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The body text of the paper ends up calling the "Hybrid metric-Palatini gravity theory" f(X) gravity. From the body text (citations and internal references omitted without notation):Hybrid metric-Palatini gravity

Salvatore Capozziello, Tiberiu Harko, Tomi S. Koivisto, Francisco S.N. Lobo, Gonzalo J. Olmo

(Submitted on 19 Aug 2015)

Recently, the phenomenology of f(R) gravity has been scrutinized motivated by the possibility to account for the self-accelerated cosmic expansion without invoking dark energy sources. Besides, this kind of modified gravity is capable of addressing the dynamics of several self-gravitating systems alternatively to the presence of dark matter. It has been established that both metric and Palatini versions of these theories have interesting features but also manifest severe and different downsides. A hybrid combination of theories, containing elements from both these two formalisms, turns out to be also very successful accounting for the observed phenomenology and is able to avoid some drawbacks of the original approaches. This article reviews the formulation of this hybrid metric-Palatini approach and its main achievements in passing the local tests and in applications to astrophysical and cosmological scenarios, where it provides a unified approach to the problems of dark energy and dark matter.

The paper closes by identifying many issues for further study and arguing that the effort to do so would be worthwhile. In this work we have presented a hybrid metric-Palatini framework for theories of gravity, and have tested the new theories it entails using a number of theoretical consistency checks and observational constraints. From the field theory perspective, we found that the f(X) class of theories, where X = R + κ^{2}T, enjoys a similar special status amongst the more general hybrid metric-Palatini theories as the f(R) theories within the narrower framework of purely metric gravity. This is so because when one excludes theories inhabited by ghost-like, superluminally propagating and otherwise pathological degrees of freedom, there is evidence that the f(X) family is singled out as the only viable form of an action one can construct using the metric (and thus the metric Levi-Civita connection) and an independent “Palatini connection”. . . . f(X) gravity represents a generic case within the one-parameter family of the Algebraic Scalar-Tensor theories at one end of which lies the pure Palatini f(R) (wherein the scalar field is a function of the stress-energy trace T) and at the other end the pure metric f(R) (where the field is a function of the metric curvature R). Furthermore, the propagating degrees of freedom have proven to be healthy also on curved backgrounds as confirmed also by our cosmological perturbation analysis. Concerning the Cauchy problem, it was shown that in this class of theories the initial value problem can always be well-formulated and well-posed depending on the adopted matter sources. Having established the theoretical consistency and interest on the hybrid metric-Palatini f(X) family of theories, we considered applications in which these theories provide gravitational alternatives to dark energy.

As shown by our post-Newtonian analysis, the hybrid theories are promising in this respect as they can avoid the local gravity constraints but modify the cosmological dynamics at large scales. This is simply because as a scalar-tensor theory, the hybrid f(X) gravity is characterised by an evolving Brans-Dicke coupling, which allows to introduce potentially large deviations from GR in the past (and future) as long as the coupling at the present epoch is strong enough to hide the field from the local gravity experiments. In contrast, in the metric f(R) models the Brans-Dicke coupling is a finite constant and one needs to invoke some of the various “screening mechanisms” (workings of which remain to be studied in the hybrid theories) in order to reconcile the Solar system experiments with cosmology.

Cosmological perturbations have been also analysed in these models up to the linear order, and the results imply that the formation of large-scale structure in the aforementioned accelerating cosmologies is viable though exhibits subtle features that might be detectable in future experiments. We derived the full perturbations equations and extracting their Newtonian limit, describing the observable scales of the matter power spectrum, the growth of matter overdensities was shown to be modified by a time-dependent effective fifth force that is expected to modify the redshift evolution of the growth rate of perturbations. We also note that numerical studies of the perturbations imply that the difference of the gravitational potentials can exhibit oscillations at higher redshifts even when the background expansion and the full lensing potential are indistinguishable from the standard ΛCDM predictions. Such features could potentially be observed in cross-correlations of the matter and lensing power spectra. . . .

At an effective level, the f(X) modifications involve both (the trace of) the matter stress energy and (the Ricci scalar of) the metric curvature, and from this point of view it appears appealing to speculate on the possible relevance of these theories to both the problems of dark energy and dark matter, in a unified theoretical framework and without distinguishing a priori matter and geometric sources. Various aspects of dark matter phenomenology from astronomical to galactic and extragalatic scales were discussed. The generalised virial theoreom can acquire, in addition to the contribution from the baryonic masses, effective contributions of geometrical origin to the total gravitational potential energy, which may account for the well-known virial theorem mass discrepancy in clusters of galaxies. In the context of galactic rotation curves, the scalar-field modified relations between the various physical quantities such as tangential velocities of test particles around galaxies, Doppler frequency shifts and stellar dispersion velocities were derived. More recently, observational data of stellar motion near the Galactic centre was compared with simulations of the hybrid gravity theory, which turned out particularly suitable to model star dynamics.

This is a classical modified GR theory, not a quantum gravity theory. But, the fact that a term in the form of a Yukawa potential naturally arises in the f(X) formulation is suggestive of the notion that f(X) gravity might be the classical limit of a quantum gravity theory with a non-trivial self-interaction term.

While f(R) gravity theories and Palatini gravity theories have each been studied extensively in isolation, the f(X) gravity hybrid is a new car that has only been on a small number of test drives.

Equations that seem to make sense have been derived to address both dark energy and dark matter phenomena with this theory (and it may be able to provide guidance on the issue of inflation as well), but no one has plugged in numbers and rigorously compared the performance of f(X) gravity to other modified gravity theories or to lamdaCDM relative to the empirical evidence. Several specific realistic to conduct observational tests of the theory have been suggested, but none have actually been conducted.

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# An interesting new paper on f(X) gravity

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