Quintessence issues can be achieved by taking into account higher order curvature invariants into the effective action of gravitational field. Such an approach is naturally related to fundamental theories of quantum gravity which predict higher order terms in loop expansion of quantum fields in curved space-times. In this framework, we obtain a class of cosmological solutions which are fitted against cosmological data. We reproduce encouraging results able to fit high redshift supernovae and WMAP observations. The age of the universe and other cosmological parameters are discussed in this context. PACS number(s): 98.80.Cq, 98.80. Hw, 04.

We investigate the possibility that the observed flatness of the rotation curves of spiral galaxies is not evidence for the existence of dark matter haloes, but rather a signal of the breakdown of General Relativity. To this aim, we consider power-law fourth-order theories of gravity obtained by replacing the scalar curvature R with f (R) = f 0 R n in the gravity Lagrangian. We show that, in the low energy limit, the gravitational potential generated by a point-like source may be written as (r) ∝ r −1 [1 + (r/r c ) β ] with β a function of the slope n of the gravity Lagrangian and r c a scalelength depending on the gravitating system properties. In order to apply the model to realistic systems, we compute the modified potential and the rotation curve for spherically symmetric and for thin disc mass distributions. It turns out that the potential is still asymptotically decreasing, but the corrected rotation curve, although not flat, is higher than the Newtonian one, thus offering the possibility to fit rotation curves without dark matter. To test the viability of the model, we consider a sample of 15 low surface brightness galaxies with combined H I and Hα measurements of the rotation curve extending in the putative dark matter dominated region. We find a very good agreement between the theoretical rotation curve and the data using only stellar disc and interstellar gas when the slope n of the gravity Lagrangian is set to the value n = 3.5 (giving β = 0.817) obtained by fitting the Type Ia supernova Hubble diagram with the assumed power-law f(R) model and no dark matter. The excellent agreement between theoretical and observed rotation curves and the values of the stellar mass-to-light ratios in agreement with the predictions of population synthesis models make us confident that R n gravity may represent a good candidate to solve both the dark energy problem on cosmological scales and the dark matter one on galactic scales with the same value of the slope n of the higher-order gravity Lagrangian.

Higher order theories of gravity have recently attracted a lot of interest as alternative candidates to explain the observed cosmic acceleration without the need of introducing any scalar field. A critical ingredient is the choice of the function f (R) of the Ricci scalar curvature entering the gravity Lagrangian and determining the dynamics of the universe. We describe an efficient procedure to reconstruct f (R) from the Hubble parameter H depending on the redshift z. Using the metric formulation of f (R) theories, we derive a third order linear differential equation for f (R(z)) which can be numerically solved after setting the boundary conditions on the basis of physical considerations. Since H(z) can be reconstructed from the astrophysical data, the method we present makes it possible to determine, in principle, what is the f (R) theory which best reproduces the observed cosmological dynamics. Moreover, the method allows to reconcile dark energy models with f (R) theories finding out what is the expression of f (R) which leads to the same H(z) of the given quintessence model. As interesting examples, we consider "quiessence" (dark energy with constant equation of state) and the Chaplygin gas. PACS numbers: 98.80.-k, 04.50+h, 98.80.Jk, 98.80.Es * Corresponding author, email: winny@na.infn.it dubbed quintessence [ 8,9]. These models typically involve scalar fields with a particular class of potentials, allowing the vacuum energy to become dominant only recently (see [10] for comprehensive reviews). Although quintessence by a scalar field is the most studied candidate for dark energy, it generally does not avoid ad hoc fine tuning to solve the coincidence problem.