Constraints on interacting dark energy model from lensed quasars: Relieving the H_0 tension from 5.3σ to 1.7σ
CConstraints on interacting dark energy model from lensed quasars: Relieving the H tension from 5.3 σ to 1.7 σ Ling-Feng Wang, Dong-Ze He, Jing-Fei Zhang, and Xin Zhang ∗
1, 3, † Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China College of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Key Laboratory of Data Analytics and Optimization for Smart Industry(Northeastern University), Ministry of Education, Shenyang 110819, China
Measurements for strong gravitational lensing time delays between multiple images of backgroundquasars can provide an independent probe to explore the expansion history of the late-time universe.In this paper, we employ the new results of the time-delay (TD) measurements for six stronggravitational lens systems to constrain interacting dark energy (IDE) model. We mainly focus onthe model of vacuum energy (with w = −
1) interacting with cold dark matter, and consider fourtypical cases of the interaction form. Our main findings include: (i) the IDE models with Q ∝ ρ de have an advantage in alleviating the H tension between the cosmic microwave background andTD observations; (ii) when the TD data are combined with the latest local distance-ladder result,the H tension can be alleviated from 5 . σ (in the standard ΛCDM cosmology) to 1 . σ in the IDEmodel with the interaction term Q = βHρ de ; (iii) the coupling parameter β in all the considered IDEmodels are preferred to be positive around 1 σ range when the late-universe measurements (TD+SN)are used to perform constraint, implying a mild preference for the case of cold dark matter decayinginto dark energy by the late-universe observations. I. INTRODUCTION
The precise measurement of the cosmic microwavebackground (CMB) anisotropies indicates the dawn ofan era of precision cosmology [1, 2]. Six base parametersin the standard ΛCDM cosmologiy have been constrainedprecisely by the CMB observation of the Planck satellite[3]. For example, the present-day cosmic expansion rate,characterized by the Hubble constant H , is derived tobe 67 . ± . − Mpc − with an accuracy of less than1% with the Planck CMB observation [3]. Nevertheless,there are some important uncertainties in understandingthe current cosmological parameter estimation, e.g., thenuisance parameter A lens used to match CMB temper-ature fluctuations and CMB-lensing data departs fromunity, implying that there may be an unknown relation-ship between CMB anisotropy and structural growth [4];there exists an inconsistency between CMB data and cos-mic shear data in the σ -Ω m plane [5]. Because the infor-mation extracted from the CMB temperature anisotropyobservation comes from the early-time universe, clearlyunderstanding these uncertainties requires some end-to-end tests, namely, using multiple cosmological probesaimed at the late-time universe.Dark energy, as an exotic form of energy with nega-tive pressure [6–14], has been proposed to explain theaccelerated expansion of the universe [15, 16]. It makesup about 68% of the total energy density, and dominatesthe evolution of the late-time universe. Because the CMBanisotropies mainly provide a measurement of the angu-lar distance to the last scattering surface, by comparing ∗ Corresponding author † Electronic address: [email protected] the angular scale of the acoustic peaks with the soundhorizon at recombination, the CMB measurement is dif-ficult to place a strong constraint on the evolution of darkenergy. Therefore, the CMB constraints on dark energyparameters are highly degenerate in cosmological mod-els with dynamical dark energy. For the sake of break-ing the cosmological parameter degeneracies inherent inthe CMB measurement, a variety of precise late-universemeasurements are required. However, in recent years, itis found that late-universe measurements on the Hubbleconstant H are in tension with the early-universe CMBobservations.For example, the SH0ES (Supernovae H for the Equa-tion of State) team uses the Cepheid-supernova distanceladder [17] to directly measure the Hubble constant andgives a result of H = 74 . ± .
42 km s − Mpc − [18].This result is in 4 . σ tension with the one inferred fromthe Planck CMB observation. Unlike the SH0ES team,the CCHP (Carnegie–Chicago Hubble Program) collab-oration uses tip of the red giant branch (TRGB), instead of Cepheids, to calibrate supernovae in the dis-tance ladder [19], and gives a result of H = 69 . ± . − Mpc − . But, subsequently, Yuan et al. showthat the extinction of TRGB by dust in the Large Mag-ellanic Cloud using the ground-based photometry canbias the calibration of TRGB, and then they give a re-determined result of H = 72 . ± . − Mpc − [20]. In addition, SH0ES team also uses the tip ofasymptotic giant branch stars (Miras) for supernovaecalibration in the distance ladder, and gives a result of H = 73 . ± . − Mpc − [21]. For other late-timemeasurements and a more detailed discussion on the H tension, see Ref. [4].The gravitational lensing effect is a marvelous predic-tion of General Relativity. In 1964, Refsdal [22] proposeda new method to determinate the Hubble constant and a r X i v : . [ a s t r o - ph . C O ] F e b masses of galaxies. Assuming that a supernova lies be-hind a galaxy, one can obtain the angular diameter dis-tance information of the lens system by measuring thearrival time difference ∆ t for the light rays from differ-ent paths. Nevertheless, the first multiple imaged su-pernova was not discovered until 2014 [23]. More re-cent researches on lensed supernovae can be found inRefs. [24–31]. Active galactic nucleus (AGN) is anotherbackground source that can provide a sufficiently vari-able luminosity to measure the time delay [32–36]. In1979, the first strongly lensed quasar with two imageswas discovered [37], and the first robust time delays weremeasured in 1997 [38].In recent years, the COSMOGRAIL (COSmologi-cal MOnitoring of GRAvItational Lenses) team moni-tors more than 20 lensed quasars, and the H0LiCOW( H Lenses in COSMOGRAIL’s Wellspring) collabora-tion [39] analyses six gravitationally lensed quasars withmeasured time delays to extract the distance informa-tion for cosmological parameter estimation. The in-ferred H value in a spatially-flat ΛCDM cosmology is73 . +1 . − . km s − Mpc − , which is in 3 . σ tension with thePlanck CMB observation. A combination of time-delaycosmography and the SH0ES distance-ladder result canyield a 5.3 σ tension with the Planck CMB determinationof H in the ΛCDM model [39].Given the tension between determinations of H fromCMB observations and late-universe probes, apart fromthe potential systematic errors in different probes, therealso exists a possibility that the underlying cosmol-ogy describing our universe may be more complex thanthe standard ΛCDM model. Thus, in Ref. [39], somecommon extended cosmological models and the con-sistency between some mainstream observational datasets are also discussed in detail. Soon afterwards,the TDCOSMO collaboration [40] employs a sampleof seven lenses to give a constraint result of H =74 . +5 . − . km s − Mpc − based on a new hierarchicalBayesian approach, in which the mass-sheet transform isonly constrained by stellar kinematics. Moreover, Den-zel et al. [41] employ eight quadruply lensing systemsto determine the Hubble constant and obtain the resultof H = 71 . +3 . − . km s − Mpc − , by adopting modellingstrategies reflecting the systematic uncertainties of lens-ing degeneracies. We refer the reader to Refs. [40–47] formore details regarding the recent works on time-delaycosmography.Although the standard ΛCDM model has been provento be in excellent agreement with current cosmologi-cal observations [3], the theoretical problems caused bythe cosmological constant, such as the “fine-tuning” and“cosmic coincidence” problems [48, 49], are still trouble-some. In addition, with the improvement of observationaccuracy, the H tension becomes more serious and grad-ually turns into a cosmological crisis. Thus, some exten-sions to the base ΛCDM cosmology are expected to benecessary not only for the development of theory itself,but also for the interpretation of experimental results. Among various extensions, a variety of models based onthe scenario of dark energy interacting with cold darkmatter have attracted lots of attention (see Ref. [50] fora recent review). This scenario could help resolve theproblem of cosmic coincidence through an attractor so-lution [51–55]. In addition, the knowledge of dark energyand dark matter is still very scarce, and whether there ex-ists a direct, non-gravitational interaction between themhas yet to be verified. Therefore, indirectly detectingthe interaction between dark sectors using cosmologicalobservations is one of the important topics in currentcosmology [50–104]. Since the interaction between darkenergy and dark matter would influence the expansionhistory of the universe and the formation of large-scalestructure [54, 57, 62, 67, 78, 82], the relevant cosmologicalobservations could place constraints on such an interac-tion in specific interacting dark energy models.As mentioned above, Ref. [39] has discussed the con-straints on some extensions to the base ΛCDM cosmol-ogy using the time-delay cosmography alone as well asthe combinations with other mainstream cosmologicalprobes, but the analysis for interacting dark energy (IDE)scenario is still absent. Thus, as a further step along thisroute, in this paper we mainly focus on the following is-sues: (i) which type of the IDE model has advantagein relieving the H tension between the time-delay cos-mography and CMB observation; (ii) what will happento the constraints on H , when the time-delay cosmog-raphy is combined with the distance ladder result; (iii)how the time-delay cosmography affects the cosmologicalconstraints on the interaction between dark energy anddark matter.The rest of this paper is organized as follows. In Sec. II,we introduce the theoretical basis of time delay cosmog-raphy, the IDE models, and the observational data setsconsidered in this work. The calculated results are shownand discussed in Sec. III. Finally, the conclusion is givenin Sec. IV. II. METHODS AND DATAA. Time-delay cosmography
When a massive object (the lens) lies between a back-ground source and an observer, the background sourcewill be gravitationally lensed into multiple images. Thelight rays emitted from the background source corre-sponding to the different image positions will travelthrough different space-time paths. Since these pathshave different gravitational potentials and lengths, lightrays will arrive at the observer at different times depend-ing on which image it corresponds to. If the source hasflux variations, time delays between multiple images canbe measured by monitoring the lens [33–35, 105, 106].If the background source and the lens are close enough,multiple images of the same source may be formed. Therewill be different excess time delays depending on whichimage the light ray is observed for. The excess time delaybetween two images is defined by∆ t ij = D ∆ t c (cid:20) ( θ i − β ) − ψ ( θ i ) − ( θ j − β ) ψ ( θ j ) (cid:21) , (1)where θ i and θ j are the positions of images i and j inthe image plane, respectively. The lens potentials at theimage positions, ψ ( θ i ) and ψ ( θ j ), and the source position β , can be determined from a mass model of the system.The time-delay distance D ∆ t [22, 107] is defined as acombination of three angular diameter distances, D ∆ t ≡ (1 + z d ) D d D s D ds , (2)where z d is the redshift of lens, D d is the angular di-ameter distance to the lens, D s is the angular diameterdistance to the source, and D ds is the angular diameterdistance between the lens and the source. If the time de-lay ∆ t ij can be measured and an accurate lens model isavailable to determine the lens potential ψ ( θ ), then thetime-delay distance can be determined. B. Interacting dark energy model
In the context of a spatially-flat Friedmann-Roberston-Walker universe, the Friedmann equation is written as3 M H = ρ de + ρ c + ρ b + ρ r , (3)where ρ crit = 3 M H is the critical density of the uni-verse, ρ de , ρ c , ρ b , and ρ r represent the energy densitiesof dark energy, cold dark matter, baryon, and radiation,respectively.In the model of interacting dark energy, the assump-tion of some direct, non-gravitational interaction betweendark energy and cold dark matter is made. Under thisassumption, in the level of phenomenological study, theenergy conservation equations for dark energy and colddark matter are given by˙ ρ de + 3 H (1 + w ) ρ de = Q, (4)˙ ρ c + 3 Hρ c = − Q, (5)where the dot denotes the derivative with respect to thecosmic time t , w is the equation of state (EoS) parameterof dark energy, and Q describes the energy transfer ratebetween dark sectors.Since we know little about the fundamental nature ofdark energy, let alone the microscopic origin of the inter-action between dark energy and dark matter, we can onlystudy the interacting dark energy in a pure phenomeno-logical way under such circumstances [53, 63, 70, 72, 77–79, 108–110]. The form of Q is usually assumed to beproportional to the energy density of dark energy or colddark matter, or some mixture of the two [111, 112]. Theproportionality coefficient Γ has the dimension of energy, and so it is with the form of Γ = βH or Γ = βH , where β is the dimensionless coupling parameter. β = 0 indi-cates no interaction between dark energy and cold darkmatter, β > β < w = −
1, in order not to introduce more extra param-eters. If there is no interaction between dark sectors, thecase of w = − w = −
1, the corre-sponding dark energy cannot serve as a pure backgroundand it actually is not a vacuum energy in essence. Herewe do not wish to study the nature of dark energy witha purely theoretical point of view, but instead we wishto study the problem concerning dark energy in a phe-nomenological way. Thus, we call the dark energy with w = − Q as typicalexamples to make an analysis for the purpose of study-ing interacting dark energy, i.e., Q = βH ρ c (IΛCDM1), Q = βH ρ de (IΛCDM2), Q = βHρ c (IΛCDM3), and Q = βHρ de (IΛCDM4).In the IDE cosmology, when considering the dark en-ergy perturbations, the early-time super-horizon cosmo-logical perturbations will occasionally diverge (in a partof the parameter space of the model), which leads to acatastrophe in cosmology as the perturbations enter thehorizon. In order to avoid such a cosmological catastro-phe caused by the interaction between dark sectors, onehas to consider some effective schemes to properly cal-culate the perturbations of dark energy, instead of usinga conventional way of treating dark energy as a perfectfluid with negative pressure.In 2014, Li, Zhang, and Zhang [108, 109] extendedthe parameterized post-Friedmann (PPF) approach [113,114] to involve the interacting dark energy, and such anextended version of PPF method (referred to as ePPF forconvenience) can successfully avoid the perturbation di-vergence problem in the IDE cosmology. In this work,we consider the IΛCDM model, and in such a modelthe “vacuum energy” is not a true background due tothe interaction, thus we also need to consider its per-turbations. Therefore, we employ the ePPF method[108, 109] to treat the cosmological perturbations in thiswork. For the applications of the ePPF method, see alsoRefs. [110, 115, 116]. C. Observational data
We use the modified version of the Markov-ChainMonte Carlo package
CosmoMC [117] to constrain the cos-mological parameters. The observational data employedin this paper include the CMB data, the baryon acous-tic oscillation (BAO) data, the type Ia supernova (SN)data, and the time-delay cosmography data. The detailsof these data sets are listed as follows. • The CMB data: We use the Planck TT, TE, EEspectra at (cid:96) ≥
30, the low- (cid:96) temperature Commanderlikelihood, and the low- (cid:96)
SimAll EE likelihood, from thePlanck 2018 data release [3]. • The BAO data: We use the measurements from6dFGS ( z eff = 0 . z eff = 0 . z eff = 0 .
38, 0.51, and 0.61) [120]. • The SN data: We use the latest Pantheon sample,which is comprised of 1048 data points from the Pantheoncompilation [121] • The time-delay cosmography: We use the time-delaydistance data of the six strong gravitational lenses listedin Ref. [39], i.e., B1608+656 (5156 +296 − Mpc) [107, 122],RXJ1131-1231 (2096 +98 − Mpc) [123, 124], HE 0435-1223(2707 +183 − Mpc) [124, 125], SDSS 1206-4332 (5769 +589 − Mpc) [126], WFI2033-4723 (4784 +399 − Mpc) [127], andPG 1115+080 (1470 +137 − Mpc) [124].
III. RESULTS AND DISCUSSIONS
In this section, we present our calculation results indetail, and further make some analyses and discussionson them. In Sec. III A, we will discuss the H ten-sion between the early- (i.e., the CMB observations fromthe Planck mission) and late-universe observations (i.e.,the time-delay cosmography) in various IDE models. InSec. III B, we combine the time-delay cosmography withthe latest local H measurement from the SH0ES teamto give late-universe constraints on the IDE models. InSec. III C, we study the joint constraints on the IDE mod-els from the combination of the time-delay cosmographyand other mainstream cosmological probes, including theobservations of CMB, BAO, and SN. A. Constraints on interacting dark energy modelsfrom time-delay cosmography
First, we constrain the Hubble constant H in the fourIDE models with the time-delay (TD) data alone. Then,we compare the TD constraint results with those con-strained from the CMB data, and calculate the tensionsbetween these two observational data sets. The con-straint results from TD and CMB are separately listedin Table I. To show the tensions more visually, the con-straint median values of H are exhibited in Fig. 1. Inthis figure, the central values and the 1 σ error bars of H from the posterior distribution are shown.As shown in Table I, the H constraint values from TDdata alone in the IΛCDM models are 74 . +1 . − . (IΛCDM1),74 . +2 . − . (IΛCDM2), 74 . +1 . − . (IΛCDM3), and 74 . +2 . − . (IΛCDM4) km s − Mpc − . Nevertheless, the values con-strained from the CMB data alone in these four IΛCDMmodels are quite different. Specifically, for the mod-els of IΛCDM2, IΛCDM3 and IΛCDM4, the values of H are close to 67 . − Mpc − , while for the caseof IΛCDM1 ( Q ∝ H ρ c ) model, the value of H is60 . ± . − Mpc − , much smaller than the value H = 67 . ± . − Mpc − inferred from the ΛCDMmodel. H (km s Mpc ) Q = H de Q = H c Q = H de Q = H c CMBTD
FIG. 1: Constraints on H (68.3% confidence level) from TDand CMB in the IΛCDM models with different forms of Q . For the sake of clearly showing which type of IDEmodel is more effective in alleviating the H tension, wealso list the tensions between the observations of TD andCMB in the last column of Table I. For the IΛCDM2and IΛCDM4 models ( Q ∝ ρ de ), we can immediatelyfind that the H tensions can be reduced to 1 . σ and1 . σ , respectively. For the IΛCDM1 and IΛCDM3 mod-els ( Q ∝ ρ c ), the H tensions are still close to 3 . σ , sim-ilar to the case in the standard ΛCDM model [39].Through contrasting these constraint results on thewhole, we find that the IDE models with Q ∝ ρ de havean advantage in alleviating the H tension between theCMB and TD data. This is one of the main conclusionsof this work.From Fig. 1, we can clearly see that, in the case of Q = βHρ c (IΛCDM3), since the CMB constraint israther tight, the tension is actually severest among thefour cases; and in the case of Q = βH ρ c (IΛCDM1),the reason for a big tension is owing to the fact that thecentral value of H from CMB constraint is too small.For the cases of Q = βH ρ de (IΛCDM2) and Q = βHρ de (IΛCDM4), we can see that the CMB constraints on H are similar, both with mild central values and relativelylarge errors, leading to reduced tensions.In Table I, the constraint results for the coupling pa-rameter β are also listed. When the TD data are em-ployed alone, the constraint values of β are 0 . +1 . − . (IΛCDM1), 0 . +0 . − . (IΛCDM2), 0 . +0 . − . (IΛCDM3),and 0 . +0 . − . (IΛCDM4). It can be seen that the TDdata cannot well constrain the coupling parameter β inall the four IDE models. However, we find that the CMBdata alone can perform a relatively tight constraint on β in the cases of Q ∝ ρ c , i.e., IΛCDM1 and IΛCDM3. TABLE I: Fitting results (68.3% confidence level) of the Hub-ble constant H (in units of km s − Mpc − ) and the couplingparameter β in the IΛCDM models from the TD and CMBdata.Model Parameter TD CMB tensionIΛCDM1 H . +1 . − . . ± . . σβ . +1 . − . − . ± . − IΛCDM2 H . +2 . − . . ± . . σβ . +0 . − . . +0 . − . − IΛCDM3 H . +1 . − . . ± . . σβ . +0 . − . − . ± . − IΛCDM4 H . +2 . − . . ± . . σβ . +0 . − . − . +0 . − . − B. Combination of the time-delay observation andthe latest local distance-ladder measurement
Since the constraints on H within the framework ofassumed cosmological models from the time-delay cos-mography are completely independent of and comple-mentary to the latest SH0ES result, in the present sub-section we combine the TD data with the latest localmeasurement of H from the SH0ES team to give alate-universe constraint on the IDE models. The lat-est local measurement of H using the distance ladderis 74 . ± .
42 km s − Mpc − . To show the tensionsbetween TD+SH0ES and CMB more visually, the con-straint median values of H are exhibited in Fig. 2 withthe form of central points and corresponding 1 σ errorbars. The constraint values from TD+SH0ES and CMBare separately listed in Table. II.From Table. II, we find that the H values con-strained by TD+SH0ES are 74 . ± . . ± . . ± . . ± . − Mpc − . The corresponding H ten-sions in the IDE models are 3 . σ (IΛCDM1), 1 . σ (IΛCDM2), 3 . σ (IΛCDM3), and 1 . σ (IΛCDM4).Compared with the 5 . σ tension in the ΛCDM modelaccording to the latest H0LICOW results [39], the H tensions in the IDE models all become smaller to someextent. Particularly, in the IΛCDM2 and IΛCDM4 model, the H tension can be alleviated from 5 . σ (in the standardΛCDM model) to 1 . σ and 1 . σ , respectively, which onceagain indicates that the IDE models with Q ∝ ρ de aremore preferred in relieving the H tension. H (km s Mpc ) Q = H de Q = H c Q = H de Q = H c CMBTD+SH0ES
FIG. 2: Constraints on H (68.3% confidence level) fromTD+SH0ES and CMB in the IΛCDM models with differentforms Q .TABLE II: Fitting results (68.3% confidence level) of the Hub-ble constant H (in units of km s − Mpc − ) in the IΛCDMmodels from the TD+SH0ES and CMB data.Model Parameter TD+SH0ES CMB tensionIΛCDM1 H . ± . . ± . . σ IΛCDM2 H . ± . . ± . . σ IΛCDM3 H . ± . . ± . . σ IΛCDM4 H . ± . . ± . . σ C. Combination of time-delay observation andother cosmological observations
Despite that the TD data are primarily sensitive to H but weakly sensitive to other cosmological parameters,the constraints from TD are still supposed to be highlycomplementary to other cosmological probes. In this sub-section, we combine the TD data with other mainstreamobservational data sets to provide joint constraints onthe IDE models. The constraint results are listed in Ta-ble. III.We first explore what would happen to the constraintson H when the SN data are combined with the TD data.The H values constrained from TD+SN are 74 . ± . . +2 . − . (IΛCDM2), 73 . +1 . − . (IΛCDM3),and 74 . +1 . − . (IΛCDM4) km s − Mpc − . With regardto the coupling parameter β , when the SN data are com-bined, the constrained values from TD+SN are 0 . +0 . − . (IΛCDM1), 0 . +0 . − . (IΛCDM2), 0 . +0 . − . (IΛCDM3),and 0 . +0 . − . (IΛCDM4). It is found that the data com-bination TD+SN slightly favors a positive β in the fourIDE models.As a high-precision probe of the early-time universe,the CMB observations from the Planck mission usheredthe era of precise cosmology. However, dark energy thatdrives the accelerating expansion of the universe mainlydominates the late-time universe, so the capability ofCMB to constrain the dark energy parameters is ratherweak, leading to strong cosmological parameter degen-eracies inherent in the CMB data. A common approachto break the degeneracies between parameters is to com-bine the CMB data with late-time observational data(e.g., BAO, SN, and TD, etc). Although in principlethere is an inconsistent between the TD data and theCMB data in constraining H , here we still give the jointconstraints from CMB+TD for completeness of the anal-ysis.When employing the data combination CMB+TD,the constraint values of H are 72 . ± . . +1 . − . (IΛCDM2), 70 . ± . . +1 . − . (IΛCDM4) km s − Mpc − . The constraint values ofthe coupling parameter β are 0 . ± .
069 (IΛCDM1),0 . +0 . − . (IΛCDM2), 0 . ± . . +0 . − . (IΛCDM4). In addition, we also combinethe TD data with the current mainstream observa-tions, i.e., the CMB, BAO, and SN data sets, to con-strain the IΛCDM models. The constraint values of H are 69 . ± .
76 (IΛCDM1), 69 . ± .
78 (IΛCDM2),68 . ± .
61 (IΛCDM3), and 69 . ± .
77 (IΛCDM4)km s − Mpc − . The constraint values of the couplingparameter β are 0 . ± .
041 (IΛCDM1), 0 . ± . . ± . . ± . H and β , the addition of the time-delay cosmography couldprovide strong complementarity that improves the cos-mological constraint accuracies especially for the param-eter β , which is also consistent with the case of theCMB+BAO+SN+TD data combination. IV. CONCLUSION
Time-delay cosmography, as a complementary cosmo-logical probe independent of the CMB observation, pro-vides an important complement to the low-redshift cos-mological observations. The light rays emitted fromthe same source could arrive at different times, due tothe gravitational lensing effect. The time-delay effect of strong gravitational lensing codes the information of thetime-delay distance D ∆ t , which is defined as a combi-nation of angular diameter distances. In this work, weinvestigate the implications of the time-delay cosmogra-phy on the IDE models. We first employed the time-delay distance data to constrain the models of IΛCDM1( Q = βH ρ c ), IΛCDM2 ( Q = βH ρ de ), IΛCDM3 ( Q = βHρ c ), and IΛCDM4 ( Q = βHρ de ). Then, we combinethe TD data with the latest SH0ES result as the late-universe measurement to make a comparison for the H tensions in different IΛCDM models. Next, we also con-sider the TD+SN data combination to demonstrate theimplications of low-redshift observational constraints onthe coupling parameter β . Finally, we discuss the con-straint results from the combination of the TD data andother current mainstream observations, i.e., the CMB,BAO and SN data sets. The main findings from ouranalyses are summarized as follows. • When the TD data alone are employed, the H ten-sions between the CMB and TD data are reduced tobe 1 . σ and 1 . σ in the IΛCDM2 and IΛCDM4models, respectively, which implies that the IDEscenarios with the interaction term Q ∝ ρ de seemto have an advantage in alleviating the H tensionbetween the CMB observation and the time-delaycosmography. • When the TD data are combined with the latestSH0ES result, the H tensions can be relieved from5 . σ (in the ΛCDM model) to about 1 . σ in theIΛCDM2 and IΛCDM4 models, which once againindicates that the IDE models with Q ∝ ρ de seemto be more preferred in relieving the H tension. • For the coupling parameter β in the IDE models,it is very poorly constrained when the TD data areemployed alone. When the TD data are combinedwith the SN data, a positive β is slightly favored inall the four IΛCDM models, which indicates thatthe late-universe observations seem to support thecase of cold dark matter decaying into vacuum en-ergy. Acknowledgments
We are very grateful to Ze-Wei Zhao for fruitful discus-sions. This work was supported by the National Natu-ral Science Foundation of China (Grants Nos. 11975072,11835009, 11875102, and 11690021), the Liaoning Revi-talization Talents Program (Grant No. XLYC1905011),the Fundamental Research Funds for the Central Univer-sities (Grant No. N2005030), and the National Programfor Support of Top-Notch Young Professionals (GrantNo. W02070050).
TABLE III: Fitting results (68.3% confidence level) of Hubble constant H (in the unit of km s − Mpc − ) and couplingparameter β in the IΛCDM models from the data combinations of CMB+TD, SN+TD, and CMB+BAO+SN+TD.Model Parameter SN+TD CMB+TD CMB+BAO+SN+TDIΛCDM1 H . +2 . − . . ± . . ± . β . +0 . − . . ± .
069 0 . ± . H . +2 . − . . +1 . − . . ± . β . +0 . − . . +0 . − . . ± . H . +1 . − . . ± . . ± . β . +0 . − . . ± . . ± . H . +1 . − . . +1 . − . . ± . β . +0 . − . . +0 . − . . ± . WMAP
Collaboration, D. Spergel et al. , “First yearWilkinson Microwave Anisotropy Probe (WMAP)observations: Determination of cosmologicalparameters,”
Astrophys. J. Suppl. (2003) 175–194, arXiv:astro-ph/0302209 .[2]
WMAP
Collaboration, C. Bennett et al. , “First yearWilkinson Microwave Anisotropy Probe (WMAP)observations: Preliminary maps and basic results,”
Astrophys. J. Suppl. (2003) 1–27, arXiv:astro-ph/0302207 .[3]
Planck
Collaboration, N. Aghanim et al. , “Planck2018 results. VI. Cosmological parameters,”
Astron.Astrophys. (2020) A6, arXiv:1807.06209[astro-ph.CO] .[4] L. Verde, T. Treu, and A. G. Riess, “Tensions betweenthe Early and the Late Universe,” in
NatureAstronomy 2019 , vol. 3, p. 891. 2019. arXiv:1907.10625 [astro-ph.CO] .[5] W. Lin and M. Ishak, “Cosmological discordances II:Hubble constant, Planck and large-scale-structure datasets,”
Phys. Rev.
D96 no. 8, (2017) 083532, arXiv:1708.09813 [astro-ph.CO] .[6] V. Sahni and A. Starobinsky, “Reconstructing DarkEnergy,”
Int. J. Mod. Phys. D (2006) 2105–2132, arXiv:astro-ph/0610026 [astro-ph] .[7] K. Bamba, S. Capozziello, S. Nojiri, and S. D.Odintsov, “Dark energy cosmology: the equivalentdescription via different theoretical models andcosmography tests,” Astrophys. Space Sci. (2012)155–228, arXiv:1205.3421 [gr-qc] .[8] S. Weinberg, “The Cosmological Constant Problem,”
Rev. Mod. Phys. (1989) 1–23. [,569(1988)].[9] P. J. E. Peebles and B. Ratra, “The CosmologicalConstant and Dark Energy,” Rev. Mod. Phys. (2003) 559–606, arXiv:astro-ph/0207347[astro-ph] . [,592(2002)]. [10] E. J. Copeland, M. Sami, and S. Tsujikawa,“Dynamics of dark energy,” Int. J. Mod. Phys. D (2006) 1753–1936, arXiv:hep-th/0603057 [hep-th] .[11] J. Frieman, M. Turner, and D. Huterer, “Dark Energyand the Accelerating Universe,” Ann. Rev. Astron.Astrophys. (2008) 385–432, arXiv:0803.0982[astro-ph] .[12] V. Sahni, “Reconstructing the properties of darkenergy,” Prog. Theor. Phys. Suppl. (2008)110–120.[13] M. Li, X.-D. Li, S. Wang, and Y. Wang, “DarkEnergy,”
Commun. Theor. Phys. (2011) 525–604, arXiv:1103.5870 [astro-ph.CO] .[14] M. Kamionkowski, “Dark Matter and Dark Energy,”in Amazing Light: Visions for Discovery: AnInternational Symposium in Honor of the 90thBirthday Years of Charles H. Townes Berkeley,California, October 6-8, 2005 . 2007. arXiv:0706.2986[astro-ph] .[15]
Supernova Search Team
Collaboration, A. G. Riess et al. , “Observational evidence from supernovae for anaccelerating universe and a cosmological constant,”
Astron. J. (1998) 1009–1038, arXiv:astro-ph/9805201 [astro-ph] .[16]
Supernova Cosmology Project
Collaboration,S. Perlmutter et al. , “Measurements of Ω and Λ from42 high redshift supernovae,”
Astrophys. J. (1999)565–586, arXiv:astro-ph/9812133 [astro-ph] .[17] A. G. Riess et al. , “A 2.4% Determination of the LocalValue of the Hubble Constant,”
Astrophys. J. no. 1, (2016) 56, arXiv:1604.01424 [astro-ph.CO] .[18] A. G. Riess et al. , “Milky Way Cepheid Standards forMeasuring Cosmic Distances and Application to GaiaDR2: Implications for the Hubble Constant,”
Astrophys. J. no. 2, (2018) 126, arXiv:1804.10655 [astro-ph.CO] . [19] W. L. Freedman et al. , “The Carnegie-Chicago HubbleProgram. VIII. An Independent Determination of theHubble Constant Based on the Tip of the Red GiantBranch,” arXiv:1907.05922 [astro-ph.CO] .[20] W. Yuan, A. G. Riess, L. M. Macri, S. Casertano, andD. Scolnic, “Consistent Calibration of the Tip of theRed Giant Branch in the Large Magellanic Cloud onthe Hubble Space Telescope Photometric System and aRe-determination of the Hubble Constant,” Astrophys.J. (2019) 61, arXiv:1908.00993 [astro-ph.GA] .[21] C. D. Huang, A. G. Riess, W. Yuan, L. M. Macri,N. L. Zakamska, S. Casertano, P. A. Whitelock, S. L.Hoffmann, A. V. Filippenko, and D. Scolnic, “HubbleSpace Telescope Observations of Mira Variables in theType Ia Supernova Host NGC 1559: An AlternativeCandle to Measure the Hubble Constant,” arXiv:1908.10883 [astro-ph.CO] .[22] S. Refsdal, “On the possibility of determining Hubble’sparameter and the masses of galaxies from thegravitational lens effect,”
Mon. Not. Roy. Astron. Soc. (1964) 307.[23] P. L. Kelly et al. , “Multiple Images of a HighlyMagnified Supernova Formed by an Early-TypeCluster Galaxy Lens,”
Science (2015) 1123, arXiv:1411.6009 [astro-ph.CO] .[24] R. M. Quimby, M. Oguri, A. More, S. More, T. J.Moriya, M. C. Werner, M. Tanaka, G. Folatelli, M. C.Bersten, and K. Nomoto, “Detection of theGravitational Lens Magnifying a Type Ia Supernova,”
Science no. 6, (2014) 396–399, arXiv:1404.6014[astro-ph.CO] .[25] C. Grillo et al. , “The Story of Supernova “refsdal”Told by Muse,”
Astrophys. J. no. 2, (2016) 78, arXiv:1511.04093 [astro-ph.GA] .[26] S. A. Rodney et al. , “SN Refsdal : Photometry andTime Delay Measurements of the First Einstein CrossSupernova,”
Astrophys. J. no. 1, (2016) 50, arXiv:1512.05734 [astro-ph.CO] .[27] P. L. Kelly et al. , “Deja Vu All Over Again: TheReappearance of Supernova Refsdal,”
Astrophys. J.Lett. no. 1, (2016) L8, arXiv:1512.04654[astro-ph.CO] .[28] R. Kawamata, M. Oguri, M. Ishigaki, K. Shimasaku,and M. Ouchi, “Precise Strong Lensing Mass Modelingof Four Hubble frontier Field Clusters and a Sample ofMagnified High-redshift Galaxies,”
Astrophys. J. no. 2, (2016) 114, arXiv:1510.06400 [astro-ph.GA] .[29] T. Treu et al. , ““Refsdal” Meets Popper: ComparingPredictions of the Re-appearance of the MultiplyImaged Supernova Behind MACSJ1149.5+2223,”
Astrophys. J. no. 1, (2016) 60, arXiv:1510.05750[astro-ph.CO] .[30] A. Goobar et al. , “iPTF16geu: A multiply imaged,gravitationally lensed type Ia supernova,”
Science (2017) 291–295, arXiv:1611.00014 [astro-ph.CO] .[31] A. More, S. H. Suyu, M. Oguri, S. More, and C.-H.Lee, “Interpreting the strongly lensed supernovaiPTF16geu: time delay predictions, microlensing, andlensing rates,”
Astrophys. J. Lett. no. 2, (2017)L25, arXiv:1611.04866 [astro-ph.CO] .[32] C. Vanderriest, J. Schneider, G. Herpe, M. Chevreton,M. Moles, and G. Wlerick, “The value of the timedelay delta T (A,B) for the ’double’ quasar 0957+561from optical photometric monitoring.,”
Astron. Astrophys. (1989) 1.[33] P. L. Schechter et al. , “The Quadruple gravitationallens PG1115+080: Time delays and models,”
Astrophys. J. Lett. (1997) L85–L88, arXiv:astro-ph/9611051 .[34] C. Fassnacht, T. Pearson, A. Readhead, I. Browne,L. Koopmans, S. Myers, and P. Wilkinson, “Adetermination of h 0 with the class gravitational lensb1608+656: I. time delay measurements with the vla,”
Astrophys. J. (1999) 498, arXiv:astro-ph/9907257 .[35] C. S. Kochanek, N. Morgan, E. Falco, B. McLeod,J. Winn, J. Dembicky, and B. Ketzeback, “The Timedelays of gravitational lens HE0435-1223: AnEarly-type galaxy with a rising rotation curve,”
Astrophys. J. (2006) 47–61, arXiv:astro-ph/0508070 .[36] A. Eigenbrod, F. Courbin, S. Dye, G. Meylan,D. Sluse, P. Saha, C. Vuissoz, and P. Magain,“Cosmograil: the cosmological monitoring ofgravitational lenses. 2. sdss j0924+0219: the redshift ofthe lensing galaxy, the quasar spectral variability andthe Einstein rings,”
Astron. Astrophys. (2006)747, arXiv:astro-ph/0510641 .[37] D. Walsh, R. F. Carswell, and R. J. Weymann, “0957+ 561 A, B - Twin quasistellar objects or gravitationallens,”
Nature (1979) 381–384.[38] T. Kundic et al. , “A Robust determination of the timedelay in 0957+561a,b and a measurement of the globalvalue of Hubble’s constant,”
Astrophys. J. (1997)75, arXiv:astro-ph/9610162 .[39] K. C. Wong et al. , “H0LiCOW – XIII. A 2.4 per centmeasurement of H0 from lensed quasars: 5.3 σ tensionbetween early- and late-Universe probes,” Mon. Not.Roy. Astron. Soc. no. 1, (2020) 1420–1439, arXiv:1907.04869 [astro-ph.CO] .[40] S. Birrer et al. , “TDCOSMO - IV. Hierarchicaltime-delay cosmography – joint inference of theHubble constant and galaxy density profiles,”
Astron.Astrophys. (2020) A165, arXiv:2007.02941[astro-ph.CO] .[41] P. Denzel, J. P. Coles, P. Saha, and L. L. R. Williams,“The Hubble constant from eight time-delay galaxylenses,” arXiv:2007.14398 [astro-ph.CO] .[42] Z. S. Greene et al. , “Improving the precision oftime-delay cosmography with observations of galaxiesalong the line of sight,”
Astrophys. J. (2013) 39, arXiv:1303.3588 [astro-ph.CO] .[43] I. Jee, E. Komatsu, S. H. Suyu, and D. Huterer,“Time-delay Cosmography: Increased Leverage withAngular Diameter Distances,”
JCAP (2016) 031, arXiv:1509.03310 [astro-ph.CO] .[44] T. E. Collett and S. D. Cunnington, “Observationalselection biases in time-delay strong lensing and theirimpact on cosmography,” Mon. Not. Roy. Astron. Soc. no. 3, (2016) 3255–3264, arXiv:1605.08341[astro-ph.CO] .[45] A. J. Shajib, T. Treu, and A. Agnello, “Improvingtime-delay cosmography with spatially resolvedkinematics,”
Mon. Not. Roy. Astron. Soc. no. 1,(2018) 210–226, arXiv:1709.01517 [astro-ph.CO] .[46] A. Yıldırım, S. H. Suyu, and A. Halkola, “Time-delaycosmographic forecasts with strong lensing and JWSTstellar kinematics,”
Mon. Not. Roy. Astron. Soc. no. 4, (2020) 4783–4807, arXiv:1904.07237[astro-ph.CO] .[47] M. Millon et al. , “TDCOSMO. I. An exploration ofsystematic uncertainties in the inference of H fromtime-delay cosmography,” Astron. Astrophys. (2020) A101, arXiv:1912.08027 [astro-ph.CO] .[48] V. Sahni and A. A. Starobinsky, “The Case for apositive cosmological Lambda term,”
Int. J. Mod.Phys. D (2000) 373–444, arXiv:astro-ph/9904398[astro-ph] .[49] R. Bean, S. M. Carroll, and M. Trodden, “Insights intodark energy: interplay between theory andobservation,” arXiv:astro-ph/0510059 .[50] B. Wang, E. Abdalla, F. Atrio-Barandela, andD. Pavon, “Dark Matter and Dark EnergyInteractions: Theoretical Challenges, CosmologicalImplications and Observational Signatures,” Rept.Prog. Phys. no. 9, (2016) 096901, arXiv:1603.08299 [astro-ph.CO] .[51] D. Comelli, M. Pietroni, and A. Riotto, “Dark energyand dark matter,” Phys. Lett. B (2003) 115–120, arXiv:hep-ph/0302080 [hep-ph] .[52] R.-G. Cai and A. Wang, “Cosmology with interactionbetween phantom dark energy and dark matter andthe coincidence problem,”
JCAP (2005) 002, arXiv:hep-th/0411025 [hep-th] .[53] X. Zhang, “Coupled quintessence in a power-law caseand the cosmic coincidence problem,”
Mod. Phys. Lett.A (2005) 2575, arXiv:astro-ph/0503072 .[54] J.-H. He and B. Wang, “Effects of the interactionbetween dark energy and dark matter on cosmologicalparameters,” JCAP (2008) 010, arXiv:0801.4233 [astro-ph] .[55] J.-H. He, B. Wang, and P. Zhang, “The Imprint of theinteraction between dark sectors in large scale cosmicmicrowave background anisotropies,”
Phys. Rev. D (2009) 063530, arXiv:0906.0677 [gr-qc] .[56] L. Amendola, “Coupled quintessence,” Phys. Rev. D (2000) 043511, arXiv:astro-ph/9908023[astro-ph] .[57] L. Amendola and D. Tocchini-Valentini, “Baryon biasand structure formation in an accelerating universe,” Phys. Rev. D (2002) 043528, arXiv:astro-ph/0111535 [astro-ph] .[58] W. Zimdahl, “Interacting dark energy andcosmological equations of state,” Int. J. Mod. Phys. D (2005) 2319–2326, arXiv:gr-qc/0505056 [gr-qc] .[59] X. Zhang, F.-Q. Wu, and J. Zhang, “A Newgeneralized Chaplygin gas as a scheme for unificationof dark energy and dark matter,” JCAP (2006)003, arXiv:astro-ph/0411221 [astro-ph] .[60] B. Wang, J. Zang, C.-Y. Lin, E. Abdalla, andS. Micheletti, “Interacting Dark Energy and DarkMatter: Observational Constraints from CosmologicalParameters,”
Nucl. Phys. B (2007) 69–84, arXiv:astro-ph/0607126 [astro-ph] .[61] Z.-K. Guo, N. Ohta, and S. Tsujikawa, “Probing theCoupling between Dark Components of the Universe,”
Phys. Rev. D (2007) 023508, arXiv:astro-ph/0702015 [ASTRO-PH] .[62] O. Bertolami, F. Gil Pedro, and M. Le Delliou, “DarkEnergy-Dark Matter Interaction and the Violation ofthe Equivalence Principle from the Abell ClusterA586,” Phys. Lett. B (2007) 165–169, arXiv:astro-ph/0703462 [ASTRO-PH] .[63] J. Zhang, H. Liu, and X. Zhang, “Statefinder diagnosisfor the interacting model of holographic dark energy,”
Phys. Lett. B (2008) 26–33, arXiv:0705.4145[astro-ph] .[64] C. G. Boehmer, G. Caldera-Cabral, R. Lazkoz, andR. Maartens, “Dynamics of dark energy with acoupling to dark matter,”
Phys. Rev. D (2008)023505, arXiv:0801.1565 [gr-qc] .[65] J. Valiviita, E. Majerotto, and R. Maartens,“Instability in interacting dark energy and dark matterfluids,” JCAP (2008) 020, arXiv:0804.0232[astro-ph] .[66] J.-H. He, B. Wang, and Y. P. Jing, “Effects of darksectors’ mutual interaction on the growth ofstructures,”
JCAP (2009) 030, arXiv:0902.0660[gr-qc] .[67] K. Koyama, R. Maartens, and Y.-S. Song, “Velocitiesas a probe of dark sector interactions,”
JCAP (2009) 017, arXiv:0907.2126 [astro-ph.CO] .[68] J.-Q. Xia, “Constraint on coupled dark energy modelsfrom observations,”
Phys. Rev. D (2009) 103514, arXiv:0911.4820 [astro-ph.CO] .[69] M. Li, X.-D. Li, S. Wang, Y. Wang, and X. Zhang,“Probing interaction and spatial curvature in theholographic dark energy model,” JCAP (2009)014, arXiv:0910.3855 [astro-ph.CO] .[70] L. Zhang, J. Cui, J. Zhang, and X. Zhang, “Interactingmodel of new agegraphic dark energy: Cosmologicalevolution and statefinder diagnostic,”
Int. J. Mod.Phys. D (2010) 21–35, arXiv:0911.2838[astro-ph.CO] .[71] H. Wei, “Cosmological Constraints on theSign-Changeable Interactions,” Commun. Theor. Phys. (2011) 972–980, arXiv:1010.1074 [gr-qc] .[72] Y. Li, J. Ma, J. Cui, Z. Wang, and X. Zhang,“Interacting model of new agegraphic dark energy:observational constraints and age problem,” Sci. ChinaPhys. Mech. Astron. (2011) 1367–1377, arXiv:1011.6122 [astro-ph.CO] .[73] J.-H. He, B. Wang, and E. Abdalla, “Testing theinteraction between dark energy and dark matter vialatest observations,” Phys. Rev. D (2011) 063515, arXiv:1012.3904 [astro-ph.CO] .[74] Y.-H. Li and X. Zhang, “Running coupling: Does thecoupling between dark energy and dark matter changesign during the cosmological evolution?,” Eur. Phys. J.C (2011) 1700, arXiv:1103.3185 [astro-ph.CO] .[75] T.-F. Fu, J.-F. Zhang, J.-Q. Chen, and X. Zhang,“Holographic Ricci dark energy: Interacting model andcosmological constraints,” Eur. Phys. J. C (2012)1932, arXiv:1112.2350 [astro-ph.CO] .[76] Z. Zhang, S. Li, X.-D. Li, X. Zhang, and M. Li,“Revisit of the Interaction between Holographic DarkEnergy and Dark Matter,” JCAP (2012) 009, arXiv:1204.6135 [astro-ph.CO] .[77] J. Zhang, L. Zhao, and X. Zhang, “Revisiting theinteracting model of new agegraphic dark energy,”
Sci.China Phys. Mech. Astron. (2014) 387–392, arXiv:1306.1289 [astro-ph.CO] .[78] Y.-H. Li and X. Zhang, “Large-scale stable interactingdark energy model: Cosmological perturbations andobservational constraints,” Phys. Rev. D no. 8,(2014) 083009, arXiv:1312.6328 [astro-ph.CO] . [79] J.-J. Geng, Y.-H. Li, J.-F. Zhang, and X. Zhang,“Redshift drift exploration for interacting darkenergy,” Eur. Phys. J. C no. 8, (2015) 356, arXiv:1501.03874 [astro-ph.CO] .[80] J.-L. Cui, L. Yin, L.-F. Wang, Y.-H. Li, and X. Zhang,“A closer look at interacting dark energy withstatefinder hierarchy and growth rate of structure,” JCAP no. 09, (2015) 024, arXiv:1503.08948[astro-ph.CO] .[81] R. Murgia, S. Gariazzo, and N. Fornengo, “Constraintson the Coupling between Dark Energy and DarkMatter from CMB data,”
JCAP no. 04, (2016)014, arXiv:1602.01765 [astro-ph.CO] .[82] A. Pourtsidou and T. Tram, “Reconciling CMB andstructure growth measurements with dark energyinteractions,”
Phys. Rev. D no. 4, (2016) 043518, arXiv:1604.04222 [astro-ph.CO] .[83] A. A. Costa, X.-D. Xu, B. Wang, and E. Abdalla,“Constraints on interacting dark energy models fromPlanck 2015 and redshift-space distortion data,” JCAP no. 01, (2017) 028, arXiv:1605.04138[astro-ph.CO] .[84] J. Solˇs € Peracaula, J. de Cruz PˇsˇSrez, andA. Gˇs ® mez-Valent, “Dynamical dark energy vs. Λ =const in light of observations,” EPL no. 3, (2018)39001, arXiv:1606.00450 [gr-qc] .[85] L. Feng and X. Zhang, “Revisit of the interactingholographic dark energy model after Planck 2015,”
JCAP no. 08, (2016) 072, arXiv:1607.05567[astro-ph.CO] .[86] D.-M. Xia and S. Wang, “Constraining interactingdark energy models with latest cosmologicalobservations,”
Mon. Not. Roy. Astron. Soc. no. 1,(2016) 952–956, arXiv:1608.04545 [astro-ph.CO] .[87] C. van de Bruck, J. Mifsud, and J. Morrice, “Testingcoupled dark energy models with their cosmologicalbackground evolution,”
Phys. Rev. D no. 4, (2017)043513, arXiv:1609.09855 [astro-ph.CO] .[88] J. Solˇs € , “Cosmological constant vis-a-vis dynamicalvacuum: bold challenging the ΛCDM,” Int. J. Mod.Phys. A no. 23, (2016) 1630035, arXiv:1612.02449[astro-ph.CO] .[89] S. Kumar and R. C. Nunes, “Echo of interactions inthe dark sector,” Phys. Rev. D no. 10, (2017)103511, arXiv:1702.02143 [astro-ph.CO] .[90] J. Solˇs € Peracaula, J. d. C. Perez, andA. Gomez-Valent, “Possible signals of vacuumdynamics in the Universe,”
Mon. Not. Roy. Astron.Soc. no. 4, (2018) 4357–4373, arXiv:1703.08218[astro-ph.CO] .[91] W. Yang, S. Vagnozzi, E. Di Valentino, R. C. Nunes,S. Pan, and D. F. Mota, “Listening to the sound ofdark sector interactions with gravitational wavestandard sirens,”
JCAP (2019) 037, arXiv:1905.08286 [astro-ph.CO] .[92] E. Di Valentino, A. Melchiorri, O. Mena, andS. Vagnozzi, “Nonminimal dark sector physics andcosmological tensions,” Phys. Rev. D no. 6, (2020)063502, arXiv:1910.09853 [astro-ph.CO] .[93] E. Di Valentino, A. Melchiorri, O. Mena, andS. Vagnozzi, “Interacting dark energy in the early2020s: A promising solution to the H and cosmicshear tensions,” Phys. Dark Univ. (2020) 100666, arXiv:1908.04281 [astro-ph.CO] . [94] R.-Y. Guo, J.-F. Zhang, and X. Zhang, “Exploringneutrino mass and mass hierarchy in the scenario ofvacuum energy interacting with cold dark matte,” Chin. Phys. C no. 9, (2018) 095103, arXiv:1803.06910 [astro-ph.CO] .[95] R.-Y. Guo, J.-F. Zhang, and X. Zhang, “Can the H tension be resolved in extensions to ΛCDMcosmology?,” JCAP (2019) 054, arXiv:1809.02340[astro-ph.CO] .[96] H.-L. Li, L. Feng, J.-F. Zhang, and X. Zhang, “Modelsof vacuum energy interacting with cold dark matter:Constraints and comparison,” Sci. China Phys. Mech.Astron. no. 12, (2019) 120411, arXiv:1812.00319[astro-ph.CO] .[97] H.-L. Li, D.-Z. He, J.-F. Zhang, and X. Zhang,“Quantifying the impacts of future gravitational-wavedata on constraining interacting dark energy,” JCAP (2020) 038, arXiv:1908.03098 [astro-ph.CO] .[98] L. Feng, D.-Z. He, H.-L. Li, J.-F. Zhang, andX. Zhang, “Constraints on active and sterile neutrinosin an interacting dark energy cosmology,” Sci. ChinaPhys. Mech. Astron. no. 9, (2020) 290404, arXiv:1910.03872 [astro-ph.CO] .[99] L. Feng, H.-L. Li, J.-F. Zhang, and X. Zhang,“Exploring neutrino mass and mass hierarchy ininteracting dark energy models,” Sci. China Phys.Mech. Astron. no. 2, (2020) 220401, arXiv:1903.08848 [astro-ph.CO] .[100] W. Yang, S. Pan, E. Di Valentino, R. C. Nunes,S. Vagnozzi, and D. F. Mota, “Tale of stableinteracting dark energy, observational signatures, andthe H tension,” JCAP (2018) 019, arXiv:1805.08252 [astro-ph.CO] .[101] S. Vagnozzi, “New physics in light of the H tension:An alternative view,” Phys. Rev. D no. 2, (2020)023518, arXiv:1907.07569 [astro-ph.CO] .[102] M. Zhang, B. Wang, J.-Z. Qi, Y. Xu, J.-F. Zhang, andX. Zhang, “Prospects for constraining interacting darkenergy model with 21 cm intensity mappingexperiments,” arXiv:2102.03979 [astro-ph.CO] .[103] L.-Y. Gao, S.-S. Xue, and X. Zhang, “Relieving the H tension with a new interacting dark energy model,” arXiv:2101.10714 [astro-ph.CO] .[104] H.-L. Li, J.-F. Zhang, and X. Zhang, “Constraints onneutrino mass in the scenario of vacuum energyinteracting with cold dark matter after Planck 2018,” Commun. Theor. Phys. no. 12, (2020) 125401, arXiv:2005.12041 [astro-ph.CO] .[105] C. Fassnacht, E. Xanthopoulos, L. Koopmans, andD. Rusin, “A Determination of H(O) with the classgravitational lens B1608+656. 3. A Significantimprovement in the precision of the time delaymeasurements,” Astrophys. J. (2002) 823–835, arXiv:astro-ph/0208420 .[106] F. Courbin et al. , “COSMOGRAIL: the COSmologicalMOnitoring of GRAvItational Lenses IX. Time delays,lens dynamics and baryonic fraction in HE 0435-1223,”
Astron. Astrophys. (2011) A53, arXiv:1009.1473[astro-ph.CO] .[107] S. Suyu, P. Marshall, M. Auger, S. Hilbert,R. Blandford, L. Koopmans, C. Fassnacht, andT. Treu, “Dissecting the Gravitational LensB1608+656. II. Precision Measurements of the HubbleConstant, Spatial Curvature, and the Dark Energy Equation of State,”
Astrophys. J. (2010) 201–221, arXiv:0910.2773 [astro-ph.CO] .[108] Y.-H. Li, J.-F. Zhang, and X. Zhang, “ParametrizedPost-Friedmann Framework for Interacting DarkEnergy,”
Phys. Rev. D no. 6, (2014) 063005, arXiv:1404.5220 [astro-ph.CO] .[109] Y.-H. Li, J.-F. Zhang, and X. Zhang, “Exploring thefull parameter space for an interacting dark energymodel with recent observations including redshift-spacedistortions: Application of the parametrizedpost-Friedmann approach,” Phys. Rev. D no. 12,(2014) 123007, arXiv:1409.7205 [astro-ph.CO] .[110] Y.-H. Li, J.-F. Zhang, and X. Zhang, “Testing modelsof vacuum energy interacting with cold dark matter,” Phys. Rev. D no. 2, (2016) 023002, arXiv:1506.06349 [astro-ph.CO] .[111] L. Amendola, “Scaling solutions in general nonminimalcoupling theories,” Phys. Rev. D (1999) 043501, arXiv:astro-ph/9904120 [astro-ph] .[112] A. P. Billyard and A. A. Coley, “Interactions in scalarfield cosmology,” Phys. Rev. D (2000) 083503, arXiv:astro-ph/9908224 .[113] W. Hu, “Parametrized Post-Friedmann Signatures ofAcceleration in the CMB,” Phys. Rev. D (2008)103524, arXiv:0801.2433 [astro-ph] .[114] W. Fang, W. Hu, and A. Lewis, “Crossing thePhantom Divide with Parameterized Post-FriedmannDark Energy,” Phys. Rev. D (2008) 087303, arXiv:0808.3125 [astro-ph] .[115] X. Zhang, “Probing the interaction between darkenergy and dark matter with the parametrizedpost-Friedmann approach,” Sci. China Phys. Mech.Astron. no. 5, (2017) 050431, arXiv:1702.04564[astro-ph.CO] .[116] L. Feng, Y.-H. Li, F. Yu, J.-F. Zhang, and X. Zhang,“Exploring interacting holographic dark energy in aperturbed universe with parameterizedpost-Friedmann approach,” Eur. Phys. J. C no. 10,(2018) 865, arXiv:1807.03022 [astro-ph.CO] .[117] A. Lewis and S. Bridle, “Cosmological parametersfrom CMB and other data: A Monte Carlo approach,” Phys. Rev. D (2002) 103511, arXiv:astro-ph/0205436 [astro-ph] .[118] F. Beutler, C. Blake, M. Colless, D. H. Jones,L. Staveley-Smith, L. Campbell, Q. Parker,W. Saunders, and F. Watson, “The 6dF GalaxySurvey: Baryon Acoustic Oscillations and the LocalHubble Constant,” Mon. Not. Roy. Astron. Soc. (2011) 3017–3032, arXiv:1106.3366 [astro-ph.CO] .[119] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden, and M. Manera, “The clustering of theSDSS DR7 main Galaxy sample ˇsC I. A 4 per centdistance measure at z = 0 . Mon. Not. Roy. Astron.Soc. no. 1, (2015) 835–847, arXiv:1409.3242[astro-ph.CO] .[120]
BOSS
Collaboration, S. Alam et al. , “The clusteringof galaxies in the completed SDSS-III BaryonOscillation Spectroscopic Survey: cosmological analysisof the DR12 galaxy sample,”
Mon. Not. Roy. Astron.Soc. no. 3, (2017) 2617–2652, arXiv:1607.03155[astro-ph.CO] .[121] D. M. Scolnic et al. , “The Complete Light-curveSample of Spectroscopically Confirmed SNe Ia fromPan-STARRS1 and Cosmological Constraints from theCombined Pantheon Sample,”
Astrophys. J. no. 2,(2018) 101, arXiv:1710.00845 [astro-ph.CO] .[122] I. Jee, S. Suyu, E. Komatsu, C. D. Fassnacht,S. Hilbert, and L. V. Koopmans, “A measurement ofthe Hubble constant from angular diameter distancesto two gravitational lenses,” arXiv:1909.06712[astro-ph.CO] .[123] S. Suyu et al. , “Cosmology from gravitational lenstime delays and Planck data,”
Astrophys. J. Lett. (2014) L35, arXiv:1306.4732 [astro-ph.CO] .[124] G. C.-F. Chen et al. , “A SHARP view of H0LiCOW: H from three time-delay gravitational lens systemswith adaptive optics imaging,” Mon. Not. Roy. Astron.Soc. no. 2, (2019) 1743–1773, arXiv:1907.02533[astro-ph.CO] .[125] K. C. Wong et al. , “H0LiCOW – IV. Lens mass modelof HE 0435 − Mon. Not. Roy.Astron. Soc. no. 4, (2017) 4895–4913, arXiv:1607.01403 [astro-ph.CO] .[126] S. Birrer et al. , “H0LiCOW - IX. Cosmographicanalysis of the doubly imaged quasar SDSS 1206+4332and a new measurement of the Hubble constant,”
Mon. Not. Roy. Astron. Soc. (2019) 4726, arXiv:1809.01274 [astro-ph.CO] .[127] C. E. Rusu et al. , “H0LiCOW XII. Lens mass model ofWFI2033 − Mon. Not. Roy. Astron.Soc. no. 1, (2020) 1440–1468, arXiv:1905.09338[astro-ph.CO]arXiv:1905.09338[astro-ph.CO]