Over the past few years much attention has been given to the study of modified gravity theories in order to find a more natural explanation for the late time acceleration of the Universe. Nevertheless, a comparison of the matter power spectrum predictions made by these theories with available data has not yet been subjected to a detailed analysis. In the context of $f(R)$ theories of gravity we study the predicted power spectra using both a dynamical systems approach for the background and solving for the matter perturbations without using the quasistatic approximation, comparing the theoretical results with several Sloan Digital Sky Survey data. The importance of studying the first order perturbed equations by assuming the correct background evolution and the relevance of the initial conditions are also stressed. We determine the statistical significance in relation to the observational data and demonstrate their conflict with existing observations.

We investigate warm inflationary scenario in which the accelerated expansion of the early Universe is driven by chameleon-like scalar fields. Due to the non-minimal coupling between the scalar field and the matter sector, the energy-momentum tensor of each fluid component is not conserved anymore, and the generalized balance equation is obtained. The new source term in the energy equation can be used to model warm inflation. On the other hand, if the coupling function varies slowly, the model reduces to the standard model used for the description of cold inflation. To test the validity of the warm chameleon inflation model, the results for warm inflationary scenarios are compared with the observational Planck2018 Cosmic Microwave Background data. In this regard, the perturbation parameters such as the amplitude of scalar perturbations, the scalar spectral index and the tensor-to-scalar ratio are derived at the horizon crossing in two approximations, corresponding to the weak and strong dissipative regimes. As a general result it turns out that the theoretical predictions of the chameleon warm inflationary scenario are consistent with the Planck 2018 observations.

The cosmological scalar perturbations of standard matter are investigated in the context of extended teleparallelf(T) gravity theories using the $$1+3$$1+3 covariant formalism. After a review of the background gravitational field equations off(T) gravity and the introduction of the covariant perturbation variables, the usual scalar and harmonic decomposition have been performed, and the analysis of the growth of the density contrasts in the quasi-static approximation for two non-interacting fluids scenarios, namely torsion-dust and torsion-radiation mixtures is presented for the genericf(T) gravity theory. Special applications to two classes off(T) gravity toy models, namely $$f(T) = \mu T_0 \Big (\frac{T}{T_0}\Big )^n$$f(T)=μT0(TT0)n and $$f(T) = T+ \mu T_0 \Big (-\frac{T}{T_0}\Big )^n$$f(T)=T+μT0(-TT0)n , have then been made within the observationally viable regions of their respective parameter spaces, and the growth of the matter density contrast for both torsion-dust and torsion-radiation epochs of the Universe has been examined. The exact solutions of the dust perturbations, with growing amplitudes in cosmic time, are obtained for some limiting cases ofn. Similarly, the long- and short-wavelength modes in the torsion-radiation case are treated, with the amplitudes either oscillating or monotonically growing with time. Overall, it is noted thatf(T) models contain a richer set of observationally viable structure growth scenarios that can be tested against up-and-coming observational data and can accommodate currently known features of the large-scale structure power spectrum in the general relativistic and $$\varLambda CDM$$ΛCDM limits.

Despite the significant accomplishments of general relativity, numerous unresolved issues persist in our understanding of the cosmos. One of the most perplexing challenges is the ongoing accelerated expansion of the Universe, which continues to elude a complete explanation. Consequently, scientists have proposed various alternative theories to general relativity in pursuit of a deeper understanding. In our analysis, we delve into the recently proposed modified f(Q) gravity, where Q represents the nonmetricity scalar responsible for gravitational effects. Specifically, we investigate a cosmological model characterized by the functional form f(Q) = Q + αQⁿ, where α (with α ≠ 0) and n serve as free parameters. Utilizing this functional form, we construct our Hubble rate, incorporating a specific equation of state to describe the cosmic fluid. Furthermore, we leverage a dataset consisting of 31 data points from Hubble measurements and an additional 1048 data points from the Pantheon dataset. These data serve as crucial constraints for our model parameters, and we employ the Markov Chain Monte Carlo (MCMC) method to explore the parameter space and derive meaningful results. With our parameter values constrained, our analysis yields several noteworthy findings. The deceleration parameter suggests a recent accelerated phase in the cosmic expansion. In addition, the EoS parameter paints a portrait of dark energy exhibiting phantom-like characteristics. Furthermore, we delve into the application of cosmological diagnostic tools, specifically the statefinder and the Om(z) diagnostics. Both of these tools align with our previous conclusions, confirming the phantom-like behavior exhibited by our cosmological model. These results collectively contribute to our understanding of the dynamic interplay between gravity, dark energy, and the expanding cosmos.