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We analyze cosmic microwave background (CMB) data taking into account the effects of a momentum dependent effective sound speed (MESS). This approach allows to study the effects of primordial entropy in a model independent way, and its implementation requires a minimal modification of existing CMB fitting numerical codes developed for single scalar field models. We adopt a phenomenological approach, and study the effects a local variation of the MESS around the scale where other analysis have shown some deviation from an approximately scale invariant curvature perturbation spectrum. We obtain a substantial improvement of the fit with respect to a model without MESS, showing that primordial entropy modeled by MESS can be an explanation of these deviations., Comment: 15 pages, 5 figures

For any physical system satisfying the Einstein's equations, the comoving curvature perturbations satisfy an equation involving the momentum-dependent effective sound speed, valid for any system with a well defined energy-stress tensor, including multi-fields models of inflation. We derive a general model-independent formula for the effective sound speed of comoving adiabatic perturbations, valid for a generic field-space metric, without assuming any approximation to integrate out entropy perturbations, but expressing the momentum-dependent effective sound speed in terms of the components of the total energy-stress tensor. As an application, we study a number of two-field models with a kinetic coupling between the fields, identifying the single curvature mode of the effective theory and showing that momentum-dependent effective sound speed fully accounts for the predictions for the power spectrum of curvature perturbations. Our results show that the momentum-dependent effective sound speed is a convenient scheme for describing all inflationary models that admit a single-field effective theory, including the effects of entropy perturbations present in multi-fields systems., Comment: 19 pages, fig. 6

The enhancement of the spectrum of primordial comoving curvature perturbation $\R$ can induce the production of primordial black holes (PBH) which could account for part of present day dark matter. As an example of the effects of the modification of gravity on the production of PBHs, we investigate the effects on the spectrum of $\R$ produced by the modification of gravity in the case of G-inflation, deriving the relation between the unitary gauge curvature perturbation $\zeta$ and the comoving curvature perturbation $\R$, and identifying a background dependent enhancement function $\E$ which can induce large differences between the two gauge invariant variables. We use this relation to derive an equation for $\R$, showing for the presence of a momentum dependent effective sound speed (MESS), associated to the intrinsic entropy which can arise in modified gravity theories, in agreement with the model independent MESS approach to cosmological perturbations. When $\zeta$ is not constant in time it is different from $\R$, for example on sub-horizon scales, or in models exhibiting an anomalous super-horizon growth of $\zeta$, but since this growth cannot last indefinitely, eventually they will coincide. We derive the general condition for super-horizon growth of $\zeta$, showing that slow-roll violation is not necessary. Since the abundance of PBHs depends on the statistics of the peaks of the comoving density contrast, which is related to the spectrum of $\R$, it is important to take into account these effects on the PBHs abundance in modified gravity theories., Comment: Version published in PLB

Local structure can have important effects on luminosity distance observations, which could for example affect the local estimation of the Hubble constant based on low red-shift type Ia supernovae. Using a spherically symmetric exact solution of the Eistein's equations and a more accurate expansion of the solution of the geodesic equations, we improve the low red-shift expansion of the monopole of the luminosity distance in terms of the curvature function. Based on this we derive the coordinate independent low red-shift expansion of the monopole of the luminosity distance in terms of the monopole of the density contrast. The advantage of this approach is that it relates the luminosity distance directly to density observations, without any dependency on the radial coordinate choice. We compare our formulae to numerical calculations, and find that the formula in terms of the density contrast is in good agreement with the numerical calculations, in the non linear regime is more accurate than the results obtained using linear perturbation theory, especially in the regime of large Laplacian of the perturbations, and is also more accurate than the formula in terms of the curvature function. Local structure can have important effects on luminosity distance observations, which could for example affect the local estimation of the Hubble constant based on low red-shift type Ia supernovae. Using a spherically symmetric exact solution of the Eistein's equations and a more accurate expansion of the solution of the geodesic equations, we improve the low red-shift expansion of the monopole of the luminosity distance in terms of the curvature function. Based on this we derive the coordinate independent low red-shift expansion of the monopole of the luminosity distance in terms of the monopole of the density contrast. The advantage of this approach is that it relates the luminosity distance directly to density observations, without any dependency on the radial coordinate choice. We compute the effects of different inhomogeneities on the luminosity distance, and find that the formulae in terms of the density contrast are in good agreement with numerical calculations, in the non linear regime are more accurate than the results obtained using linear perturbation theory, and are also more accurate than the formulae in terms of the curvature function.

We show that in single field inflationary models the super-horizon evolution of curvature perturbations on comoving slices &calR;, which can cause the production of primordial black holes (PBH), is not due to entropy perturbations, but to the background evolution effect on the conversion between entropy and curvature perturbations. We derive a general relation between the time derivative of comoving curvature perturbations and entropy perturbations, in terms of a conversion factor depending on the background evolution. Contrary to previous results derived in the uniform density gauge assuming the gradient term can be neglected on super-horizon scales, the relation is valid on any scale for any minimally coupled single scalar field model, also on sub-horizon scales where gradient terms are large. We apply it to the case of quasi-inflection inflation, showing that while entropy perturbations are decreasing, &calR; can grow on super-horizon scales, due to a large increase of the conversion factor. This happens in the time interval during which a sufficiently fast decrease of the equation of state w transforms into a growing mode what in slow-roll models would be a decaying mode. The same mechanism also explains the super-horizon evolution of &calR; in globally adiabatic systems, for which entropy perturbations vanish on any scale, such as ultra slow roll inflation and its generalizations. We show that in single field inflationary models the super-horizon evolution of curvature perturbations on comoving slices $\mathcal{R}$, which can cause the production of primordial black holes (PBH), is not due to entropy perturbations, but to the background evolution effect on the conversion between entropy and curvature perturbations. We derive a general relation between the time derivative of comoving curvature perturbations and entropy perturbations, in terms of a conversion factor depending on the background evolution. Contrary to previous results derived in the uniform density gauge assuming the gradient term can be neglected on super-horizon scales, the relation is valid on any scale for any minimally coupled single scalar field model, also on sub-horizon scales where gradient terms are large. We apply it to the case of quasi-inflection inflation, showing that while entropy perturbations are decreasing, $\mathcal{R}$ can grow on super-horizon scales, due to a large increase of the conversion factor. This happens in the time interval during which a sufficiently fast decrease of the equation of state $w$ transforms into a growing mode that in slow-roll models would be a decaying mode. The same mechanism also explains the super-horizon evolution of $\mathcal{R}$ in globally adiabatic systems, for which entropy perturbations vanish on any scale, such as ultra slow-roll inflation and its generalizations.

We derive the equation for the evolution of the curvature perturbation on the comoving time slice, $\mathcal{R}_c$, in the presence of anisotropic and non-adiabatic terms in the energy–momentum tensor of matter fields. The equation is obtained by manipulating the perturbed Einstein’s equations in the comoving time slice. It could be used to study the evolution of the comoving curvature perturbations for systems with an anisotropic energy–momentum tensor, such as in the presence of vector fields, in the presence of entropy, such as in a multi-field system, or in modified gravity theories. As a simple application, after checking that the comoving time slice for a multi-field system does not coincide with the uniform field time slice in general, we use the equation in the case of two minimally coupled scalar fields and derive a closed set of equations for the curvature and entropy perturbations on the comoving time slice. We derive the equation for the evolution of the curvature perturbation on the comoving time slice, $\mathcal{R}_c$, in the presence of anisotropic and non-adiabatic terms in the energy-momentum tensor of matter fields. The equation is obtained by manipulating the perturbed Einstein's equations in the comoving time slice. It could be used to study the evolution of the comoving curvature perturbations for systems with an anisotropic energy-momentum tensor, such as in the presence of vector fields, in the presence of entropy, such as in a multi-field system, or in modified gravity theories. As a simple application, after checking that the comoving time slice for a multi-field system does not coincide with the uniform field time slice in general, we use the equation in the case of two minimally coupled scalar fields and derive a closed set of equations for the curvature and entropy perturbations on the comoving time slice.