Remedy of some cosmological tensions via effective phantom-like behavior of interacting vacuum energy
RRemedy of some cosmological tensions via effective phantom-like behavior ofinteracting vacuum energy
Suresh Kumar
1, 2, 3, ∗ Department of Mathematics, Indira Gandhi University, Meerpur, Haryana-122502, India Department of Mathematics, National Institute of Technology, Kurukshetra, Haryana-136119, India Department of Mathematics, BITS Pilani, Pilani Campus, Rajasthan-333031, India
Since physics of the dark sector components of the Universe is not yet well-understood, the phe-nomenological studies of non-minimal interaction in the dark sector could possibly pave the wayto theoretical and experimental progress in this direction. Therefore, in this work, we intend toexplore some features and consequences of a phenomenological interaction in the dark sector. Weuse the Planck 2018, BAO, JLA, KiDS and HST data to investigate two extensions of the baseΛCDM model, viz., (i) we allow the interaction among vacuum energy and dark matter, namelythe IΛCDM model, wherein the interaction strength is proportional to the vacuum energy densityand expansion rate of the Universe, and (ii) the IΛCDM scenario with free effective neutrino massand number, namely the ν IΛCDM model. We also present comparative analyses of the interactionmodels with the companion models, namely, ΛCDM, ν ΛCDM, w CDM and νw CDM. In both theinteraction models, we find non-zero coupling in the dark sector up to 99% CL with energy trans-fer from dark matter to vacuum energy, and observe a phantom-like behavior of the effective darkenergy without actual “phantom crossing”. The well-known tensions on the cosmological param-eters H and σ , prevailing within the ΛCDM cosmology, disappear in these models wherein the ν IΛCDM model shows consistency with the standard effective neutrino mass and number. Both theinteraction models find a better fit to the combined data compared to the companion models underconsideration.
I. INTRODUCTION
The standard model of the modern cosmology, namelythe ΛCDM (cosmological constant Λ + cold dark mat-ter) model, exhibits an excellent agreement with vari-ety of observational data such as the cosmic microwavebackground (CMB) observations from Planck [1], bary-onic acoustic oscillations (BAO) [2], Supernovae type Ia[3] are to mention a few. Despite the great success ofthe ΛCDM model in representing the accelerated expan-sion and large scale structure (LSS) of the Universe inline with the observations, there are some potential andcompelling problems with this model which motivate theinvestigation of its extension or alternative cosmologicalmodels. For, a profound problem is the well-known “cos-mological constant fine tuning problem”, which refers tothe fact that most quantum field theories predict a hugecosmological constant with energy density ρ Λ ∼ GeV from the energy of the quantum vacuum at present,which is about 118 orders of magnitude larger than thecosmological upper bounds yielding ρ Λ ∼ − GeV ,leading to an extreme fine-tuning problem. Among theproblems in representing the small scale structures in theUniverse encountered by the ΛCDM model, there existsthe “missing satellite problem” which refers to the dis-crepancy between the predicted number of subhalos inN-body simulations and the observed number of subha-los. It is further complicated by the “Too Big To Fail”problem because the ΛCDM predicted satellites are toomassive and dense compared to the observed ones (see[4, 5] for recent reviews and references therein).Further, recent measurements of the present Hubbleconstant H are in serious disagreement with the onesinferred from the Planck CMB experiments within theΛCDM cosmology. To be more precise, the Hubble Space ∗ [email protected] Telescope (HST) observations of the Large MagellanicCloud Cepheids yield: H = 74 . ± .
42 km s − Mpc − which finds 4.4 σ level tension with Planck CMB ΛCDMvalue: H = 67 . ± . − Mpc − , raising the discrep-ancy beyond a plausible level of chance [6]. The measure-ments from other approaches also prefer higher values of H showing tensions between 4 σ to 5 . σ when comparedto the H value derived from Planck CMB ΛCDM [7].Thus, H tension is a compelling and potential tension inthe ΛCDM cosmology.The measurements of the r.m.s. fluctuation of densityperturbation at 8 h − Mpc scale, characterized by σ orthe parameter S = σ (cid:112) Ω m / .
3, from the cosmic shearsurveys also show inconsistency at a significant level withthe one given by Planck CMB data within the ΛCDMcosmology. For instance, the recent cosmic shear analy-sis of the fourth data release of the Kilo-Degree Survey(KiDS-1000) has reported the value S = 0 . +0 . − . ,which is in 3 σ tension with the prediction of the PlanckCMB ΛCDM cosmology. For more details, we refer thereader to [8], and references therein, where one can inferthat S does not fully capture the degeneracy directionin all the analyses. In fact, there are varying degreesof S tension from different surveys whereas a combinedanalysis from KiDS and GAMA galaxy clustering leavesno tension on S [9]. So the S tension may not be awell defined compared to the H tension possibly due tothe large systematics in the measurements from the cos-mic shear surveys. Nonetheless, various cosmic surveyspredict lower values of S compared to the Planck CMBprediction.In the ΛCDM model, dark matter (DM) and dark en-ergy (DE) are assumed to behave as separate fluids with-out any interactions beyond the gravitational ones. How-ever, DM and DE are the major energy ingredients inthe energy budget of the Universe, and a non-minimalinteraction among these two sources of energy could bea natural possibility. There are numerous studies in the a r X i v : . [ a s t r o - ph . C O ] F e b literature where the interaction between DM and DE isstudied in different contexts [10–135]. It is argued thatthe above mentioned H and S tensions could possiblydue to systematics in the data or some physics beyondthe standard ΛCDM model [136, 137]. Considering thepossibility of new physics, it is shown in the literaturethat the tensions on H and S can be alleviated by al-lowing the interaction among vacuum energy and DMwherein the interaction strength is proportional to thevacuum energy density and expansion rate of the Uni-verse (see [99, 122, 127, 131, 134] for some recent studies,and references therein). In the present study, we reassessthis model, and in addition, we investigate constraints onthe effective neutrino mass and neutrino number withinthis interacting vacuum energy scenario with the obser-vational data (see Section III) wherein the motivation forstudying effective neutrino mass and neutrino number isdeferred to the next section. The sole and main objectiveof this work is to explore and highlight some interestingphysics and features of the interaction in the dark sec-tor of the Universe via this phenomenological model ofinteraction between vacuum energy and DM. The math-ematical details of the models are described in Section II.Observational constraints and discussions are presentedin Section III while findings of this study are summarizedin Section IV. II. MODELS OF INTERACTION IN THE DARKSECTOR
The expansion rate of the Universe is given by Hub-ble parameter H = ˙ a/a , where a is scale factor andan overdot denote cosmic time derivative. We as-sume a spatially-flat Friedmann-Robertson-Walker Uni-verse filled with photons, neutrinos, baryons, cold DMand DE. In such a Universe, the evolution of Hubble pa-rameter is given by the Friedmann equation:3 H = 8 πG ( ρ γ + ρ ν + ρ b + ρ dm + ρ de ) , (1)where ρ γ , ρ ν , ρ b , ρ dm , and ρ de are the energy densitiesof photons, neutrinos, baryons, cold DM and DE, respec-tively. We allow the interaction between DM and DE viaa coupling function Q given by˙ ρ dm + 3 Hρ dm = Q = − ˙ ρ de − Hρ de (1 + w de ) . (2)Following our previous work [122], we assume that thecoupling term is proportional to the Hubble parameterand DE density, that is, Q ∝ Hρ de . Thus, we use thefollowing form of Q : Q = ξHρ de , (3)where ξ is the coupling parameter that characterizes thecoupling strength between DM and DE. Solving (2) and(3), the energy densities of DM and DE are obtained as ρ dm = ρ dm0 a − + ξρ de0 w de + ξ (cid:104) a − − a − w de ) − ξ (cid:105) , (4) ρ de = ρ de0 a − w de ) − ξ . (5) We see that ξ = 0 (no interaction) yields the standardevolution of DM and DE, as expected. Further, we noticethat ξ < ξ > w de = −
1. Then the effective EoSparameters of DM and DE read as follows [24]: w effdm = − ξρ de ρ dm , (6) w effde = − ξ . (7)The present Universe may well be dominated by aphantom-like behavior of DE. Such a behavior can beexplained by invoking a phantom field. However, a phan-tom field suffers from some undesirable problems such asthe instabilities and violation of the null energy condition.So, it could be interesting to realize an effective phantom-like behavior without introducing a phantom field. Notethat effective phantom-like behavior of DE means thatthe effective DE energy density is positive and increaseswith time, wherein the effective EoS parameter of DEstays less than −
1. A phantom-like behavior of DE, with-out invoking any phantom fields, is studied in the normalbranch of the DGP cosmological solution in [138–142].Phantom behavior via cosmological creation of particlesis studied in [143]. From eq.(7), we notice that the condi-tion for effective phantom behavior of DE is ξ <
0. Thus,it is readily possible to realize phantom-like behavior inthe framework of interacting vacuum energy model.As described in [144], the neutrino flavour oscillationexperiments allow normal mass hierarchy with the min-imal mass (cid:80) m ν = 0 .
06 eV. On the other hand, manyneutrino mass models are consistent with current obser-vations, and one does not find strong preference over theothers with compelling theoretical reasons. So consider-ing the fact that the (cid:80) m ν is non-zero, extension of thebase model with free (cid:80) m ν is naturally a well-motivatedextension of the base model. Next, many extensions ofthe Standard Model of particle physics allow the existenceof new light particles while it is usual to parametrize thedark radiation density in the early Universe by N eff , viz.,well after electron-positron annihilation the total rela-tivistic energy density in neutrinos and any other darkradiation reads ρ ν = N eff (cid:18) (cid:19) / ρ γ . (8)The standard model of particle physics predicts, N eff = 3 . N eff could well correspond to a fully thermalizedsterile neutrino which decoupled at T (cid:46)
100 MeV. Sohereafter we will refer (cid:80) m ν and N eff to as effectiveneutrino mass and number, respectively. In this work,we intend to investigate the effects of the interactionin the dark sector on (cid:80) m ν and N eff . In summary, weinvestigate the following two models in this work. I Λ CDM Model:
We consider a model where the inter-action between vacuum energy and DM is mediated bythe coupling term as mentioned in (3), and we assumethree active neutrinos as in the case of standard model.The base parameters set for this model is P = { ω b , ω dm , θ s , A s , n s , τ reio , ξ } . Here the first six parameters are the baseline parametersof the standard ΛCDM model [1]. ν I Λ CDM Model:
We consider the IΛCDM Model withthe effective neutrinos and mass number of relativisticspecies as a free parameter, i.e, as free parameters, thatis, IΛCDM+ N eff + (cid:80) m ν . The base parameters set forthis mode is therefore P = { ω b , ω dm , θ s , A s , n s , τ reio , ξ, N eff , (cid:88) m ν } . We adopt the evolution of linear perturbations as de-scribed in [122].
III. OBSERVATIONAL CONSTRAINTS ANDDISCUSSIONS
We analyze the IΛCDM and ν IΛCDM models incontrast with their respective counterparts ΛCDM and ν ΛCDM. We also present the analyses of w CDM and νw CDM models in contrast with the IΛCDM and ν IΛCDM models. In our analyses, we use the follow-ing data sets. (i) Planck: Planck-2018 [1] CMB tem-perature and polarization data comprising of the low- l temperature and polarization likelihoods at l ≤ l ≥
30, polarization (EE) powerspectra, and cross correlation of temperature and polar-ization (TE), also including the Planck-2018 CMB lens-ing power spectrum likelihood [145], (ii) BAO: the lat-est BAO measurements from SDSS collaboration com-piled in Table 3 in [146] (iii) JLA: the compilation ofJoint Light-curve Analysis (JLA) supernova Ia data [147](iv) KiDS: the measurements of the weak gravitationallensing shear power spectrum from the Kilo Degree Sur-vey [148], and (iv) R19: the new local value of Hub-ble constant H = 73 . ± .
74 km s − Mpc − [6] fromHST. First, we analyze the IΛCDM and ν IΛCDM mod-els with Planck+R19 data in order to explore the fea-tures of interaction in the dark sector. Then we as-sess these models by considering the combined data set:Planck+BAO+JLA+KiDS+R19. We present these anal-yses in a comprehensive way while doing a comparativeanalysis of the six models under consideration: IΛCDM, ν IΛCDM, ΛCDM, ν ΛCDM, w CDM and νw CDM.We use the publicly available Boltzmann code
CLASS [149] with the parameter inference code
Montepython+MultiNest [150, 151] to obtain correlated MonteCarlo Markov Chain (MCMC) samples. We analyzethe MCMC smaples using the python package
GetDist .In all analyses performed here, we use uniform priorson the model parameters: ω b ∈ [0 . , . ω cdm ∈ [0 . , . θ s ∈ [1 . , . A s ) ∈ [3 . , . n s ∈ [0 . , . τ reio ∈ [0 . , . ξ ∈ [ − , w de ∈ [ − , N eff ∈ [1 , (cid:80) m ν ∈ [0 , ν IΛCDM models in contrast with the parameters ofthe ΛCDM, ν ΛCDM, w CDM and νw CDM models, using the Planck+R19 and Planck+BAO+JLA+KiDS+R19data. We see tight constraints on the parametersof all the six models from the combined data set:Planck+BAO+JLA+KiDS+R19. Also, we observe tightconstraints on the parameters of IΛCDM, ΛCDM and w CDM models compared to the ν IΛCDM, ν ΛCDM and νw CDM models, as expected, due to the presence of twoadditional free parameters (cid:80) m ν and N eff in the laterthree models. In the following, we analyze our resultsto highlight and discuss some key features of the darksector interaction. (i) Effective phantom-like behavior ofDE: In the IΛCDM and ν IΛCDM models,we find ξ = − . +0 . − . (cid:0) − . +0 . − . (cid:1) and ξ = − . +0 . − . (cid:0) − . +0 . − . (cid:1) , respectively, both at 99% CLfrom the Planck+R19(Planck+BAO+JLA+KiDS+R19)data. Thus, in both these models, the coupling param-eter is non-zero and negative, implying the interactionbetween DM and vacuum energy at 99% CL whereinthe energy transfer takes place from DM to vacuumenergy. Further, we find w effde = − . +0 . − . (cid:0) − . +0 . − . (cid:1) and w effde = − . +0 . − . (cid:0) − . +0 . − . (cid:1) , both at 99% CLfrom the Planck+R19(Planck+BAO+JLA+KiDS+R19)data, for the IΛCDM and ν IΛCDM models, respectively.Thus, an effective phantom-like behavior of DE prevailsin the two models of interaction without “phantomcrossing” at 99% CL. It should be noted that thiseffective phantom-like behavior of DE is accompaniedby effective positive pressure of DM (see eq.(6)) at 99%CL for the IΛCDM and ν IΛCDM models. Also, inour analyses, we obtain phantom behavior of DE inthe w CDM and νw CDM models at 99% CL from thePlanck+R19 and Planck+BAO+JLA+KiDS+R19 data. (ii) Alleviation of the H and S tensions: In Fig.1, 2D posteriors for S and H are shown for all themodels under consideration where we overlay 2 σ bandsfor the measurements H = 74 . ± .
42 km s − Mpc − [6] and S = 0 . ± .
058 [148]. One may see thatthe H and S tensions are alleviated in the IΛCDMand ν IΛCDM models constrained with Planck+R19and Planck+BAO+JLA+KiDS+R19 data. Here, itis interesting to note that the coupling parameter ξ finds strong correlations with H and S , which maybe observed from Fig. 1, and more explicitly fromthe plot of correlation matrices in Fig. 2. The strongnegative correlation of ξ with H implies that lowervalues of ξ correspond to higher values of H . It followsfrom a natural physical interpretation of the interactionbetween DM and vacuum energy. For, ξ < H compared to theΛCDM model. On the other hand, the H tension inthe ΛCDM model is relaxed neither with Planck+R19data nor with Planck+BAO+JLA+KiDS+R19 databut H gets bit larger values in ν IΛCDM model due tothe larger values of N eff . The H tension is relieved inboth w CDM and νw CDM models with Planck+R19 aswell as Planck+BAO+JLA+KiDS+R19 data becauseof the phantom behavior of DE in these models. We
TABLE I. Constraints (68% and 95% CLs) on the free and some derived parameters of the IΛCDM, ν IΛCDM, ΛCDM, ν ΛCDM, w CDM and νw CDM models from Planck+R19 and Planck+BAO+JLA+KiDS+R19 data. The parameter H is measured inthe units of km s − Mpc − , and (cid:80) m ν in eV (95% CL). Planck+R19 Planck+BAO+JLA+KiDS+R19Parameter IΛCDM ν IΛCDM IΛCDM ν IΛCDMΛCDM ν ΛCDM ΛCDM ν ΛCDM w CDM νw CDM w CDM νw CDM10 ω b . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . ω cdm . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . θ s . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . ln 10 A s . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . n s . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . τ reio . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . ξ − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . w effde − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − − − − − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . N eff .
046 2 . +0 . . − . − . .
046 3 . +0 . . − . − . .
046 3 . +0 . . − . − . .
046 3 . +0 . . − . − . .
046 2 . +0 . . − . − . .
046 3 . +0 . . − . − . (cid:80) m ν . < .
15 0 . < . . < .
08 0 . < . . < .
35 0 . < . m . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . H . +1 . . − . − . . +1 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +1 . . − . − . . +0 . . − . − . . +0 . . − . − . . +1 . . − . − . . +1 . . − . − . . +0 . . − . − . . +0 . . − . − . S . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . notice that the S tension is not relaxed in any of thefour companion models: ΛCDM, ν ΛCDM, w CDM and νw CDM under consideration. . . . . S H IVCDMΛCDM w CDM . . . . S H ν IVCDM ν ΛCDM νw CDM . . . . S H IVCDMΛCDM w CDM . . . . S H ν IVCDM ν ΛCDM νw CDM
FIG. 1. 2D posteriors for S and H from Planck+R19 (upper panels) and Planck+BAO+JLA+KiDS+R19 data (lower panels)illustrating how the interacting dark sector can remedy cosmological tensions compared to the other models under consideration.We overlay 2 σ bands for the measurements H = 74 . ± .
42 km s − Mpc − [6] and S = 0 . ± .
058 [148]. (iii) A possible solution to the small scale prob-lems:
From Table I, we notice that the matter den-sity parameter Ω m and S get lower values in IΛCDMand ν IΛCDM models as compared to ΛCDM, ν ΛCDM, w CDM and νw CDM models. In Fig. 3, we notice lesspower on all the linear scales in the matter power spec-trum of IΛCDM model when compared to the ΛCDM.A plausible physical explanation could be that effectivephantom like behavior of DE in the interaction modelsleads to faster expansion of the late Universe, which di-lutes the matter and decrease the structure formationon linear and non-linear scales. These observations indi-cate that allowing the interaction among the dark sectorcomponents could pave the way for resolving the smallscale problems associated with the ΛCDM model. How-ever, in order to assess the effects of DM-DE couplingon non-linear scales, one needs to perform N-body sim-ulations of the interaction scenario. For instance, theresults of a series of high-resolution hydrodynamical N-body simulations of DM-DE coupling scenario are pre-sented in [23, 94], wherein the authors carried out a basicanalysis of the formation of non-linear structures, andfound that a larger coupling strength leads to a loweraverage halo concentration. Further, they deduced thatthe halo density profiles get shallower in the inner part ofmassive halos with increasing value of the coupling, andthe halo concentrations at z = 0 are significantly reducedwith respect to ΛCDM, proportionally to the value of the coupling. They concluded that such effects alleviate thetensions between observations and the ΛCDM model onsmall scales, implying that the coupled DE models couldbe viable alternatives to the standard ΛCDM model. (iv) Consistency with the standard neutrinos andmass number: From Table I, we notice that the con-straints on (cid:80) m ν and N eff in the interaction models areconsistent with the standard model predictions. This ob-servation is interesting in the sense that higher values H usually correspond to higher values of N eff becauseof the positive correlation between these two parameters[111, 113], leading to larger values of H . But in the ν IΛCDM model, we find no correlation between H and N eff . Thus, the ν IΛCDM model yields higher values of H and smaller values S while being consistent with thestandard model predictions of the neutrinos and massnumber. We notice that νw CDM model is also consis-tent with the standard model predictions of the neutrinosand mass number but this model does not relax the S tension. (v) Better fit to the data: The Bayes’ factor B M a , M b of two competing models M a and M b is given by [152,153] B M a , M b = E M a E M b , (9)where E M a and E M b are respectively the Bayesianevidences of the models M a and M b . It is interpreted as ξ Ω m H S ξ Ω m H S − . − . − . − . . . . . . ξ Ω m H S ξ Ω m H S − . − . − . − . . . . . . ξ Ω m H S ξ Ω m H S − . − . − . − . . . . . . ξ Ω m H S ξ Ω m H S − . − . − . − . . . . . . FIG. 2. Upper-left and upper-right panels respectively show the correlation matrices of some selected parameters of IΛCDMand ν IΛCDM models from Planck+R19 data while the lower panles show the corresponding correlation matrices fromPlanck+BAO+JLA+KiDS+R19 data. k [h/Mpc] P ( k ) [ M p c / h ] CDMIVCDM
FIG. 3. Matter power spectra of the IΛCDM and ΛCDM mod-els with the best-fit mean values of the parameters displayedin Table I. per the Jeffery’s scale in four different cases dependingon the value of | ln B M a , M b | , viz., values lying in theranges [0,1), [1,3), [3,5) and [5 , ∞ ) imply the strengthof the evidence to be weak, positive, strong and the strongest, respectively. From our analyses, we findln B ΛCDM , IΛCDM = − .
58, ln B ΛCDM , wCDM = − . B ν ΛCDM ,ν IΛCDM = − .
95, ln B ν ΛCDM ,ν wCDM = − . B ΛCDM , IΛCDM = − .
49, ln B ΛCDM , wCDM = − . B ν ΛCDM ,ν IΛCDM = − .
65, ln B ν ΛCDM ,ν wCDM = − . ν IΛCDM, w CDM and νw CDM modelsfind better fit to the Planck+R19 data when comparedto their respective counterparts ΛCDM and ν ΛCDMwherein we observe strong evidence in each case as per theJeffery’s scale. One may notice that the ΛCDM modeldoes not accommodate/accept the R19 prior in the fit,and hence shows a big tension with the H values givenby R19 prior. The interaction models find better fit tothe Planck+BAO+JLA+KiDS+R19 data compared tothe other models under consideration. IV. FINAL REMARKS
We have used the Planck 2018, BAO, JLA, KiDSand HST data to investigate two extensions of thebase ΛCDM model, viz., IΛCDM and ν IΛCDM models,wherein the non-minimal interaction strength betweenvacuum energy and CDM is proportional to the vacuumenergy density and expansion rate of the Universe. Wehave compared the results of these models with the com-panion ΛCDM, ν ΛCDM, w CDM and νw CDM models inorder to show some features and consequences of the phe-nomenological interaction in the dark sector by consider-ing two combinations of data sets, namely, Planck+R19and Planck+BAO+JLA+KiDS+R19. In both the inter-action models, we find non-zero coupling in the dark sec-tor up to 99% CL with energy transfer from dark matterto vacuum energy, and observe a phantom-like behav-ior of the effective dark energy without actual “phantomcrossing”. The well-known tensions on the cosmologicalparameters H and σ , prevailing within the ΛCDM cos-mology, disappear in these models wherein the ν IΛCDMmodel shows consistency with the standard effective neu-trino mass and number. The νw CDM model is also foundto be consistent with the standard model predictions ofthe neutrinos and mass number but this model does notrelax the S tension. We observe phantom-like behaviorof DE and the relaxation of H tension in the w CDMand νw CDM models. We notice that interaction in thedark sector could pave the way to resolving the smallscale problems associated with the ΛCDM model. Boththe interaction models find a better fit to the combineddata when compared to the companion ΛCDM, ν ΛCDM, w CDM and νw CDM models. Overall, in the contextof the cosmological tensions and fit to the observationaldata, we find that the interaction models do a better jobthan the companion models under consideration. How-ever, the choice of the coupling term in these interactionmodels is purely phenomenological for which a natural guidance from the fundamental physics does not exist,and therefore it could be worthwhile to investigate thedark sector interaction using some model-independentapproach such as the Gaussian process (e.g. [154, 155]and references therein). Such a model-independent ap-proach could also be useful to measure H . For in-stance, in a recent study [156], a model-independent esti-mate of H is obtained using the Gaussian process, viz., H = 73 . ± .
84 km s − Mpc − , which is consistentwith H = 73 . ± .
74 km s − Mpc − , the measurementfrom HST [6]. Finally, it is worth mentioning that theinteraction among dark sector components of the Uni-verse could have very interesting and far reaching con-sequences in the contemporary particle physics and cos-mology. Since physics of the dark sector components isnot yet well-understood, the phenomenological studies ofinteraction in the dark sector, like the ones carried out inthe present work, can reveal interesting and worthwhilefeatures and consequences of the interaction in the darksector, which in turn could possibly pave the way to the-oretical and experimental progress in this direction. Inthe next decade or so, the cosmology community is gear-ing up for solution to various problems associated withthe ΛCDM model such as the Hubble constant tensionand others [157–160]. With the future CMB experimentssuch as CMB-S4, and the cosmic surveys such as Euclidand LSST, one may expect the H estimate within anuncertainty of ∼ . Acknowledgments
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