Cosmological perturbations in Palatini formalism

Abstract

We investigate cosmological perturbations of scalar-tensor theories in Palatini formalism. First we introduce an action where the Ricci scalar is conformally coupled to a function of a scalar field and its kinetic term and there is also a k-essence term consisting of the scalar and its kinetic term. This action has three frames that are equivalent to one another: the original Jordan frame, the Einstein frame where the metric is redefined, and the Riemann frame where the connection is redefined. For the first time in the literature, we calculate the quadratic action and the sound speed of scalar and tensor perturbations in three different frames and show explicitly that they coincide. Furthermore, we show that for such action the sound speed of gravitational waves is unity. Thus, this model serves as dark energy as well as an inflaton even though the presence of the dependence of the kinetic term of a scalar field in the non-minimal coupling, different from the case in metric formalism. We then proceed to construct the L3 action called Galileon terms in Palatini formalism and compute its perturbations. We found that there are essentially 10 different (inequivalent) definitions in Palatini formalism for a given Galileon term in metric formalism. We also see that, in general, the L3 terms have a ghost due to Ostrogradsky instability and the sound speed of gravitational waves could potentially deviate from unity, in sharp contrast with the case of metric formalism. Interestingly, once we eliminate such a ghost, the sound speed of gravitational waves also becomes unity. Thus, the ghost-free L3 terms in Palatini formalism can still serve as dark energy as well as an inflaton, like the case in metric formalism.

(3+1)-Formulation for gravity with torsion and non-metricity: The stress-energy-momentum equation

“…A detailed and complete explanation of MAG could be found in [7,8], while a detailed account of f (R) and Metric-Affine f (R) could be found in [1,10,11]. Some of the latest progress on these theories include the mathematical and geometrical analysis [12,13], ghost-free higher-order curvature gravity [14][15][16], metric-affine cosmology [17][18][19][20][21], and the (3+1) formulation of special cases of MAG [22][23][24][25][26][27], which become the main interest in this article.…”

Cosmology of Quadratic Metric-Affine Gravity

“…Under the perspective described above, MAG theories result to be rather general and powerful [11,12,[15][16][17][18][19][20][21][22][23], and can possibly lead to diverse scenarios beyond GR, which may have relevant consequences, for instance, in the cosmological context. For some relevant results already obtained in MAG cosmology we refer the reader to, e.g., [24][25][26][27][28][29][30][31]. In particular, in this paper we are interested in the cosmological aspects, in a Friedmann-Lemaître-Robertson-Walker (FLRW) background, of the most general MAG theory in n spacetime dimensions involving all possible parity preserving quadratic terms in torsion and non-metricity, in the presence of a cosmological hyperfluid.…”

Riemann Tensor and Gauss-Bonnet density in Metric-Affine Cosmology

“…This Metric-Affine framework has received a renewed attention over the last years [11][12][13][14][15][16][17][18][19][20][21][22][23] especially in regards to its cosmological applications [5,[24][25][26][27][28][29][30][31]. It therefore becomes apparent that it is of great importance to have a complete setup for both the geometrical and the matter structures on this non-Riemannian Cosmology.…”