Directly comparing the 6 expansion rate measured by type Ia supernovae data and the lower bound on the expansion rate set by the strong energy conditions or the null hypothesis that there never exists cosmic acceleration, we see 3σ direct evidence of cosmic acceleration and the Rh=ct model is strongly excluded by the type Ia supernovae data. We also use Gaussian process method to reconstruct the expansion rate and the deceleration parameter from the 31 cosmic chronometers data and the 6 data points on the expansion rate measured from type Ia supernoave data, the direct evidence of cosmic acceleration is more than 3σ and we find that the transition redshift zt=0.60−0.12+0.21 at which the expansion of the Universe underwent the transition from acceleration to deceleration. The Hubble constant inferred from the cosmic chronometers data with the Gaussian process method is H0=67.46±4.75 Km/s/Mpc. To understand the properties of cosmic acceleration and dark energy, we fit two different two-parameter models to the observational data, and we find that the constraints on the model parameters from either the full distance modulus data by the Pantheon compilation or the compressed expansion rate data are very similar, and the derived Hubble constants are consistent with the Planck 2018 result. Our results confirm that the 6 compressed expansion rate data can replace the full 1048 distance modulus data from the Pantheon compilation. We derive the transition redshift zt=0.61−0.16+0.24 by fitting a simple q(z) model to the combination of cosmic chronometers data and the Pantheon compilation, the result is consistent with that obtained from the reconstruction with Gaussian process. By fitting the observational data by the SSLCPL model which approximates the dynamics of general thawing scalar fields over a large redshift range, we obtain that H0=66.8± 1.4, Ωϕ 0=0.69± 0.01 and w0=−1.03± 0.07. The result shows that ΛCDM model is consistent with the observational data.

Inflation predicts that the Universe is spatially flat. The Planck 2018 measurements of the cosmic microwave background anisotropy favour a spatially closed universe at more than 2σ confidence level. We use model independent methods to study the issue of cosmic curvature. The method reconstructs the Hubble parameter H(z) from cosmic chronometers data with the Gaussian process method. The distance modulus is then calculated with the reconstructed function H(z) and fitted by type Ia supernovae data. Combining the cosmic chronometers and type Ia supernovae data, we obtain Ωk0h2 = 0.102 ± 0.066 which is consistent with a spatially flat universe at the 2σ confidence level. By adding the redshift space distortions data to the type Ia supernovae data with a proposed novel model independent method, we obtain $\Omega _{k0}h^2=0.117^{+0.058}_{-0.045}$ and no deviation from ΛCDM model is found.

We derive the reconstruction formulae for the inflation model with the non-minimal derivative coupling term. If reconstructing the potential from the tensor-to-scalar ratio r, we could obtain the potential without using the high friction limit. As an example, we reconstruct the potential from the parameterization r=8α/(N+β)γ, which is a general form of the α-attractor. The reconstructed potential has the same asymptotic behavior as the T- and E-model if we choose γ=2 and α≪1. We also discuss the constraints from the reheating phase by assuming the parameter wre of state equation during reheating is a constant. The scale of big-bang nucleosynthesis could put an upper limit on ns if wre=2/3 and a low limit on ns if wre=1/6.

Taking advantage of Gaussian process (GP), we obtain an improved estimate of the Hubble constant, H0 = 70.41 ± 1.58 km s−1 Mpc−1, using Hubble parameter [H(z)] from cosmic chronometers (CCH) and expansion rate function [E(z)], extracted from type Ia supernovae, data. We also use CCH data, including the ones with full covariance matrix, and E(z) data to obtain a determination of $H_0=72.34_{-1.92}^{+1.90}$ km s−1 Mpc−1, which implies that the involvement of full covariance matrix results in higher values and uncertainties of H0. These results are higher than those obtained by directly reconstructing CCH data with GP. In order to estimate the potential of future CCH data, we simulate two sets of H(z) data and use them to constrain H0 by either using GP reconstruction or fitting them with E(z) data. We find that simulated H(z) data alleviate H0 tension by pushing H0 values higher towards ∼70 km s−1 Mpc−1. We also find that joint H(z) + E(z) data favor higher values of H0, which is also confirmed by constraining H0 in the flat concordance model and 2-order Taylor expansion of H(z). In summary, we conclude that more and better-quality CCH data as well as E(z) data can provide a new and useful perspective on resolving H0 tension.

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