Slow-roll Conditions Research Articles

Consistency of the tachyon warm inflationary universe models

This study concerns the consistency of the tachyon warm inflationary models. A linear stability analysis is performed to find the slow-roll conditions, characterized by the potential slow-roll (PSR) parameters, for the existence of a tachyon warm inflationary attractor in the system. The PSR parameters in the tachyon warm inflationary models are redefined. Two cases, an exponential potential and an inverse power-law potential, are studied, when the dissipative coefficient Γ = Γ0 and Γ = Γ(ϕ), respectively. A crucial condition is obtained for a tachyon warm inflationary model characterized by the Hubble slow-roll (HSR) parameter ϵH, and the condition is extendable to some other inflationary models as well. A proper number of e-folds is obtained in both cases of the tachyon warm inflation, in contrast to existing works. It is also found that a constant dissipative coefficient (Γ = Γ0) is usually not a suitable assumption for a warm inflationary model.

On squeezed limits in single-field inflation. Part I

The n-point correlation functions in single-field inflation obey a set of consistency conditions in the exact squeezed limit which are not present in multi-field models, and thus are powerful tools to distinguish between the two. However, these consistency conditions may be violated for a finite range of scales in single-field models, for example by departures from the Bunch-Davies state. These excited states may be the consequence of interactions during inflation, or may be a remnant of the era that preceded inflation. In this paper we analyze the bispectrum, and show that in the regime of theoretical control the resulting signal in the squeezed limit remains undetectably small in all known models which continuously excite the state. We also show that the signal remains undetectably small if the initial state is related to the Bunch-Davies state by a Bogoliubov transformation and the energy density in the state is small enough so that the usual slow-roll conditions are obeyed. Bogoliubov states that lead to violations of the slow-roll conditions, as well as more general excited states, require more careful treatment and will be discussed in a separate publication.

δNformalism

Precise understanding of nonlinear evolution of cosmological perturbations during inflation is necessary for the correct interpretation of measurements of non-Gaussian correlations in the cosmic microwave background and the large-scale structure of the Universe. The ``$\ensuremath{\delta}N$ formalism'' is a popular and powerful technique for computing nonlinear evolution of cosmological perturbations on large scales. In particular, it enables us to compute the curvature perturbation $\ensuremath{\zeta}$ on large scales without actually solving perturbed field equations. However, people often wonder why this is the case. In order for this approach to be valid, the perturbed Hamiltonian constraint and matter-field equations on large scales must, with a suitable choice of coordinates, take on the same forms as the corresponding unperturbed equations. We find that this is possible when (1) the unperturbed metric is given by a homogeneous and isotropic Friedmann-Lema\^{\i}tre-Robertson-Walker metric; and (2) on large scales and with a suitable choice of coordinates, one can ignore the shift vector (${g}_{0i}$) as well as time dependence of tensor perturbations to ${g}_{ij}/{a}^{2}(t)$ of the perturbed metric. While the first condition has to be assumed a priori, the second condition can be met when (3) the anisotropic stress becomes negligible on large scales. However, in order to explicitly show that the second condition follows from the third condition, one has to use gravitational field equations, and thus this statement may depend on the details of the theory of gravitation. Finally, as the $\ensuremath{\delta}N$ formalism uses only the Hamiltonian constraint and matter-field equations, it does not a priori respect the momentum constraint. We show that the error in the momentum constraint only yields a decaying mode solution for $\ensuremath{\zeta}$, and the error vanishes when the slow-roll conditions are satisfied.

Constraining cosmologies with fundamental constants – I. Quintessence and K-essence

Many cosmological models invoke rolling scalar fields to account for the observed acceleration of the expansion of the universe. These theories generally include a potential V(phi) which is a function of the scalar field phi. Although V(phi) can be represented by a very diverse set of functions, recent work has shown the under some conditions, such as the slow roll conditions, the equation of state parameter w is either independent of the form of V(phi) or is part of family of solutions with only a few parameters. In realistic models of this type the scalar field couples to other sectors of the model leading to possibly observable changes in the fundamental constants such as the fine structure constant alpha and the proton to electron mass ratio mu. This paper explores the limits this puts on the validity of various cosmologies that invoke rolling scalar fields. We find that the limit on the variation of mu puts significant constraints on the product of a cosmological parameter w+1 times a new physics parameter zeta_mu^2, the coupling constant between mu and the rolling scalar field. Even when the cosmologies are restricted to very slow roll conditions either the value of zeta_mu must be at the lower end of or less than its expected values or the value of w+1 must be restricted to values vanishingly close to 0. This implies that either the rolling scalar field is very weakly coupled with the electromagnetic field, small zeta_mu, very weakly coupled with gravity, w+1 ~ 0 or both. These results stress that adherence to the measured invariance in mu is a very significant test of the validity of any proposed cosmology and any new physics it requires. The limits on the variation of mu also produces a significant tension with the reported changes in the value of alpha.

Strong multifield slowroll condition and spiral inflation

We point out the existing confusions about the slowroll parameters and conditions for multifield inflation. If one requires the fields to roll down the gradient flow, we find that only articles adopting the Hubble slowroll expansion are on the right track, and a correct condition can be found in a recent book by Liddle and Lyth. We further analyze this condition and show that the gradient flow requirement is stronger than just asking for a slowly changing, quasi-de Sitter solution. Therefore it is possible to have a multifield slowroll model that does not follow the gradient flow. Consequently, it no longer requires the gradient to be small. It even bypasses the first slowroll condition and some related no-go theorems from string theory. We provide the "spiral inflation" as a generic blueprint of such inflation model and show that it relies on a monodromy locus---a common structure in string theory effective potentials.

Decoupling survives inflation: a critical look at effective field theory violations during inflation

We investigate the validity of effective field theory methods and the decoupling ofheavy fields during inflation.Considering models of inflation in which the inflaton is coupled to a heavy (super-Hubble) degree of freedominitially in its vacuum state, we find that violations of decoupling are absent unlessthere is a breakdown of the slow-roll conditions. Next we allow for atemporary departure from inflation resulting in a period of non-adiabaticity during which effective field theory methodsare known to fail.We find that the locality of the event and energy conservation lead to a tight bound on the size of the effects of the heavy field. We discuss the implications for the power spectrum and non-gaussianity, and comment on the connection with recent studies of the dynamics of multi-field inflation models. Our results further motivate the use of effective field theory methods to characterize cosmic inflation,and focus the question of observability of additional degrees of freedom during inflation to near the Hubble scale or below —as anticipated from the Wilsonian notions of decoupling and naturalness.

Predictions of just-enough inflation

We find the best-fit cosmological parameters for a scenario of inflation with only the sufficient amount of accelerated expansion for the $\lambda\phi^4$ potential. While for the simplest scenario of chaotic inflation all observable primordial fluctuations cross the Hubble horizon during the slow-roll epoch, for the scenario of just-enough inflation the slow-roll conditions are violated at the largest length scales. Performing a numerical mode-by-mode integration for the perturbations on the largest scales and comparing the predicted anisotropies of the cosmic microwave background to results from the WMAP 7-yr data analysis, we find the initial conditions in agreement with current cosmological data. In contrast to the simplest chaotic model for the quartic potential, the just-enough inflation scenario is not ruled out. Although this scenario naturally gives rise to a modification of the first multipoles, for a quartic potential it cannot explain the lack of power at the largest angular scales.

Phenomenology of the minimal inflation scenario: inflationary trajectories and particle production

We discuss the phenomenology of the minimal inflation scenario. We concentrate on two aspects: inflationary trajectories and particle production. Our findings can be summarized in two main results: first, that inflationary trayectories that are very flat and provide enough number of e-foldings are natural in the scenario without fine tunning. We present a general formalism toidentify attractors in multi-field inflation regardless of trajectories fulfilling the slow-roll conditions. We then explore particle production inthe model and show how the inflaton naturally transmutes into a dark matter particle. One interesting feature of our model is that it provides a novel mechanism to generate particles and entropy in the universe: the filling of the Fermi sphere up to a given momentum pF due to the sea of goldstinos that are an important part of thematter generated after inflation. With this mechanism in hand we predict that the gravitino should have a mass of > 100−1000 TeV. Another interesting feature of our model is that the predicted level of gravity waves is r = 0.1−0.001, which is in the range of detectability from Planck and upcoming CMB polarization experiments.

Generalized non-commutative inflation

Non-commutative geometry indicates a deformation of the energy–momentum dispersion relation for massless particles. This distorted energy–momentum relation can affect the radiation-dominated phase of the universe at sufficiently high temperature. This prompted the idea of non-commutative inflation by Alexander et al (2003 Phys. Rev. D 67 081301) and Koh and Brandenberger (2007 JCAP06(2007)021 and JCAP11(2007)013). These authors studied a one-parameter family of a non-relativistic dispersion relation that leads to inflation: the α family of curves f(E) = 1 + (λE)α. We show here how the conceptually different structure of symmetries of non-commutative spaces can lead, in a mathematically consistent way, to the fundamental equations of non-commutative inflation driven by radiation. We describe how this structure can be considered independently of (but including) the idea of non-commutative spaces as a starting point of the general inflationary deformation of . We analyze the conditions on the dispersion relation that leads to inflation as a set of inequalities which plays the same role as the slow-roll conditions on the potential of a scalar field. We study conditions for a possible numerical approach to obtain a general one-parameter family of dispersion relations that lead to successful inflation.

The scalar bi-spectrum in the Starobinsky model: the equilateral case

While a featureless, nearly scale invariant, primordial scalar power spectrum fits the most recent Cosmic Microwave Background (CMB) data rather well, certain features in the spectrum are known to lead to a better fit to the data (although, the statistical significance of such results remains an open issue). In the inflationary scenario, one or more periods of deviations from slow roll are necessary in order to generate features in the scalar perturbation spectrum. Over the last couple of years, it has been recognized that such deviations from slow roll inflation can also result in reasonably large non-Gaussianities. The Starobinsky model involves the canonical scalar field and consists of a linear inflaton potential with a sudden change in the slope. The change in the slope causes a brief period of departure from slow roll which, in turn, results in a sharp rise in power, along with a burst of oscillations in the scalar spectrum for modes that leave the Hubble radius just before and during the period of fast roll. The hallmark of the Starobinsky model is that it allows the scalar power spectrum to be evaluated analytically in terms of the three parameters that describe the model, viz. the two slopes that describe the potential on either side of the discontinuity and the Hubble scale at the time when the field crosses the discontinuity. In this work, we evaluate the bi-spectrum of the scalar perturbations in the Starobinsky model in the equilateral limit. Remarkably, we find that, just as the power spectrum, all the different contributions to the the bi-spectrum too can be evaluated completely analytically and expressed in terms of the three paramaters that describe the model. We show that the quantity fNL, which characterizes the extent of non-Gaussianity, can be expressed purely in terms of the ratio of the two slopes on either side of the discontinuity in the potential. Further, we find that, for certain values of the parameters, fNL in the Starobinsky model can be as large as the mean value that has been arrived at from the analysis of the recent CMB data. We also demonstrate that the usual hierarchy of contributions to the bi-spectrum can be altered for certain values of the parameters. Altogether, we find that the Starobinsky model represents a unique scenario wherein, even when the slow roll conditions are violated, the background, the perturbations as well as the corresponding two and three point correlation functions can be evaluated completely analytically. As a consequence, the Starobinsky model can also be used to calibrate numerical codes aimed at computing the non-Gaussianities.

Non-Gaussianity in single field models without slow-roll conditions

We investigate non-Gaussianity in general single field models without assuming slow-roll conditions or the exact scale-invariance of the scalar power spectrum. The models considered include general single field inflation (e.g. DBI and canonical inflation) as well as bimetric models. We compute the full non-Gaussian amplitude, its size fnl, its shape, and the running with scale n_{NG}. In doing so we show that observational constraints allow significant violations of slow roll conditions and we derive explicit bounds on slow-roll parameters for fast-roll single field scenarios. A variety of new observational signatures is found for models respecting these bounds. We also explicitly construct concrete model implementations giving rise to this new phenomenology.

The Effect of Varying Sound Velocity on Primordial Curvature Perturbations

We study the effects of sudden change in the sound velocity on primordial curvature perturbation spectrum in inflationary cosmology, assuming that the background evolution satisfies the slow-roll condition throughout. It is found that the power spectrum acquires oscillating features which are determined by the ratio of the sound speed before and after the transition and the wavenumeber which crosses the sound horizon at the transition, and their analytic expression is given. In some values of those parameters, the oscillating primordial power spectrum can better fit the observed Cosmic Microwave Background temperature anisotropy power spectrum than the simple power-law power spectrum, although introduction of such a new degree of freedom is not justified in the context of Akaike's Information Criterion.

Ultraviolet-Protected Inflation

In natural inflation, the inflaton is a pseudo-Nambu-Goldstone boson which acquires a mass by explicit breaking of a global shift symmetry at scale f. In this case, for small field values, the potential is flat and stable under radiative corrections. Nevertheless, slow roll conditions enforce f ≫ M(p), making the validity of the whole scenario questionable. In this Letter, we show that a coupling of the inflaton kinetic term to the Einstein tensor allows f ≪ M(p) by enhancing the gravitational friction acting on the inflaton during inflation. This new unique interaction (a) keeps the theory perturbative in the whole inflationary trajectory, (b) preserves the tree-level shift invariance of the pseudo-Nambu-Goldstone boson, and (c) avoids the introduction of any new degrees of freedom with respect to standard natural inflation.

Inflation driven by scalar field with non-minimal kinetic coupling with Higgs and quadratic potentials

We study a scalar field with non-minimal kinetic coupling to itself and to the curvature. The slow rolling conditions allowing an inflationary background have been found. The quadratic and Higgs type potentials have been considered, and the corresponding values for the scalar fields at the end of inflation allows to recover the connection with particle physics.

A minimal inflation scenario

We elaborate on a minimal inflation scenario based entirely on the general propertiesof supersymmetry breaking in supergravity models. We identify the inflaton as the scalar component of the Goldstino superfield. We write plausible candidates for the effective action describing this chiral superfield. In particular the theory depends (apart from parameters of O(1)) on a single free parameter: the scale of supersymmetry breaking. This can be fixed using the amplitude of CMB cosmological perturbations and we therefore obtain the scale of supersymmetry breaking to be 1012−14 GeV. The model also incorporates explicit R-symmetry breaking in order to satisfy the slow roll conditions. In our model the η−problem is solved without extra fine-tuning. We try to obtain as much information as possible in a model independent way using general symmetry properties of the theory's effective action, this leads to a new proposal on how to exit the inflationary phase and reheat the Universe.

Nonminimally coupled hybrid inflation

We discuss the hybrid inflation model where the inflaton field is nonminimally coupled to gravity. In the Jordan frame, the potential contains $\phi^4$ term as well as terms in the original hybrid inflation model. In our model, inflation can be classified into the type (I) and the type (II). In the type (I), inflation is terminated by the tachyonic instability of the waterfall field, while in the type (II) by the violation of slow-roll conditions. In our model, the reheating takes place only at the true minimum and even in the case (II) finally the tachyonic instability occurs after the termination of inflation. For a negative nonminimal coupling, inflation takes place in the vacuum-dominated region, in the large field region, or near the local minimum/maximum. Inflation in the vacuum dominated region becomes either the type (I) or (II), resulting in blue or red spectrum of the curvature perturbations, respectively. Inflation around the local maximum can be either the type (I) or the type (II), which results in the red spectrum of the curvature perturbations, while it around the local minimum must be the type (I), which results in the blue spectrum. In the large field region, to terminate inflation, potential in the Einstein frame must be positively tilted, always resulting in the red spectrum. We then numerically solve the equations of motion to investigate the whole dynamics of inflaton and confirm that the spectrum of curvature perturbations changes from red to blue ones as scales become smaller.

VECTOR FIELD AND INFLATION

We have investigated if the vector field can give rise to an accelerating phase in the early universe. We consider a timelike vector field with a general quadratic kinetic term in order to preserve an isotropic background spacetime. The vector field potential is required to satisfy the three minimal conditions for successful inflation: i) ρ > 0, ii) ρ + 3P < 0, and iii) the slow-roll conditions. As an example, we consider a massive vector potential and a small-field type potential as in scalar driven inflation.

The slow-roll condition and the amplitude of the primordial spectrum of cosmic fluctuations: contrasts and similarities of the standard account and the ‘collapse scheme’

The inflationary paradigm enjoys a very wide acceptance in the cosmological community, due largely to the fact that it is said to ‘naturally account’ for a nearly scale-independent power primordial spectrum of fluctuations which is in very good agreement with the observations. The expected overall scale of the fluctuations in most models turns out to be too large because it is inversely proportional to the slow-roll parameter, which is expected to be very small. This fact requires the fine tuning of the inflaton potential. In series of recent works it has been argued that the success of the inflationary picture is not fully justified in terms of the rules of quantum theory as applied to the cosmological setting and that an extra element, something akin to a self-induced collapse of the wavefunction, is required. There, it was suggested that the incorporation of such a collapse in the treatment might avoid the need for fine tuning of the potential that afflicts most inflationary models. In this paper we will discuss in detail the manner in which one obtains the estimation of the magnitude of the perturbations in the new scheme and that of the standard accounts, comparing one of the most popular among the latter and that corresponding to the new proposal. We will see that the proposal includes a collapse scheme that bypasses the problem, but we will see that the price seems to be a teleological, and thus unphysical, fine tuning of the characteristics of collapse.

Inflation in models with conformally coupled scalar fields: An application to the noncommutative spectral action

Slow-roll inflation is studied in theories where the inflaton field is conformally coupled to the Ricci scalar. In particular, the case of Higgs field inflation in the context of the noncommutative spectral action is analyzed. It is shown that while the Higgs potential can lead to the slow-roll conditions being satisfied once the running of the self-coupling at two-loops is included, the constraints imposed from the CMB data make the predictions of such a scenario incompatible with the measured value of the top quark mass. We also analyze the role of an additional conformally coupled massless scalar field, which arises naturally in the context of noncommutative geometry, for inflationary scenarios.

Chaotic inflation in Jordan frame supergravity

We consider the inflationary scenario with non-minimal coupling in 4D Jordan frame supergravity.We find that there occurs a tachyonic instability along the direction of the accompanying non-inflaton field in generic Jordan frame supergravity models. We propose a higher order correction to the Jordan frame function for solving the tachyonic mass problem and show that the necessary correction can be naturally generated by the heavy thresholds without spoiling the slow-roll conditions. We discuss the implication of the result on the Higgs inflation in NMSSM.