We explore the scalar field obtained under the conformal transformation of the spacetime metric gμν from the Jordan frame to the Einstein frame in f(R) gravity. This scalar field is the result of the modification in the gravitational part of the Einstein's general relativistic theory of gravity. For f(R)=R1+δ/Rcδ, we find the effective potential of the scalar field and calculate the mass of the scalar field particle “scalaron”. It is shown that the mass of the scalaron depends upon the energy density of standard matter in the background (in solar system, mϕ∼ 10−16 eV) . The interaction between standard matter and scalaron is weak in the high curvature regime. This linkage between the mass of the scalaron and the background leads to the physical effects of dark matter and is expected to reflect the anisotropic propagation of scalaron in moving baryonic matter fields as in merging clusters (Bullet cluster, the Abell 520 system, MACS etc.). Such scenario also satisfies the local gravity constraints of f(R) gravity. We further calculate the equation of state of the scalar field in the action-angle variable formalism and show its distinct features as the dark matter and dark energy with respect to energy density of the scalar field at different values of the model parameter δ.

An attempt has been made to explore the geometric effects of f(R) action on the galactic dynamics under the weak field approximation. The rotational velocity is calculated beyond the Einstein’s geometric theory of gravity. It is inspired by the cosmological geometric relation obtained in the power-law f(R) gravity model in vacuum. We analyse the action with a small positive deviation from the Einstein–Hilbert gravity action (taking R as $$f(R)\propto R^{1+\delta }$$f(R)∝R1+δ) at the galactic scales for the explanation of the flatness paradox associated with the clustered galactic dark matter. We obtain the contribution of a dynamical f(R) cosmological background geometry on accelerating the test mass. Furthermore, the integrated effective acceleration of the test mass due to a massive spherically symmetric source in f(R) background is calculated via the study of geodesics for the suitable spacetime metric and an equation for the effective rotational velocity has been developed. We test the viability of the proposed model by tracing the motion of a test mass far from the disk of galactic matter for smaller $$\delta $$δ. The possible galactic rotational velocity curves in f(R) background are discussed for the formula obtained with $$\delta<< 1$$δ<<1. We also obtain constraints on $$\delta $$δ$$O(10^{-6})$$O(10-6) confirmed by observations.

We explore a new realisation of the galactic scale dynamics via gravitational lensing phenomenon in power-law f(R) gravity theory of the type $$f(R)\propto R^{1+\delta }$$ f ( R ) ∝ R 1 + δ with $$\delta<<1$$ δ < < 1 for interpreting the clustered dark matter effects. We utilize the single effective point like potential (Newtonian potential + f(R) background potential) obtained under the weak field limit to study the combined observations of galaxy rotation curve beyond the optical disk size and their lensing profile in f(R) frame work. We calculate the magnitude of light deflection angle with the characteristic length scale (because of Noether symmetry in f(R) theories) appearing in the effective f(R) rotational velocity profile of a typical galaxy with the model parameter $$\delta \approx O(10^{-6})$$ δ ≈ O ( 10 - 6 ) constrained in previous work. For instance, we work with the two nearby controversial galaxies NGC 5533 and NGC 4138 and explore their galactic features by analysing the lensing angle profiles in f(R) background. We also contrast the magnitudes of f(R) lensing angle profiles and the relevant parameters of such galaxies with the generalised pseudo-isothermal galaxy halo model and find consistency.

It is shown that the structures in the universe can be interpreted to show a closed wheel of time, rather than a straight arrow. An analysis in f (R) gravity model has been carried out to show that due to local observations a small arc at any given spacetime point would invariably indicate an arrow of time from past to future, though on a quantum scale it is not a linear flow but a closed loop, a fact that can be examined through future observations. PACS numbers: 98.80.-k, 95.36.+x, 04.50.-h

We solve the field equations of modified gravity for f (R) model in metric formalism. Further, we obtain the fixed points of the dynamical system in phase space analysis of f (R) models, both with and without the effects of radiation. Stability of these points is studied against perturbations in a smooth spatial background by applying the conditions on the eigenvalues of the matrix obtained in the linearized first-order differential equations. Following this, these fixed points are used for analysing the dynamics of the system during the radiation, matter and acceleration dominated phases of the universe. Certain linear and quadratic forms of f (R) are determined from the geometrical and physical considerations and the behaviour of the scale factor is found for those forms. Further, we also determine the Hubble parameter H(t), Ricci scalar R for these cosmic phases. We show the emergence of an asymmetry of time from the dynamics of the scalar field exclusively owing to the f (R) gravity in the Einstein frame that may lead to an arrow of time at a classical level.PACS numbers: 98.80.-k, 95.36.+x, 04.50.-h