eq. (4.13) can be used to estimate the parameter s for known values of q and j , s = − q j − (3 n + 2) j + 2 q (3 n − ,n = j − q − . (5.5)In particular, in LCDM j = 1 and the relation (4.18) is transformed into s = − q − .It is easy to see that the cosmographic parameters of the LCDM q = − Ω m ,s = 1 − Ω m , (5.6)exactly satisfy this relationship.6. The method presents a simple test for analyzing the compatibility of different models.The analysis consists of two steps. In the first step, the model parameters are expressedthrough cosmological parameters. The second step consists in finding the intervalsof cosmological parameter changes that can be realized within the framework of theconsidered model. Since the cosmological parameters are universal, only in the case ofa nonzero intersection of the obtained intervals, the models are compatible.– 9 – eferences [1] Riess, A. G., Filippenko, A. V., Challis, P., et al. Observational evidence from supernovae foran accelerating universe and a cosmological constant. The Astronomical Journal, 116(3):1009,1998.[2] Perlmutter, S., Aldering, G., Goldhaber, G., et al. Measurements of and from 42 high-redshiftsupernovae. The Astrophysical Journal, 517(2):565, 1999.[3] M Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, et al., Improved cosmological constraintsfrom a joint analysis of the SDSS-II and SNLS supernova samples, A & A 568, A22[4] Planck Collaboration, Ade, P. A. R., Aghanim, N., et al., Planck 2015 results. XIII.Cosmological parameters, 2016, A&A, 594, A13[5] Yu. L. Bolotin, D. A. Erokhin, O. A. Lemets, Expanding Universe: slowdown or speedup?,Phys. Usp. 55 (2012) 876918.[6] . N. A. Bahcall, J. P. Ostriker, S. Perlmutter, and P. J. Steinhardt, The Cosmic triangle:Assessing the state of the universe, Science 284 (1999) 1481-1488, arXiv:astro-ph/9906463[7] J. Ostriker and P. J. Steinhardt, ”Cosmic concordance”, arXiv:astro-ph/9505066 [astro-ph].[8] S. Green, R. M. Wald, How well is our universe described by an FLRW model? Classical andQuantum Gravity 31 (23), 234003[9] Cheng Cheng, Qing-Guo Huang, The Dark Side of the Universe after Planck, Phys. Rev. D 89,043003 [arXiv:1306.4091][10] M. Lopez-Corredoira, Tests and problems of the standard model in Cosmology, Foundations ofPhysics June 2017, Volume 47, Issue 6, pp 711âĂŞ768 [arXiv:1701.08720][11] W. Lin, M. Ishak , Cosmological discordances: a new measure, marginalization effects, andapplication to geometry vs growth current data sets , arXiv:1705.05303[12] A. Joyce, B. Jain, J. Khoury, and M. Trodden, Beyond the cosmological standard model, Phys.Rep. 568, 1-98 (2015), [arXiv:1407.0059][13] M. Raveri, Is there concordance within the concordance LCDM model?, Phys. Rev. D 93,043522 (2016), [arXiv:1510.00688[14] A. Del Popolo, M. Le Delliou, Small Scale Problems of the LCDM Model: A Short Review,Galaxies 2017, 5(1), 17, [arXiv:1606.07790][15] Freese K and Lewis M, Cardassian Expansion: a Model in which the Universe is Flat, MatterDominated, and Accelerating, 2002 Phys. Lett. B 540 1 [astro-ph/0201229][16] Gondolo P and Freese K, Fluid Interpretation of Cardassian Expansion, 2003 Phys. Rev. D 68063509 [hep-ph/0209322][17] M. Visser, Cosmography: Cosmology without the Einstein equations, Gen. Relat. Grav. 37(2005) 1541 [gr-qc/0411131].[18] M. Visser, Jerk, snap, and the cosmological equation of state, Class. Quantum Grav. 21 (2004)2603 [gr-qc/0309109].[19] S. Capozziello, V.F. Cardone, V. Salzano, Cosmography of f(R) gravity, Phys. Rev. D 78(2008) 063504 [arxiv:0802.1583].[20] P. K. S. Dunsby, O. Luongo, On the theory and applications of modern cosmography, Int. J.Geom. Methods Mod. Phys. 13, 1630002 (2016) [42 pages], [arXiv:1511.06532][21] A. Aviles, C. Gruber, O. Luongo, and H.Quevedo, Cosmography and constraints on theequation of state of the Universe in various parametrizations, Phys.Rev. D86, 123516 (2012) – 10 –
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