super-Hubble Scales Research Articles

Local non-Gaussianity from inflation

The non-Gaussian distribution of primordial perturbations has the potential to reveal the physical processes at work in the very early universe. Local models provide a well-defined class of non-Gaussian distributions that arise naturally from the nonlinear evolution of density perturbations on super-Hubble scales starting from Gaussian field fluctuations during inflation. I describe the δN formalism used to calculate the primordial density perturbation on large scales and then review several models for the origin of local primordial non-Gaussianity, including the curvaton, modulated reheating and ekpyrotic scenarios. I include an appendix with a table of sign conventions used in specific papers.

Super-Hubble de Sitter fluctuations and the dynamical RG

Perturbative corrections to correlation functions for interacting theories in de Sitter spacetime often grow secularly with time, due to the properties of fluctuations on super-Hubble scales. This growth can lead to a breakdown of perturbation theory at late times. We argue that Dynamical Renormalization Group (DRG) techniques provide a convenient framework for interpreting and resumming these secularly growing terms. In the case of a massless scalar field in de Sitter with quartic self-interaction, the resummed result is also less singular in the infrared, in precisely the manner expected if a dynamical mass is generated. We compare this improved infrared behavior with large-N expansions when applicable.

Matter bounce in Hořava-Lifshitz cosmology

Horava-Lifshitz gravity, a recent proposal for a UV-complete renormalizable gravity theory, may lead to a bouncing cosmology. In this note we argue that Horava-Lifshitz cosmology may yield a concrete realization of the matter bounce scenario, and thus give rise to an alternative to inflation for producing a scale-invariant spectrum of cosmological perturbations. In this scenario, quantum vacuum fluctuations exit the Hubble radius in the pre-bounce phase and the spectrum is transformed into a scale-invariant one on super-Hubble scales before the bounce because the long wavelength modes undergo squeezing of their wave-functions for a longer period of time than shorter wavelength modes. The scale-invariance of the spectrum of curvature fluctuations is preserved during and after the bounce. A distinctive prediction of this scenario is the amplitude and shape of the bispectrum.

Do observations offer evidence for cosmological-scale extra dimensions?

We present a case that current observations may already indicate new gravitational physicson cosmological scales. The excess of power seen in the Lyman-α forest and small-scale CMB experiments, the anomalously largebulk flows seen both in peculiar velocity surveys and in kinetic SZ, and the higher ISW cross-correlation all indicate thatstructure may be more evolved than expected from ΛCDM. While these anomalies may constitute a statistical fluke or may eventually disappear as systematic effects are better understood, we argue that they can be explained in models with infinite-volume (or, at least, cosmological-size) extra dimensions, where the graviton is a resonance with a tiny width.The longitudinal mode of the graviton mediates an extra scalar force which speeds up structure formation at late times,thereby accounting for the above anomalies. The required graviton Compton wavelength is relatively small compared to the present Hubble radius,of order 300-600 Mpc. Moreover, with certain assumptions about the behavior of the longitudinal mode on super-Hubble scales, our modified gravity framework canalso alleviate the tension with the low quadrupole and the peculiar vanishing of the CMB correlation function on large angular scales, seen both in COBE and WMAP. This relies on a novel mechanism that cancels a late-time ISW contribution against the primordial Sachs-Wolfe amplitude.

Statistical anisotropy in the inflationary universe

During cosmological inflation, quasiclassical perturbations are permanently generated on super-Hubble spatial scales, their power spectrum being determined by the fundamental principles of quantum field theory. By the end of inflation, they serve as primeval seeds for structure formation in the universe. At early stages of inflation, such perturbations break homogeneity and isotropy of the inflationary background. In the present paper, we perturbatively take into account this quasiclassical background inhomogeneity of the inflationary universe while considering the evolution of small-scale (sub-Hubble) quantum modes. As a result, the power spectrum of primordial perturbations develops statistical anisotropy, which can subsequently manifest itself in the large-scale structure and cosmic microwave background. The statistically anisotropic contribution to the primordial power spectrum is predicted to have almost scale-invariant form dominated by a quadrupole. Theoretical expectation of the magnitude of this anisotropy depends on the assumptions about the physics in the trans-Planckian region of wave numbers.

Adiabatic fluctuations from cosmic strings in a contracting universe

We show that adiabatic, super-Hubble, and almost scale invariant density fluctuations are produced by cosmic strings in a contracting universe. An essential point is that isocurvature perturbations produced by topological defects such as cosmic strings on super-Hubble scales lead to a source term which seeds the growth of curvature fluctuations on these scales. Once the symmetry has been restored at high temperatures, the isocurvature seeds disappear, and the fluctuations evolve as adiabatic ones in the expanding phase. Thus, cosmic strings may be resurrected as a mechanism for generating the primordial density fluctuations observed today.

Two-field Kähler moduli inflation in large volume moduli stabilization

In this paper we present a two-field inflation model, which is distinctive in having anon-canonical kinetic Lagrangian and comes from the large volume approach to the modulistabilization in flux compactification of type IIB superstring on a Calabi–Yau orientifold withh(1,2)>h(1,1)≥4. The Kähler moduli are classified as the volume modulus, heavy moduli and two light moduli.The axion–dilaton, complex structure moduli and all heavy Kähler moduli including thevolume modulus are frozen by a non-perturbatively corrected flux superpotential and theα′-corrected Kähler potential in the large volume limit. The minimum of the scalar potentialat which the heavy moduli are stabilized provides the dominant potential energy for thesurviving light Kähler moduli. We consider a simplified case where the axionic componentsin the light Kähler moduli are further stabilized at the potential minimum and onlythe geometrical components are taken as scalar fields to drive an assisted-likeinflation. For a certain range of moduli stabilization parameters and inflation initialconditions, we obtain a nearly flat power spectrum of the curvature perturbation, withns≈0.96 at Hubble exit, and an inflationary energy scale of3 × 1014 GeV. In our model, there is significant correlation between the curvature and isocurvatureperturbations on super-Hubble scales, so at the end of inflation a great deal of thecurvature power spectrum originates from this correlation.

Decoherence of inflationary primordial fluctuations

We study the process whereby quantum cosmological perturbations become classical within inflationary cosmology. By setting up a master-equation formulation we show how quantum coherence for super-Hubble modes can be destroyed by its coupling to the environment provided by sub-Hubble modes. We identify what features the sub-Hubble environment must have in order to decohere the longer wavelengths, and identify how the onset of decoherence (and how long it takes) depends on the properties of the sub-Hubble physics which forms the environment. Our results show that the decoherence process is largely insensitive to the details of the coupling between the sub- and super-Hubble scales. They also show how locality implies, quite generally, that the decohered density matrix at late times is diagonal in the field representation (as is implicitly assumed by extant calculations of inflationary density perturbations). Our calculations also imply that decoherence can arise even for couplings which are as weak as gravitational in strength.

A BACK-REACTION INDUCED LOWER BOUND ON THE TENSOR-TO-SCALAR RATIO

There are large classes of inflationary models, particularly popular in the context of string theory and braneworld approaches to inflation, in which the ratio of linearized tensor-to-scalar metric fluctuations is very small. In such models, however, gravitational waves produced by scalar modes cannot be neglected. We derive the lower bound on the tensor-to-scalar ratio by considering the back-reaction of the scalar perturbations as a source of gravitational waves. These results show that no cosmological model that is compatible with a metric scalar amplitude of ≈10-5 can have a ratio of the tensor-to-scalar power spectra less than ≈10-8 at recombination and that higher-order terms leads to logarithmic growth for r during radiation domination. Our lower bound also applies to non-inflationary models which produce an almost scale-invariant spectrum of coherent super-Hubble scale metric fluctuations.

Density perturbations inf(R)gravity theories in metric and Palatini formalisms

We make a detailed study of matter density perturbations in both metric and Palatini formalisms. Considering general theories whose Lagrangian density is a general function, $f(R)$, of the Ricci scalar $R$, we derive the equation of matter density perturbations in each case, in a number of gauges, including comoving, longitudinal and uniform density gauges. We show that for viable $f(R)$ models that satisfy cosmological and local gravity constraints (LGC), matter perturbation equations derived under a subhorizon approximation are valid even for super-Hubble scales provided the oscillating mode (scalaron) does not dominate over the matter-induced mode. Such approximate equations are especially reliable in the Palatini formalism because of the absence of scalarons. Using these equations we make a comparative study of the behavior of matter density perturbations as well as gravitational potentials for a number of classes of $f(R)$ theories. In the metric formalism the quantity $m=R{f}_{,RR}/{f}_{,R}$ that characterizes the deviation from the $\ensuremath{\Lambda}\mathrm{CDM}$ model is constrained to be very small during a matter era in order to ensure compatibility with LGC, but the models in which $m$ grows to the order of ${10}^{\ensuremath{-}1}$ around the present epoch can be allowed. These models also suffer from an additional fine-tuning due to the presence of scalaron oscillating modes which are absent in the Palatini case. In Palatini formalism LGC and background cosmological constraints provide only weak bounds on $|m|$ by constraining it to be smaller than $\ensuremath{\sim}0.1$. This is in contrast to matter density perturbations which, on galactic scales, place far more stringent constraints on the present deviation parameter $m$ of the order of $|m|\ensuremath{\lesssim}{10}^{\ensuremath{-}5}--{10}^{\ensuremath{-}4}$. This is due to the peculiar evolution of matter perturbations in the Palatini case, which exhibits a rapid growth or a damped oscillation depending on the sign of $m$.

Non-Gaussianities from ekpyrotic collapse with multiple fields

We compute the non-Gaussianity of the curvature perturbation generated by ekpyroticcollapse with multiple fields. The transition from the multi-field scaling solution to asingle-field dominated regime converts initial isocurvature field perturbations to an almostscale invariant comoving curvature perturbation. In the specific model of two fields,ϕ1 andϕ2, with exponentialpotentials, −Viexp(−ciϕi), we calculate the bispectrum of the resulting curvature perturbation. We find that thenon-Gaussianity is dominated by non-linear evolution on super-Hubble scales and hence isof the local form. The non-linear parameter of the curvature perturbation is given byfNL = 5cj2/12,where cj is the exponent of the potential for the field which becomes subdominant at late times. Sincecj2 must be large, in order to generate an almost scale invariant spectrum, the non-Gaussianityis inevitably large. Thus, the model is strongly constrained by observational bounds on thespectral index and non-Gaussianity.

Cosmological perturbations on a bouncing brane

The cosmological perturbations on a bouncing brane are studied. The brane is movinginside a Klebanov–Strassler throat where the infrared region of the geometry is smoothlycut off. For an observer confined to the world volume of the brane, this results in anon-singular bouncing mirage cosmology. We have calculated the scalar perturbationscorresponding to the normal displacements of the brane. This is performed in the probebrane limit where the gravitational back-reaction of the brane on the bulk throat is absent.Our model provides a framework for studying the transfer of fluctuations from acontracting to an expanding phase. We find that the spectral index of the dominantmode of the metric fluctuation is unchanged, unlike what is obtained by gluingcontracting to expanding Einstein universes with the help of the usual matchingconditions. Assuming that the fluctuations start off in a vacuum state on sub-Hubblescales during the contracting phase, it is shown that the resulting spectral indexns on super-Hubble scales in the expanding phase has a large blue tilt. When the brane is movingslowly inside the throat and its kinetic energy is negligible compared to its rest mass, one findsns = 3. For a fast-rolling brane with a large kinetic energy, the spectral index isns = 2.3. This may put severe constraints on models of mirage cosmology.

Evolution of gravitational perturbations in non-commutative inflation

We consider the non-commutative inflation model in which it is the unconventionaldispersion relation for regular radiation which drives the accelerated expansion of space. Inthis model, we study the evolution of linear cosmological perturbations through thetransition between the phase of accelerated expansion and the regular radiation-dominatedphase of standard cosmology, a transition which is analogous to the reheating period inscalar field-driven models of inflation. If matter consists of only a single non-commutativeradiation fluid, then the curvature perturbations are constant on super-Hubble scales. Onthe other hand, if we include additional matter fields which oscillate during the transitionperiod, e.g. scalar moduli fields, then there can be parametric amplification of the amplitudeof the curvature perturbations. We demonstrate this explicitly by numerically solvingthe full system of perturbation equations in the case where matter consists ofboth the non-commutative radiation field and a light scalar field which undergoesoscillations. Our model is an example where the parametric resonance of the curvaturefluctuations is driven by the oscillations not of the inflaton field, but of the entropymode.

On the evolution of tachyonic perturbations at super-Hubble scales

In the slow-roll inflationary scenario, the amplitude of the curvature perturbationsapproaches a constant value soon after the modes leave the Hubble radius. However,relatively recently, it was shown that the amplitude of the curvature perturbations inducedby the canonical scalar field can grow at super-Hubble scales if there is either a transitionto fast roll inflation or if inflation is interrupted for some period of time. In this work, weextend the earlier analysis to the case of a non-canonical scalar field described by theDirac–Born–Infeld action. With the help of a specific example, we show thatthe amplitude of the tachyonic perturbations can be enhanced or suppressed atsuper-Hubble scales if there is a transition from slow roll to fast roll inflation. We alsoillustrate as to how the growth of the entropy perturbations during the fast rollregime proves to be responsible for the change in the amplitude of the curvatureperturbations at super-Hubble scales. Furthermore, following the earlier analysis for thecanonical scalar field, we show that the power spectrum evaluated in the longwavelength approximation matches the exact power spectrum obtained numericallyvery well. Finally, we briefly comment on an application of this phenomenon.

Stochastic gravitoelectromagnetic inflation

Abstract Gravitoelectromagnetic inflation was recently introduced to describe, in an unified manner, electromagnetic, gravitatory and inflaton fields in the early (accelerated) inflationary universe from a 5D vacuum state. In this Letter, we study a stochastic treatment for the gravitoelectromagnetic components A B = ( A μ , φ ) , on cosmological scales. We focus our study on the seed magnetic fields on super-Hubble scales, which could play an important role in large scale structure formation of the universe.

Precision measurement of the mean curvature

Very small mean curvature is a robust prediction of inflation worth rigorous checking. Since current constraints are derived from determinations of the angular-diameter distance to the CMB last-scattering surface, which is also affected by dark energy, they are limited by our understanding of the dark energy. Measurements of luminosity or angular-diameter distances to redshifts in the matter-dominated era can greatly reduce this uncertainty. With a 1% measurement of the distance to $z=3$, combined with the CMB data expected from Planck, one can achieve $\ensuremath{\sigma}({\ensuremath{\Omega}}_{k}{h}^{2})\ensuremath{\sim}{10}^{\ensuremath{-}3}$. A nonzero detection at this level would be evidence against inflation or for unusually large curvature fluctuations on super-Hubble scales.

Non-Gaussian perturbations from multi-field inflation

We show how primordial bispectrum of density perturbations from inflationmay be characterized in terms of manifestly gauge-invariant cosmologicalperturbations at second order. The primordial metric perturbation,ζ, describing the perturbed expansion of uniform-density hypersurfaces on large scales isrelated to scalar field perturbations on unperturbed (spatially flat) hypersurfaces at firstand second order. The bispectrum of the metric perturbation is thus composed of (i) alocal contribution due to the second-order gauge transformation, and (ii) theintrinsic bispectrum of the field perturbations on spatially flat hypersurfaces. Wegeneralize previous results to allow for scale dependence of the scalar field powerspectra and correlations that can develop between fields on super-Hubble scales.

Gauge-invariant perturbation theory for trans-Planckian inflation

The possibility that the scale-invariant inflationary spectrum may be modified due to the hidden assumptions about the Planck scale physics -- dubbed as trans-Planckian inflation -- has received considerable attention. To mimic the possible trans-Planckian effects, among various models, modified dispersion relations have been popular in the literature. In almost all the earlier analyzes, unlike the canonical scalar field driven inflation, the trans-Planckian effects are introduced to the scalar/tensor perturbation equations in an ad hoc manner -- without calculating the stress-tensor of the cosmological perturbations from the covariant Lagrangian. In this work, we perform the gauge-invariant cosmological perturbations for the single-scalar field inflation with the Jacobson-Corley dispersion relation by computing the fluctuations of all the fields including the unit time-like vector field which defines a preferred rest frame. We show that: (i) The non-linear effects introduce corrections only to the perturbed energy density. The corrections to the energy density vanish in the super-Hubble scales. (ii) The scalar perturbations, in general, are not purely adiabatic. (iii) The equation of motion of the Mukhanov-Sasaki variable corresponding to the inflaton field is different than those presumed in the earlier analyzes. (iv) The tensor perturbation equation remains unchanged. We perform the classical analysis for the resultant system of equations and also compute the power-spectrum of the scalar perturbations in a particular limit. We discuss the implications of our results and compare with the earlier results.

Conservation of second-order cosmological perturbations in a scalar field dominated universe

We discuss second-order cosmological perturbations on super-Hubble scales, in a scalar field dominated universe, such as during single field inflation. In this contest we show that the gauge-invariant curvature perturbations defined on uniform density and comoving hypersurfaces coincide and that perturbations are adiabatic in the large scale limit. Since it has been recently shown that the uniform density curvature perturbation is conserved on large scales if perturbations are adiabatic, we conclude that both the uniform density and comoving curvature perturbations at second order, in a scalar field dominated universe, are conserved. Finally, in the light of this result, we comment on the variables recently used in the literature to compute non-Gaussianities.

Publisher's Note: Cosmological density fluctuations and gravity waves: A covariant approach to gauge-invariant nonlinear perturbation theory [Phys. Rev. D70, 103524 (2004)

We present a new approach to gauge-invariant cosmological perturbations at second order, which is also covariant. We examine two cases in particular for a dust Friedman-Lemaitre-Robertson-Walker model of any curvature: we investigate gravity waves generated from clustering matter, that is, induced tensor modes from scalar modes; and we discuss the generation of density fluctuations induced by gravity waves -- scalar modes from tensor perturbations. We derive a linear system of evolution equations for second-order gauge-invariant variables which characterise fully the induced modes of interest, with a source formed from variables quadratic in first-order quantities; these we transform into fully-fledged second-order gauge-invariant variables. Both the invariantly defined variables and the key evolution equations are considerably simpler than similar gauge-invariant results derived by other methods. By finding analytical solutions, we demonstrate that non-linear effects can significantly amplify or dampen modes present in standard linearised cosmological perturbation theory, thereby providing an important source of potential error in, and refinement of, the standard model. Moreover, these effects can dominate at late times, and on super-Hubble scales.