The $\Lambda$CDM model provides a good fit to a large span of cosmological data but harbors areas of phenomenology. 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 $4-6\sigma$ disagreement between predictions of the Hubble constant $H_0$ by early time probes with $\Lambda$CDM 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 demand a hypothesis with enough rigor to explain multiple observations--whether these invoke new physics, unexpected large-scale structures or multiple, unrelated errors. We present a thorough review of the problem, including a discussion of recent Hubble constant estimates and a summary of the proposed theoretical solutions. Some of the models presented are formally successful, improving the fit to the data in light of their additional degrees of freedom, restoring agreement within $1-2\sigma$ between {\it Planck} 2018, using CMB power spectra data, BAO, Pantheon SN data, and R20, the latest SH0ES Team measurement of the Hubble constant ($H_0 = 73.2 \pm 1.3{\rm\,km\,s^{-1}\,Mpc^{-1}}$ at 68\% confidence level). Reduced tension might not simply come from a change in $H_0$ but also from an increase in its uncertainty due to degeneracy with additional physics, 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.[Abridged]

https://iopscience.iop.org/article/10.1088/1361-6382/ac086d/pdf

In the realm of the Hubble tension - a review of solutions

We compile a list of 28 independent measurements of the Hubble parameter between redshifts 0.07 ≤ z ≤ 2.3 and use this to place constraints on model parameters of constant and time-evolving dark energy cosmologies. These H(z) measurements by themselves require a currently accelerating cosmological expansion at about, or better than, 3σ confidence. The mean and standard deviation of the six best-fit model deceleration-acceleration transition redshifts (for the three cosmological models and two Hubble constant priors we consider) are z da = 0.74 ± 0.05, in good agreement with the recent Busca et al. determination of z da = 0.82 ± 0.08 based on 11 H(z) measurements between redshifts 0.2 ≤ z ≤ 2.3, almost entirely from baryon-acoustic-oscillation-like data.

https://iopscience.iop.org/article/10.1088/2041-8205/766/1/L7/pdf

Hubble parameter measurement constraints on the cosmological deceleration-acceleration transition redshift

In this paper, we reconstruct cosmological models in the framework of f(R,T) gravity, where R is the Ricci scalar and T is the trace of the stress-energy tensor. We show that the dust fluid reproduces ΛCDM, phantom–non-phantom era and phantom cosmology. Further, we reconstruct different cosmological models, including the Chaplygin gas, and scalar field with some specific forms of f(R,T). Our numerical simulation for the Hubble parameter shows good agreement with the BAO observational data for low redshifts, z<2.

Reconstruction of some cosmological models in f( R, T ) cosmology

In this paper, we reconstruct cosmological models in the framework of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy tensor. We show that the dust fluid reproduces $\Lambda $CDM, phantom-non-phantom era and the phantom cosmology. Further, we reconstruct different cosmological models including, Chaplygin gas, scalar field with some specific forms of $f(R,T)$. Our numerical simulation for Hubble parameter shows good agreement with the BAO observational data for low redshifts $z<2$.

Reconstruction of some cosmological models in f(R,T) gravity

Motivated by recent attempts to solve the cosmological constant problem, we examine the observational consequences of a vacuum energy which decays in time. In both radiation and matter dominated eras, the ratio of the vacuum to the total energy density of the universe must be small. Although the vacuum cannot provide the “missing mass” required to close the universe today, its presence earlier in the history of the universe could have important consequences. Element abundances from primordial nucleosynthesis require the ratio x = ϱ vac /( ϱ vac + ϱ rad ) ⩽ 0.1 of neutrino (or equivalent light) species to exceed N ν > 4, a case ruled out in the standard cosmological model. If the vacuum decays into low energy photons, the lack of observed spectral distortions in the microwave background gives tighter bounds, x −4 . In the matter-dominated era, the presence of a vacuum term may allow more time for growth of protogalactic perturbations.

Cosmology with Decaying Vacuum Energy

We use the Markov Chain Monte Carlo method to investigate a global constraints on the modified Chaplygin gas (MCG) model as the unification of dark matter and dark energy from the latest observational data: the Union2 dataset of type supernovae Ia (SNIa), the observational Hubble data (OHD), the cluster X-ray gas mass fraction, the baryon acoustic oscillation (BAO), and the cosmic microwave background (CMB) data. In a flat universe, the constraint results for MCG model are, ${\Omega_{b}h^{2}\,{=}\,0.02263^{+0.00184}_{-0.00162} (1\sigma)^{+0.00213}_{-0.00195} (2\sigma)}$, ${B_{s}\,{=}\,0.7788^{+0.0736}_{-0.0723}(1\sigma)^{+0.0918}_{-0.0904} (2\sigma)}$, ${\alpha\,{=}\,0.1079^{+0.3397}_{-0.2539} (1\sigma)^{+0.4678}_{-0.2911} (2\sigma)}$, ${B\,{=}\,0.00189^{+0.00583}_{-0.00756}(1\sigma)^{+0.00660}_{-0.00915} (2\sigma)}$, and ${H_{0}=70.711^{+4.188}_{-3.142} (1\sigma)^{+5.281}_{-4.149}(2\sigma)}$.

Combined constraints on modified Chaplygin gas model from cosmological observed data: Markov Chain Monte Carlo approach

A modified Chaplygin gas (MCG) model of unifying dark energy and dark matter is considered in this paper. Concretely, the evolution of such a unified dark sector model is studied and the statefinder diagnostic to the MCG model is performed in our model. By analysis, it is shown that the state parameter of dark energy can cross the so-called phantom divide ω = -1, the behavior of MCG will be like ΛCDM in the future and therefore our universe will not end up with Big Rip in the future. In addition, we plot the evolution trajectory of the MCG model in the statefinder parameter r–s plane and show the discrimination between this scenario and other dark energy models.

The Modified Chaplygin Gas as a Unified Dark Sector Model

A modified f(G) gravity model with coupling between matter and geometry is proposed, which is described by the product of the Lagrange density of the matter and an arbitrary function of the Gauss-Bonnet term. The field equations and the equations of motion corresponding to this model show the non-conservation of the energy-momentum tensor, the presence of an extra-force acting on test particles and the non-geodesic motion. Moreover, the energy conditions and the stability criterion at de Sitter point in the modified f(G) gravity models with curvature-matter coupling are derived, which can degenerate to the well-known energy conditions in general relativity. Furthermore, in order to get some insight on the meaning of these energy conditions, we apply them to the specific models of f(G) gravity and the corresponding constraints on the models are given. In addition, the conditions and the candidate for late-time cosmic accelerated expansion in the modified f(G) gravity are studied by means of conditions of power-law expansion and the equation of state of matter less than -1/ 3 .

Modified f(G) gravity models with curvature-matter coupling

The evolution of the universe in Brans-Dicke (BD) theory is discussed in this paper. Considering a parameterized scenario for BD scalar field φ = φ
 0
 a
 α
 which plays the role of gravitational constant G, we apply the Markov Chain Monte Carlo method to investigate a global constraints on BD theory with a self-interacting potential according to the current observational data: the Union2 dataset of type-Ia supernovae (SNIa), the high-redshift Gamma-Ray Bursts (GRBs) data, the observational Hubble data (OHD), the cluster X-ray gas mass fraction, the baryon acoustic oscillation (BAO), and the cosmic microwave background (CMB) data. It is shown that an expanded universe from deceleration to acceleration is given in this theory, and the constraint results of dimensionless matter density Ω
 0m
 and parameter α are, Ω
 0m
 = 0.286
 −0.039−0.047
 +0.037+0.050
 and α = 0.0046
 −0.0171−0.0206
 +0.0149+0.0171
 which is consistent with the result of current experiment exploration, |α| ≤ 0.132124. In addition, we use the geometrical diagnostic method, jerk parameter j, to distinguish the BD theory and the cosmological constant model in Einstein’s theory of general relativity.

Does accelerating universe indicate Brans-Dicke theory?

A modified f(G) gravity model with coupling between matter and geometry is proposed, which is described by the product of the Lagrange density of the matter and an arbitrary function of the Gauss–Bonnet term. The field equations and the equations of motion corresponding to this model show the non-conservation of the energy-momentum tensor, the presence of an extra force acting on test particles and non-geodesic motion. Moreover, the energy conditions and the stability criterion at the de Sitter point in modified f(G) gravity models with curvature–matter coupling are derived, which can degenerate to the well-known energy conditions in general relativity. Furthermore, in order to get some insight in the meaning of these energy conditions, we apply them to the specific models of f(G) gravity and the corresponding constraints on the models are given. In addition, the conditions and the candidate for late-time cosmic accelerated expansion in modified f(G) gravity are studied by means of conditions of power-law expansion and the equation of state of matter ω smaller than $-\frac{1}{3}$
 .

Yabo Wu

Papers

Combined constraints on modified Chaplygin gas model from cosmological observed data: Markov Chain Monte Carlo approach

The Modified Chaplygin Gas as a Unified Dark Sector Model

Modified f(G) gravity models with curvature-matter coupling

Does accelerating universe indicate Brans-Dicke theory?

Modified f(G) gravity models with curvature–matter coupling