Superhorizon Scales Research Articles

Scale-invariant perturbations from null-energy-condition violation: A new variant of Galilean genesis

We propose a novel branch of the Galilean Genesis scenario as an alternative to inflation, in which the universe starts expanding from Minkowski in the asymptotic past with a gross violation of the null energy condition (NEC). This variant, described by several functions and parameters within the Horndeski scalar-tensor theory, shares the same background dynamics with the existing Genesis models, but the nature of primordial quantum fluctuations is quite distinct. In some cases, tensor perturbations grow on superhorizon scales. The tensor power spectrum can be red, blue, or scale invariant, depending on the model, while scalar perturbations are nearly scale invariant. This is in sharp contrast to typical NEC-violating cosmologies, in which a blue tensor tilt is generated. Though the primordial tensor and scalar spectra are both nearly scale invariant as in the inflationary scenario, the consistency relation in our variant of Galilean Genesis is non-standard.

General background conditions for K-bounce and adiabaticity

We study the background conditions for a bounce uniquely driven by a single scalar field model with a generalized kinetic term K(X), without any additional matter field. At the background level we impose the existence of two turning points where the derivative of the Hubble parameter H changes sign and of a bounce point where the Hubble parameter vanishes. We find the conditions for K(X) and the potential which ensure the above requirements. We then give the examples of two models constructed according to these conditions. One is based on a quadratic K(X), and the other on a K(X) which is avoiding divergences of the second time derivative of the scalar field, which may otherwise occur. An appropriate choice of the initial conditions can lead to a sequence of consecutive bounces, or oscillations of H. In the region where these models have a constant potential they are adiabatic on any scale and because of this they may not conserve curvature perturbations on super-horizon scales. While at the perturbation level one class of models is free from ghosts and singularities of the classical equations of motion, in general gradient instabilities are present around the bounce time, because the sign of the squared speed of sound is opposite to the sign of the time derivative of H. We discuss how this kind of instabilities could be avoided by modifying the Lagrangian by introducing Galilean terms in order to prevent a negative squared speed of sound around the bounce.

Cumulative effects in inflation with ultra-light entropy modes

In multi-field inflation one or more non-adiabatic modes may become light, potentially inducing large levels of isocurvature perturbations in the cosmic microwave background. If in addition these light modes are coupled to the adiabatic mode, they influence its evolution on super horizon scales. Here we consider the case in which a non-adiabatic mode becomes approximately massless (``ultralight") while still coupled to the adiabatic mode, a typical situation that arises with pseudo-Nambu-Goldstone bosons or moduli. This ultralight mode freezes on super-horizon scales and acts as a constant source for the curvature perturbation, making it grow linearly in time and effectively suppressing the isocurvature component. We identify a Stückelberg-like emergent shift symmetry that underlies this behavior. As inflation lasts for many e-folds, the integrated effect of this source enhances the power spectrum of the adiabatic mode, while keeping the non-adiabatic spectrum approximately untouched. In this case, towards the end of inflation all the fluctuations, adiabatic and non-adiabatic, are dominated by a single degree of freedom.

Numerical evaluation of the bispectrum in multiple field inflation—the transport approach with code

We present a complete framework for numerical calculation of the power spectrum and bispectrum in canonical inflation with an arbitrary number of light or heavy fields. Our method includes all relevant effects at tree-level in the loop expansion, including (i) interference between growing and decaying modes near horizon exit; (ii) correlation and coupling between species near horizon exit and on superhorizon scales; (iii) contributions from mass terms; and (iv) all contributions from coupling to gravity. We track the evolution of each correlation function from the vacuum state through horizon exit and the superhorizon regime, with no need to match quantum and classical parts of the calculation; when integrated, our approach corresponds exactly with the tree-level Schwinger or `in-in' formulation of quantum field theory. In this paper we give the equations necessary to evolve all two- and three-point correlation functions together with suitable initial conditions. The final formalism is suitable to compute the amplitude, shape, and scale dependence of the bispectrum in models with |fNL| of order unity or less, which are a target for future galaxy surveys such as Euclid, DESI and LSST. As an illustration we apply our framework to a number of examples, obtaining quantitatively accurate predictions for their bispectra for the first time. Two accompanying reports describe publicly-available software packages that implement the method.

Ultra slow-roll G inflation

The conventional slow-roll approximation is broken in the so-called ``ultra slow-roll'' models of inflation, for which the inflaton potential is exactly (or extremely) flat. The interesting nature of (canonical) ultra slow-roll inflation is that the curvature perturbation grows on superhorizon scales, but has a scale-invariant power spectrum. We study the ultra slow-roll inflationary dynamics in the presence of noncanonical kinetic terms of the scalar field, namely ultra slow-roll $G$ inflation. We compute the evolution of the curvature perturbation and show that the primordial power spectrum follows a broken power law with an oscillation feature. It is demonstrated that this could explain the lack of large-scale power in the cosmic microwave background temperature anisotropies. We also point out that the violation of the null energy condition is prohibited in ultra slow-roll $G$ inflation, and hence a blue tensor tilt is impossible as long as inflation is driven by the potential. This statement is, however, not true if the energy density is dominated by the kinetic energy of the scalar field.

Consistency relations and conservation of ζ in holographic inflation

It is well known that, in single clock inflation, the curvature perturbation ζ is constant in time on superhorizon scales. In the standard bulk description this follows quite simply from the local conservation of the energy momentum tensor in the bulk. On the other hand, in a holographic description, the constancy of the curvature perturbation must be related to the properties of the RG flow in the boundary theory. Here, we show that, in single clock holographic inflation, the time independence of correlators of ζ follows from the absence of the anomolous dimension of the energy momentum tensor in the boundary theory, and from the so-called consistency relations for vertex functions with a soft leg.

Cosmic magnetization out from the vacuum

The large-scale magnetic fields we observe today in galaxies and galaxy clusters could be the result of a pure quantum effect taking place during inflation, to wit, the creation of particles (photons) out from the vacuum in a curved spacetime. We show that, whenever the conformal invariance of electromagnetism is broken during inflation, the actual magnetic field spectrum, in the classical limit, is given by [Formula: see text], where [Formula: see text] is the number of created photons with wave number [Formula: see text]. In particular, a scale-invariant magnetic field of order of 10[Formula: see text] G can emerge in the simplest model of cosmic magnetogenesis, the one where the inflaton is kinetically coupled to the photon. Moreover, and contrarily to the general belief, we show that such a model is free from the so-called strong-coupling and backreaction problems. This conclusion follows, indirectly, from the observation that post-inflationary electric currents, which in the literature are incorrectly supposed to freeze superhorizon magnetic fields after inflation, are indeed vanishing on superhorizon scales due to causality.

Global adiabaticity and non-Gaussianity consistency condition

In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, $\R_c$, on super-horizon scales is one of the assumptions necessary to derive the consistency condition between the squeezed limit of the bispectrum and the spectrum of the primordial curvature perturbation. However, the conservation of $\R_c$ holds only after the perturbation has reached the adiabatic limit where the constant mode of $\R_c$ dominates over the other (usually decaying) mode. In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, $\delta P_{nad}\equiv\delta P-c_w^2\delta\rho$ where $c_w^2=\dot P/\dot\rho$, usually becomes also negligible on superhorizon scales. Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity. This is in fact not true. In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of $\R_c$ on super-horizon scales. In this paper, we consider models that satisfy $\delta P_{nad}=0$ on all scales, which we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where $c_s$ is the phase velocity of the propagation of the perturbation. A known example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$. In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor. Applying this method we show that there indeed exists a wide class of GA models with $c_w^2=c_s^2$, which allows $\R_c$ to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition.

Primordial black holes as a novel probe of primordial gravitational waves. II. Detailed analysis

Recently we have proposed a novel method to probe primordial gravitational waves from upper bounds on the abundance of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations generated in the early Universe is fairly large, they induce substantial scalar perturbations due to their second-order effects. If these induced scalar perturbations are too large when they reenter the horizon, then PBHs are overproduced, their abundance exceeding observational upper limits. That is, primordial tensor perturbations on superhorizon scales can be constrained from the absence of PBHs. In our recent paper we have only shown simple estimations of these new constraints, and hence in this paper, we present detailed derivations, solving the Einstein equations for scalar perturbations induced at second order in tensor perturbations. We also derive an approximate formula for the probability density function of induced density perturbations, necessary to relate the abundance of PBHs to the primordial tensor power spectrum, assuming primordial tensor perturbations follow Gaussian distributions. Our new upper bounds from PBHs are compared with other existing bounds obtained from big bang nucleosynthesis, cosmic microwave background, LIGO/Virgo and pulsar timing arrays.

FIRST-ORDER COSMOLOGICAL PERTURBATIONS ENGENDERED BY POINT-LIKE MASSES

ABSTRACT In the framework of the concordance cosmological model, the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The expressions found for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at the locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant, this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.

Quality of the Peccei-Quinn symmetry in the aligned QCD axion and cosmological implications

We show that the required high quality of the Peccei-Quinn symmetry can be naturally explained in the aligned QCD axion models where the QCD axion arises from multiple axions with decay constants much smaller than the axion window, e.g., around the weak scale. Even in the presence of general Planck-suppressed Peccei-Quinn symmetry breaking operators, the effective strong CP phase remains sufficiently small in contrast to the standard axion models without the alignment. The QCD axion potential has small or large modulations due to the symmetry breaking operators, which can significantly affect the axion cosmology. When the axions are trapped in different minima, domain walls appear and their scaling behavior suppresses the axion isocurvature perturbations at super-horizon scales. Our scenario predicts many axions and saxions coupled to gluons, and they may be searched for at collider experiments. In particular, the recently found diphoton excess at 750 GeV could be due to one of such (s)axions.

Causality, initial conditions, and inflationary magnetogenesis

The post-inflationary evolution of inflation-produced magnetic fields, conventional or not, can change dramatically when two fundamental issues are accounted for. The first is causality, which demands that local physical processes can never affect superhorizon perturbations. The second is the nature of the transition from inflation to reheating and then to the radiation era, which determine the initial conditions at the start of these epochs. Causality implies that inflationary magnetic fields dot not freeze into the matter until they have re-entered the causal horizon. The nature of the cosmological transitions and the associated initial conditions, on the other hand, determine the large-scale magnetic evolution after inflation. Put together, the two can slow down the adiabatic magnetic decay on superhorizon scales throughout the universe's post-inflationary evolution and thus lead to considerably stronger residual magnetic fields. This is "good news" for both the conventional and the non-conventional scenarios of cosmic magnetogenesis. Mechanisms operating outside standard electromagnetism, in particular, do not need to enhance their fields too much during inflation, in order to produce seeds that can feed the galactic dynamo today. In fact, even conventionally produced inflationary magnetic fields might be able to sustain the dynamo.

Revisiting metric perturbations in tensor-vector-scalar theory

I revisit cosmological perturbations in Bekenstein's tensor-vector-scalar theory (TeVeS). Considering only scalar modes in the conformal Newtonian gauge, the extra degrees of freedom are expressed in a way suitable for studying modifications at the level of the metric potentials. Assuming a universe in the matter-dominated phase, I discuss the mechanism responsible for boosting structure growth, and confirm the vector field as its key ingredient. Using a semi-analytic approach, I further characterize the evolution of density perturbations and the potentials on sub- and superhorizon scales.

Dark energy homogeneity in general relativity: Are we applying it correctly?

Thus far, there does not appear to be an agreed (or adequate) definition of homogeneous dark energy (DE). This paper seeks to define a valid, adequate homogeneity condition for DE. Firstly, it is shown that as long as $w_x \neq -1$, DE must have perturbations. It is then argued, independent of $w_x$, that a correct definition of homogeneous DE is one whose density perturbation vanishes in comoving gauge: and hence, in the DE rest frame. Using phenomenological DE, the consequence of this approach is then investigated in the observed galaxy power spectrum - with the power spectrum being normalized on small scales, at the present epoch z = 0. It is found that for high magnification bias, relativistic corrections in the galaxy power spectrum are able to distinguish the concordance model from both a homogeneous DE and a clustering DE - on super-horizon scales.

Gravitational-wave implications for structure formation: A second-order approach

Gravitational waves are propagating undulations in the spacetime fabric, which interact very weakly with their environment. In cosmology, gravitational-wave distortions are produced by most of the inflationary scenarios and their anticipated detection should open a new window to the early universe. Motivated by the relative lack of studies on the potential implications of gravitational radiation for the large-scale structure of the universe, we consider its coupling to density perturbations during the post-recombination era. We do so by assuming an Einstein-de Sitter background cosmology and by employing a second-order perturbation study. At this perturbative level and on superhorizon scales, we find that gravitational radiation adds a distinct and faster growing mode to the standard linear solution for the density contrast. Given the expected weakness of cosmological gravitational waves, however, the effect of the new mode is currently subdominant and it could start becoming noticeable only in the far future. Nevertheless, this still raises the intriguing possibility that the late-time evolution of large-scale density perturbations may be dictated by the long-range (the Weyl), rather than the local (the Ricci) component of the gravitational field.

Superhorizon magnetic fields

[Abridged] We analyze the evolution of superhorizon-scale magnetic fields from the end of inflation till today. Whatever is the mechanism responsible for their generation during inflation, we find that a given magnetic mode with wavenumber $k$ evolves, after inflation, according to the values of $k\eta_e$, $n_{\mathbf{k}}$, and $\Omega_k$, where $\eta_e$ is the conformal time at the end of inflation, $n_{\mathbf{k}}$ is the number density spectrum of inflation-produced photons, and $\Omega_k$ is the phase difference between the two Bogolubov coefficients which characterize the state of that mode at the end of inflation. For any realistic inflationary magnetogenesis scenario, we find that $n_{\mathbf{k}}^{-1} \ll |k\eta_e| \ll 1$, and three evolutionary scenarios are possible: ($i$) $|\Omega_k \mp \pi| = \mathcal{O}(1)$, in which case the evolution of the magnetic spectrum $B_k(\eta)$ is adiabatic, $a^2B_k(\eta) = \mbox{const}$, with $a$ being the expansion parameter; ($ii$) $|\Omega_k \mp \pi| \ll |k\eta_e|$, in which case the evolution is superadiabatic, $a^2B_k(\eta) \propto \eta$; ($iii$) $|k\eta_e| \ll |\Omega_k \mp \pi| \ll 1$ or $|k\eta_e| \sim |\Omega_k \mp \pi| \ll 1$, in which case an early phase of adiabatic evolution is followed, after a time $\eta_\star \sim |\Omega_k \mp \pi|/k$, by a superadiabatic evolution. Once a given mode reenters the horizon, it remains frozen into the plasma and then evolves adiabatically till today. As a corollary of our results, we find that inflation-generated magnetic fields evolve adiabatically on all scales and for all times in conformal-invariant free Maxwell theory, while they evolve superadiabatically after inflation on superhorizon scales in the non-conformal-invariant Ratra model. The latter result supports our recent claim that the Ratra model can account for the presence of cosmic magnetic fields.

Adiabaticity and gravity theory independent conservation laws for cosmological perturbations

We carefully study the implications of adiabaticity for the behavior of cosmological perturbations. There are essentially three similar but different definitions of non-adiabaticity: one is appropriate for a thermodynamic fluid $\delta P_{nad}$, another is for a general matter field $\delta P_{c,nad}$, and the last one is valid only on superhorizon scales. The first two definitions coincide if $c_s^2=c_w^2$ where $c_s$ is the propagation speed of the perturbation, while $c_w^2=\dot P/\dot\rho$. Assuming the adiabaticity in the general sense, $\delta P_{c,nad}=0$, we derive a relation between the lapse function in the comoving sli\-cing $A_c$ and $\delta P_{nad}$ valid for arbitrary matter field in any theory of gravity, by using only momentum conservation. The relation implies that as long as $c_s\neq c_w$, the uniform density, comoving and the proper-time slicings coincide approximately for any gravity theory and for any matter field if $\delta P_{nad}=0$ approximately. In the case of general relativity this gives the equivalence between the comoving curvature perturbation $R_c$ and the uniform density curvature perturbation $\zeta$ on superhorizon scales, and their conservation. We then consider an example in which $c_w=c_s$, where $\delta P_{nad}=\delta P_{c,nad}=0$ exactly, but the equivalence between $R_c$ and $\zeta$ no longer holds. Namely we consider the so-called ultra slow-roll inflation. In this case both $R_c$ and $\zeta$ are not conserved. In particular, as for $\zeta$, we find that it is crucial to take into account the next-to-leading order term in $\zeta$'s spatial gradient expansion to show its non-conservation, even on superhorizon scales. This is an example of the fact that adiabaticity (in the thermodynamic sense) is not always enough to ensure the conservation of $R_c$ or $\zeta$.

Disformal invariance of curvature perturbation

We show that under a general disformal transformation the linear comoving curvature perturbation is not identically invariant, but is invariant on superhorizon scales for any theory that is disformally related to Horndeski's theory. The difference between disformally related curvature perturbations is found to be given in terms of the comoving density perturbation associated with a single canonical scalar field. In General Relativity it is well-known that this quantity vanishes on superhorizon scales through the Poisson equation that is obtained on combining the Hamiltonian and momentum constraints, and we confirm that a similar result holds for any theory that is disformally related to Horndeski's scalar-tensor theory so long as the invertibility condition for the disformal transformation is satisfied. We also consider the curvature perturbation at full nonlinear order in the unitary gauge, and find that it is invariant under a general disformal transformation if we assume that an attractor regime has been reached. Finally, we also discuss the counting of degrees of freedom in theories disformally related to Horndeski's.

Large-scale anomalies in the cosmic microwave background as signatures of non-Gaussianity

We derive a general expression for the probability of observing deviations from statistical isotropy in the cosmic microwave background (CMB) if the primordial fluctuations are non-Gaussian and extend to superhorizon scales. The primary motivation is to properly characterize the monopole and dipole modulations of the primordial power spectrum that are generated by the coupling between superhorizon and subhorizon perturbations. Unlike previous proposals for generating the hemispherical power asymmetry, we do not assume that the power asymmetry results from a single large superhorizon mode. Instead, we extrapolate the observed power spectrum to superhorizon scales and compute the power asymmetry that would result from a specific realization of non-Gaussian perturbations on scales larger than the observable universe. Our study encompasses many of the scenarios that have been put forward as possible explanations for the CMB hemispherical power asymmetry. We confirm our analytic predictions for the probability of a given power asymmetry by comparing them to numerical realizations of CMB maps. We find that non-local models of non-Gaussianity and scale-dependent local non-Gaussianity produce scale-dependent modulations of the power spectrum, thereby potentially producing both a monopolar and a dipolar power modulation on large scales. We then provide simple examples of finding the posterior distributions for the parameters of the bispectrum from the observed monopole and dipole modulations.

Primordial black holes as a novel probe of primordial gravitational waves

We propose a novel method to probe primordial gravitational waves by means of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations on comoving scales much smaller than those relevant to Cosmic Microwave Background is very large, it induces scalar perturbations due to second-order effects substantially. If the amplitude of resultant scalar perturbations becomes too large, then PBHs are overproduced to a level that is inconsistent with a variety of existing observations constraining their abundance. This leads to upper bounds on the amplitude of initial tensor perturbations on super-horizon scales. These upper bounds from PBHs are compared with other existing bounds.