Revisiting Metastable Dark Energy and Tensions in the Estimation of Cosmological Parameters
Xiaolei Li, Arman Shafieloo, Varun Sahni, Alexei A. Starobinsky
DDraft version November 28, 2019
Typeset using L A TEX twocolumn style in AASTeX62
Revisiting Metastable Dark Energy and Tensions in the Estimation of Cosmological Parameters
Xiaolei Li,
1, 2, 3
Arman Shafieloo,
2, 4
Varun Sahni, and Alexei A. Starobinsky
6, 7 Department of Physics, Hebei Normal University, Shijiazhuang 050024, China Korea Astronomy and Space Science Institute, Daejeon 34055, Korea Quantum Universe Center, Korean Institute of Advanced Studies, Hoegiro 87, Dongdaemun-gu, Seoul 130-722, Korea University of Science and Technology, Yuseong-gu 217 Gajeong-ro, Daejeon 34113, Korea Inter-University Centre for Astronomy and Astrophysics, Pune, India L. D. Landau Institute for Theoretical Physics, Moscow 119334, Russia National Research University Higher School of Economics, Moscow 101000, Russia (Dated: November 28, 2019)
ABSTRACTWe investigate constraints on some key cosmological parameters by confronting metastable darkenergy models with different combinations of the most recent cosmological observations. Along withthe standard ΛCDM model, two phenomenological metastable dark energy models are considered:(i) DE decays exponentially, (ii) DE decays into dark matter. We find that: (1) when consideringthe most recent supernovae and BAO data, and assuming a fiducial ΛCDM model, the inconsistencyin the estimated value of the Ω m , h parameter obtained by either including or excluding PlanckCMB data becomes very much substantial and points to a clear tension (Sahni et al. 2014; Zhaoet al. 2017); (2) although the two metastable dark energy models that we study provide greaterflexibility in fitting the data, and they indeed fit the SNe Ia+BAO data substantially better thanΛCDM, they are not able to alleviate this tension significantly when CMB data are included; (3)while local measurements of the Hubble constant are significantly higher relative to the estimatedvalue of H in our models (obtained by fitting to SNe Ia and BAO data), the situation seems to berather complicated with hints of inconsistency among different observational data sets (CMB, SNeIa+BAO and local H measurements). Our results indicate that we might not be able to remove thecurrent tensions among different cosmological observations by considering simple modifications of thestandard model or by introducing minimal dark energy models. A complicated form of expansionhistory, different systematics in different data and/or a non-conventional model of the early Universemight be responsible for these tensions. Keywords:
Cosmology: observational - Dark Energy - Methods: statistical INTRODUCTIONThe nature of Dark energy (DE) is a key open issue inmodern cosmology. The presence of DE may be requiredto explain an accelerating universe as suggested by ob-servations of Type Ia Supernovae (SNe Ia) (Riess et al.1998; Perlmutter et al. 1999), and supported by mea-surements of large scale structure (LSS) and the cosmicmicrowave background radiation (CMB) (Spergel et al.2003; Abazajian et al. 2004; Tegmark et al. 2004; Eisen-
Corresponding author: Arman Shafielooshafi[email protected] stein et al. 2005; Komatsu et al. 2009). The simplest andbest known candidate for dark energy is the cosmologicalconstant Λ whose value remains unchanged as the Uni-verse expands. While current observational data are inagreement with the standard ΛCDM cosmology, GeneralRelativity (GR) with a cosmological constant, thoughbeing completely consistent intrinsically at the classi-cal level and having no more problems than GR itselfat the quantum level, faces some well-known theoreticaldifficulties, such as the “fine-tuning” and “cosmic coin-cidence” problems, when trying to relate the observedsmall and positive value of the cosmological constant toparameters of the Standard Model of elementary parti- a r X i v : . [ a s t r o - ph . C O ] N ov cles and its generalizations like the string theory (Sahni& Starobinsky 2000; Bean et al. 2005). Recent papershave also drawn attention to some other difficulties facedby ΛCDM including tension between the value of H es-timated by fitting to the acoustic peaks in the Planck
CMB power spectrum (Collaboration et al. 2014; Adeet al. 2016; Aghanim et al. 2018) and that obtainedfrom distance scale estimates (Sahni et al. 2014; Dinget al. 2015; Zheng et al. 2016; Sol`a et al. 2017; Alamet al. 2017b; Shanks et al. 2018). In order to alleviatethese problems, different solutions such as dark energymodels beyond ΛCDM model, modifications to generalrelativity theory and other physically-motivated possi-bilities like modifications to the dark matter sector havebeen put forward (Ko et al. 2017; Raveri et al. 2017; Ku-mar & Nunes 2017; Di Valentino et al. 2017; Renk et al.2017; Sol`a et al. 2017; Di Valentino et al. 2018; Khosraviet al. 2019; Poulin et al. 2019; Vattis et al. 2019; Li &Shafieloo 2019; Pan et al. 2019).A new class of ‘Metastable DE’ models was introducedin Shafieloo et al. (2017). In these models DE decaysinto other dark sector components of the Universe suchas dark matter or dark radiation. The rate of decayof DE depends only upon its ‘intrinsic’ properties andnot on extrinsic considerations such as the rate of ex-pansion of the universe, etc. The metastable DE modelwas largely inspired by the radioactive decay of heavynuclei into lighter elements. A total of three metastableDE models were considered, namely, i) DE decays expo-nentially, ii) DE decays into non-baryonic Dark Matter(DM), iii) DE decays into Dark Radiation. We shouldnote that from a theoretical perspective one can achievemetastable behaviour of dark energy from an interme-diate phase of quantum vacuum decay(Szyd(cid:32)lowski et al.2017, 2018). It was found that model II showed lesstension between CMB and QSO based H (2 .
34) BAOdata than that faced by ΛCDM. Clearly in order tounderstand DE, one has to turn to cosmological obser-vations. In previous work, DE models have been dis-cussed in the context of different kinds of cosmologi-cal observations (Cao et al. 2015; Li et al. 2016; Zhenget al. 2017; Shafieloo et al. 2017; Li et al. 2017). Formetastable DE models, Shafieloo et al. (2017) used 580SNe Ia from the Union-2.1 compilation (Suzuki et al.2012) and four BAO data sets in combination with CMBshift parameters R , l a , to place constraints on the DEparameters. Since then more precise data sets have beenreleased. In this work, we present constraints on twometastable DE models using the SNe Ia Pantheon sam-ple (Scolnic et al. 2017), latest BAO data from the 6dFGalaxy Survey (6dFGS) (Beutler et al. 2011), the SDSSDR7 main galaxies sample (MGS) (Ross et al. 2015), the BOSS DR12 galaxies (Alam et al. 2017a), newlyreleased eBOSS DR14 (Zhao et al. 2018) and high red-shift measurement from complete SDSS-III Ly α -quasarcross-correlation function at z = 2 . Planck measurements of theCMB anisotropies Aghanim et al. (2018); Chen et al.(2018). The aim of our analysis is to place constraintson metastable DE models using the latest data, comparemetastable DE with ΛCDM, and check whether the H tension has been alleviated.This paper is organized as follows, in section 2 webriefly introduce the Friedmann equations for our model.The observational data to be used including SNe Ia,BAO and distance prior from CMB are presented insection 3. Section 4 contains our main results and somediscussion. We summarize our results in section 5. COSMOLOGICAL MODELSIn this work, we test two metastable DE models: (i)in the first DE decays exponentially, (ii) in the secondDE decays into non-baryonic dark matter. For compari-son, we also place constraints on ΛCDM using the samedata sets. We assume that the Friedmann - Lemaˆitre -Robertson - Walker (FLRW) metric is spatially flat thatis strongly supported by recent observations (L’Huillier& Shafieloo 2017; Shafieloo et al. 2018; Aghanim et al.2018). Under this assumption, the angular diameter dis-tance D A ( z ) at redshift z can be written as D A ( z ) = cH (1 + z ) (cid:90) z dz (cid:48) E ( z (cid:48) ) (1)where E ( z ) = H ( z ) /H is the expansion rate and H is the current value of Hubble parameter.2.1. ΛCDMΛCDM model is perhaps the simplest of all dark en-ergy models. In it the cosmological constant Λ playsthe role of DE. The Hubble parameter in ΛCDM hasthe form H ( z ) = H (cid:2) Ω m, (1 + z ) + Ω DE (cid:3) (2)where Ω m, is the current matter density parameter andΩ DE = Λ3 H is the density parameter associated withdark energy.2.2. Model I: Exponentially decaying DE
In this model, DE decays exponentially as˙ ρ DE = − Γ ρ DE (3)where Γ is the only free parameter in this equation andΓ > < H ( z ) = H (cid:2) Ω m, (1 + z ) + (1 − Ω m, )exp (cid:18) Γ H (cid:90) z dz (cid:48) E ( z (cid:48) )(1 + z (cid:48) ) (cid:19)(cid:21) (4)2.3. Model II: DE decays into DM
In this model dark energy decays into non-baryonicdark matter as follows:˙ ρ DE = − Γ ρ DE (5)˙ ρ DM + 3 Hρ DM = Γ ρ DE (6)This model is effectively an interacting DE-DM modelsince when Γ (cid:54) = 0, energy is exchanged between DMand DE. The Hubble parameter for this model can bewritten as H ( z ) = H (cid:2) Ω DE ( z ) + Ω DM ( z ) + Ω b, (1 + z ) (cid:3) (7)Here Ω b, is the baryon density. Since in this model DEinteracts with non-baryonic DM, we need to separateDM density from baryon density. While for metastableDE model I, dark matter and baryon matter can betreated as a whole, e.g., Ω m , = Ω DM , + Ω b, .For the metastable DE models, the cosmological pa-rameters to be constrained are { Ω m, , H , Γ } . Bothmodel I and model II become ΛCDM when Γ = 0.We refer the reader to Shafieloo et al. (2017) for moredetails about these two DE models. DATA AND ANALYSISIn this work, we consider the combination of threedifferent kinds of cosmological probes to put constraintson DE models, including SNe Ia as standard candles andBAO together with CMB as standard rulers.3.1.
Type Ia Supernovae
In their role as standard candles, SNe Ia have been ofgreat importance to measure cosmological distances. Inour analysis, we use the new ”Pantheon” sample (Scolnicet al. 2017), which is the largest combined sample of SNIa and consists of 1048 data with redshifts in the range0 . < z < .
3. In order to reduce the impact of cali-bration systematics on cosmology, the Pantheon compi-lation used cross-calibration of the photometric systemsof all the subsamples used to construct the final sample. 3.2.
Baryonic Acoustic Oscillations
The second data set used in our analysis is BAO. Itincludes lower redshift BAO measurements from galaxysurveys and higher redshift BAO measurement fromLyman- α forest (Ly α ) data. For the lower redshiftBAO observations, we turn to the latest measurementsof acoustic-scale distance ratio from the 6-degree FieldGalaxy Survey (6dFGS) (Beutler et al. 2011), the SDSSData Release 7 Main Galaxy sample (MGS) (Ross et al.2015), the BOSS DR12 galaxies (Alam et al. 2017a) andthe eBOSS DR14 quasars (Zhao et al. 2018), while thehigher redshift BAO measurement is derived from thecomplete SDSS-III Ly α quasar cross-correlation func-tion at z = 2 . Cosmic Microwave Background
We include CMB into our analysis by using the CMBdistance prior, the acoustic scale l a and the shift param-eter R together with the baryon density Ω b h . The shiftparameter is defined as R ≡ (cid:113) Ω m H r ( z ∗ ) /c (8)and the acoustic scale is l a ≡ πr ( z ∗ ) /r s ( z ∗ ) (9)where r ( z ∗ ) is the comoving distance to the photon-decoupling epoch z ∗ . We use the distance priors fromthe finally release Planck
TT, TE, EE +low E data in2018 (Chen et al. 2018), which makes the uncertainties40% smaller than those from
Planck
TT+low P.When using SNe Ia and BAO as cosmological probes,we use a conservative prior for Ω b h based on the mea-surement of D/H by Cooke et al. (2018) and standardBBN with modelling uncertainties. The constraint re-sults are obtained with Markov Chain Monte Carlo(MCMC) estimation using CosmoMC (Lewis & Bridle2002).In our analysis, four kinds of combined data sets areconsidered: 1) Pantheon compilation in combinationwith BAO data from 6dFGS, MGS and BOSS DR12.2) We add BAO data from eBOSS DR14 to the firstdata set. 3) Adding high redshift BAO measurementfrom Ly α to the second data combination. 4) Finally,we include the CMB distance prior to the full combina-tion of data sets. RESULTS AND DISCUSSIONWe first show the constraints for the ΛCDM model inFig. 1 where the different colors denote the results from
Table 1.
BAO measurements used in our analysis. Here r d is the comoving sound horizon at the baryon drag epoch z drag , and D V = (cid:2) (1 + z ) D ( z ) cz/H ( z ) (cid:3) / , D M = (1 + z ) D A , where D A is the angular diameter distance defined in equation (1). Thefiducial comoving sound horizon for BOSS DR12, eBOSS DR14 and Ly α is r d , fid = 147 .
78 Mpc. In practice, our analysis usesthe full covariance matrix for BAO measurements from Alam et al. (2017a); Zhao et al. (2018); Des Bourboux et al. (2017). z D v /r d D M × ( r d , fid /r d )(Mpc) H × ( r d /r d , fid )(km/s/Mpc) D A × ( r d , fid /r d )(Mpc) Ref.0.106 3 . ± .
137 - - - Beutler et al. (2011)0.150 4 . ± .
168 - - - Ross et al. (2015)0.38 - 1512 ±
24 81 . ± . ±
30 90 . ± . ±
37 99 . ± . . ± .
63 1586 . ± . . ± .
42 1769 . ± .
67 Zhao et al. (2018)1.526 - - 148 . ± .
75 1768 . ± . . ± .
79 1586 . ± . . ± . . ± . m, 0 h Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB H m , H = 73.45 ± 1.74(km/s/Mpc)Riess et al. (2016) Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB
Figure 1.
Observed constrains on standard ΛCDM. Left plot gives the 1D likelihood for Ω m , h and right plots shows the 1 σ and 2 σ contours for Ω m , vs H . The cyan shadow in the right plot give H results from Riess et al. (2016) and we show theconstrain results from different data sets in different color. Table 2.
The best fit of cosmological parameters (the first row in each parameter row) for ΛCDM and its mean value togetherwith its marginalized 1 σ uncertainties (the second row in each parameter row) as well as their χ value.data Pantheon Pantheon Pantheon Pantheon+BAO(6dF+MGS+DR12) +BAO(6dF+MGS+DR12 +BAO(6dF+MGS+DR12 +BAO(6dF+MGS+DR12parameters +DR14) +DR14+Ly α ) +DR14+Ly α )+CMBΩ m, .
312 0 .
299 0 .
276 0 . . +0 . − . . +0 . − . . +0 . − . . +0 . − . H .
10 67 .
03 66 . . . +1 . − . . +1 . − . . +1 . − . . +0 . − . Ω m, h .
145 0 .
134 0 . . . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ different data combinations. The left plot shows the 1D marginalized results for Ω m, h and the right plotpresents the 2D marginalized 1 σ and 2 σ contours of Ω m, vs H . In the right plot, we also show the H constraintsfrom Riess et al. (2016) in the cyan shadow. The bestfit for cosmological parameters of ΛCDM and their 1 σ uncertainties are summarized in Table 2. We also showthe χ from each data combination in Table 2. FromFig. 1 and Table 2, we can clearly see that, adding BAOmeasurements from eBOSS DR14 pushes the best fit ofΩ m, and H towards a lower value (the green curves),and including BAO measurement from Ly α makes thebest fit of Ω m, and H even lower (the red curves).However, a higher matter density and a higher Hubbleconstant are obtained when using the combined data ofPantheon+BAO(6dF+MGS+DR12+DR14+Ly α )+CMB(the blue curves), which makes the best fits of Ω m , h in excellent agreement with the results from Pan-theon+BAO(6dF+MGS+DR12). One might note thatthere is a clear tension between the results obtained fromadding high redshift Ly α BAO measurement and theresults obtained from including CMB. We should notethat this tension has been reported earlier by Sahniet al. (2014); Shafieloo et al. (2017). It is importantto emphasis here that one of our main aims is to seewhether we can alleviate this tension by analyzing themetastable DE models using current data sets.In Fig. 2 we show the results for the metastable DEmodel I, in which DE decays exponentially. We showthe 1D likelihoods for Ω m, h and Γ /H in the upperplots and the 2D marginalized 1 σ and 2 σ contours inthe lower plots. As before, different colors imply dif-ferent data combinations. The two left plots should becompared with Fig. 1. The best fit for the cosmologicalparameters of model I and the marginalized 1 σ uncer-tainties as well as the χ of each data combination arepresented in Table 3. Compared with ΛCDM, the con-fidence contours are much larger. However, the H ten-sion between higher redshift BAO measurements fromLy α and CMB increases. Lower matter density andlower Hubble parameter are favoured by adding highredshift BAO measurements from Ly α .Moreover, from the two right plots of Fig. 2 wecan see that the constraint on Γ /H obtained withPantheon+BAO(6dF+MGS+DR12) (grey curves) sup-port Γ < < >
0, which means that Pan-theon+BAO(6dF+MGS+DR12) data suggest that theDE density is increasing, while for Pantheon in combina-tion with BAO(6dF+MGS+DR12+DR14) the best fitson Γ don’t show any preference for the DE density to beeither increasing or decreasing. However, adding Ly α BAO data into the analysis gives Γ > H ( z ), the equationof state of dark energy as a function of redshift w ( z ), the Om diagnostic Om ( z ) = ( h ( z ) − / [(1 + z ) −
1] andthe deceleration parameter q ( z ) = − ˙ H/H − σ range ofthe MCMC chains corresponding to different data sets.Fig. 7 shows the corresponding results for themetastable DE model II. The upper two plots showthe 1D likelihoods for Ω m , h (left) and Γ /H (right)obtained from different data combinations. The lowerplots show the 2D marginalized 1 σ and 2 σ regions forΩ m , vs H (left plot) and Ω DE vs Γ /H (right plot).The details of the best fits and 1 σ uncertainties for pa-rameters of model II are summarized in Table 4. Fromthe left bottom plot and Table 4, we can see that addingBAO measurement from Ly α makes the best fit of Ω m , larger than it obtained without BAO measurement fromLy α . While the constrain results for H become lowerwhen including BAO measurement from Ly α . However,adding CMB distance prior to the data set pushes theresults back to higher H and lower Ω m , . The H ten-sion still exists between CMB and BAO measurementfrom Ly α . However, as can be seen from the upper leftplot, Ω m , h agrees well between CMB and BAO mea-surement from Ly α since the degeneracy of contours forΩ m , and H changes.Now let’s look at the two right plots, which focus onthe constraints on Γ from observations. As mentionedearlier, Γ > < α favours the opposite. Including CMBdistance prior to the analysis gives Γ (cid:39)
0, which meansthat the DE energy density remains unchanged.In Fig. 8, Fig. 9 and Fig. 10, we also show the Hubbleparameter as a function of redshift H ( z ), the Om diag- m, 0 h Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB / H Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB H m , H = 73.45 ± 1.74(km/s/Mpc)Riess et al. (2016) Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB / H D E Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB
Figure 2.
Observed constrains on model I. The upper two plots show the marginalized 1D likelihood for Ω m, h (left) andΓ /H (right). The lower two plots show the marginalized 1 σ and 2 σ contours for matter density vs Hubble constant (left) andΩ DE vs Γ /H (right). Different color denotes for the constraint results from different data sets. Table 3.
The best fit of cosmological parameters (the first row in each parameter row) for the metastable DE model I and itsmean value together with its marginalized 1 σ uncertainties (the second row in each parameter row) as well as their χ value.data Pantheon Pantheon Pantheon Pantheon+BAO(6dF+MGS+DR12) +BAO(6dF+MGS+DR12 +BAO(6dF+MGS+DR12 +BAO(6dF+MGS+DR12parameters +DR14) +DR14+Ly α ) +DR14+Ly α )+CMBΩ m, .
360 0 .
276 0 .
256 0 . . +0 . − . . +0 . − . . +0 . − . . +0 . − . H .
09 64 .
03 61 .
98 68 . . +4 . − . . +4 . − . . +1 . − . . +0 . − . Ω m, h .
203 0 .
113 0 .
098 0 . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Γ /H − .
57 0 .
25 0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . χ nostic Om ( z ) = ( h ( z ) − / [(1+ z ) −
1] and the deceler-ation parameter q ( z ) = − ˙ H/H − H ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z H ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model I
Figure 3.
The Hubble parameter H ( z ) for the metastable DE model I obtained with different data combinations. The solidblack lines and the dashed black lines show H ( z ) from the best fit of the ΛCDM model and the metastable DE model I withsame data set, respectively. w ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z w ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model I
Figure 4.
The equation of state of dark energy as a function of redshift for the metastable DE model I obtained with differentdata combinations. The solid black lines and the dashed black lines show w ( z ) from the best fit of the ΛCDM model and themetastable DE model I with same data set, respectively. O m ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z O m ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model I
Figure 5.
The Om diagnostic as a function of redshift for the metastable DE model I obtained with different data combinations.The solid black lines and the dashed black lines show Om ( z ) from the best fit of the ΛCDM model and the metastable DEmodel I with the same data set, respectively. q ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z q ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model I
Figure 6.
The deceleration parameter as a function of redshift for the metastable DE model I obtained with different datacombinations. The solid black lines and the dashed black lines show q ( z ) from the best fit of the ΛCDM model and themetastable DE model I with the same data set, respectively. m, 0 h Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB / H Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB H m , H = 73.45 ± 1.74(km/s/Mpc)Riess et al. (2016) Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB / H D E Pantheon+BAO(6dF+MGS+DR12)Pantheon+BAO(6dF+MGS+DR12+DR14)Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB
Figure 7.
The constrain results for Model II. The upper two plots show the marginalized 1D likelihood for Ω m, h (left) andΓ /H (right). The lower two plots show the marginalized 1 σ and 2 σ regions for matter density vs Hubble constant (left) andΩ DE vs Γ /H (right). Different color denotes for the constraint results from different data sets. Table 4.
The best fit of cosmological parameters (the first row in each parameter row) and its mean value together with itsmarginalized 1 σ uncertainties (the second row in each parameter row) for metastable DE model II obtained from different datacombination. The last row show the χ value of each data combination.data Pantheon Pantheon Pantheon Pantheon+BAO(6dF+MGS+DR12) +BAO(6dF+MGS+DR12 +BAO(6dF+MGS+DR12 +BAO(6dF+MGS+DR12parameters +DR14) +DR14+Ly α ) +DR14+Ly α )+CMBΩ m, .
253 0 .
314 0 .
367 0 . . +0 . − . . +0 . − . . +0 . − . . +0 . − . H .
61 66 .
74 62 .
30 67 . . +4 . − . . +2 . − . . +1 . − . . +0 . − . Ω m, h .
133 0 .
137 0 .
142 0 . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Γ /H − .
78 0 .
03 0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . χ parameters randomly chosen from within 2 σ range of theMCMC chains corresponding to different data sets. In Fig. 11, we show the H ( z ) samples within 2 σ con-fidence level for different cosmological models obtained0 H ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z H ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model I
Figure 8.
The Hubble parameter H ( z ) for the metastable DE model II obtained with different data combinations. The solidblack lines and the dashed black lines show H ( z ) from the best fit of the ΛCDM model and the metastable DE model II withthe same data set, respectively. O m ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z O m ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model II
Figure 9.
The Om diagnostic as a function of redshift for the metastable DE model II obtained with different data combinations.The solid black lines and the dashed black lines show Om ( z ) from the best fit of the ΛCDM model and the metastable DEmodel II with the same data set, respectively. q ( z ) Pantheon+BAO(6dF+MGS+DR12)
Pantheon+BAO(6dF+MGS+DR12+DR14) z q ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly ) z Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB best fit CDMbest fit model II
Figure 10.
The deceleration parameter as a function of redshift for the metastable DE model II obtained with differentdata combinations. The solid black lines and the dashed black lines show q ( z ) from the best fit of the ΛCDM model and themetastable DE model II with the same data set, respectively. z H ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB z H ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB z H ( z ) Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )Pantheon+BAO(6dF+MGS+DR12+DR14+Ly )+CMB
Figure 11.
Hubble parameter for different cosmological models constrained with two different data sets described above. Fromleft to right, we show Hubble parameter as a function of redshift for standard ΛCDM model, metastable model I and model II,respectively. H tension between CMB and BAOmeasurements from Ly α existing in ΛCDM model (seeFig. 1) become larger when compared with the resultsobtained from previous data sets (see Fig. 1 in Shafielooet al. (2017)) and metastable DE models cannot re-duce this H tension. Including the CMB distance priorto our analysis shows that both model I and model IIare consistent with ΛCDM model, while without CMBdata, the results from Pantheon in combination withBAO(6dF+MGS+DR12+DR14+Ly α ) support that DEdensity is decaying (exponentially for the model I andinto dark matter density for the model II). SUMMARYIn this work, we revisit two metastable DE modelsproposed in Shafieloo et al. (2017) confronting themwith the Pantheon SNe Ia sample, BAO measurementsderived from 6dFGS, the SDSS DR7 MGS sample, theBOSS DR12, the eBOSS DR14 and high redshift BAOmeasurement from the Ly α forest in combination withthe CMB distance prior from the final Planck release in2018.In the metastable DE models, the DE density decaysexponentially in the model I and decays into dark matterin the model II (the reverse process DM → DE is alsopermitted). The specific feature of these two modelsis that the decay rate is a constant and depends onlyon intrinsic properties of dark energy and not on otherfactors such as cosmological expansion, etc.We estimate some key cosmological parameters as-suming standard ΛCDM on the one hand, and the twometastable DE models on the other. We find that withcurrent data sets, the Ω m h tension between CMB andhigh redshift BAO measurement from Ly α becomes sig-nificant in ΛCDM and also in the model I. The modelII shows slightly better consistency. We should notethat the degeneracy direction for Ω m , vs H for themodel II is different from the model I, that makes con-straints on the derived parameter Ω m , h to agree bet-ter with CMB and high redshift BAO measurements.Marginalised probability distribution function for H in the metastable dark energy models including super- novae and all BAO data (except the Ly α BAO data)shows clear consistency with the results including PlanckCMB constraints. However, including Ly α BAO data(and without Planck CMB measurements) changes theconstraints on H dramatically, lowering it to a cen-tral value of 62 km/sec/Mpc. Since local measurementsof the Hubble constant place its value to be around73 km/sec/Mpc, the situation seems to be very conflict-ing. In fact Ly α BAO data, Planck CMB data and thelocal measurement of H , each pull our models to a dif-ferent region in parameter space. This could be due totension between different data sets. Possible resolutionsof this dilemma might lie in systematics in some of thedata, a more complicated form of the expansion history(which needs to be reconstructed carefully to satisfy allobservations) or an unconventional model of the earlyUniverse (Hazra et al. 2019).We should note here that from a statistical point ofview and to compare the analysed metastable dark en-ergy models in this paper with ΛCDM model, we donot expect that these models perform better than thestandard ΛCDM model by estimating the Bayes factor(as done in Pan et al. (2019), analysing the model pro-posed in Li & Shafieloo (2019)). The fact is that havingan extra degree of freedom in these metastable models,we have not achieved a substantial improvements in thelikelihood estimations (as shown in Table 2, 3, 4). Insuch a situation, the Bayesian analysis prefers a modelwith lower degrees of freedom (ΛCDM model). However,future data with higher precision and better control ofsystematics might change the current situation and wemight get substantially different likelihoods for thesemodels in comparison to the standard ΛCDM model.This might not be surprising as the standard ΛCDMmodel has problems fitting the low and high redshiftdata simultaneously, and with higher precision data thelikelihood for this model might get substantially worsen.This is the main reason why we should continue to studymodels that might not be currently favoured comparedto the standard model while they may have interestingphenomenological or theoretical properties.By the time we were finalising our work, a recent workby Riess et al. (2019) came out and in their work theyshowed a larger H tension between locally measurementand the value inferred from Planck
CMB and ΛCDM.It is therefore extremely important to understand thenature of these tensions.X. Li thanks Ryan E. Keeley and Benjamin L’Huillierfor valuable suggestions. X. Li thanks JingzhaoQi for kindly instructions about
Cosmomc . X. Liand A.S. would like to acknowledge the support of3the National Research Foundation of Korea (NRF-2016R1C1B2016478). X. Li was supported by the fundof Hebei Normal University. X. Li was supported bythe Strategic Priority Research Program of the ChineseAcademy of Sciences, Grant No. XDB23000000. A.A.S. was partly supported by the program KP19-270 ”Ques-tions of the origin and evolution of the Universe” of thePresidium of the Russian Academy of Sciences. Thiswork benefits from the high performance computingclusters Polaris and Seondeok at the Korea Astronomyand Space Science Institute.REFERENCES
Abazajian, K., Adelman-McCarthy, J. K., Ag¨ueros, M. A.,et al. 2004, The Astronomical Journal, 128, 502Ade, P. A., Aghanim, N., Arnaud, M., et al. 2016,Astronomy & Astrophysics, 594, A13Aghanim, N., Akrami, Y., Ashdown, M., et al. 2018, arXivpreprint arXiv:1807.06209Alam, S., Ata, M., Bailey, S., et al. 2017a, Monthly Noticesof the Royal Astronomical Society, 470, 2617Alam, U., Bag, S., & Sahni, V. 2017b, Physical Review D,95, 023524Bean, R., Carroll, S., & Trodden, M. 2005, arXiv preprintastro-ph/0510059Beutler, F., Blake, C., Colless, M., et al. 2011, MonthlyNotices of the Royal Astronomical Society, 416, 3017Cao, S., Biesiada, M., Gavazzi, R., Pi´orkowska, A., & Zhu,Z.-H. 2015, The Astrophysical Journal, 806, 185Chen, L., Huang, Q.-G., & Wang, K. 2018, arXiv preprintarXiv:1808.05724Collaboration, P., Ade, P., Aghanim, N., Armitage-Caplan,C., et al. 2014, Astron. Astrophys, 571, A16Cooke, R. J., Pettini, M., & Steidel, C. C. 2018, TheAstrophysical Journal, 855, 102Des Bourboux, H. D. M., Le Goff, J.-M., Blomqvist, M.,et al. 2017, Astronomy & Astrophysics, 608, A130Di Valentino, E., Linder, E. V., & Melchiorri, A. 2018,Physical Review D, 97, 043528Di Valentino, E., Melchiorri, A., & Mena, O. 2017, PhysicalReview D, 96, 043503Ding, X., Biesiada, M., Cao, S., Li, Z., & Zhu, Z.-H. 2015,The Astrophysical Journal Letters, 803, L22Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, TheAstrophysical Journal, 633, 560Hazra, D. K., Shafieloo, A., & Souradeep, T. 2019, Journalof Cosmology and Astroparticle Physics, 2019, 036Khosravi, N., Baghram, S., Afshordi, N., & Altamirano, N.2019, Physical Review D, 99, 103526Ko, P., Nagata, N., & Tang, Y. 2017, Physics Letters B,773, 513Komatsu, E., Dunkley, J., Nolta, M., et al. 2009, TheAstrophysical Journal Supplement Series, 180, 330Kumar, S., & Nunes, R. C. 2017, Physical Review D, 96,103511 Lewis, A., & Bridle, S. 2002, Physical Review D, 66, 103511L’Huillier, B., & Shafieloo, A. 2017, Journal of Cosmologyand Astroparticle Physics, 2017, 015Li, X., Cao, S., Zheng, X., et al. 2017, The EuropeanPhysical Journal C, 77, 677Li, X., & Shafieloo, A. 2019, The Astrophysical JournalLetters, 883Li, X.-L., Cao, S., Zheng, X.-G., Li, S., & Biesiada, M.2016, Research in Astronomy and Astrophysics, 16, 084Pan, S., Yang, W., Di Valentino, E., Shafieloo, A., &Chakraborty, S. 2019, arXiv preprint arXiv:1907.12551Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999,The Astrophysical Journal, 517, 565Poulin, V., Smith, T. L., Karwal, T., & Kamionkowski, M.2019, Physical Review Letters, 122, 221301Raveri, M., Hu, W., Hoffman, T., & Wang, L.-T. 2017,Physical Review D, 96, 103501Renk, J., Zumalac´arregui, M., Montanari, F., & Barreira,A. 2017, Journal of Cosmology and AstroparticlePhysics, 2017, 020Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., &Scolnic, D. 2019, arXiv preprint arXiv:1903.07603Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, TheAstronomical Journal, 116, 1009Riess, A. G., Macri, L. M., Hoffmann, S. L., et al. 2016,The Astrophysical Journal, 826, 56Ross, A. J., Samushia, L., Howlett, C., et al. 2015, MonthlyNotices of the Royal Astronomical Society, 449, 835Sahni, V., Shafieloo, A., & Starobinsky, A. A. 2014, TheAstrophysical Journal Letters, 793, L40Sahni, V., & Starobinsky, A. 2000, International Journal ofModern Physics D, 9, 373Scolnic, D., Jones, D., Rest, A., et al. 2017, arXiv preprintarXiv:1710.00845Shafieloo, A., Hazra, D. K., Sahni, V., & Starobinsky, A. A.2017, Monthly Notices of the Royal AstronomicalSociety, 473, 2760Shafieloo, A., L’Huillier, B., & Starobinsky, A. A. 2018,Physical Review D, 98, 083526Shanks, T., Hogarth, L., & Metcalfe, N. 2018, arXivpreprint arXiv:1810.025954