The $\Lambda$CDM model provides a good fit to most astronomical observations but harbors large areas of phenomenology and ignorance. With the improvements in the precision and number of observations, discrepancies between key cosmological parameters of this model have emerged. Among them, the most notable tension is the 4$\sigma$ to 6$\sigma$ deviation between the Hubble constant ($H_0$) estimations measured by the local distance ladder and the cosmic microwave background (CMB) measurement. In this review, we revisit the $H_0$ tension based on the latest research and sort out evidence from solutions to this tension that might imply new physics beyond the $\Lambda$CDM model. The evidence leans more towards modifying the late-time universe.

The current discrepancy between the Hubble constant, H₀, derived from the local distance ladder and from the cosmic microwave background is one of the most crucial issues in cosmology, as it may possibly indicate unknown systematics or new physics. Here, we present a novel non-parametric method to estimate the Hubble constant as a function of redshift. We establish independent estimates of the evolution of Hubble constant by diagonalizing the covariance matrix. From type Ia supernovae, baryon acoustic oscillation data and the observed Hubble parameter data, a decreasing trend in the Hubble constant with a significance of a 5.6σ confidence level is found. At low redshift, its value is dramatically consistent with that measured from the local distance ladder and it drops to the value measured from the cosmic microwave background at high redshift. Our results may relieve the Hubble tension, with a preference for recent solutions, especially with respect to novel physics.

Gamma-ray bursts (GRBs), as a possible probe to extend the Hubble diagram to high redshifts, have attracted much attention recently. In this paper, we select two samples of GRBs that have a plateau phase in X-ray afterglow. One is short GRBs (SGRBs) with plateau phases dominated by magnetic dipole (MD) radiations. The other is long GRBs (LGRBs) with gravitational wave (GW) dominated plateau phases. These GRBs can be well standardized using the correlation between the plateau luminosity L₀ and the end time of plateau t_b. The so-called circularity problem is mitigated by using the observational Hubble parameter data and Gaussian process method. The calibrated L₀ – t_b correlations are also used to constrain Lambda cold dark matter (ΛCDM) and w(z) = w₀ models. Combining the MD–LGRBs sample from Wang et al. ( 2021) and the MD–SGRBs sample, we find |$\Omega _{\mathrm{ m}} = 0.33_{-0.09}^{+0.06}$| and Ω_Λ = |$1.06_{-0.34}^{+0.15}$| excluding systematic uncertainties in the non-flat ΛCDM model. Adding Type Ia supernovae from Pantheon sample, the best-fitting results are w₀ = |$-1.11_{-0.15}^{+0.11}$| and Ω_m = |$0.34_{-0.04}^{+0.05}$| in the w = w₀ model. These results are in agreement with the ΛCDM model. Our result supports that selection of GRBs from the same physical mechanism is crucial for cosmological purposes.

Gamma-ray bursts (GRBs) are the most luminous explosions in and can be detectable out to the edge of the universe. They have long been thought to be able to extend the Hubble diagram to very high redshifts. Several correlations between temporal or spectral properties and GRB luminosities have been proposed to make GRBs cosmological tools. However, those correlations cannot be properly standardized. In this paper, we select a long-GRB sample with X-ray plateau phases produced by electromagnetic dipole emissions from central newborn magnetars. A tight correlation is found between the plateau luminosity and the end time of the plateau in the X-ray afterglows out to the redshift z = 5.91. We standardize these long-GRB X-ray light curves to a universal behavior through this correlation, with a luminosity dispersion of 0.5 dex. The derived distance–redshift relation of GRBs is in agreement with the standard ΛCDM model both at low and high redshifts. The evidence for an accelerating universe from this GRB sample is 3σ, which is the highest statistical significance from GRBs to date.

The discrepancy between the value of the Hubble constant H₀ measured from the local distance ladder and from the cosmic microwave background is the most serious challenge to the standard ΛCDM model. Various models have been proposed to solve or relieve it, but no satisfactory solution has been given until now. Here, we report a late-time transition of H₀, i.e. H₀ changes from a low value to a high one from early to late cosmic time, by investigating the Hubble parameter H(⁠|$\mathit{ z}$|⁠) data based on the Gaussian process (GP) method. This finding effectively reduces the Hubble crisis by 70 per cent. Our results are also consistent with the descending trend of H₀ measured using time-delay cosmography of lensed quasars at 1 σ confidence level, and support the idea that the Hubble crisis arises from new physics beyond the standard ΛCDM model. In addition, in the ΛCDM model and |$w$|CDM model, there is no transition behaviour of H₀.

The cosmological principle is fundamental to the standard cosmological model. It assumes that the Universe is homogeneous and isotropic on very large scales. As the basic assumption, it must stand the test of various observations. In this work, we investigated the properties of the Pantheon+ sample, including redshift distribution and position distribution, and we give its constraint on the flat ΛCDM model: Ω_m = 0.36 ± 0.02 and H₀ = 72.83 ± 0.23 km s⁻¹ Mpc⁻¹. Then, using the region fitting (RF) method, we mapped the all-sky distribution of cosmological parameters (Ω_m and H₀) and find that the distribution significantly deviates from isotropy. A local matter underdensity region exists toward (308.4°_−48.7^+47.6, −18.2°_−28.8^+21.1) as well as a preferred direction of the cosmic anisotropy (313.4°_−18.2^+19.6, −16.8°_−10.7^+11.1) in galactic coordinates. Similar directions may imply that local matter density might be responsible for the anisotropy of the accelerated expansion of the Universe. Results of statistical isotropy analyses including Isotropy and Isotropy with real-data positions (RP) show high confidence levels. For the local matter underdensity, the statistical significances are 2.78σ (isotropy) and 2.34σ (isotropy RP). For the cosmic anisotropy, the statistical significances are 3.96σ (isotropy) and 3.15σ (isotropy RP). The comparison of these two kinds of statistical isotropy analyses suggests that inhomogeneous spatial distribution of real sample can increase the deviation from isotropy. The similar results and findings are also found from reanalyses of the low-redshift sample (lp+) and the lower screening angle (θ_max = 60°), but with a slight decrease in statistical significance. Overall, our results provide clear indications for a possible cosmic anisotropy. This possibility must be taken seriously. Further testing is needed to better understand this signal.