Extended Theories of Gravity and their Cosmological and Astrophysical Applications

TL;DR

This review analyzes Extended Theories of Gravity (ETGs) in both metric and Palatini formalisms, focusing on $f(R)$-type models and their cosmological and astrophysical implications. It clarifies how conformal transformations can map higher-order or non-minimally coupled theories to Einstein gravity with scalar fields, and how Palatini dynamics induce a bi-metric structure linking geometry to matter through a structural equation. The authors discuss observational constraints, lookback-time analyses, and the potential for curvature effects to mimic dark energy, while also showing that modified gravity can address galactic rotation curves and halo profiles, potentially reducing the need for dark matter. They further highlight stochastic gravitational waves as a novel benchmark for discriminating ETGs from GR and for constraining the theory space with upcoming GW experiments. Overall, ETGs emerge as a promising, testable framework that may unify explanations for dark energy and dark matter through geometric corrections to gravity, pending rigorous structure-formation and solar-system tests.

Abstract

We review Extended Theories of Gravity in metric and Palatini formalism pointing out their cosmological and astrophysical application. The aim is to propose an alternative approach to solve the puzzles connected to dark components.

Figures (11)

• Figure 1: The CMBR anisotropy spectrum for different values of $w$. Data points are the WMAP measurements and the best fit is obtained for $w\simeq -1$. If $w\neq -1$ the clustering of dark energy has been considered in this plot.
• Figure 2: Best fit curve to the SNeIa Hubble diagram for the power law Lagrangian model. Only data of "Gold" sample of SNeIa have been used.
• Figure 3: The Hubble diagram of 20 radio galaxies together with the "Gold" sample of SNeIa, in term of the redshift as suggested in daly. The best fit curve refers to the $R^n$ - gravity model without dark matter (left), while in the right panel it is shown the difference between the luminosity distances calculated without dark matter and in presence of this component in term of redshift. It is evident that the two behaviors are quite indistinguishable.
• Figure 4: Contour plot in the plane ($q_0\,,\ n$) describing the Universe age as induced by $R^n$ - gravity model without dark matter. The contours refer to age ranging from 11 Gyr to 16 Gyr from up to down. The dashed curves define the $1-\sigma$ region relative to the best fit Universe age suggested by the last WMAP release ($13.73_{-0.17}^{+0.13}$ Gyr) in the case of $\Lambda$-CDM model wmap2. At the best fit $n\simeq 3.5$ for SNeIa, the measured $q_0\simeq -0.5$ gives a rather short age (about $11.5$ Gyr) with respect to the WMAP constraint. This is an indication that the $f(R)$ model has to be further improved.
• Figure 5: Scale factor evolution of the growth index : ( left) modified gravity, in the case $\Omega_m\,=\,\Omega_{bar}\,\sim 0.04$, for the SNeIa best fit model with $n\,=\,3.46$, ( right) the same evolution in the case of a $\Lambda$CDM model. In the case of $R^n$ - gravity it is shown also the dependence on the scale $k$. The three cases $k\,=\,0.01,\ 0.001,\ 0.0002$ have been checked. Only the latter case shows a very small deviation from the leading behavior. Clearly, the trend is that the growth law saturates to ${\cal F}=1$ for higher redshifts (i.e. $a\sim 0.001$ to $0.01$). This behavior agrees with observations since we know that comparing CMB anisotropies and LSS, we need roughly $\delta \propto a$ between recombination and $z\sim 5$ to generate the present LSS from the small fluctuations at recombination seen in the CMB.