Looking for interactions in thecosmological dark sector

M. Benettia,b,c W. Mirandad H. A. Borgese C. Pigozzoe S.Carneiroe J. S. Alcanizc

aDipartimento di Fisica “E. Pancini”, Universita di Napoli “Federico II”, Via Cinthia, I-80126, Napoli, ItalybIstituto Nazionale di Fisica Nucleare (INFN), sez. di Napoli, Via Cinthia 9, I-80126 Napoli,ItalycDepartamento de Astronomia, Observatorio Nacional, Rio de Janeiro, RJ, BrasildInstituto Federal da Bahia, Paulo Afonso, BA, BrasileInstituto de Fısica, Universidade Federal da Bahia, Salvador, BA, Brasil

E-mail: benettim@na.infn.it, mirandawelber@gmail.com, humberto@ufba.br,cpigozzo@ufba.br, saulo.carneiro.ufba@gmail.com, alcaniz@on.br

Abstract. We study observational signatures of non-gravitational interactions between thedark components of the cosmic fluid, which can be either due to creation of dark particlesfrom the expanding vacuum or an effect of the clustering of a dynamical dark energy. In par-ticular, we analyse a class of interacting models (Λ(t)CDM), characterised by the parameterα, that behaves at background level like cold matter at early times and tends to a cosmo-logical constant in the asymptotic future. In our analysis we consider both background andprimordial perturbations evolutions of the model. We use Cosmic Microwave Background(CMB) data together with late time observations, such as the Joint Light-curve Analysis(JLA) supernovae data, the Hubble Space Telescope (HST) measurement of the local valueof the Hubble-Lemaıtre parameter, and primordial deuterium abundance from Lyα systemsto test the observational viability of the model and some of its extensions. We found thatthere is no preference for values of α different from zero (characterising interaction), even ifthere are some indications for positive values when the minimal Λ(t)CDM model is analysed.When extra degrees of freedom in the relativistic component of the cosmic fluid are consid-ered, the data favour negative values of α, which means an energy flux from dark energy todark matter.

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Contents

1 Introduction 1

2 Parametrising the interactions 22.1 Background 3

2.1.1 Implication in the matter power spectrum 42.2 Primordial perturbations 6

3 Data 7

4 Analysis and Results 7

5 Final Remarks 8

1 Introduction

The standard cosmological model, also known as Λ - Cold Dark Matter (ΛCDM), providesa successful description of the structure and evolution of the universe, requiring half a dozenparameters. However, in spite of its observational successes, many open questions remain,including the very nature of the dark matter and dark energy components, which drive thecurrent cosmic evolution. Furthermore, as the accuracy of cosmological observations increases– some of the current constraints on the ΛCDM parameters can reach sub-percent level –tensions between different data sets have also emerged.

This is the case for instance of the discrepancy involving current measurements of theHubble-Lemaıtre parameter, in which the value obtained from CMB data assuming theΛCDM model, H0 = 66.93 ± 0.62 km/s/Mpc [1], differs by ' 4.4σ from the value mea-sured using distance measurements of galaxies in the local Universe calibrated by Cepheidvariables and type Ia Supernovae (SNe Ia), H0 = 74.03 ± 1.42 km/s/Mpc [2] (see e.g. [3]for a recent discussion). Another ongoing issue concerns the preference of the Planck CMBangular spectra for a large amplitude of the lensing signal, Al. The current CMB data fur-nish Al = 1.18± 0.14 at 95% C. L. [1], which is about 3σ off from the ΛCDM value Al = 1.This discrepancy is particularly challenging given that the lensing signal obtained from thePlanck angular trispectrum is consistent with the standard cosmology [1]. Furthermore, a' 2σ difference in the Ωm − σ8 plane can be inferred by comparing CMB and cosmic sheardata [4] (see also [5]), where Ωm is the matter density parameter and σ8 is the matter fluc-tuation amplitude on scales of 8h−1Mpc. Currently, one of the main arguments is that suchdiscrepancies may be an indication of new physics beyond ΛCDM, and many analyses haveinvestigated alternative scenarios that could reconcile the data sets. Attempts to solve theH0-tension problem for instance include extensions of the standard model, such as the exis-tence of new relativistic particles [6, 7], early dark energy [8], primordial gravitational waves[9], small spatial curvature [10], among others (see also [11, 12] and references therein). Ingeneral, it is not straightforward to reconcile both the H0 and σ8 discrepancies in the sametheoretical framework (see e.g. [12]).

On the other hand, there has been a growing interest in models with interaction betweenthe dark components [13–23]. A basic hypothesis of standard cosmology is that pressureless

– 1 –

matter is conserved, which is mathematically expressed by the well-known evolution lawρm ∝ a−3, where ρm is the matter density and a the cosmological scale factor. This hypothesisis in fact a tautology if dark energy is formed by a cosmological constant. However, if darkenergy is a dynamical field, it can in principle cluster. In this case, it is impossible toknow which part of the clustering energy is made by matter and which part is made byclustered dark energy. This is a manifestation of the so called dark degeneracy [24–27]. Ifwe identify the clustered energy as pressureless matter, the clustering of the dynamical darkenergy will be interpreted as matter production. At the same time, the smooth part of darkenergy, responsible for accelerating the expansion, will decrease. This is a scenario in whichan interaction between dark matter (defined as clustering energy) and dark energy (definedas the smooth component responsible for the acceleration) would appear [28–33]. Anothersituation is that dark particles are indeed created from the expanding vacuum, a processgenerally present in expanding spacetimes. Since we know very little about the properties ofdark particles (as their masses and couplings), their creation from vacuum cannot, a priori,be ruled out.

From the observational viewpoint, analyses of these models have shown that the currentmatter density derived from Large-Scale Structure (LSS) observations are systematicallylower than the values obtained through SNe Ia data. The ΛCDM best-fit for the 2dFGRSdata, for instance, is Ωm0 ≈ 0.23 [34], while a value Ωm0 ≈ 0.24 was obtained with thelinear-range data of the SDSS galaxy catalogue [35, 36]∗. On the other hand, from type Iasupernovae (SNe Ia) samples larger densities are obtained. For example, the JLA compilationfurnishes Ωm0 ≈ 0.3 [38] or even Ωm0 ≈ 0.4, depending on the light-curve calibration methodused [39]. Larger values are also derived from CMB observations [1]. This difference maybe understood as a signature of matter creation since the matter power spectrum dependssensibly on the matter density at the time of matter-radiation equality, whereas cosmologicaldistances determinations are more dependent on the present matter density. Therefore, ifmatter is created in the late-time expansion, but we assume matter conservation in LSS anddistance tests, the former will lead to a lower present density as compared to the latter. TheH0-tension mentioned earlier has also been discussed in the context of interacting models ofdark energy (see e.g. [40–42]).

Here, we study a class of interacting model that, at background level, can be associatedwith a Generalised Chaplygin Gas (GCG) model. The observational viability of the modelis analysed at both background and perturbative levels using the current data of SNe Iaand CMB, along with measurements of H0 and primordial deuterium abundance from Lyαsystems†. We organise this paper as follows. In Section 2 we introduce the theory of themodel, discussing the background evolution as well as the primordial perturbations formula.Details of the data used and the analysis performed are given in Section 3 and Section 4 alongwith the results of our statistical analysis. Finally, discussions and conclusions are presentedin Section 5.

2 Parametrising the interactions

In a FLRW spacetime the energy-momentum tensor has the general form

Tµν = (ρ+ p)uµuν − pgµν , (2.1)

∗When non-linear scales are included, one finds Ωm0 ≈ 0.29 [36, 37].†Soon after the completion of the analyses presented in this work the latest CMB data from the Planck

Collaboration was made available.

– 2 –

where ρ is the energy density, p = ω(ρ)ρ is the pressure, and u is the cosmic fluid 4-velocity.One may decompose this perfect fluid as

Tµν = ρmuµuν + Λgµν , (2.2)

by defining Λ = −p = −ωρ and ρm = (1+ω)ρ. Such a decomposition resolves the degeneracydiscussed above, provided we can show that the vacuum-type component Λ does not clusterif we identify ρm as the observed cold matter. For this purpose, let us express the covariantconservation equation Tµν;ν = 0 in the form

Tµνm;ν = Qµ, (2.3)

TµνΛ;ν = −Qµ, (2.4)

where Tµνm = ρmuµuν and TµνΛ = Λgµν . Here, Qµ is the energy-momentum transfer between

the two components, which we decompose as Qµ = Quµ + Qµ, with Qµuµ = 0. For acomoving observer, it is easy to see that Q represents the energy transfer. The second term,Qµ, represents the momentum transfer, null in the isotropic background. From (2.4) we have

Q = −Λ,νuν , (2.5)

Qµ = Λ,ν(uµuν − gµν). (2.6)

A linear perturbation of the above equations leads to δQ0 = 0 and

δQ = −(δΛ),0 +Λ

2δg00, (2.7)

δQi = (δΛ + Λθ),i ≡ δΛc,i, (2.8)

where θ is the dark fluid velocity potential. The last equation plays an important role inour analysis [27]. The left-hand side represents the momentum transfer between the twocomponents in the perturbed spacetime. If the matter component is non-relativistic, themomentum transfer must be negligible, and matter will follow geodesics [43–45]. We thenobtain δΛc ≈ 0. This quantity represents the covariant, comoving perturbation of the Λcomponent [46], which is, therefore, smooth. From (2.7) we also see that δQ ≈ 0 in asynchronous comoving gauge. If, in addition, there is no energy transfer between the twocomponents, Λ is constant and matter is conserved‡. In this case, observational cosmologywould be reduced to a precise determination of the ΛCDM free parameters. However, thereis no reason to assume such a prior, which demands for observational confirmation.

2.1 Background

With the above decomposition of the cosmic fluid, the Friedmann and conservation equationsassume the form

3H2 = ρm + Λ, (2.9)

ρm + 3Hρm = Γρm = −Λ, (2.10)

where the unknown function Γ is the rate of matter creation. Note that the second equalityis quite general and can be seen as a definition of the creation rate. In the late-time universe

‡The reverse is also true: If we assume that clustering cold matter is conserved, observations will favor acosmological constant among dark energy candidates.

– 3 –

there are two natural possibilities for fixing Γ. The first is a constant creation rate [47], anatural choice for the creation of non-relativistic particles in a low-energy expansion. Thesecond is a creation rate proportional to the expansion rate H. The former (latter) is aparticular (limiting) case of a parametrisation given by the ansatz

Λ = σH−2α, (2.11)

with constants α > −1 and σ = 3(1−Ωm0)H2(α+1)0 . From (2.9) and (2.10) it is easy to derive

Γ = −ασH−(2α+1), (2.12)

and, including a conserved radiation component, we obtain the Hubble function of our model,that hereafter we call Λ(t)CDM model, as

E(z) = H(z)/H0 =

√[(1− Ωm0) + Ωm0(1 + z)3(1+α)

] 1(1+α) + ΩR0(1 + z)4. (2.13)

The ΛCDM model corresponds to α = 0. Negative values of α means creation of mat-ter, while for α = −1/2 we have a constant creation rate Γ = 3HdS/2, where H−1

dS is theasymptotic future de Sitter horizon. The reader may identify Eq. (2.13) with a generalisedChaplygin gas (GCG) [48–52], that behaves like cold matter at early times and tends to acosmological constant in the asymptotic future. This identification, however, is only validat the background level. Our ansatz (2.11) is, actually, equivalent to a decomposed, non-adiabatic GCG [53–56]. This parametrisation, of course, does not include all the possibleforms for the interaction term. Nevertheless, it is general enough for our purpose, namely tolook for signatures of interactions in the current observational data. From Eq. (2.13) it iseasy to see that, for high redshifts, the matter density scales as

ρm(z) = 3H20 Ω

11+α

m0 z3 (z 1). (2.14)

Therefore, owing to matter creation, we do not have the standard relation between the matterdensity at high redshifts and at present. This difference affects some expressions used in fittingformulae and numerical codes, as for example equation (2.18) below. Furthermore, in order tohave the correct density at high redshifts (preserving in this way the CMB spectrum profile),we will have today a density higher or smaller than the standard model value (depending onthe sign of α).

2.1.1 Implication in the matter power spectrum

Since the only clustering component is pressureless matter, for sub-horizon scales there is noscale-dependence in the perturbation equations governing the evolution of the matter linearcontrast. Therefore, all the observed modes evolve in the same way, and the matter powerspectrum retains at present the profile it has at the beginning of the matter era. It can bewritten as [57]

P (k) = P0knsT 2(k/keq), (2.15)

where T (x ≡ k/keq) is the transfer function. The normalisation constant P0 is determinedin the fitting and can be related to σ8 by [58]

σ28 =

1

2π2

∫ ∞0

k2W 2(kR)P (k)dk, (2.16)

– 4 –

10

0 2 4 6 8 100.000

0.001

0.002

0.003

0.004

0.005

0.006

a

D+

0.0 0.5 1.0 1.5 2.00.30

0.35

0.40

0.45

0.50

z

f rsds8

FIG. 4: Left panel: Evolution of the density contrast with the scale factor for the ΛCDM model (black curve) and for model(I) with α = −0.5: for dark matter (blue) and for baryons (red). Right panel: Evolution of the baryonic frsdσ8 versus z forthe ΛCDM model (black) and for model (I) with α = −0.5 (red).

Hence, for baryonic matter we have, instead of (59),

frsd(z)σ8(z) = f b1(z)σ8(0)DN

b+(z). (61)

This does not mean that the baryonic velocity field does not coincide with the dark matter one. Indeed, sincewe are assuming that vacuum is not perturbed (i.e. δQ = 0), there is no momentum transfer to dark matter, andthen it follows geodesics like baryons. This can be verified by taking the perturbed Euler equation for a pressurelessinteracting component i, which expresses its momentum conservation. In the Newtonian gauge it is written as

(ρiϑi)′ + 4Hρiϑi + ρiΦ = aQiϑi, (62)

where Qi is the interaction term, and Φ is the gravitational potential. Making use of the background conservationequation

ρ′i + 3Hρi = −aQi, (63)

we can obtain, for both baryonic and dark matter components,

ϑ′i + Hϑi = −Φ. (64)

Therefore, given the same initial conditions, dark matter and baryons follow the same geodesics. Note also that, as(64) is valid for any Qi, it is also valid for total matter, as should be.

Relations (55) and (60) can be re-derived in the Newtonian gauge, in the limit of sub-horizon scales k ≫ H. In thisgauge, the energy conservation equation for a pressureless interacting component is given by

δρ′i + 3Hδρi − 3ρiΦ

′ + ρiϑi = −aQiΦ. (65)

Using the background conservation equation (63), we obtain

δ′i − aQi

ρiδi + ϑi = −aQi

ρiΦ + 3Φ′. (66)

For the gravitational potential one can obtain, from the perturbed Einstein equations,

k2Φ = −a2

2δρm − 3Ha2ρm

2k2ϑm, (67)

where ϑm and ρm are, respectively, the velocity potential and density of total matter. Taking the limit k ≫ H, wehave the Poisson’s equation

k2Φ = −a2

2δρm (k ≫ H). (68)

Figure 1. Left panel: Evolution of the matter contrast in the ΛCDM model (black curve), and ofthe total matter (blue) and baryonic (red) contrasts for α = −1/2 [61]. Right panel: Evolution of thebaryonic frsdσ8 for ΛCDM (black) and for α = −1/2 (red) [63].

with a spherical top-hat filter defined as

W (kR) = 3

[sin(kR)

(kR)3− cos(kR)

(kR)2

], (2.17)

and R = 8 Mpc. The spectrum turnover occurs at keq, the mode which enters in the horizonat the time of matter-radiation equality. By using (2.14) for the matter density at highredshifts and ρR ≈ ρR0z

4 for the radiation density, we obtain

keq =H(zeq)

zeq= 0.073 Mpc−1h2 Ω

11+α

m0 , (2.18)

where zeq is the redshift of matter-radiation equality, and h = H0/(100km/s/Mpc). Forα = 0 we recover the standard expression for keq [59].

Until now we referred to the total matter density Ωm0 but, in order to study the growingfunction, that is, the evolution of the observed spectrum with time, baryons must be explicitlyconsidered. The linear contrast of dark matter suffers a late-time suppression owing to darkmatter creation. This does not affect the spectrum profile, which, as discussed above, ispreserved during the matter era. It only affects its present normalisation, which is correctlyfixed by the data. However, it does lead to a suppression in the dark matter growing rate ascompared to the standard model [60]. Nevertheless, we should have in mind that observationsof the linear power spectrum and redshift space distortions (RSD) refer to visible matter and,as baryons are conserved, there is no late-time suppression in their linear contrast§.

As an example, let us take again the particular case α = −1/2. The baryonic contrast

§One may argue that weak lensing and CMB are sensitive to both baryonic and dark matter distributions.Actually, these observations are sensitive to the gravitational potential generated by total matter. Despitethe suppression in the dark matter contrast, the gravitational potential is less affected [53, 61], because it isproportional to δρm, not to δm.

– 5 –

δb can be obtained by a numerical integration of the perturbation equations¶ [61]

a2H2δ′′m + aH(aH ′ + 3H + Γ

)δ′m + 2ΓHδm =

ρmδm2

, (2.19)

a2H2δ′′b + aH(aH ′ + 3H

)δ′b =

ρmδm2

, (2.20)

where a prime means derivative w.r.t. the scale factor a, and the index m stands for totalmatter. It is straightforward to generalise them to any α. The corresponding growing rateis given by

fb(z) =δ′b(z)

Haδb(z), (2.21)

and RSD can be tested by using the bias-independent combination

frsd(z)σ8(z) = fb(z)σ8DNb+(z), (2.22)

where DNb+(z) = δb(z)/δb(0) and σ8 = σ8(0) ≈ 0.83 [62]. The results are shown in Fig. 1.

In the left panel we can see the evolution of total matter contrast in both standard andinteracting models, together with the baryonic contrast evolution in the latter, with sameinitial conditions [61]. In the right panel we plot frsdσ8 for the standard and interactingmodels, which differ at most 12% [63]. For the interacting model we have used Ωm0 = 0.45,while for the standard model we have taken Ωm0 = 0.23, the best-fit values for the 2dFGRSdata in each case [34, 47].

2.2 Primordial perturbations

The Boltzmann equations for conserved baryons and radiation are the same as in the ΛCDMmodel. For the Λ(t)CDM model analysed here, assuming that there is no momentum transferin the dark matter rest frame, the Poisson and dark matter perturbation equations can beobtained in the longitudinal gauge as‖,

θ′dm +Hθdm − k2Φ = 0, (2.23)

δ′dm − 3Φ′ + θdm = − aQρdm

[δdm −

1

k2

(k2Φ +

Q′

Qθdm

)], (2.24)

− k2Φ =a2

2(ρdmδdm + ρbδb)−

(a3Q

2− 3a2

2Hρm

)θdmk2

, (2.25)

where H = aH, Q = Γρm = −Λ, a prime means derivative with respect to conformal time,θdm is the dark matter velocity potential, and φ is the gravitational potential. In the limitof small scales, k H, these equations are reduced to

θ′dm +Hθdm − k2Φ = 0, (2.26)

δ′dm + θdm = − aQρdm

δdm, (2.27)

− k2Φ =a2

2(ρdmδdm + ρbδb). (2.28)

¶The constant Γ in (2.19) is the creation rate of total matter, not dark matter. The resulting differencein the background solution is negligible, because baryons are sub-dominant [61].

‖We adopt here the signature (−,+,+,+), with φ = ψ.

– 6 –

3 Data

In our analysis, we use the Joint Light-curve Analysis (JLA) supernovae data [38] togetherwith the second release of Planck data [64] (hereafter TT+lowP), namely the high-` Plancktemperature data (in the range of 30 < ` < 2508) from the 100-, 143-, and 217- GHz half-mission TT cross-spectra, and the low-P data by the joint TT, EE, BB and TE likelihood(in the range of 2 < ` < 29). It is worth mentioning that, compared to other recent SNeIa compilations e.g. the Pantheon compilation [65], the JLA sample has the advantageof allowing the light-curve recalibration with the model under consideration, which is animportant issue when testing alternative cosmologies. We also consider current measurementsof the local value of the Hubble-Lemaıtre parameter [2] and observations of D/H abundance[66] as Gaussian priors on the H0 and Ωb0h

2 parameters. In the analysis which Al is left freeto vary, we also consider CMB lensing data [62].

4 Analysis and Results

We implemented the above sets of background and perturbation equations in the numericalCosmic Linear Anisotropy Solving System (CLASS) code [67] to generate the theoretical spec-tra of the model, while Monte Carlo Markov Chains (MCMC) analyses for the cosmologicalparameters constraints are obtained with Monte Python [68] code.

In our analysis, we vary the usual cosmological parameters, namely, the physical baryondensity, ωb = Ωb0h

2, the physical cold dark matter density, ωcdm = Ωdm0h2, the optical

depth, τreio, the primordial scalar amplitude, As, the primordial spectral index, ns, the localHubble-Lemaıtre parameter value H0, in addition to the interaction parameter, α. Also,we consider the lensing amplitude, Al, and the number of effective degrees of freedom atdecoupling era, Neff. Indeed, it cannot be assumed a priori that interaction mechanismsallow for the same lensing scenario as expected from the standard cosmological model, i.e.Al = 1. Actually, as mentioned earlier, observational evidence seem to prefer large values ofthe lensing amplitude, which can be explained by allowing a closed curvature of the universeor interactions in the dark sector with dark matter production [42, 69]. At the same time,we also analyse an extension of the minimal model leaving Neff as a free parameter, in orderto test how such a dark interaction can be degenerate with it. In our analysis we vary thenuisance foreground parameters [64] and consider purely adiabatic initial conditions. Wechoose to work with the Newtonian gauge, and we set the sum of neutrino masses fixed to0.06 eV. We work with flat priors for the cosmological parameters, also limiting the analysisto scalar perturbations with pivot scale k0 = 0.05 Mpc−1.

Our main results are shown in Table 1, where the derived constraints on the cosmologicalparameters are displayed. In order to have a clearer comparison, we also show in Fig. 2 thecorresponding confidence intervals at 68% and 95% and the posterior probability distributionfor the most interesting behaviours. For instance, when the effective number of relativisticspecies is fixed in the standard value, Neff = 3.04, a slightly positive mean value is preferredfor the interaction parameter. This is in agreement with CMB joint analyses performed withdifferent datasets [70], highlighting a contrast with the preference for negative values foundwith background and LSS probes [71, 72].

The value of the Hubble-Lemaıtre parameter is still in good agreement with the valueconstrained by the standard model using CMB data only, whereas for the Λ(t)CDM modeland its extensions both the mean value and the best fit are slightly shifted to higher values, in

– 7 –

Table 1. 68% confidence limits for the cosmological parameters using TT+lowP+JLA+ priors(Riess+Cooke). When the Al parameter is considered as a free parameter in the analysis, we alsoadd CMB lensing data.

Λ CDM Λ(t)CDM Λ(t)CDM + Al Λ(t)CDM + Al + Neff

Parameter mean best fit mean best fit mean best fit mean best fit

100 Ωbh2 2.245± 0.018 2.250 2.243± 0.019 2.249 2.260± 0.021 2.269 2.265± 0.017 2.272

Ωch2 0.1174± 0.0019 0.1185 0.1115± 0.0079 0.1094 0.1073± 0.0100 0.1063 0.1230± 0.0088 0.1245

τ 0.087± 0.017 0.104 0.088± 0.018 0.089 0.079± 0.015 0.088 0.074± 0.018 0.063ln 1010As

∗ 3.102± 0.032 3.140 3.106± 0.034 3.104 3.087± 0.027 3.101 3.083± 0.036 3.057ns 0.9709± 0.0056 0.9679 0.9704± 0.0057 0.9692 0.9727± 0.0059 0.9735 0.9782± 0.0062 0.9851H0 68.43± 0.82 68.10 69.12± 1.14 69.60 69.95± 1.27 70.40 69.57± 1.14 70.25α - - 0.037± 0.050 0.045 0.059± 0.066 0.060 −0.018± 0.047 -0.030Al - - - - 1.18± 0.07 1.14 1.14± 0.07 1.23Neff - - - - - - 3.22± 0.14 3.30σ8 0.827± 0.014 0.845 0.841± 0.024 0.841 0.839± 0.026 0.849 0.817± 0.027 0.798χ2/2 − 5977.0 − 5976.4 − 5977.8 − 5973.6

better agreement with local measurements [2]. For instance, for the combination of data setsused in our analysis, the discrepancy between the ΛCDM prediction and the value obtainedfrom local distance estimators is ' 3.41σ while for the Λ(t)CDM model one finds ' 2.65σ.The best concordance is obtained for the model Λ(t)CDM + Al, i.e., ' 2.25σ. Besides, thepositive correlation between the Hubble-Lemaıtre parameter and the interaction parameter,α, seems to indicate a possibility to relax the H0 tension even more at the cost of assumingan energy flux from matter to dark energy (α > 0).

When Neff is a free parameter of the model, the allowed interval of values shows a goodagreement with previous ΛCDM analyses [6, 73]. An important aspect worth mentioningis that the Λ(t)CDM + Al + Neff model allows at the same time for lower values of σ8

and slightly higher values of H0 with respect to the standard model and, therefore, offers apossibility to alleviate the H0 - σ8 tension discussed earlier (see eg. [12]). This behaviouris better visualised in Fig. 3, where the plane H0 - α is shown with coloured σ8 parametervalues. Note that for negative values of α, values of H0 > 70 km/s/Mpc and σ8 < 0.82 areallowed at 1σ level.

5 Final Remarks

In this work we have considered a generalised Chaplygin gas model which provides a primeexample for the class of unified dark sector models through the interacting parameter α.While the standard ΛCDM model is recovered for α = 0, positive (or negative) values of suchan interaction parameter indicate a creation of dark energy (or dark matter) by the darksector.

Our results show that, although the α parameter value is always compatible with zero at1σ, there is a preference for positive values when the CMB data are used. This is contrary tothe results of previous works, where negative values of α were preferred at 2σ level using LSS(2dFGRS), SNe Ia (JLA) and the position of the first peak of CMB [71]. At the same time,our results are in good agreement with more recent results where values of the interactionparameter close to zero were obtained from LSS and CMB data [6, 19, 70].

We have also shown (see Table 1) that negative values of α are obtained at the cost of ahigher number of relativistic degrees of freedom (with respect to the standard Neff = 3.046).In this case, we note an interesting behaviour in the H0 − σ8 plane, since lower values of theamplitude of matter density fluctuations are allowed for higher values of H0, relaxing both

– 8 –

0.75 0.80 0.85 0.90 0.958

0.08

0.10

0.12

0.14

cdm

65.0

67.5

70.0

72.5

75.0

H0

0.15

0.00

0.15

0.30

Lt

2.16 2.20 2.24 2.28 2.3210 2

b

0.75

0.80

0.85

0.90

0.95

8

0.08 0.10 0.12 0.14cdm

65.0 67.5 70.0 72.5 75.0H0

0.15 0.00 0.15 0.30Lt

(t)CDM+Al+Neff

(t)CDM+Al

(t)CDMCDM

Figure 2. Comparison between the constraints on the ΛCDM and Λ(t)CDM models usingTT+lowP+JLA+ priors (Riess+Cooke). The confidence intervals are shown in Table 1.

the H0 and σ8 tensions. It is important to emphasise that this is achieved not by breakingdegeneracy between the two parameters but by relaxing the constraint on them, i.e., at 1σ,the error on the two parameters is about twice what has been obtained in the context of thestandard cosmological model using the same dataset. Finally, the Λ(t)CDM model and itsextensions, Λ(t)CDM + Al and Λ(t)CDM + Al + Neff, reduce the discrepancy between thelocal and CMB H0 measurements to 2.65σ, 2.25σ and 2.45σ, respectively.

Acknowledgements

MB thanks support of the Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estadodo Rio de Janeiro (FAPERJ - fellowship Nota 10 ), and Istituto Nazionale di Fisica Nu-

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66 68 70 72H0

0.10

0.05

0.00

0.05

0.10

Lt

0.78

0.80

0.82

0.84

0.86

0.88

8

Figure 3. Samples in the H0 - α plane for the ΛCDM + Al + Neff model using TT+lowP+JLA+priors (Riess+Cooke), coloured by the value of the σ8 parameter. The confidence intervals are givenin the last column of Table 1.

cleare (INFN), sezione di Napoli, iniziative specifiche QGSKY. SC is supported by CNPq(Brazil) with grant no. 307467/2017-1. JSA acknowledges support from CNPq (grants no.310790/2014-0 and 400471/2014-0) and FAPERJ (grant no. 204282). The authors thank theuse of CLASS and Monte Python codes. We also acknowledge the use of the High PerformanceComputing Centre at the Universidade Federal do Rio Grande do Norte (NPAD/UFRN) andthe Observatorio Nacional (ON) computational support.

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