Frederico Arroja

We compute analytically the dominant contribution to the tree-level bispectrum in the Starobinsky model of inflation. In this model, the potential is vacuum energy dominated but contains a subdominant linear term which changes the slope abruptly at a point. We show that on large scales compared with the transition scale $k_0$ and in the equilateral limit the analogue of the non-linearity parameter scales as $(k/k_0)^2$, that is its amplitude decays for larger and larger scales until it becomes subdominant with respect to the usual slow-roll suppressed corrections. On small scales we show that the non-linearity parameter oscillates with angular frequency given by $3/k_0$ and its amplitude grows linearly towards smaller scales and can be large depending on the model parameters. We also compare our results with previous results in the literature.

We study an inflationary model driven by a single minimally coupled standard kinetic term scalar field with a step in its mass modeled by an Heaviside step function. We present an analytical approximation for the mode function of the curvature perturbation, obtain the power spectrum analytically and compare it with the numerical result. We show that, after the scale set by the step, the spectrum contains damped oscillations that are well described by our analytical approximation. We also compute the dominant contribution to the bispectrum in the equilateral and the squeezed limits and find new shapes. In the equilateral and squeezed limits the bispectrum oscillates and it has a linear growth envelope towards smaller scales. The bispectrum size can be large depending on the model parameters.

In this short note we clarify the role of the boundary terms in the calculation of the leading order tree-level bispectrum in a fairly general minimally coupled single field inflationary model, where the inflaton's Lagrangian is a general function of the scalar field and its first derivatives. This includes k-inflation, DBI-inflation and standard kinetic term inflation as particular cases. These boundary terms appear when simplifying the third order action by using integrations by parts. We perform the calculation in the comoving gauge obtaining explicitly all total time derivative interactions and show that a priori they cannot be neglected. The final result for the bispectrum is equal to the result present in the literature which was obtained using the field redefinition.

In this short note, we obtain the necessary and sufficient condition for a class of non-canonical single scalar field models to be exactly equivalent to barotropic perfect fluids, under the assumption of an irrotational fluid flow. An immediate consequence of this result is that the non-adiabatic pressure perturbation in this class of scalar field systems vanishes exactly at all orders in perturbation theory and on all scales. The Lagrangian for this general class of scalar field models depends on both the kinetic term and the value of the field. However, after a field redefinition, it can be effectively cast in the form of a purely kinetic K-essence model.

We confirm the claim by Blas et al. [arXiv:0909.3525] that, in the infrared limit of Ho\v{r}ava-Lifshitz gravity, the scalar graviton becomes a ghost if the sound speed squared is positive on the flat de Sitter and Minkowski background. In order to avoid the ghost and tame the instability, the sound speed squared should be negative and very small, which means that the flow parameter $\lambda$ should be very close to its General Relativity (GR) value. We calculate the cubic interactions for the scalar graviton which are shown to have a similar structure with those of the curvature perturbation in k-inflation models. The higher order interactions become increasing important for a smaller sound speed squared, that is, when the theory approaches GR. This invalidates any linearized analysis and any predictability is lost in this limit as quantum corrections are not controllable. This pathological behaviour of the scalar graviton casts doubt on the validity of the projectable version of the theory.

We compute the second-order matching conditions for tensor metric perturbations at an abrupt change in the equation of state. For adiabatic perturbations on large scales the matching hypersurface coincides with a uniform-density hypersurface. We show that in the uniform-density gauge both the tensor perturbation and its time-derivative are continuous in this case. For non-adiabatic perturbations, the matching hypersurface need not coincide with a uniform-density hypersurface and the tensor perturbation in the uniform-density gauge may be discontinuous. However, we show that in the Poisson gauge both the tensor perturbation and its time-derivative are continuous for adiabatic or non-adiabatic perturbations. As an application we solve the evolution equation for second-order tensor perturbations on large scales for a constant equation of state and we use the matching conditions to evolve the solutions through the transition from an inflationary era to a radiation era. We show that in the radiation era the resulting free part of the large-scale tensor perturbation (constant mode) is slow-roll suppressed in both the uniform-density and Poisson gauges. Thus, we conclude that second-order gravitational waves from slow-roll inflation are suppressed.

We compute the leading order connected four-point function of the primordial curvature perturbation coming from the four-point function of the fields in multi-field DBI inflation models. We confirm that the consistency relations in the squeezed limit and in the counter-collinear limit hold as in single field models thanks to special properties of the DBI action. We also study the momentum dependence of the trispectra coming from the adiabatic, mixed and purely entropic contributions separately and we find that they have different momentum dependence. This means that if the amount of the transfer from the entropy perturbations to the curvature perturbation is significantly large, the trispectrum can distinguish multi-field DBI inflation models from single field DBI inflation models. A large amount of transfer $T_{\mathcal{RS}} \gg 1 $ suppresses the tensor to scalar ratio $r \propto T_{\mathcal{RS}}^{-2}$ and the amplitude of the bispectrum $f_{NL}^{equi} \propto T_{\mathcal{RS}}^{-2}$ and so it can ease the severe observational constraints on the DBI inflation model based on string theory. On the other hand, it enhances the amplitude of the trispectrum $\tau_{NL}^{equi} \propto T_{\mathcal{RS}}^2 f_{NL}^{equi 2}$ for a given amplitude of the bispectrum.

We show that higher-order actions for cosmological perturbations in the multi-field DBI-inflation model are obtained by a Lorentz boost from the rest frame of the brane to the frame where the brane is moving. We confirm that this simple method provides the same third- and fourth- order actions at leading order in slow-roll and in the small sound speed limit as those obtained by the usual ADM formalism. As an application, we compute the leading order connected four-point function of the primordial curvature perturbation coming from the intrinsic fourth-order contact interaction in the multi-field DBI-inflation model. At the third order, the interaction Hamiltonian arises purely by the boost from the second-order action in the rest frame of the brane. The boost acts on the adiabatic and entropy modes in the same way thus there exists a symmetry between the adiabatic and entropy modes. But at fourth order this symmetry is broken due to the intrinsic fourth-order action in the rest frame and the difference between the Lagrangian and the interaction Hamiltonian. Therefore, contrary to the three-point function, the momentum dependence of the purely adiabatic component and the components including the entropic contributions are different in the four-point function. This suggests that the trispectrum can distinguish the multi-field DBI-inflation model from the single field DBI-inflation model.

We compute the tree-level connected four-point function of the primordial curvature perturbation for a fairly general minimally coupled single field inflationary model, where the inflaton's Lagrangian is a general function of the scalar field and its first derivatives. This model includes K-inflation and DBI-inflation as particular cases. We show that, at the leading order in the slow-roll expansion and in the small sound speed limit, there are two important tree-level diagrams for the trispectrum. One is a diagram where a scalar mode is exchanged and the other is a diagram where the interaction occurs at a point, i.e. a contact interaction diagram. The scalar exchange contribution is comparable to the contact interaction contribution. For the DBI-inflation model, in the so-called equilateral configuration, the scalar exchange trispectrum is maximized when the angles between the four momentum vectors are equal and in this case the amplitude of the trispectrum from the scalar exchange is one order of magnitude higher than the contact interaction trispectrum.

Models with extra dimensions have attracted much interest recently because they may provide the solution for long standing problems in physics. One interesting and very attractive idea is that our visible universe is confined to a four-dimensional hypersurface in a higher-dimensional spacetime. This membrane like universe was dubbed brane-world. The main goal of this thesis is the study of the four-dimensional (4D) effective theories and their observational consequences in the brane-world universe. After introducing the brane-world idea with some detail we shall use the gradient expansion method to obtain the 4D effective theories of gravity for several higher-dimensional theories with different numbers of extra-dimensions. In the second half of the thesis, after introducing the concept of brane-inflation we will focus on some observational consequences of these low energy effective theories. In particular, the last two chapters before the conclusion are devoted to the study of non-Gaussianities in general models of inflation.

We study the non-gaussianity from the bispectrum in multi-field inflation models with a general kinetic term. The models include the multi-field K-inflation and the multi-field Dirac-Born-Infeld (DBI) inflation as special cases. We find that, in general, the sound speeds for the adiabatic and entropy perturbations are different and they can be smaller than 1. Then the non-gaussianity can be enhanced. The multi-field DBI-inflation is shown to be a special case where both sound speeds are the same due to a special form of the kinetic term. We derive the exact second and third order actions including metric perturbations. In the small sound speed limit and at leading order in the slow-roll expansion, we derive the three point function for the curvature perturbation which depends on both adiabatic and entropy perturbations. The contribution from the entropy perturbations has a different momentum dependence if the sound speed for the entropy perturbations is different from the adiabatic one, which provides a possibility to distinguish the multi-field models from single field models. On the other hand, in the multi-field DBI case, the contribution from the entropy perturbations has the same momentum dependence as the pure adiabatic contributions and it only changes the amplitude of the three point function. This could help to ease the constraints on the DBI-inflation models.

We compute the fourth order action in perturbation theory for scalar and second order tensor perturbations for a minimally coupled single field inflationary model, where the inflaton's lagrangian is a general function of the field's value and its kinetic energy. We obtain the fourth order action in two gauges, the comoving gauge and the uniform curvature gauge. Using the comoving gauge action we calculate the trispectrum at leading order in slow-roll, finding agreement with a previously known result in the literature. We point out that in general to obtain the correct leading order trispectrum one cannot ignore second order tensor perturbations as previously done by others. The next-to-leading order corrections may become detectable depending on the shape and we provide the necessary formalism to calculate them.

We derive the low energy effective theory on a brane in six-dimensional chiral supergravity. The conical 3-brane singularities are resolved by introducing cylindrical codimension one 4-branes whose interiors are capped by a regular spacetime. The effective theory is described by the Brans-Dicke (BD) theory with the BD parameter given by $\omega_{\rm BD}=1/2$. The BD field is originated from a modulus which is associated with the scaling symmetry of the system. If the dilaton potentials on the branes preserve the scaling symmetry, the scalar field has an exponential potential in the Einstein frame. We show that the time dependent solutions driven by the modulus in the four-dimensional effective theory can be lifted up to the six-dimensional exact solutions found in the literature. Based on the effective theory, we discuss a possible way to stabilize the modulus to recover standard cosmology and also study the implication for the cosmological constant problem.

We present a systematic way to derive the four-dimensional effective theories for warped compactifications with fluxes and branes in the ten-dimensional type IIB supergravity. The ten-dimensional equations of motion are solved using the gradient expansion method and the effective four-dimensional equations of motions are derived by imposing the consistency condition that the total derivative terms with respect to the six-dimensional internal coordinates vanish when integrated over the internal manifold. By solving the effective four-dimensional equations, we can find the gravitational backreaction to the warped geometry due to the dynamics of moduli fields, branes and fluxes.

We consider a 5D BPS dilatonic two brane model which reduces to the Randall-Sundrum model or the Horava-Witten theory for a particular choice of parameters. Recently new dynamical solutions were found by Chen et al., which describe a moduli instability of the warped geometry. Using a 4D effective theory derived by solving the 5D equations of motion, based on the gradient expansion method, we show that the exact solution of Chen et. al. can be reproduced within the 4D effective theory and we identify the origin of the moduli instability. We revisit the gradient expansion method with a new metric ansatz to clarify why the 4D effective theory solution can be lifted back to an exact 5D solution. Finally we argue against a recent claim that the 4D effective theory allows a much wider class of solutions than the 5D theory and provide a way to lift solutions in the 4D effective theory to 5D solutions perturbatively in terms of small velocities of the branes.