2021-H_0 Odyssey: Closed, Phantom and Interacting Dark Energy Cosmologies
Weiqiang Yang, Supriya Pan, Eleonora Di Valentino, Olga Mena, Alessandro Melchiorri
22021- H Odyssey: Closed, Phantom and Interacting Dark Energy Cosmologies
Weiqiang Yang, ∗ Supriya Pan, † Eleonora Di Valentino, ‡ Olga Mena, § and Alessandro Melchiorri ¶ Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, UK IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain Physics Department and INFN, Universit`a di Roma “La Sapienza”, Ple Aldo Moro 2, 00185, Rome, Italy
Up-to-date cosmological data analyses have shown that (a) a closed universe is preferred by thePlanck data at more than 99% CL, and (b) interacting scenarios offer a very compelling solutionto the Hubble constant tension. In light of these two recent appealing scenarios, we consider herean interacting dark matter-dark energy model with a non-zero spatial curvature component and afreely varying dark energy equation of state in both the quintessential and phantom regimes. Whenconsidering Cosmic Microwave Background data only, a phantom and closed universe can perfectlyalleviate the Hubble tension, without the necessity of a coupling among the dark sectors. Accountingfor other possible cosmological observations compromises the viability of this very attractive scenarioas a global solution to current cosmological tensions, either by spoiling its effectiveness concerningthe H problem, as in the case of Supernovae Ia data, or by introducing a strong disagreement inthe preferred value of the spatial curvature, as in the case of Baryon Acoustic Oscillations.
1. INTRODUCTION
With the development of observational cosmology, ourUniverse is becoming more complex and mysterious. Thedark energy issue is already an unsolved problem in mod-ern cosmology and some recent observational evidencesclaim that our Universe is closed, despite its widely-believed flatness nature. Analyses of the latest CosmicMicrowave Background (CMB) measurements from thePlanck 2018 legacy release with the official
Plik likelihoodpoint towards the possibility of a closed Universe at morethan three standard deviations [1–4]. This observationaloutcome, undoubtedly, is one of the most significant re-sults at present times, and puts a question mark on thestandard ΛCDM scenario and the inflationary predictionof a flat Universe. Even though a flat Universe cannotbe excluded, accordingly to other complementary resultsperformed with the
CamSpec alternative likelihood [5, 6],the obtained marginalized constraints using this alter-native likelihood still prefer a closed model at a highersignificance level, above 99% CL, see Ref. [6].Such a closed universe has been found to exacerbateprevious tensions in some cosmological parameters. Forexample, the existing 4 . σ tension on the Hubble con-stant value between Planck CMB data [1] (within theminimal ΛCDM model) and the SH0ES collaboration [7](R19) is increased to the 5 . σ [3] level . Additionally, ifthe curvature parameter is allowed to freely vary, thereis an increase in the tensions between Planck CMB and ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: omena@ific.uv.es ¶ Electronic address: [email protected] See also [8, 9] for a recent overview.
Baryon Acoustic Oscillation (BAO) observations [2, 3]and between Planck CMB and the full-shape galaxypower spectrum measurements [10].Therefore, the evidence for a closed universe has raisedsome unavoidable questions that need to be answered. Apartial resolution of these problems can be achieved bythe introduction a phantom dark energy component [4],where the discrepancy in the H measurements betweenPlanck and SH0ES collaborations can be solved in theclosed Universe model, but the tension with the BAOdata still persists. Consequently, possible solutions mayreside on the dark sector microphysics. In this regard,looking for some physically motivated dark energy mod-els that could lead to an effective phantom dark energyequation of state in a closed Universe model may pro-vide the key to fully address the current cosmologicaltensions. Interacting scenarios originally proposed to ex-plain the cosmological constant first [11] and coincidenceproblem later [12] got serious attention since the end of90’s with some appealing outcomes [13–47] (see also tworeviews [48, 49]). Indeed, it was pointed out that whenthere is a coupling between the dark matter and the darkenergy sectors, the system may naturally resemble an ef-fective phantom w x < − w x = −
1. Recently, interacting dark sectorshave received plenty of attention in the literature dueto their effectiveness in alleviating the Hubble constanttension [45, 54–67]. We extend our recent work [68] byallowing the dark energy equation of state to freely varyin the quintessence and phantom regimes.The article has been organized as follows. Section 2describes the basic equations of a general interacting sce-nario assuming a non-flat background of our Universe. a r X i v : . [ a s t r o - ph . C O ] J a n Section 3 presents the observational data that are used toconstrain the model. In Sec. 4 we discuss the constraintson the scenario explored here and finally in Sec. 5 webriefly summarize the main conclusions.
2. INTERACTING DARK ENERGY IN ACURVED UNIVERSE
Assuming a Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) universe with a non-zero spatial curvature, weconsider a generalized cosmological scenario where thedark fluids of the universe, namely the pressureless (orcold) DM and the DE interact with each other by al-lowing a transfer of energy and/or momentum betweenthem. The other fluids of the Universe, instead, suchas radiation and baryons, do not take part in the inter-action process. The energy densities of the pressurelessDM and DE are respectively denoted by ρ c and ρ x andtheir corresponding pressure components are p c , whichis zero, and p x . We further assume that DE has a con-stant equation-of-state parameter w x = p x /ρ x differentfrom w x = −
1. The conservation equations of the DMand DE fluids at the background level are coupled via anarbitrary function, known as the interacting function Q :˙ ρ c + 3 Hρ c = − Q ; (1)˙ ρ x + 3 H (1 + w x ) ρ x = Q, (2)where the dot represents derivatives with respect to cos-mic time and H ≡ ˙ a/a is the Hubble rate. The signof the interaction function Q determines the direction ofthe energy and/or momentum flow. Here, Q >
Q < Q isprescribed, then, by solving the conservation equations(1) and (2), either analytically or numerically, togetherwith the equation for the Hubble rate evolution, it is pos-sible to know the evolution of the Universe. Thus, thechoice of the interaction function is essential to determinethe dynamical evolution of the Universe. In the presentpaper we shall work with a very well known form for theinteraction function [69–73]: Q = 3 Hξρ x , (3)where ξ is a dimensionless coupling parameter. Apartfrom the modifications at the background level, the pres-ence of the coupling also affects the equations at the levelof perturbations, and plenty of work has been devoted toestablish the stability conditions for their evolution incosmic time. While the interaction function in Eq. (3)was initially motivated from a pure phenomenological Parameter prior (phantom) prior (quintessence)Ω b h [0 . , .
1] [0 . , . c h [0 . , .
99] [0 . , . θ MC [0 . ,
10] [0 . , τ [0 . , .
8] [0 . , . n S [0 . , .
3] [0 . , . A s ] [1 . , .
0] [1 . , . K [ − ,
2] [ − , ξ [ − ,
0] [0 , w x [ − , −
1] [ − , perspective, a recent investigation shows that using amulti-scalar field action [74], the coupling function (3)can be derived . Therefore, the interaction model of theform given by Eq. (3) also benefits from a solid theoreti-cal motivation following some action principle.
3. OBSERVATIONAL DATA
In this section we discuss the observational data andthe statistical methodology that we use to constrain theinteracting scenarios of our interest. In what follows wedescribe the main observational data sets: • Planck 2018 CMB data : we consider the CosmicMicrowave Background (CMB) measurements fromthe Planck 2018 legacy release [1, 75]. • BAO : various measurements of the Baryon Acous-tic Oscillations (BAO) from different galaxy sur-veys, such as 6dFGS [76], SDSS-MGS [77], andBOSS DR12 [78], as used by the Planck collabo-ration [1], have been used. • Pantheon : we include the Pantheon sample of theSupernovae Type Ia distributed in the redshift in-terval z ∈ [0 . , .
3] [79]. • R19 : we also include the measurement of theHubble constant by the SH0ES collaboration in2019 [7], yielding H = 74 . ± .
42 km/s/Mpcat 68% CL.To constrain this interacting scenario, we have used amodified version of the well known cosmological package Additionally, the scalar field theory is not the only motivationto generate interaction functions of the form considered here.Another detailed investigation in Ref. [43] shows that some ex-isting cosmological theories with a Lagrangian description canalso return such interaction functions.
CosmoMC [80, 81], which is publicly available . This pack-age is equipped with a convergence diagnostic based onthe Gelman-Rubin criterion [82] and supports the Planck2018 likelihood [75]. The flat priors on the free parame-ters of the scenario explored here are displayed in Tab. I.
4. OBSERVATIONAL RESULTS
The stability criteria for interacting dark sector modelsgoverned by the interaction function given by Eq. (3)requires to analyze separately the two allowed regions,namely, (i) ξ < w x < −
1, named as
IDEp , and (ii) ξ > w x > −
1, referred to as
IDEq . In the following weshall describe the results arising from the analyses of thedifferent observational data sets considered here withinthese two regimes. ξ < , w x < − The results for the cosmological parameters within thisphantom and interacting dark energy model are shownin Tab. II and Fig. 1 for the different observational datasets.From the analyses of the CMB data alone, as shownin the second column of Tab. II, we find that the in-teraction is perfectly consistent with ξ = 0 within 1 σ .The preference for a closed universe found in previousanalyses in the literature still persists, and it is con-firmed at more than two standard deviations. We reporthere a value of Ω K = − . +0 . − . at 95% CL. Notethat the Hubble constant has a very low mean value, H = 66 +7 − km/s/Mpc at 68% CL. The very large errorbars on H allow to solve the tension with R19 withinone standard deviation. In this case, the ameliorationof the H tension is mainly driven by an increase of thevolume in the parameter space, rather than being due toan increase of the mean value of the Hubble constant.The reason for the lower value of H is due to the stronganti-correlation between Ω m and H , see Fig. 1: the(negative) DM-DE interaction ξ , implies a flux of en-ergy from the DE to the DM sector, producing a largervalue of Ω m and consequently a lower value of H . Thestrong geometrical degeneracy between the DE equationof state, w x , and H relaxes the error bars on the Hub-ble constant. The DE equation of state is in agreementwith the cosmological constant value w x = − σ .Consequently, the CMB alone data still prefers a closeduniverse. Also, the fact that w x is allowed to freely varyin the phantom region helps in alleviating the H tension. http://cosmologist.info/cosmomc/ When BAO observations are added to the CMB ones,we observe some changes in the constraints, due to theirstrong disagreement when the spatial curvature is a freeparameter [2, 3]. The curvature becomes perfectly consis-tent with a spatially flat universe (equivalently, Ω K =0), and there is a very mild preference both for a cou-pling in the dark sector, ξ = − . +0 . − . at 68% CL,and for a phantom model, w x = − . +0 . − . at 68% CL.In this case, because of the positive correlation betweenΩ K and H , see Fig. 1, the Hubble constant value isincreased to H = 69 . +1 . − . km/s/Mpc at 68% CL, alle-viating the tension with R19 at 2 . σ . However, note thatthese two data sets, i.e. CMB and BAO, are in tensionwhen considering closed cosmologies.A completely different result arises from the com-bination of the Pantheon Supernovae Ia compilationand CMB measurements: in this case, the preferencefor a closed universe is increased. We obtain Ω K = − . +0 . − . at 95% CL, shifting the value of H to-wards much lower values than those quoted above: H =61 . +2 . − . km/s/Mpc at 68% CL. The tension with R19 hasa statistical significance of 4 . σ . While the combinationof Planck+Pantheon data sets is perfectly in agreementwith no interaction in the dark sector, a strong indica-tion for a phantom universe appears at more than 2 σ , w x = − . +0 . − . at 95% CL. Therefore, the inclusionPantheon data to CMB observations increases both theHubble tension and the preference for a phantom closeduniverse (see also Refs. [4, 83, 84]).We then move to the CMB+R19 combination shownin the fifth column of Tab. II. We note that here we cansafely add these two data sets because they are not intension. The addition of the H prior will lead to a verynegative value for the phantom DE equation of state, w x = − . +0 . − . at 68% CL, ruling out the cosmolog-ical constant hypothesis with a high significance. ThisCMB+R19 combination also prefers a closed universe atmore than 2 standard deviations, Ω K = − . +0 . − . at95% CL, but it does not prefer an interaction. Conse-quently, the CMB+R19 data set is completely in agree-ment with R19 at the price of a phantom closed universe.Finally, we consider the last two possible combinationsof the data sets exploited here, CMB+BAO+Pantheonand CMB+BAO+Pantheon+R19, shown in the last twocolumns of Tab. II. We find that for both combinations,due to the presence of the BAO data, the preference for aphantom closed universe is diluted, the Hubble constanttension is not solved, and the preference for a dark sectorcoupling is either very mild or completely insignificant. ξ > , w x > − The results for the quintessential scenario are summa-rized in Tab. III and Fig. 2 considering CMB from Planck2018 and its combination with other external data sets.
Parameters CMB CMB+BAO CMB+Pantheon CMB+R19 CMB+BAO+Pantheon CMB+BAO+Pantheon+R19Ω c h . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . Ω b h . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . θ MC . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . τ . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . n s . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . ln(10 A s ) 3 . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . w x > − . > − . − . +0 . − . > − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . − . > − . − . +0 . . − . − . ξ > − . > − . − . +0 . − . > − . > − . > − . > − . > − . − . +0 . − . > − . > − . > − . K − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . Ω m . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . σ . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . H +7+24 − − . +1 . . − . − . . +2 . . − . − . . +1 . . − . − . . +0 . . − . − . . +0 . . − . − . S . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . r drag . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . TABLE II: 68% and 95% CL constraints on several free and derived parameters of the interacting scenario for w x < − r drag − . − . ξ − . Ω K . . . . Ω m . . . . S H w x . . r d r ag ξ Ω K Ω m S
45 75 105 H IDEp: CMBIDEp: CMB+BAOIDEp: CMB+PantheonIDEp: CMB+R19IDEp: CMB+BAO+PantheonIDEp: CMB+BAO+Pantheon+R19
FIG. 1: One dimensional posterior distributions and two dimensional joint contours for the interacting scenario with w x < − As before, we shall start by discussing the results ob-tained from CMB data alone, which are shown in the sec-ond column of Tab. III. Notice that there is a mild pref-erence for a non-zero dark sector interaction at one stan-dard deviation, ξ = 0 . +0 . − . at 68% CL. The preference for a closed universe is stronger, surpassing the 2 σ signif-icance, Ω K = − . +0 . − . at 95% CL. However, the H tension with R19 is above 3 σ , because we obtain a valueof the Hubble constant of H = 47 . +7 . − . km/s/Mpc at68% CL, even if H has very large error bars. In this case, Parameters CMB CMB+BAO CMB+Pantheon CMB+R19 CMB+BAO+Pantheon CMB+BAO+Pantheon+R19Ω c h . +0 . . − . − . . +0 . . − . − . . +0 . − . < . < . < .
059 0 . +0 . − . < .
11 0 . +0 . − . < . b h . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . θ MC . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . τ . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . n s . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . ln(10 A s ) 3 . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . w x − . +0 . − . unconstr. < − . < − . < − . < − . < − . < − . < − . < − . < − . < − . ξ . +0 . − . < .
49 0 . +0 . − . < .
26 0 . +0 . . − . − . . +0 . . − . − . . +0 . − . < .
25 0 . +0 . . − . − . Ω K − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . Ω m . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . σ . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +1 . . − . − . . +0 . . − . − . . +0 . . − . − . H . +7 . − . − . +1 . . − . − . . +2 . . − . − . . +1 . . − . − . . +0 . . − . − . . +0 . . − . − . S . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . r drag . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . TABLE III: 68% and 95% CL constraints on several free and derived parameters of the interacting scenario for w x > − r drag . . . ξ − . − . Ω K . . . . Ω m . . . S H w x . . r d r ag ξ Ω K Ω m S
30 40 50 60 70 H IDEq: CMBIDEq: CMB+BAOIDEq: CMB+PantheonIDEq: CMB+R19IDEq: CMB+BAO+PantheonIDEq: CMB+BAO+Pantheon+R19
FIG. 2: As Fig. 1 but within the quintessential stable region, i.e. ξ > w x > − the strong negative degeneracy between w x and H leadsto very low values for H , due to the mean preferred valueof the DE equation of state, w x = − . +0 . − . at 68% CL,which differs from the cosmological constant picture atmore than 1 σ . CMB alone data prefers a quintessentialinteracting closed universe, and the tension with R19 is quite significant.The inclusion of BAO observations pushes the Hubbleconstant to higher values, H = 68 . +1 . − . km/s/Mpc at68% CL, reducing the tension with R19 to 2 . σ . ForCMB+BAO, the curvature parameter becomes perfectlyconsistent with Ω K = 0, there is only a mild indicationfor the presence of a coupling in the dark sector ( ξ =0 . +0 . − . at 68% CL), and the DE equation of state is inperfect agreement with the cosmological constant case, w x < − .
88 at 68% CL.In analogy to the phantom closed interacting scenario,completely different results are obtained when combiningthe Pantheon compilation data with the CMB observa-tions. In this case, we find again a larger preference fora closed universe, Ω K = − . ± .
019 at 95% CL,and the value of H is also shifted towards lower values H = 60 . +2 . − . km/s/Mpc at 68% CL, in strong disagree-ment with R19. However, in this quintessence scenario,the Planck+Pantheon combination leads to an indicationfor an interaction at more than 2 σ , ξ = 0 . +0 . − . at 95%CL, and to a DE equation of state in perfect agreementwith the cosmological constant scenario. Therefore, theCMB+Pantheon data set is in disagreement with R19,but prefers a closed and interacting universe.The R19 data set is added in two different combina-tions and the results are shown in the fifth and sev-enth columns of Tab. III for completeness, but noticethat these results should be regarded as not fully reliabledue to the strong tension among CMB and R19 in the w x > − w x = − H value, there is still adisagreement with R19 with a significance of 3 . σ .
5. CONCLUDING REMARKS
Cosmologies with non-gravitational interactions be-tween dark matter and dark energy have been extensivelystudied in the literature for their ability to address theHubble constant tension – a discrepancy in the estima-tion of H between high and low redshift measurements.These interacting scenarios are very general, since theyallow for an energy exchange mechanism between thedark components. A further generalization can be re-alized if the flatness condition is relaxed. This possibil-ity is strongly motivated by the claimed preference for a closed universe in a large number of recent and inde-pendent analyses in the literature [1–4]. Whether or notnon-flat interacting cosmologies can alleviate the H ten-sion has been discussed before, see Ref. [68]. However,the framework was not as general and complete as possi-ble since the dark energy equation of state was supposedto mimic the vacuum energy case and in interacting cos-mologies this assumption may no longer be valid. In thecurrent article, we have allowed the dark energy equationof state to freely vary. To ensure the stability of the inter-acting models, we have separately discussed two distinctcases where the dark energy equation of state lies in thephantom region ( w x < − IDEp ) or in the quintessenceregion ( w x > − IDEq ). Both scenarios have been con-fronted against the latest cosmological observations, andthe results have been summarized in Tab. II and Fig. 1and in Tab. III and Fig. 2 for the
IDEp and the
IDEq scenarios, respectively.While a phantom closed universe can provide a com-pelling solution to the H tension without the necessityof a coupling between the dark matter and dark energysectors, a strong disagreement appears when consider-ing Baryon Acoustic Oscillation data. Future indepen-dent estimates of the Hubble constant, as, for instance,those from gravitational-wave standard siren measure-ments [85–89] may shed light on the current, 2021 cos-mological tensions. Acknowledgements
WY was supported by the National Natural ScienceFoundation of China under Grants No. 11705079 andNo. 11647153. SP gratefully acknowledges the Sci-ence and Engineering Research Board, Govt. of India,for their Mathematical Research Impact-Centric Sup-port Scheme (File No. MTR/2018/000940). EDV ac-knowledges the support of the Addison-Wheeler Fellow-ship awarded by the Institute of Advanced Study atDurham University. OM is supported by the Spanishgrants FPA2017-85985-P, PROMETEO/2019/083 andby the European ITN project HIDDeN (H2020-MSCA-ITN-2019//860881-HIDDeN). [1] N. Aghanim et al. (Planck), Astron. Astrophys. , A6(2020), 1807.06209.[2] W. Handley, arXiv:1908.09139 (2019), 1908.09139.[3] E. Di Valentino, A. Melchiorri, and J. Silk, Nature As-tron. , 196 (2019), 1911.02087.[4] E. Di Valentino, A. Melchiorri, and J. Silk,arXiv:2003.04935 (2020), 2003.04935.[5] G. Efstathiou and S. Gratton (2020), 2002.06892.[6] G. Efstathiou and S. Gratton (2019), 1910.00483.[7] A. G. Riess, S. 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