Dark sector interactions and the curvature of the Universe in light of Planck's 2018 data
Micol Benetti, Humberto Borges, Cassio Pigozzo, Saulo Carneiro, Jailson Alcaniz
PPrepared for submission to JCAP
Dark sector interactions and thecurvature of the Universe in light ofPlanck’s 2018 data
M. Benetti a,b
H. A. Borges c C. Pigozzo c S. Carneiro c,d
J. S.Alcaniz e a Dipartimento di Fisica “E. Pancini”, Universit`a di Napoli “Federico II”, Via Cinthia, I-80126 Napoli, Italy b Istituto Nazionale di Fisica Nucleare (INFN), sez. di Napoli, Via Cinthia 9, I-80126 Napoli,Italy c Instituto de F´ısica, Universidade Federal da Bahia, 40210-340 Salvador, BA, Brasil d PPGCosmo, CCE, Universidade Federal do Esp´ırito Santo, 29075-910 Vit´oria, ES, Brasil e Departamento de Astronomia, Observat´orio Nacional, 20921-400 Rio de Janeiro, RJ, BrasilE-mail: [email protected], [email protected], [email protected],[email protected], [email protected]
Abstract.
We investigate the observational viability of a class of interacting dark energy(iDE) models in the light of the latest Cosmic Microwave Background (CMB), type Ia su-pernovae (SNe) and SH0ES Hubble parameter measurements. Our analysis explores theassumption of a non-zero spatial curvature, the correlation between the interaction parame-ter α and the current expansion rate H , and updates the results reported in [1]. Initially,assuming a spatially flat universe, the results show that the best-fit of our joint analysisclearly favours a positive interaction, i.e., an energy flux from dark matter to dark energy,with α ≈ .
2, while the non-interacting case, α = 0, is ruled out by more than 3 σ confidencelevel. On the other hand, considering a non-zero spatial curvature, we find a slight preferencefor a negative value of the curvature parameter, which seems to relax the correlation betweenthe parameters α and H , as well as between H and the normalization of the matter powerspectrum on scales of 8 h − Mpc ( σ ). Finally, we discuss the influence of considering theSH0ES prior on H in the joint analyses, and find that such a choice does not change con-siderably the standard cosmology predictions but has a significant influence on the results ofthe iDE model. a r X i v : . [ a s t r o - ph . C O ] F e b ontents The increasing tension between the local measurements of the current expansion rate [2, 3]and that derived from the temperature anisotropies in the Cosmic Microwave Background(CMB) [4] assuming the Λ-Cold Dark Matter (ΛCDM) model has motivated the investigationof generalized models of the dark sector, physics beyond the standard model, and alternativegravity theories [5–19]. In the case of dynamical dark energy models, an important aspectworth considering is how to properly treat the dark energy (DE) perturbations, which inprinciple can be done in two different ways. The first one is to explicitly include DE per-turbations in the perturbed equations and assume a sound speed to DE. If a luminal soundspeed is assumed DE perturbations can be always neglected. An alternative procedure isto decompose the dynamical dark energy into a pressureless, clustering component and avacuum-type term with an Equation-of-State (EoS) parameter, w = − α of a particular class of iDEmodels to be slightly positive, corroborating results of similar studies [23, 24]. Moreover, astrong correlation was found between α , the Hubble constant ( H ), and the normalization ofthe matter power spectrum on scales of 8 h − Mpc ( σ ), i.e., while a positive interaction pa-rameter favours higher values of H , a negative α favours lower values of σ . This correlationis particularly important in the study of iDE models as a possible solution of the current H and σ tensions.The aim of this paper is twofold: first, to perform an updated analysis of [1] withthe Planck (2018) likelihoods [4], where the CMB polarization has been taken into account.Second, to explore the influence of a non-zero spatial curvature in the determinations of theother model parameters, motivated by the recent study of [11]. The present analysis alsopays special attention to the role played by the H prior from local measurements on theparameter estimates. In particular, we find that such a prior has a significant influence onthe determination of the iDE model parameters, given the α - H correlation reported in [1].– 1 – Parametrising the interaction
In a FLRW universe fulfilled with a pressureless component interacting with a vacuum-liketerm, the Friedmann and conservation equations assume the form3 H = ρ m + Λ , (2.1)˙ ρ m + 3 Hρ m = Γ ρ m = − ˙Λ , (2.2)where Γ is defined as the rate of matter creation (not necessarily constant). We will use aparametrisation for the vacuum term evolution given byΛ = σH − α , (2.3)where α ( > −
1) is the interaction parameter, and σ = 3(1 − Ω m ) H α +1)0 . From (2.1) and(2.2) we can show that Γ = − ασH − (2 α +1) . (2.4)By including a conserved radiation component, we obtain the Hubble function E ( z ) = H ( z ) /H = (cid:113)(cid:2) (1 − Ω m ) + Ω m (1 + z ) α ) (cid:3) α ) + Ω R (1 + z ) . (2.5)A negative α corresponds to creation of matter, while a positive one means that dark matteris annihilated. As one may check, for α = 0 the standard ΛCDM model is recovered. Eq. (2.5)is the Hubble function of a generalized Chaplygin gas (GCG) [26–30], which behaves like coldmatter at early times and as a cosmological constant in the asymptotic future. Actually, it isequivalent to a non-adiabatic Chaplygin gas [22, 23, 31, 32], because the vacuum componentdoes not cluster and, consequently, there is no pressure term in the perturbation equations.For this reason, the power spectrum does not suffer from oscillations and instabilities presentin the adiabatic version. Note as well that conserved baryons are included in the termproportional to (1 + z ) in the binomial expansion of the square brackets. A small spatialcurvature can also be included by adding a term Ω k (1 + z ) into the square root. The Boltzmann equations for conserved baryons and radiation are the same as in the standardmodel. For the dark sector, assuming that there is no momentum transfer in the dark matterrest frame, the Poisson and dark matter perturbation equations in the conformal Newtoniangauge are [1, 25] θ (cid:48) dm + H θ dm − k Φ = 0 , (2.6) δ (cid:48) dm − (cid:48) + θ dm = − aQρ dm (cid:20) δ dm − k (cid:18) k Φ + Q (cid:48) Q θ dm (cid:19)(cid:21) , (2.7) − k Φ = a ρ dm δ dm + ρ b δ b ) − (cid:18) a Q − a H ρ m (cid:19) θ dm k , (2.8)where H = aH , Q = Γ ρ m = − ˙Λ, a prime indicates derivative w.r.t. the conformal time, θ dm is the dark matter velocity potential, and Φ is the gravitational potential. In the sub-horizonlimit k (cid:29) H , these equations assume the form θ (cid:48) dm + H θ dm − k Φ = 0 , (2.9)– 2 – (cid:48) dm + θ dm = − aQρ dm δ dm , (2.10) − k Φ = a ρ dm δ dm + ρ b δ b ) . (2.11)For the vacuum term we have δ Λ = − aQθ dm /k and δQ = Q (cid:48) θ dm /k , that are negligible forsub-horizon modes. The vacuum term velocity remains undetermined. Let us show now that the inclusion of spatial curvature in the dynamical equations at latetime is only relevant at the background level. For this, consider the components of Einstein’sequations in the longitudinal gauge, ψ − φ = a π, (2.12)3 H ( ψ (cid:48) + H φ ) + k ψ − κφ = − a δρ, (2.13) ψ (cid:48)(cid:48) + 2 H ψ (cid:48) + H φ (cid:48) + (2 H (cid:48) + H − κ ) φ = a δp − k π ) , (2.14) ψ (cid:48) + H φ = − a ρ + p ) v, (2.15)where the spatial curvature can assume values κ = 0 , +1 ou −
1. In the absence of anisotropicstress π = 0, the gravitational potential and curvature perturbation are equal, φ = ψ . Theperturbation in the fluid pressure is δp = c s δρ + ( c s − c a ) ρ (cid:48) v, (2.16)where c s and c a = p (cid:48) /ρ (cid:48) are, respectively, the entropic and adiabatic sound speed, and v isthe matter velocity potential.With the help of the continuity equation, ρ (cid:48) = − H ( ρ + p ) , (2.17)we are able to rewrite the equation (2 .
15) as a ρ (cid:48) v = 3 H ( φ (cid:48) + H φ ) . (2.18)Substituting (2 .
16) into (2 .
14) and eliminating δρ and ( a / ρ (cid:48) v through equations (2 .
13) and(2 . φ (cid:48)(cid:48) + 3 H (1 + c a ) φ (cid:48) + [2 H (cid:48) + (1 + 3 c a ) H + c s ( k − κ ) − κ ] φ = 0 . (2.19)Combining (2 .
15) with (2 .
13) and using the continuity equation (2 .
17) we obtain the Poissonequation − k − κ ) φ = a ρδ c , (2.20)where δρ c = δρ + ρ (cid:48) v is a comoving gauge invariant quantity.For any interacting model with ρ (cid:48) m + 3 H ρ m = aQ, (2.21)– 3 –he adiabatic sound speed that appear in equation (2 .
19) is directly related to the energytransfer function as c a = − aQ H ρ m . (2.22)As in the flat case, it is easy to show that the vacuum energy component does not clusterin sub-horizon scales, and therefore there is no entropic sound speed, i.e. c s ∝ δρ c Λ = 0 inthe perturbation equation (2 . .
20) into (2 .
19) we can find a second orderdifferential equation for the evolution of the density contrast, δ c (cid:48)(cid:48) m + (cid:18) H + aQρ m (cid:19) δ c (cid:48) m + (cid:20)(cid:18) aQρ m (cid:19) (cid:48) + aQρ m H + H (cid:48) − H − κ (cid:21) δ cm = 0 . (2.23)From the Friedmann and Raychaudhuri equations H + κ = a ρ m + ρ Λ ) , (2.24) H (cid:48) + H + κ = a ρ m + 4 ρ Λ ) , (2.25)we derive the following relation, H (cid:48) − H − κ = − a ρ m . (2.26)Note that the spatial curvature parameter κ that appears in the last term of equation (2 . .
13) and (2 . . δ cm in the final form δ c (cid:48)(cid:48) m + (cid:18) H + aQρ m (cid:19) δ c (cid:48) m + (cid:20)(cid:18) aQρ m (cid:19) (cid:48) + aQρ m H − a ρ m (cid:21) δ cm = 0 . (2.27)This same differential equation for total matter can be derived by combining equations (2 . . δ m at sub-horizon scales is affected by spatial curvature only through its back-ground solutions. For our analysis, we updated the CMB data set used in the previous work ∗ , joining the Plik“TT,TE,EE+lowE” CMB Planck (2018) likelihood (by combination of temperature powerspectra and cross correlation TE and EE over the range (cid:96) ∈ [30 , (cid:96) temperatureCommander likelihood, and the low- (cid:96) SimAll EE likelihood) [34], its lensing reconstructionpower spectrum † [34, 35] and SNe data from the Joint Light-curve sample [36]. The latter ∗ In Ref.[1] we have used the second release of Planck data [33] “TT+lowP” (2015), namely the high- (cid:96)
Planck temperature data (in the range 30 < (cid:96) < < (cid:96) < † As shown by the Planck Collaboration, lensing data is needed both to resolve the tension of the CMB datawith a flat universe prediction, and to reduce the tension of the CMB with data from Redshift Space Distorsionand Weak Lensing. Let us stress that the Planck “TT,TE,EE+lowE + lensing” data set combination isconsidered by the Planck Collaboration as the most robust of the current release [4]. – 4 – able 1 . 68% confidence limits for the model parameters. We call as “base data set” the set ofPlanck(2018)+lensing+JLA+Ω b h prior. The ∆ χ best = ∆ χ CDM − ∆ χ model refers to the best fit ofthe model (negative value means a better χ of the reference model, ΛCDM). Parameter ΛCDM Λ(t)CDMbase dataset base dataset + SH0ES base dataset base dataset + SH0ES100 Ω b h . ± .
012 2 . ± .
012 2 . ± .
012 2 . ± . cdm h . ± . . ± . . ± . . ± . α − − . ± .
06 0 . ± . H . ± .
50 67 . ± .
48 67 . ± .
22 70 . ± . χ − − . . Table 2 . 68% confidence limits for the model parameters, using the “base dataset”. The ∆ χ best =∆ χ CDM − ∆ χ model refers to the best fit of the model (negative value means a better χ of thereference model, ΛCDM). Parameter ΛCDM + Ω k Λ(t)CDM + Ω k
100 Ω b h . ± .
012 2 . ± . cdm h . ± . . ± . k . ± . − . ± . α − . ± . H . ± .
60 66 . ± . χ − − . . < z < .
3. This sample allowsfor light-curve recalibration with the model under consideration, which is an important issuewhen testing alternative cosmologies [1, 37]. Although SNe data have lower statistical powerwith respect to Planck, they are useful for fixing the background cosmology at low redshiftsin models involving dark energy evolution and modified gravity. We include a prior on Ω b h in order to take into account the observations of D/H abundance [38], and we call such adata set Planck(2018)+lensing+JLA+Ω b h prior as “base data set”. Also, we consider theHubble constant of SH0ES collaboration, H = 74 . ± .
42 km/s/Mpc [2], that is in tensionat 4.4 σ with CMB estimations within the minimal cosmological model, to discuss the changesin the parameters constraining due to the assumption of this prior.We modify the numerical Cosmic Linear Anisotropy Solving System (CLASS) code [39]according with the theory discussed in the previous sections, and we use the Monte Python [40]code to perform Monte Carlo Markov Chains (MCMC) analyses. We build our theory by let-ting free the usual cosmological parameters, namely, the physical baryon density, ω b = Ω b h ,the physical cold dark matter density, ω cdm = Ω cdm h , the optical depth, τ reio , the primordialscalar amplitude, A s , the primordial spectral index, n s , the Hubble constant H , in additionto the interaction parameter, α .We present our results in Table 1 (flat universe) and Table 2 (non-zero curvature).Within the standard model framework, we first note that the base data set fully supportsthe flat hypothesis. At the same time, when a Gaussian prior on H centered in the SH0ESvalue is considered, we note a shift in the constrained value of the cold dark matter density,although the two values remain within 1 σ agreement. This behavior is not observed when we– 5 – .72 0.78 0.84 0.90 0.96 k α H cdm k α
60 64 68 72 H (t)CDM + k base dataset(t)CDM base dataset(t)CDM base dataset+ SH0ES Figure 1 . Comparison between flat and curved Λ(t)CDM models, using as “base dataset” the Planck2018 likelihood combined with SNe JLA sample and Ω b h prior. analyse the Λ(t)CDM model, where the H prior has a significant influence both on the peakshift of ω cdm to lower values (removing such a prior it is fully compatible with that predictedby the standard model) and on the α constraint. In this latter, using the SH0ES prior thestandard model is discarded at 3 σ . These results are shown in Fig. 1, where we show theΛ(t)CDM model with (red line) and without (green line) the SH0ES prior, and without theassumption of flat universe (light blue line).The analysis considering a curved space in the context of the Λ(t)CDM model showsan anti-correlated behaviour between the curvature density and the α parameter. A flat andnon-interacting universe is still compatible at 1 σ even if the data show preference for slightnegative curvature values and slight positive interaction parameter. Noteworthy, it seemsthat a spatially curved universe relaxes the degeneracy between α and H , and also between– 6 – and σ .Finally, in order to comment our results in the light of the previous ones [1], let usnow compare the results obtained for both Λ(t)CDM and standard models using both 2015and 2018 Planck likelihoods, combined with JLA + Ω b h prior + SH0ES prior. We remindthat the analysis using Planck 2015 (dashed black curve) was presented in [1], where weused the most robust combination at the time, namely “TT + lowP” [33], while for the newanalysis of this work we have chosen the combination currently considered more reliable,that is “TT,TE,EE + lowE” [34]. The main difference between the two Planck likelihoodsessentially lies in: (i) the use at high- (cid:96) of polarization modes, and (ii) a different treatmentof EE polarization at low multipoles [34], that implies a stronger constraint on the opticaldepth parameter (for a detailed discussion we refer the reader to Ref. [4]). Due to the cor-relation between cosmological parameters, this also determines a preference, in the standardcosmological model context, for lower values of A s and the late-time fluctuation amplitudeparameter, σ ‡ , than those estimated by Planck (2015) likelihood. This can be seen in Fig. 2,comparing the standard model constrained with Planck data from the 2015 release (dashedgray curve) and the 2018 release (solid blue curve). At the same time, we also note a signifi-cant different constraint on the cold dark matter density, comparing the Λ(t)CDM (red solidcurve) and ΛCDM model using 2018 CMB data. On the other hand, the Hubble parameteris compatible with that predicted by ΛCDM at only 2 . σ , relaxing the tension with the valuemeasured by SH0ES to 1.9 σ . We also emphasize that α = 0 (the value of the interactionparameter required to recover the standard model) is excluded by the analysis using the 2018CMB data, with a clear preference for a positive α , that is, an energy flux from dark matterto dark energy.As noted in the previous work [1], there exist a positive correlation between the values of α and H , which implies (due to the additional degree of freedom) that the Λ(t)CDM modelcan predict values of H in better agreement with the local measurements than the standardΛCDM model. Our present analysis not only confirms this correlation but also shows thatit is even stronger when the Planck (2018) data are used. On the other hand, given theanti-correlation between these two latter parameters and ω cdm , a significant shift of the colddark matter density to smaller values are obtained for the iDE model when compared withthe standard model prediction. It is worth noticing that these results are obtained using aprior on H given by the SH0ES collaboration. By removing this prior from the analysis theconstraints on these parameters are compatible with those of the ΛCDM model, as shown inFig. 1. In this paper we have not only updated the previous results of [1] with the Planck 2018polarization data, but also explored the influence of non-zero spatial curvature, and theweight of the priors choice in the analysis, particularly the use of the SH0ES prior on thelocal value of the Hubble parameter, when constraining the cosmological parameters withPlanck data. Taking such a H prior in combination with CMB data has in fact opened upnumerous debates on whether it is statistically valid or not to perform analyses of models bycombining data in tension [41–47]. ‡ The σ parameter roughly corresponds to the primordial scalar amplitude, A s , converted into presentfluctuations amplitude. – 7 – .75 0.80 0.85 0.90 0.95 n s r e i o α H cdm n s reio α H (t)CDM CMB18(t)CDM CMB15CDM CMB18CDM CMB15 ΛΛΛΛ
Figure 2 . Comparison between ΛCDM and Λ(t)CDM models using Planck likelihood 2015 andPlanck likelihood 2018 combined with SNe JLA sample, as well as Ω b h and SH0ES priors. The choice to use this prior must therefore be seriously considered and the resultsobtained carefully analysed. We have shown that, without such a prior, the current CMBdata are not capable of discerning an interaction in the dark sector, even in combination withSNe Ia data. On the other hand, when the SH0ES prior is taken into account, the best-fit ofthe α parameter is clearly positive, whereas the standard model ( α = 0) is excluded with ≈ σ confidence level. This results from the fact that the interaction and the Hubble parametersare directly correlated, as shown by the analysis with Planck data only. Therefore, a priorthat prefers a higher value for the latter will also naturally lead to a higher value to theformer.We have also considered the role of spatial curvature in the data analysis. We note thatthe best-fits of the cosmological parameters are not substantially altered, and no robust signof the presence of curvature can be concluded. In this respect, the curvature has a weakerinfluence on the analysis as compared to the number of relativistic species, that leads to anegative interaction parameter if left free, as shown in [1]. Our general conclusion is that the– 8 –ignature of interaction, if exists, is too weak to be found with the present data set. On theother hand, the present analysis also suggests that the H tension observed in the contextof the ΛCDM model seems to be more fundamental, refusing solutions based exclusively ongeneralizations of the dark sector. Acknowledgements
We are thankful to Joel Carvalho for an insightful discussion on the proper use of theSH0ES prior. MB thanks support of the Istituto Nazionale di Fisica Nucleare (INFN),sezione di Napoli, iniziative specifiche QGSKY. SC is supported by CNPq (Brazil) withgrant 307467/2017-1. JSA acknowledges support from CNPq (grants no. 310790/2014-0 and400471/2014-0) and FAPERJ (grant no. 204282). The authors thank the use of CLASS andMonte Python codes. We also acknowledge the use of the High Performance Data Center(DCON) at the Observat´orio Nacional for providing the computational facilities to run ouranalysis.
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