The simplest ΛCDM model provides a good fit to a large span of cosmological data but harbors large areas of phenomenology and ignorance. With the improvement of the number and the accuracy of observations, discrepancies among key cosmological parameters of the model have emerged. The most statistically significant tension is the 4σ to 6σ disagreement between predictions of the Hubble constant, H 0, made by the early time probes in concert with the ‘vanilla’ ΛCDM cosmological model, and a number of late time, model-independent determinations of H 0 from local measurements of distances and redshifts. The high precision and consistency of the data at both ends present strong challenges to the possible solution space and demands a hypothesis with enough rigor to explain multiple observations—whether these invoke new physics, unexpected large-scale structures or multiple, unrelated errors. A thorough review of the problem including a discussion of recent Hubble constant estimates and a summary of the proposed theoretical solutions is presented here. We include more than 1000 references, indicating that the interest in this area has grown considerably just during the last few years. We classify the many proposals to resolve the tension in these categories: early dark energy, late dark energy, dark energy models with 6 degrees of freedom and their extensions, models with extra relativistic degrees of freedom, models with extra interactions, unified cosmologies, modified gravity, inflationary models, modified recombination history, physics of the critical phenomena, and alternative proposals. Some are formally successful, improving the fit to the data in light of their additional degrees of freedom, restoring agreement within 1–2σ between Planck 2018, using the cosmic microwave background power spectra data, baryon acoustic oscillations, Pantheon SN data, and R20, the latest SH0ES Team Riess, et al (2021 Astrophys. J. 908 L6) measurement of the Hubble constant (H 0 = 73.2 ± 1.3 km s−1 Mpc−1 at 68% confidence level). However, there are many more unsuccessful models which leave the discrepancy well above the 3σ disagreement level. In many cases, reduced tension comes not simply from a change in the value of H 0 but also due to an increase in its uncertainty due to degeneracy with additional physics, complicating the picture and pointing to the need for additional probes. While no specific proposal makes a strong case for being highly likely or far better than all others, solutions involving early or dynamical dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, and modified gravity provide the best options until a better alternative comes along.

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We investigate the observational consequences of a novel class of stable interacting dark energy (IDE) models, featuring interactions between dark matter (DM) and dark energy (DE). In the first part of our work, we start by considering two IDE models which are known to present earlytime linear perturbation instabilities. Applying a transformation depending on the dark energy equation of state (EoS) to the DM-DE coupling, we then obtain two novel stable IDE models. Subsequently, we derive robust and accurate constraints on the parameters of these models, assuming a constant EoS wx for the DE fluid, in light of some of the most recent publicly available cosmological data. These include Cosmic Microwave Background (CMB) temperature and polarization anisotropy measurements from the Planck satellite, a selection of Baryon Acoustic Oscillation measurements, Supernovae Type-Ia luminosity distance measurements from the JLA sample, and measurements of the Hubble parameter up to redshift 2 from cosmic chronometers. Our analysis displays a mild preference for the DE fluid residing in the phantom region (wx < −1), with significance up to 95% confidence level, while we obtain new upper limits on the coupling parameter between the dark components. The preference for a phantom DE suggests a coupling function Q < 0, thus a scenario where energy flows from the DE to the DM. We also examine the possibility of addressing the H0 and σ8 tensions, finding that only the former can be partially alleviated. Finally, we perform a Bayesian model comparison analysis to quantify the possible preference for the two IDE models against the standard concordance ΛCDM model, finding that the latter is always preferred with the strength of the evidence ranging from positive to very strong. 98.80.Cq, 95.35.+d, 95.36.+x, 98.80.Es. *

We study large-scale inhomogeneous perturbations and instabilities of interacting dark energy (IDE) models. Past analysis of large-scale perturbative instabilities, has shown that we can only test IDE models with observational data when its parameter ranges are either wx ≥ −1 and ξ ≥ 0, or wx ≤ −1 and ξ ≤ 0, where wx is the dark energy equation of state (EoS), and ξ is a coupling parameter governing the strength and direction of the energy transfer. We show that by adding a factor (1 + wx) to the background energy transfer, the whole parameter space can be tested against all the data and thus, the instabilities in such interaction models can be removed. We test three classes of interaction model using the latest astronomical data from different sources. Precise constraints are found. Our analysis shows that a very small but non-zero deviation from pure Λcosmology is suggested by the observational data while the no-interaction scenario can be recovered at the 68.3% confidence-level. In particular, for three IDE models, identified as IDE 1, IDE 2, and IDE 3, the 68.3% CL constraints on the interaction coupling strengths are, ξ = 0.0360 +0.0091 −0.0360 (IDE 1), ξ = 0.0433 +0.0062 −0.0433 (IDE 2), ξ = 0.1064 +0.0437 −0.1064 (IDE 3). In addition, we find that the dark energy EoS tends towards the phantom region taking the 68.3% CL constraints, wx = −1.0230 +0.0329 −0.0257 (IDE 1), wx = −1.0247 +0.0289 −0.0302 (IDE 2), and wx = −1.0275 +0.0228 −0.0318 (IDE 3). However, the possibility of wx > −1 is also not rejected by the astronomical data used here. Moreover, we find in all IDE models that, as the value of Hubble constant decreases, the behavior of the dark energy EoS shifts from phantom to quintessence type with its EoS very close to that a simple cosmological constant at the present time.PACS numbers: 95.36.+x, 95.35.+d, 98.80.Es Originally, the possibility that dark energy might interact with dark matter was introduced to justify the very small value of the cosmological constant by Wetterich [5,6]. However, when dynamical models were introduced as alternatives to a simple (non-interacting) cosmological constant, it was found that interactions between dark energy and dark matter might provide a simple explanation for the cosmic coincidence problem [7]. If one views this interaction from the particle physics perspective, then it is natural that the two fields should interact with each other * Electronic address: d11102004@163.com †

The phenomenological parametrizations of dark-energy (DE) equation of state can be very helpful, since they allow for the investigation of its cosmological behavior despite the fact that its underlying theory is unknown. However, although there has been a large amount of research on DE parametrizations which involve two or more free parameters, the one-parameter parametrizations seem to be underestimated. We perform a detailed observational confrontation of five one-parameter DE models, with observational data from cosmic microwave background (CMB), Joint light-curve analysis sample from Supernovae Type Ia observations (JLA), baryon acoustic oscillations (BAO) distance measurements, and cosmic chronometers (CC). We find that all models favor a phantom DE equation of state at present time, while they lead to H0 values in perfect agreement with its direct measurements and therefore they offer an alleviation to the H0-tension. Finally, performing a Bayesian analysis we show that although ΛCDM cosmology is still favored, one-parameter DE models have similar or better efficiency in fitting the data comparing to two-parameter DE parametrizations, and thus they deserve a thorough investigation.

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The interacting dark energy model could propose a effective way to avoid the coincidence problem. In this paper, dark energy is taken as a fluid with a constant equation of state parameter wx. In a general gauge, we could obtain two sets of different perturbation equations when the momentum transfer potential is vanished in the rest frame of dark matter or dark energy. There are many kinds of interacting forms from the phenomenological considerations, here, we choose Q = 3Hξxρx which owns the stable perturbations in most cases. Then, according to the Markov Chain Monte Carlo method, we constrain the model by currently available cosmic observations which include cosmic microwave background radiation, baryon acoustic oscillation, type Ia supernovae, and f σ8(z) data points from redshift-space distortion. Jointing the geometry tests with the large scale structure information, the results show a tighter constraint on the interacting model than the case without f σ8(z) data. We find the interaction rate in 3σ regions: ξx = 0.00372 +0.000768+0.00655+0.0102 −0.00372−0.00372−0.00372 . It means that the recently cosmic observations favor a small interaction rate between the dark sectors, at the same time, the measurement of redshift-space distortion could rule out a large interaction rate in the 1σ region.