Does Chaplygin gas have salvation?
Juliano Pereira Campos, Júlio César Fabris, Rafael Perez, Oliver F. Piattella, Hermano Velten
aa r X i v : . [ a s t r o - ph . C O ] M a r Eur. Phys. J. C manuscript No. (will be inserted by the editor)
Does Chaplygin gas have salvation?
Juliano P. Campos a,1 , J´ulio C. Fabris b,2 ,Rafael Perez c,2 , Oliver F. Piattella d,2 ,Hermano Velten e,3 Centro de Ciˆencias Exatas e Tecnol´ogicas - UFRB, Campus Cruz das Almas, 710,44.380-000, Cruz das Almas, BA, Brazil Departamento de F´ısica - CCE - UFES, Campus Goiabeiras, 514, 29075-910 Vit´oria,ES, Brazil Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, GermanyReceived: date / Accepted: date
Abstract
We investigate the unification scenario provided by the generalisedChaplygin gas model (a perfect fluid characterized by an equation of state p = − A/ρ α ). Our concerns lie with a possible tension existing between backgroundkinematic tests and those related to the evolution of small perturbations. Weanalyse data from the observation of the differential age of the universe, type Iasupernovae, baryon acoustic oscillations and the position of the first peak of theangular spectrum of the cosmic background radiation. We show that these testsfavour negative values of the parameter α : we find α = − . +0 . − . at the 2 σ level and that α < α = 0: this is usually consideredto be undistinguishable from the standard Λ CDM model, but we show that theevolution of small perturbations, governed by the M´esz´aros equation, is indeeddifferent and the formation of sub-horizon GCG matter halos may be importantlyaffected in comparison with the Λ CDM scenario.
Keywords
Dark Matter · Dark Energy · Chaplygin gas · Physics beyond theStandard Model · type Ia Supernovae · Baryon Acoustic Oscillations · Rastall’stheory of gravity. a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] e e-mail: [email protected] Under the assumption of homogeneity and isotropy of the universe at large scales,and that General Relativity is the correct theory of gravity, the accelerated ex-pansion [1–4] is due to an exotic component, with negative pressure, called darkenergy (DE). Moreover, structures such as galaxies and clusters of galaxies seemto require the existence of another exotic component, dubbed dark matter (DM)[5, 6], in order to form. DE and DM constitute the so-called dark sector of the uni-verse, accounting for about 95% of the total cosmic energy budget. Their natureis still a mystery, but they clearly do not fit in the standard model of elementaryparticles [5, 6].One important proposal describing the dark sector of the universe is the uni-fication scenario, where DM and DE are considered different manifestations of aunique component. The paradigm of such idea is the Chaplygin gas (GC) model[7], where an equation of state inspired by string theory [8] allows to obtain a dy-namic behaviour mimicking DM in the past and DE in current time, as structureformation and the accelerated expansion require. Later, the CG has been phe-nomenologically generalized, leading to the so-called generalized Chaplygin gas(GCG) [9, 10].Though the idea is very appealing, the GCG faces crucial drawbacks whenthe model undergoes observational tests. This has been first indicated in reference[11], concerning the behaviour of small perturbations and structure formation, andpointed out again in [12–17]. One important aspect is that the so-called kinematictests (those based on the background expansion and the calculation of distances)seem to favour values for the GCG equation of state parameters that imply nega-tive square speed of sound [18, 19] (though they do not exclude positive values).If this is really the case, up to our knowledge in the literature there are only twoways out: i ) to introduce ad hoc entropic perturbations [20, 21] which nullify (ormake positive) the effective speed of sound or ii ) to modify the gravity theory, bysupposing e.g. a non-conservative theory of gravity [22].Our goal in this work is to understand to which extent the results of [18, 19]are really robust from the point of view of the kinematic tests. This question hasalready been addressed in [19], using the differential age of the universe (i.e. the H ( z ) test) and type Ia Supernova (SNIa). We extend that analysis by enlargingthe number of observational data for the H ( z ) test and by including the position ofthe first acoustic peak of the cosmic microwave background (CMB) spectrum andthe baryonic acoustic oscillation (BAO) tests. We revisit the constraints placed bythese datasets, but we do not touch delicate issues such as the calibration problemof the SNIa data [23]. Our results seem to reveal the robustness of the kinematictests for the GCG model: we find α = − . +0 . − . (at the 2 σ level). Therefore,we not only confirm the previous results, but stress that negative values of α seemto be favoured over the positive ones (for comparison, see Table 1 of [19]).Note that we cannot report a real ”tension“ between background tests and theones involving perturbations, since at 2 σ we find that positive values of α are stillallowed ( α < . | α | . − , for perturbations),but we can speculate how the Chaplygin gas could be ”saved“ if such tension wouldappear in the future, given the fact that many experiments are ongoing and othersforthcoming (such as EUCLID), thus the precision on the observational data isincreasing day after day. Within the ”tension“ scenario, if the GCG is indeed the matter componentbehind the dark sector, it must get its origin from a different theory of gravityrather than GR. For example, non-standard theories of gravity such as Rastall’s[24–28], f ( R ) [29] or Horava-Lifshitz [30, 31] could possibly improve the status ofthe GCG. Our discussion in Sec. V is devoted to open a new window for GCGbased cosmological models where standard GR is replaced by Rastall’s theory.We also stress that, in general, the accepted idea is that the GCG (based on GRtheory) works only if its parameter space is reduced to the Λ CDM one, in particularif α = 0, see Eq. (1) and Eq. (4) [32]. We discuss this issue in Sec. VI. We point outthat the reduction α = 0 implies that the GCG does not behave as a pressurelessfluid at high redshifts, but as a fluid with a small negative constant pressure. Hencethe analysis of the growth of sub-horizon GCG dark matter halos during the matterdominated epoch and the “M´esz´aros effect” could be different from the standard Λ CDM picture, possibly leading to a different non-linear clustering pattern at smallscales and to a modification of the transfer function. This argument agrees withthe conclusions of [33]. For a opposite point of view, i.e. that the perturbations inthe GCG with α = 0 are the same (at all perturbative orders) as in the Λ CDM,see [34].The paper is organized as follows. In Sec. 2, we review briefly the possibletensions within GCG model tests. In Sec. 3 we describe the observational tests weperform, while in Sec. 4 we carry out the statistical analysis based on such tests.In Sec. 5 we investigate a Rastall’s theory approach which could possibly save theGCG. In Sec. 6 we discuss the general relativistic limit α → The GCG model is characterized by the equation of state [7, 9, 10] p c = − Aρ αc , (1)where A and α are free parameters. The original Chaplygin gas model, which issomehow connected with the Nambu-Goto action of string theory, implies α =1 [8], the case α = 1 being a phenomenological generalization. Integrating theconservation equation for the fluid, dρ c da + 3 a ρ c (1 + w c ) = 0 , with EoS parameter w c ≡ p c ρ c = − Aρ αc , (2)leads to the following expression for the GCG density as function of the scalefactor: ρ c ( a ) = ρ c (cid:20) ¯ A + (1 − ¯ A ) a α ) (cid:21) α , (3)where we have defined the new parameter ¯ A ≡ A/ρ α +10 and present-time quanti-ties are indicated by the subscript 0. From Eq. (3) one can see that the originalmotivation behind the GCG, i.e. a fluid that evolves from the matter behaviourat early times to a constant density at late times, occurs only if α ≥ − When one consider a model where the energy content is given by radiation andbaryons, besides the Chaplygin gas, Friedmann’s equation becomes H ( z ) H = (cid:20) Ω b (1 + z ) + Ω r (1 + z ) + Ω c (cid:16) ¯ A + (1 − ¯ A )(1 + z ) α ) (cid:17) (cid:21) α , (4) Ω c = 1 − Ω b − Ω r , (5)where H is the Hubble constant and z is the redshift, which is related to the scalefactor by a = 1 / (1+ z ). We have also assumed a spatially flat background, accordingto the recent results of WMAP7 [3]. The Hubble parameter today can be expressedas H = 100 h km s − Mpc − , where h ≈ . ≤ h < ≤ E ( z ) ≡ H ( z ) /H , for futureconvenience.The idea of DM-DE unification into a single fluid in expression (4), faces dif-ficulties from different points of view. From the theoretical side, if vacuum energyis not responsible for DE, it should be explained how it contributes to gravity andtherefore how it enters Eq. (4). From the observational point of view, introducingfrom Eq. (1) the GCG speed of sound c s ( a ) = − αw c ( a ) , if a = 1 → c s = α Aρ α +1 c = α ¯ A , (6)one can see that if background tests favour negative values of the parameter α , then c s is negative. This is not only an undesirable feature of the model, but dependingon how much it is negative a “tension” between background and perturbativetests may arise. These problems can be alleviated if, for example, positive entropicperturbations σ = ∂p/∂S > α . Another possibility of achieving negative values of α is to implement a scalarversion of the GCG model using a non-conservative theory of gravity, like Rastall’stheory. This has been done in reference [22], and in fact negative values of α seemstill to be favoured at the level of perturbations.This problem concerns not only the confrontation with the power spectrumdata, but also the analysis of the anisotropy of the cosmic microwave backgroundradiation (CMB). In references [12, 13], the confrontation of the GCG model withthe CMB spectrum has been performed, indicating that the most favoured scenarioimplies α →
0, that is, the GCG model reduces to the Λ CDM model. This resultis one of the reasons for speculating that the GCG model should be be ruled out.
There are four main background tests for a cosmological model:1. The differential age of old galaxies, given by H ( z ).2. The SN Ia data.3. The position of the first CMB acoustic peak.4. The peak position of the baryonic acoustic oscillations (BAO). For the differential age data, connected with the evaluation of the age of oldgalaxies that have evolved passively, leading to values of H ( z ) for specific redshifts,there are 13 observational data [35–40]. Recently, a compilation of 21 data pointshas been considered [41]. The fundamental relation is H ( z ) = −
11 + z dzdt . (7)The value of the Hubble parameter today obtained by the HST ( H = 72 km s − Mpc − ) could also be added to this sample, but this would imply a prior on thefinal parameter estimation and, therefore, will not be included here. The analysiswith the sample of 13 data points for the Chaplygin gas model has been madein reference [19], leading to the conclusion that slightly negative values for α arefavoured. In our analysis below we use the set of 21 data points compiled in ref.[41].The SN Ia test is based on the luminosity distance, given by µ = m − M = 5 log D L , (8) D L = cH (1 + z ) Z z dz ′ E ( z ′ ) . (9)For the SN Ia data, we have two main problems. The first one concerns the choiceof the sample. There are many different SN Ia data set, obtained with differenttechniques. In some cases, these different samples may give very different results.The second point is the existence of two different calibration methods: one usingcosmology and which takes into account SN with high z (Salt2); the other usingastrophysics methods, valid for small z (MLCS2k2) [23]. In some case, the em-ployment of different calibrations can lead to different results also. All this, makethe SN Ia analysis very delicate. Here, however, we use the Union sample [42],calibrated by the Salt2 method. This choice is motivated by looking for a contactwith previous results in the literature, including those of reference [19].The position of the first peak of the CMB spectrum is a more complex test. Itis linked to oscillations of the baryon-photon plasma at the recombination periodand is given by l = l A (1 − φ ) . (10) A detailed numerical analysis leading to the following fitting formula for parame-ters of this fundamental quantity [43–46]: φ = 0 . (cid:18) Ω r z ls . Ω m + Ω b ) (cid:19) . , (11) g = 0 . (cid:18) ω − . b (cid:19)(cid:18) . ω . b (cid:19) − , (12) g = 0 . (cid:18) . ω . b (cid:19) − , (13) z ls = 1048 (cid:18) . ω − . b (cid:19)(cid:18) g ω g m (cid:19) , (14) l A = π I I , (15) I = Z z ls dzE ( z ) , I = Z ∞ z ls c s ( z ) E ( z ) , (16) c s = (cid:18) Ω b Ω γ (1 + z ) (cid:19) − / . (17)We will use the following values for the different density parameter: ω i = Ω i h , Ω r h = 4 . × − , (18) Ω b h = 0 . , ω m h = Ω dm + Ω b . (19)From observation: l = 220 . ± . z = 1090 (itis of course the same physics which produces the acoustic peaks structure in theCMB, but now this effect is observed in the baryonic distribution). This effect isquantified by the following expression [48]: A = √ Ω m E ( z ) (cid:18) z b Z z b dzE ( z ) (cid:19) . (20)In this work we use data from the WiggleZ Dark Energy Survey [49].The GCG behaves as dust in the past. Hence, we should identify an effectiveDM component in the GCG in order to use the above formulas for the analysis.There are different prescriptions in this sense in the literature. We will adopt thedecomposition proposed in [47], where the effective DM component is given by, Ω m = Ω b + Ω c (cid:18) − ¯ A (cid:19) α . (21) We perform a Bayesian statistic analysis in comparing the theoretical predictionswith the observational data. First, for each dataset, we compute χ = N X i =1 ( µ thi − µ ob ) σ i , (22) where µ thi is the theoretical prediction for the quantity µ , whereas µ obi is thecorresponding observational datum, with an error estimation σ i . From the chi-squared, we build the probability distribution function (PDF), as follows: P ( h, α, ¯ A ) = C exp (cid:18) − χ (cid:19) , (23)where we assumed the data to be independent and normally distributed; C is anormalization constant. As indicated, the PDF depends on three free parameters: h , α and ¯ A , characterizing a three dimensional function. Two and one dimensionalPDF can be computed integrating (marginalizing) on the remaining parameters.As already said, we will compute the χ separately for each dataset. The total χ , for the set of the four observational tests, shall be the sum of the separated χ . Accordingly to the exponential form of the PDF, in Eq. (23), the total PDFshall be the product of the single ones.The intervals considered for each of the free parameters are crucial for the finalestimations. For ¯ A , all values 0 ≤ ¯ A ≤ h we consider bothdelta and constant priors i.e., for the former prior we fix values for h while for thelatter we integrate in the interval 0 ≤ h ≤
1. For α , the situation is more complex,since we can consider two possibilities: α > −
1, in order to assure a transitionfrom dust in the past to a cosmological constant in the future, or α >
0, in orderto assure a positive square speed of sound. Our goal here is to show that negativevalues for α are preferred. Therefore, we focus our analysis on the prior α > − α free to vary to arbitrarily negative and positive values.We evaluate the one dimensional PDF for ¯ A and α for the four independenttests listed before and combining all of them. The results are shown in Fig. 1 - 6.4.1 Delta prior over hIn Fig. 1, 2 we show the one-dimensional probability distributions for α and ¯ A for specific choices of h = h ⋆ . The values chosen for h ⋆ are shown in the fourpanels. Each panel corresponds to a data set. All the curves have been normalizedfollowingPDF( α ) = R P ( h ⋆ , α, ¯ A ) d ¯ A R R ∞− P ( h ⋆ , α, ¯ A ) dα d ¯ A and PDF( ¯ A ) = R ∞− P ( h ⋆ , α, ¯ A ) dα R R ∞− P ( h ⋆ , α, ¯ A ) dα d ¯ A . (24)Except for the H(z) data, we see that the maximum likelihoods occur for hyper-surfaces with negative values of α . h to vary in the range 0 < h <
1. The final one-dimensionalPDF for α and ¯ A are calculated formally asPDF( α ) = R R P ( h, α, ¯ A ) dh d ¯ A R R ∞− R P ( h, α, ¯ A ) dh dα d ¯ A and PDF( ¯ A ) = R R ∞− P ( h, α, ¯ A ) dh dα R R ∞− R P ( h, α, ¯ A ) dh dα d ¯ A . (25)It is crucial to restrict the interval for α . If we consider the standard scenariofor structure formation, in the GR context without entropic perturbations, it isrequired that α ≥
0. However, if the only constraint is to impose an accelerationto deceleration transition, then the restriction is α > −
1, since for α < − α free in order to test the consistency of the backgroundtests in the context of the GCG model. In what follows, we will consider two cases: α > − α free.For both the cases, the general behaviour follows similar features, that canbe summarized as follows. The H ( z ) and the baryonic acoustic oscillations testspredict a maximum for α slightly negative, while the position of the first peakindicates a maximum for a small positive value of α . For these three tests the PDFdecreases as the value α = − α can be significant for α < −
1. It is important to stress that, for the baryonicacoustic oscillations and for the CMB first peak, the PDF becomes essentially zerofor α < −
1. This is due to the decomposition into a ”dark matter” componentemployed in Eq. (21): for α < −
1, the behaviour for ”dark matter” is reversed,due to the change of sign of the exponent in Eq. (21).We show the results of the analysis in Fig. 3 and Fig. 4, for α > −
1, and in Fig. 5and Fig. 6, for α free. The 2 σ estimation for α in the first case is α = − . +0 . − . ,while in the second case is α = − . +0 . − . . This result shows that α > − α < Some ideas on how to avoid a possible “tension” and then saving the Chaplygin gasappeared recently and concern a modified theory of gravity: Rastall’s theory [24].In the latter the Einstein-Hilbert action still holds and the modification resides inthe conservation law of the matter stress-energy momentum, i.e. T µν ; µ = κR ; ν , (26)where κ is a parameter and R is the Ricci scalar curvature. The above modificationcan be interpreted in various ways. To us the most interesting and significant oneis that Rastall’s idea may be viewed as a kind of semi-classical formulation ofquantum phenomena, which we expect to appear when the curvature (which entersas R on the right hand side of Eq. (26)) becomes important. Of course, one couldchoose other scalars rather than R in order to represent the Riemann tensor, butperhaps R is the most natural choice.Since Eq. (26) must fit into Bianchi identities, one can find the following mod-ified Einstein equations R µν − g µν R = 8 πG (cid:18) T µν − γ − g µν T (cid:19) , (27) T µν ; µ = γ − T ; ν . (28)Then, it is clear that, when assuming a FLRW background, Friedmann equationshall be modified with terms proportional to γ .We choose, for our discussion in the present section, a single-fluid componentwith density ρ and pressure p . In Rastall’s theory, Friedmann equation becomes H = 8 πG ρ (cid:20) − γ w γ − (cid:21) , (29)where note that, differently from GR, now the pressure contributes to the expan-sion (via the equation of state parameter w ). Moreover, the continuity equationbecomes: ˙ ρ + 3 H ( ρ + p ) = γ −
12 ( ˙ ρ − p ) , (30)or, trading the cosmic time for the scale factor, and rearranging the derivatives:3 − γ dρda + 3 a ( ρ + p ) = − γ − dpda . (31)Clearly, since the energy conservation equation has changed, we cannot use againthe GCG equation of state and hope to obtain again the same result as in Eq. (3).What we can do is to take advantage of the foregoing analysis and assume a formfor H given by Eq. (3). Then, we can derive the equation of state and speed ofsound of the corresponding fluid in Rastall’s theory and investigate its stabilityproperties.Using E ≡ H /H , it is easy, by simple inspection of Eqs. (29) and (31) to find − a ( ρ + p ) = dEda , (32) which is a relation which holds true also for GR. Solving Eq. (29) for p andsubstituting in the above equation, one finds ρ = γ − − γ ) a dEda + 13 − γ E , (33)and for the pressure p = γ − − γ ) a dEda − − γ E . (34)Now, as said, we assume E = (cid:20) ¯ A + (1 − ¯ A ) a α ) (cid:21) α , (35)and then it is straightforward to compute density and pressure. In particular, theequation of state parameter and the speed of sound take the form w = − Aa α +3 + ( γ − − ¯ A )¯ A (2 a α +3 + 3 γ − − γ + 5 , (36) c s = ¯ A (cid:8) γ (cid:2) ( α + 1) a α +3 − (cid:3) − (3 α + 1) a α +3 + 1 (cid:9) + γ − A { γ [( α + 1) a α +3 − − (3 α + 5) a α +3 + 5 } + 3 γ − , (37)where we have defined the speed of sound as c s = dp/dρ = ( dp/da ) / ( dρ/da ), i.e. asif it were adiabatic. We can formally invert the function ρ ( a ) and then obtain a p = p ( ρ ), i.e. a barotropic equation of state. We plot in Fig. 7 the above w and c s for the choice α = − . A = 0 . γ varying about unity.In the range chosen for γ , the speed of sound is monotonically decreasing and,by inspection, positive if γ . .
8. Let us investigate briefly the asymptotic: in thefar past, for α →
0, we have w → γ − γ − , a → , (38) c s → γ − γ − , a → , (39)and for γ = 1 we reproduce the known GR result for the GCG. Now, in the remotefuture, for a → ∞ : w → − , a → ∞ , (40) c s → γ − α ( γ − γ − α ( γ − , a → ∞ , (41)where again we reproduce correctly the limit c s → α when γ = 1. By construction,the final stage of the evolution is a de Sitter one. Of course, in all the abovecalculations we have assumed that α ≥ −
1. Asking for the asymptotic speed ofsound of Eq. (41) to be positive, we find the following ranges for γ : γ < α + 11 + α , γ > − α − α ) . (42)For α = −
1, this amounts to γ < / γ > / T µν = ρu µ u ν + ph µν , (43)where h µ ν = g µ ν + u µ u ν . More explicitly, the background components of (43) are T = − ρ , T i = T i = 0 , T ij = pδ ij = − Aρ α δ ij . (44)We assume a conformal Newtonian gauge line element for scalar perturbations ds = a ( η ) h − (1 + 2 Φ ) dη + (1 − Ψ ) δ ij dx i dx j i , (45)where we introduced the conformal time η . The perturbations of the fluid 4-velocityup to first order are given by u = 1 a (1 − Φ ) , u = − a (1 + Φ ) . (46)In the absence of anisotropic stresses the spatial off-diagonal Einstein equationimplies Φ = Ψ . Assuming the above perturbed metric, we can find the Einsteinequation for the potential Φ for the Rastall’s theory, namely ∆Φ − a ′ a (cid:18) a ′ a Φ + Φ ′ (cid:19) = 4 πGa (cid:20) δρ − γ −
12 ( δρ − δp ) (cid:21) , (47) Φ ′′ + 3 a ′ a Φ ′ + " (cid:18) a ′ a (cid:19) ′ + (cid:18) a ′ a (cid:19) Φ = 4 πGa (cid:20) δp + γ −
12 ( δρ + δρ − δp ) (cid:21) , (48)where the prime denotes derivation wrt the conformal time. Introducing the speedof sound (assuming adiabaticity, i.e. δp/δρ = dp/dρ ): ∆Φ − a ′ a (cid:18) a ′ a Φ + Φ ′ (cid:19) = 4 πGa δρ (cid:18) − γ γ − c s (cid:19) , (49) Φ ′′ + 3 a ′ a Φ ′ + " (cid:18) a ′ a (cid:19) ′ + (cid:18) a ′ a (cid:19) Φ = 4 πGa δρ (cid:18) γ −
12 + 5 − γ c s (cid:19) . (50)Changing the conformal time for the scale factor and combining the two aboveequations, we find an equation for the gravitational potential: Φ aa + (cid:18) H a H + 4 a (cid:19) Φ a + (cid:18) H a H a + 1 a (cid:19) Φ = γ − − γ ) c s − γ + 3( γ − c s a (cid:18) − k H Φ − Φ − aΦ a (cid:19) , (51)where the subscript a denotes derivation wrt the scale factor, H ≡ a ′ /a and wehave introduced a plane-wave expansion. Note the “Rastall factor” of Eq. (51): Rf ≡ γ − − γ ) c s − γ + 3( γ − c s . (52)Since it multiplies the wavenumber k , it may be considered as an “effective” speedof sound, different from the adiabatic c s we introduced. In this sense, Rastall’s theory seems to call into play a sort of “geometric entropy”. Using Eq. (37) inorder to reduce the “Rastall factor” one finds Rf = ¯ Aα ¯ A + (1 − ¯ A ) a − α +1) , (53)which is exactly the GCG speed of sound in GR! Rastall’s theory seems to be ableto reproduce the same evolution of perturbations of a fluid in GR which providesa given background expansion. Therefore, we may conclude that even framing theGCG in Rastall’s theory does not save the model from being ruled out due to thebehaviour of small perturbations.Just to check. We choose as initial conditions Φ = − Φ a = 0 in a = 0 . α = − . A = 0 .
7. In Fig. 8 we plotthe evolution of Φ and δ as function of the scale factor for a representative scale k = 0 . − .Clearly, the results are catastrophic. A remaining possibility of salvation is toadopt a scalar field description for the GCG, as done in [22]. As shown in [53](where the authors do not mention Rastall’s theory and investigate just the γ = 2case) and [27] in a scalar field approach the effective speed of sound in the restframe of the field (i.e. where T i = 0) can be written as c s = 2 − γγ , (54)which evidently vanishes for γ = 2. Since the scalar field description possessesone degree of freedom more than the fluid description (encoded in the scalar fieldpotential) the background expansion can be fixed without any drawbacks on theevolution of perturbations, which for γ = 2 may be identical to the one of apressureless fluid in GR (e.g. CDM). The results of our background tests do not impose negative values for α with ahigh confidence level. This could mean that the GCG has indeed α = 0 meaningthat the dark sector of the universe is actually an ordinary adiabatic fluid with asmall, constant and negative pressure. Hence, we can argue that at high redshiftsthe GCG does not look exactly like standard CDM ( p cdm = 0).Our goal in this section is to revisit the issue concerning the equivalence ofGCG with α = 0 and the Λ CDM model. Obviously both cases produces the samebackground expansion. However, does this equivalence holds at first order in theperturbations?We present an analysis of the M´esz´aros effect which describes the formation ofdark matter structures during the first stages of the matter dominated epoch. Weassume a single fluid description, i.e. we neglect the contribution of baryons andradiation. This is a reasonable approximation if we want to track the growth ofsub-horizon perturbations during the matter dominated epoch. For the technicaldetails, we follow the analysis of [50].For the linear perturbations of (43) we define the velocity scalar v , which isassociated with the peculiar velocity by δu i,i ≡ kv/a . At first order, the (0-0) component of the Einstein equation reads − k Ψ − H Ψ ′ − H Ψ = 32 H δ tot , (55)where δ tot is the total density contrast, i.e. δ tot ≡ δρ tot /ρ tot . During the timeinterval between the kinetic decoupling of DM particles from the primordial plasmaand the epoch of matter radiation equality ( z eq ≈ Λ CDM model) thesub-horizon DM perturbations grow only logarithmically with the scale factor.After z eq the DM perturbations obey to δ ∝ a . This is the main result behind theso called M´esz´aros effect [52].We study here an example in which the GCG with α = 0 behaves differentlyfrom the Λ CDM. We show that the growth of sub-horizon GCG matter pertur-bations is different from the standard CDM. In order to obtain a M´esz´aros-likeequation for the GCG we make use of the covariant conservation of the energy-momentum tensor ( T µν ; µ = 0). At first-order we find δ ′ = − H δ (cid:16) c s − w c (cid:17) − (1 + w c ) (cid:0) kv − Ψ ′ (cid:1) , (56)and v ′ = −H (1 − w c ) v − w ′ c w c v + kΨ + kc s w c δ , (57)for the energy and momentum balances of each single component (which we assumeconserving separately), respectively.For small scales we can neglect the term Ψ ′ in Eq. (56) and we also take thesub-horizon limit of the Poisson equation Eq. (55). Together with Eq. (57) andwith the fact that δ tot = δ gcg , we find a M´esz´aros-like equation for the GCG with α = 0, i.e. c s = 0: a d δ gcg da + (cid:20) − w c ) + aH d Hda (cid:21) a dδ gcg da + (cid:20) − − w c + 9 w c − aw c H dHda − a dw c da (cid:21) δ gcg = 0 . (58)The standard equation for CDM in the Λ CDM model can be obtained in the sameway, but remembering that now δ tot = ρ cdm ρ cdm + ρ Λ δ cdm = Ω cdm δ cdm , (59)since the Λ CDM model is basically a two-fluid model. Therefore, we have a d δ cdm da + (cid:20) aH d Hda (cid:21) a dδ cdm da − Ω cdm δ cdm = 0 . (60)Identifying the background expansion of the two instances, Λ CDM and GCG, onecan write Ω cdm = 1 + w c and thus a d δ cdm da + (cid:20) aH d Hda (cid:21) a dδ cdm da −
32 (1 + w c ) δ cdm = 0 . (61) Equations (58) and (61) are clearly different (note that H and w c have the sameevolution), therefore δ gcg and δ cdm evolve differently. Indeed, one can show that δ gcg = (1 + w c ) δ cdm . This is also a result of [34], see also [54–56].We show in Fig. 9 the evolution of a typical sub-horizon scale after the equality( z eq ≈ k = 0 . − with help of the CAMB code [57]. We then solve numericallyEq. 58 for ¯ A = 0 . , . .
90. The perturbations in the GCG fluid are stronglysuppressed in comparison to the CDM case. This result can also be appreciated inFig. 1 of [56].The main conclusion of [34] is that the Λ CDM model and the α = 0 GCGare indistinguishable. We agree on this at the linear perturbations level since, be-ing the speed of sound vanishing, the evolution of the gravitational potential Ψ is determined univocally and it is the same in the two models (it depends on thebackground evolution only, which is identical in the two models, by construction).This imply that the evolution of δ tot is also the same and therefore, if the com-ponents conserve independently, the evolution of the density contrasts of baryonsand radiation are also respectively identical in the two models. Since the lattertwo are the only observable components, one may conclude that the Λ CDM modeland the α = 0 GCG are indistinguishable, from gravity only. No difference can beexpected for the e.g. integrated Sachs-Wolfe effect or for the baryonic large scalestructure power spectrum (which is indeed the one we infer from observation).On the other hand, the density contrast δ gcg seems unable to reach the non-linearregime δ ≈
1, see Fig. 9. We wonder if this would not have a dramatic effect on thehalo formation. The absence of the latter would be in contrast with the observationof the almost flat velocity curves of galaxies and with lensing experiments. Prob-ably, only numerical simulations would provide the correct matter distribution,but semi-analytical studies touching the non-linear regime [10, 51] could supplyadditional information about the final fate of the α = 0 GCG. We performed a Bayesian analysis of the background behaviour of the GCGmodel using H ( z ), SNIA, CMB and BAO datasets. We focused particularly onthe parameter α , on which a huge literature already exists. Our result is that α = − . +0 . − . , at the 2 σ level, i.e. negative values of α seem to be favouredover the positive ones. Indeed α is negative with 85% of confidence. The uncer-tainty is still too large for us to claim that a “tension” with perturbative tests(which constrain | α | . − at 2 σ ) does exist, but we speculate on the conse-quences of this occurrence, which is not far from being settled, given the ongoingand forthcoming observational programs which collect more precise data day af-ter day. That kind of inconsistency would immediately rule out the GCG model,since it is unacceptable to obtain two different set of parameters depending on theobservational test applied. We assume this to occur and figure out how to possi-bly save the GCG unification paradigm. We introduce a fluid model in Rastall’stheory which exactly behaves as the GCG at the background level (i.e. it providesthe same evolution for H ( z )) and discuss its perturbative properties. It turns outthat small and positive speeds of sound are compatible with α <
0. On the other hand, we show that the evolution of perturbations is ruled by an effective speedof sound which is identical to the GCG one in GR, as if Rastall’s theory intro-duced a sort of “geometric entropy”. Therefore, as one should expect, the resultsare catastrophic, with the density contrast and the gravitational potential grow-ing too fast (for α <
0) for being in agreement with observation. As a possiblesalvation, we indicate a scalar field description of the GCG in Rastall’s theory,where the background expansion and the effective speed of sound can be fixedindependently (and the latter to zero).We also address the issue of the α → Λ CDM model at the linear perturbative regime, but we show that the smallnegative pressure of the GCG affects the evolution of the GCG density contrastduring the matter dominated phase, where it is expected to behave as CDM andprovides a different evolution for the sub-horizon matter perturbations. This anal-ysis shows that sub-horizon structures (e.g. dark halos) may not form as in thestandard Λ CDM case. Thus, it could be very interesting to perform numericalsimulations for matter fluids with a hydrodynamical evolution distinct from thestandard pressureless case. This analysis could clarify if the α → Λ CDM model.
Acknowledgements
This work was supported by CNPq (Brazil). HV acknowledges supportfrom the DFG within the Research Training Group 1620 “Models of Gravity”. We would liketo thank Alejandro Aviles and Pedro Avelino for enlightening remarks and suggestions.
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473 (2000) - - Α P D F H H z L datah = = = - Α P D F CMB datah = = = - Α P D F SN data h = = = - - - - - - Α P D F BAO datah = = = Fig. 1
One-dimensional PDF for the parameter α under the restriction α > − H ( z )(upper left panel), CMB (upper right panel), SNIa (lower left panel) and BAO (lower rightpanel). We have used here delta priors of h .0 A - P D F H H z L datah = = = A - P D F CMB datah = = = A - P D F SN data h = = = A - P D F BAO datah = = = Fig. 2
One-dimensional PDF for the parameter ¯ A under the restriction α > − H ( z )(upper left panel), CMB (upper right panel), SNIa (lower left panel) and BAO (lower rightpanel). We have used here delta priors of h .1 - Α P D F - Α P D F - - - - - Α P D F - - Α P D F - - - Α P D F Fig. 3
One-dimensional PDF for the parameter α under the restriction α > − H ( z )(upper left panel), CMB (upper central panel), SNIa (upper right panel), BAO (lower leftpanel) and the combination of these four tests (lower right panel). We have used a flat priorfor h and marginalized over it. A P D F A P D F A P D F A P D F A P D F - - - Α A Fig. 4
One-dimensional PDF for the parameter ¯ A under the restriction α > − H ( z )(upper left panel), CMB (upper central panel), SNIa (upper right panel), BAO (lower leftpanel) and the combination of these four tests (lower central panel). In the lower right panel,we present the contour plots (1, 2, 3 σ ) for the total PDF in the ( α , ¯ A ) plane. We have useda flat prior for h and marginalized over it.2 - - Α P D F Α P D F - - - Α P D F - - Α P D F - - - Α P D F Fig. 5
One-dimensional PDF for the parameter α , with no restriction, using H ( z ) (upperleft panel), CMB (upper central panel), SNIa (upper right panel), BAO (lower left panel) andthe combination of these four tests (lower right panel). We have used a flat prior for h andmarginalized over it. A P D F A P D F A P D F A P D F A P D F - - - Α A Fig. 6
One-dimensional PDF for the parameter ¯ A , with no restriction on α , using H ( z ) (upperleft panel), CMB (upper central panel), SNIa (upper right panel), BAO (lower left panel) andthe combination of these four tests (lower central panel). In the lower right panel, we presentthe contour plots (1, 2, 3 σ ) for the total PDF in the ( α , ¯ A ). We have used a flat prior for h and marginalized over it.3 - - - - w - - c s Fig. 7
Evolution of the equation of state parameter w and of the speed of sound c s given inEqs. (36) and (37) as functions of the scale factor. The parameter γ has been chosen γ = 0 . γ = 1 (i.e. the GR limit, red dashed lines) and γ = 1 . - F a ∆ Fig. 8
Evolution of Φ and δ as functions of the scale factor. The GCG parameters have beenfixed as α = − . A = 0 .
7. The parameter γ has been chosen γ = 0 . γ = 1 (i.e. the GR limit, red dashed lines) and γ = 2 (blue dash-dotted line). For comparison,the Λ CDM lines are depicted as the magenta dotted ones.4 - a ∆ k = - L CDM A - = A - = A - = a ∆ k = - L CDM A - = A - = A - = Fig. 9