Recently it has been shown that the cosmological dynamics of covariant f(Q) gravity depend on different affine connections. In this paper, two specific f(Q) models are investigated with SNe+CC+BAO+QSO observational data, and the spatial curvature of the universe is studied in covariant f(Q) gravity. It is found that the parameters $${\mathcal {X}}_0$$X 0 and $${\mathcal {X}}'_0$$X 0 ′ characterizing affine connections significantly affect the behavior of the effective equation of state $$w_Q$$w Q and may drive it across the phantom divide line. These results imply some inertial effects of the universe change the cosmic dynamics. However based on the Bayesian evidence, the zero inertial effect is more favored in the flat universe. Moreover, a closed universe is favored not only in the $$\Lambda $$Λ CDM model but also in covariant f(Q) gravity. The f(Q) models have less support evidence than the $$\Lambda $$Λ CDM model in the non-flat universe.

In this paper, we investigate the potential-driven inflation models with a disformal coupling to Einstein gravity, to find out the effects of such a coupling on these models. We consider a simple coupling form which introduces only one parameter, and three inflation models, namely the chaotic inflation, the Higgs inflation, and the monodromy inflation. We find that the disformal coupling can have some modifications to the observational variables of these models such as the power spectrum, the spectral index as well as the tensor/scalar ratio, although not too large due to the constraints on the disformal coupling parameter. With these modifications, one has the opportunity of improving models that lie on the edge of the favorable regions of Planck observational data, such as monodromy inflation. Moreover, the nontrivial sound speed of tensor perturbations (gravitational waves) may come out, due to the coupling of gravity and kinetic terms of the field.

There have been thousands of cosmological models for our early Universe proposed in the literature, and many of them claimed to be able to give rise to scale-invariant power spectrum as was favored by the observational data. It is thus interesting to think about whether there are some relations among them, e.g., the duality relation. In this paper, we investigate duality relations between cosmological models in framework of general relativity, not only with varying slow-roll parameter $\epsilon$ but also with sound speed $c_s$, which can then be understood as adiabatic duality. Several duality relationships are formulated analytically and verified numerically. We show that models with varying $\epsilon$ and constant $c_s$ can be dual in the scalar spectral index, but not the tensor one. On the other hand, allowing both $\epsilon$ and $c_s$ to vary can make models dual in both scalar and tensor spectral indices.

We study spherically symmetric solutions with a scalar field in the shift-symmetric subclass of the Horndeski theory. Constructing an effective energy-momentum tensor of the scalar field based on the two-fluid model, we decompose the scalar field into two components: dark matter and dark energy. We find the dark-matter fluid is pressureless, and its distribution of energy density obeys the inverse-square law. We show the scalar field dark matter can explain the galaxy rotation curve and discuss the time evolution of the dark matter in the cosmic background.

In this paper, we investigate the potential-driven inflation models with a disformal coupling to Einstein Gravity, to find out the effects of such a coupling on these models. We consider a simple coupling form which introduces only one parameter, and three inflation models, namely the chaotic inflation, the Higgs inflation, and the monodromy inflation. We find that the disformal coupling can have some modifications to the observational variables of these models such as the power spectrum, the spectral index as well as the tensor/scalar ratio, although not too large due to the constraints on the disformal coupling parameter. With these modifications, one has the opportunity of improving models that lie on the edge of the favorable regions of Planck observational data, such as monodromy inflation. Moreover, the non-trivial sound speed of tensor perturbations (gravitational waves) may come out, due to the coupling of gravity and kinetic terms of the field.