The primordial spectra are calculated using dispersion relations which deviate from the relativistic one above a certain energy scale Lambda. We determine the properties of the leading modifications with respect to the standard spectra when Lambda is much greater than H, the Hubble scale during inflation. To be generic, we parameterize the lowest order deviation from the relativistic law by alpha, the power of P/Lambda where P is the proper momentum. When working in the asymptotic vacuum, the leading modification scales as (H/Lambda)^alpha for all alpha, except for a discrete set where the power is higher. Moreover, this modification is robust against introducing higher order terms in the dispersion relation. We then algebraically deduce the modifications of scalar and tensor power spectra in slow roll inflation from modifications calculated in de Sitter space. The modifications do not exhibit oscillations unless the dispersion relation induces some non-adiabaticity near a given scale. Finally, we explore the much less studied regime where H and Lambda are comparable. Our results indicate that the project of reconstructing the inflaton potential cannot be pursued without making some hypothesis about the dispersion relation of the fluctuation modes.
We consider a more general initial condition satisfying the minimal uncertainty relationship. We calculate the power spectrum of a simple model in inflationary cosmology. The results depend on perturbations generated below a fundamental scale, e.g. the Planck scale.
We provide a class of QFTs which exhibit dissipation above a threshold energy, thereby breaking Lorentz invariance. Unitarity is preserved by coupling the fields to additional degrees of freedom (heavy fields) which introduce the rest frame. Using the Equivalence Principle, we define these theories in arbitrary curved spacetime. We then confront the trans-Planckian question of inflationary cosmology. When dissipation increases with the energy, the quantum field describing adiabatic perturbations is completely damped at the onset of inflation. However it still exists as a composite operator mad…
In this Letter we study the dependence of the spectrum of fluctuations in inflationary cosmology on possible effects of trans-Planckian physics, using the Corley/Jacobson dispersion relations as an example. We compare the methods used in previous work [1] with the WKB approximation, give a new exact analytical result, and study the dependence of the spectrum obtained using the approximate method of Ref. [1] on the choice of the matching time between different time intervals. We also comment on recent work subsequent to Ref. [1] on the trans-Planckian problem for inflationary cosmology.
We consider a minimum uncertainty vacuum choice at a fixed energy scale Lambda as an effective description of trans-Planckian physics, and discuss its implications for the linear perturbations of a massless scalar field in power-law inflationary models. We find possible effects with a magnitude of order H/Λin the power spectrum, in analogy with previous results for de-Sitter space-time.
The accelerated expansion of space during the period of cosmological inflation leads to trans-Planckian issues which need to be addressed. Most importantly, the physical wavelength of fluctuations which are studied at the present time by means of cosmological observations may well originate with a wavelength smaller than the Planck length at the beginning of the inflationary phase. Thus, questions arise as to whether the usual predictions of inflationary cosmology are robust considering our ignorance of physics on trans-Planckian scales, and whether the imprints of Planck-scale physics are at …
In the present paper we discuss how trans-Planckian physics affects inflationary vacuum fluctuations and primordial density perturbations. The trans-Planckian problem during inflation has been widely discussed in literature, but it is still under debate. We reconsider this problem by using the two-point correlation function of the non-minimally coupled scalar fields and constructing the effective potential with the adiabatic (WKB) regularization or approximation. First, we clearly show that the cut-off divergence of the quantum fluctuations does not drastically change during inflation under re…
We study the implications of the recently proposed Trans-Planckian Censorship Conjecture (TCC) for early universe cosmology and in particular inflationary cosmology. The TCC leads to the conclusion that if we want inflationary cosmology to provide a successful scenario for cosmological structure formation, the energy scale of inflation has to be lower than $10^9$ GeV. Demanding the correct amplitude of the cosmological perturbations then forces the generalized slow-roll parameter $ε$ of the model to be very small ($<10^{-31}$). This leads to the prediction of a negligible amplitude of primordi…
We study the constraint on the inflationary energy scale from the recently proposed Trans-Planckian Censorship Conjecture (TCC). We find a universal upper bound on the inflationary Hubble expansion rate $H_\text{inf}$ which is solely determined by the reheating temperature $T_\text{rh}$: $H_\text{inf}/M_\text{Pl}\lesssim T_0/T_\text{rh}$, where $T_0$ is the photon temperature today. The upper limit can be saturated by a post-inflationary oscillatory stage with the critical equation-of-state parameter $w\approx-1/3$, or by an inflation model with multiple stages. In the lowest reheating tempera…
Due to the quasi-exponential red-shifting which occurs during an inflationary period in the very early Universe, wavelengths which at the present time correspond to cosmological lengths are in general sub-Planckian during the early stages of inflation. This talk discusses two approaches to addressing this issue which both indicate that the standard predictions of inflationary cosmology - made using classical general relativity and weakly coupled scalar matter field theory - are not robust against changes in the physics on trans-Planckian scales. One approach makes use of modified dispersion re…
The recently proposed Trans-Planckian Censorship Conjecture (TCC) can be used to constrain the energy scale of inflation. The conclusions however depend on the assumptions about post-inflationary history of the Universe. E.g. in the standard case of a thermal post-inflationary history in which the Universe stays radiation dominated at all times from the end of inflation to the epoch of radiation matter equality, TCC has been used to argue that the Hubble parameter during inflation, $H_{\inf}$, is below ${\cal O}(0.1) ~{\rm GeV}$. Cosmological scenarios with a non-thermal post-inflationary hist…
I review the "Trans-Planckian Censorship Conjecture" (TCC) and its implications for cosmology, in particular for the inflationary universe scenario. Whereas the inflationary scenario is tightly constrained by the TCC, alternative early universe scenarios are not restricted.
We investigate the insensitivity of the predictions of inflationary models with respect to modifications of Planck energy physics. The modification we consider consists in replacing the usual dispersion relation by nonlinear ones. This way of addressing the problem has recently recieved attention and contradictory results were found. Our main result is to show that the adiabaticity of the mode propagation and the separation of two scales of interest, the Planck scale and the cosmological horizon scale, are sufficient conditions for the predictions to be unchanged. We then show that almost all …
We calculate the power spectrum of metric fluctuations in inflationary cosmology starting with initial conditions which are imposed mode by mode when the wavelength equals some critical length $\ell_{_{\rm C}}$ corresponding to a new energy scale $M_{_{\rm C}}$ at which trans-Planckian physics becomes important. In this case, the power spectrum can differ from what is calculated in the usual framework (which amounts to choosing the adiabatic vacuum state). The fractional difference in the results depends on the ratio $σ_0$ between the Hubble expansion rate $H_{\rm inf}$ during inflation and th…
There are good indications that fundamental physics gives rise to a modified space-momentum uncertainty relation that implies the existence of a minimum length scale. We implement this idea in the scalar field theory that describes density perturbations in flat Robertson-Walker space-time. This leads to a non-linear time-dependent dispersion relation that encodes the effects of Planck scale physics in the inflationary epoch. Unruh type dispersion relations naturally emerge in this approach, while unbounded ones are excluded by the minimum length principle. We also find red-shift induced modifi…
We investigate the implication of Trans-Planckian Censorship Conjecture (TCC) for the initial state of primordial perturbations. It is possible to set the state of perturbation modes in the infinite past as the Minkowski vacuum, only if the pre-inflationary era is past-complete. We calculate the evolution of the perturbation modes in such a pre-inflationary era and show that at the beginning of inflation the perturbation modes with wavelengths much shorter than the Hubble scale (but still larger than the Planck length scale) will behave as they are in the Bunch-Davis state. Therefore, a past-c…
We consider an inflationary universe model in which the phase of accelerated expansion was preceded by a non-singular bounce and a period of contraction which involves a phase of deceleration. We follow fluctuations which exit the Hubble radius in the radiation-dominated contracting phase as quantum vacuum fluctuations, re-enter the Hubble radius in the deflationary period and re-cross during the phase of inflationary expansion. Evolving the fluctuations using the general relativistic linear perturbation equations, we find that they exit the Hubble radius during inflation not with a scale-inva…
In this paper we put forward the idea that the comoving Hubble horizon undergoes multiple stages of contraction (a.k.a. inflationary phase) and expansion. The observable inflation, that produces the CMB anisotropies and generates primordial gravitational waves, follows and is followed by multiple early and late inflations. The trans-Planckian censorship conjecture restricts the duration of each inflationary phases and determines their Hubble rates. Early inflations could start immediately after the universe emerges from the Planck era. It alleviates the initial condition problem for the lower-…
It is investigated if predictions of the inflationary scenario regarding spectra of scalar and tensor perturbations generated from quantum vacuum fluctuations are robust with respect to a modification of the dispersion law for frequencies beyond the Planck scale. For a large class of such modifications of special and general relativity, for which the WKB condition is not violated at super-high frequencies, the predictions remain unchanged. The opposite possibility is excluded by the absence of large amount of created particles due to the present Universe expansion. Creation of particles in the…
We explore the bound of the trans-Planckian censorship conjecture on an inflation model with multiple stages. We show that if the first inflationary stage is responsible for the primordial perturbations in the cosmic microwave background window, the $e$-folding number of each subsequent stage will be bounded by the energy scale of the first stage. This seems to imply that the lifetime of the current era of accelerated expansion (regarded as one of the multiple inflationary stages) might be a probe for distinguishing inflation from its alternatives. We also present a multistage inflation model …
In most current models of inflation based on a weakly self-coupled scalar matter field minimally coupled to gravity, the period of inflation lasts so long that, at the beginning of the inflationary period, the physical wavelengths of comoving scales which correspond to the present large-scale structure of the Universe were smaller than the Planck length. Thus, the usual computations of the spectrum of fluctuations in these models involve extrapolating low energy physics (both in the matter and gravitational sector) into regions where this physics is not applicable. In this paper we demonstrate…
Planck scale physics may influence the evolution of cosmological fluctuations in the early stages of cosmological evolution. Because of the quasi-exponential redshifting, which occurs during an inflationary period, the physical wavelengths of comoving scales that correspond to the present large-scale structure of the Universe were smaller than the Planck length in the early stages of the inflationary period. This trans-Planckian effect was studied before using toy models. The Horava-Lifshitz (HL) theory offers the chance to study this problem in a candidate UV complete theory of gravity. In th…
We investigate the Trans-Planckian Censorship Conjecture (TCC) and the arising bounds on the inflationary cosmology caused by that conjecture. In that investigation, we analyze TCC bounds for both Jordan and Einstein frames in the presence of a generic non-minimal coupling (to gravity) term. That term allows us to use the functional freedom it brings to the inflationary Lagrangian as an effective Planck mass. In this sense, we argue one should consider the initial field value of the effective Planck mass for the TCC. We show that as a result, one can remove the TCC upper bounds without the nee…
In this short note, we pointed out that Warm inflationary scenario remains to be favoured over its cold counterpart by the recently proposed conjectures, which aim to overrule de Sitter like constructions in String Landscapes. On the other hand, the canonical cold (single-field slow-roll) inflationary models, which were in tension with the previously proposed de Sitter conjecture, have now become even more unlikely to realise in String Landscapes as these scenarios fail to cope up with both the conjectures, de Sitter and trans-Planckian censorship, at one go.
The trans-Planckian censorship conjecture (TCC) puts an upper bound on the life-time of de Sitter spacetimes. It has immediate consequences for inflationary cosmology. In the standard paradigm, the universe has experienced a single stage of inflation and follows a thermal history. Then, the TCC puts an upper bound on the Hubble parameter during inflation $H_{\rm inf}$ around $0.1$ GeV. Consequently, it implies a severe fine-tuning in initial condition for inflation and non-detection of primordial gravitational waves. In this note, we study non-standard cosmological paradigms with non-thermal h…
In this paper I introduce a precise constraint on primordial magnetogenesis, for a generic class of single-field inflationary model followed by small field excursion below the Planck scale. I also establish a connection between the magnetic field at the present epoch and primordial gravity waves ($r$) via non-vanishing CP asymmetry parameter ($ε_{\bf CP}$), which triggers the leptogenesis scenario. Finally, I explore various hidden cosmophenomenological features of theoretical CMB B-mode polarization spectra, which can be treated as a significant probe to put further stringent constraint on lo…
Due to the tremendous red-shift that occurs during the inflationary epoch in the early universe, it has been realized that trans-Planckian physics may manifest itself at energies much lower than the Planck energy. The presence of a fundamental scale suggests that local Lorentz invariance may be violated at sufficiently high energies. Motivated by this possibility, recently, different models that violate Lorentz invariance locally have been used to evaluate the trans-Planckian corrections to the inflationary density perturbation spectrum. However, certain astrophysical observations seem to indi…
A Remedy of the Trans-Planckian Censorship Problem with Smooth Slow-roll to Power-law Inflation Transitions in Scalar Field Theory
Institute of Space Sciences (ICE, CSIC)（空间科学研究所）；ICREA（加泰罗尼亚高级研究学院）；L.N. Gumilyov Eurasian National University - Astana（欧亚国立大学）；Department of Physics, Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学物理系）
The constraint of trans-Planckian censorship conjecture on brane inflation model is considered. The conjectures put an upper bound on the main parameter including temperature, inflation time, potential, and the tensor-to-scalar ratio parameter $r$. It is determined that the resulting constraint could be more stronger than what we have for the standard inflationary models. The constraint in general depends on the brane tension and it is concluded that the conjecture also confined the value of brane tension in order to have consistency for the model. Confining the brane tension turns into a dete…
In this paper, we calculate the spectrum of scalar field fluctuations in a bouncing, asymptotically flat Universe, and investigate the dependence of the result on changes in the physics on length scales shorter than the Planck length which are introduced via modifications of the dispersion relation. In this model, there are no ambiguities concerning the choice of the initial vacuum state. We study an example in which the final spectrum of fluctuations depends sensitively on the modifications of the dispersion relation without needing to invoke complex frequencies. Changes in the amplitude and …
The recently proposed trans-Planckian censorship conjecture (TCC) imposes a strong constraint on the inflationary Hubble scale, of which the upper bound could be largely relaxed by considering a noninstantaneous reheating history. In this paper we will show that, if the primordial black holes (PBHs) are formed at reentry in the radiation-dominated era from the enhanced curvature perturbations at small scales, the TCC would impose a lower bound on the PBH mass $M_\mathrm{PBH}>γ(H_\mathrm{end}/10^9\,\mathrm{GeV})^2\,M_\odot$ regardless of the details for reheating history, where $γ$ is the colla…
We reconsider trans-Planckian corrections to inflationary spectra by taking into account a physical effect which has been overlooked and which could have important consequences. We assume that the short length scale characterizing the new physics is endowed with a finite width, the origin of which could be found in quantum gravity. As a result, the leading corrections responsible for superimposed osillations in the CMB temperature anisotropies are generically damped by the blurring of the UV scale. To determine the observational ramifications of this damping, we compare it to that which effect…
We calculate the spectrum of the relic gravitational wave due to the trans-Planckian effect in which the standard linear dispersion relations may be modified. Of the modified dispersion relations suggested in literatures which have investigated the trans-Planckian effect, we especially use the Corley-Jacobson dispersion relations. The Corley-Jacobson type modified dispersion relations can be obtained from Hořava-Lifshitz gravity which is non-relativistic and UV complete. Although it is not clear how the transitions from Hořava-Lifshitz gravity in the UV regime to Einstein gravity in the IR lim…
We show that the constraints which follow from the {\it Trans-Planckian Censorship Conjecture} for inflationary cosmology can be strengthened if the pre-inflationary universe was dominated by radiation. The resulting upper bound on the energy scale of inflation is $η\sim 10^4 {\rm GeV}$, close to the scale accessible to accelerator experiments.
To reconcile the BICEP2 measurement on the tensor-to-scalar ratio $r$ with Planck constraint, a large negative running of scalar spectral index $n_s$ is needed. So the inflationary observable such as $n_s$ should be expanded at least to the second-order slow-roll parameters for single-field inflationary models. The large value of $r$ and the Lyth bound indicate that it is impossible to obtain the sub-Planckian excursion for the inflaton. However, we derive an absolutely minimal bound $Δϕ/M_{\rm Pl}>\sqrt{r/2}$ on the inflaton excursion for single-field inflationary models, which can be applied…
The implications of the recently proposed Trans-Planckian censorship conjecture (TCC) are analyzed in the context of warm inflation. It is found that for a single-stage accelerated expansion the constraints imposed by the censorship are roughly the same as for cold inflation. Next, we study how a two-stage inflationary expansion with an intermediate radiation-dominated era can alleviate the bounds imposed by the censorship. For a demonstrative toy model we found $r<10^{-23}$, but can be $r<10^{-5}$ for a weaker form of TCC for the later stages of expansion, while still satisfying the other swa…
In the framework of inflation, scales which nowadays correspond to large scale structures were smaller than the Planck length at the beginning of inflation. Therefore, measurements of CMBR anisotropy or surveys of galaxies and of clusters of galaxies could help us to probe physics beyond the Planck scale. In these proceedings, we study how the inflationary observables depend on trans-Planckian physics. It is found that there exist cases where adiabaticity can be violated and the primordial spectrum of the quantum cosmological perturbations modified. These cases correspond to rather drastic alt…
We derive the semiclassical evolution of massless minimally coupled scalar matter in the de Sitter space-time from the Born-Oppenheimer reduction of the Wheeler-DeWitt equation. We show that the dynamics of trans-Planckian modes can be cast in the form of an effective modified dispersion relation and that high energy corrections in the power spectrum of the cosmic microwave background radiation produced during inflation remain very small if the initial state is the Bunch-Davies vacuum.
A new family of inflationary models is introduced and analysed. The behaviour of the parameters characterising the models suggest preferred values, which generate the most interesting testable predictions. Results are further improved if late reheating and/or a subsequent period of thermal inflation is taken into account. Specific model realisations consider a sub-Planckian inflaton variation or a potential without fine-tuning of mass scales, based on the Planck and grand unified theory scales. A toy model realisation in the context of global and local supersymmetry is examined and results fit…
We provide strong evidence for universality of the inflationary field range: given an accurate measurement of $(n_s,r)$, one can infer $Δϕ$ in a model-independent way in the sub-Planckian regime for a range of universality classes of inflationary models. Both the tensor-to-scalar ratio as well as the spectral tilt are essential for the field range. Given the Planck constraints on $n_s$, the Lyth bound is strengthened by two orders of magnitude: whereas the original bound gives a sub-Planckian field range for $r \lesssim 2 \cdot 10^{-3}$, we find that $n=0.96$ brings this down to $r \lesssim 2 …
In this paper we will analyse the constraints on a sub-Planckian excursion of a single inflaton field, which would yield a large tensor to scalar ratio, while explaining the temperature anisotropy of the cosmic microwave background (CMB) radiation. In particular, our attempt will be to reconstruct the inflationary potential by constraining, $V(ϕ_0), V^{\prime}(ϕ_0), V^{\prime\prime}(ϕ_0), V^{\prime\prime\prime}(ϕ_0)$ and $V^{\prime\prime\prime\prime}(ϕ_0)$, in the vicinity of the field, $ϕ_0\ll M_p$, and the field displacement, $Δϕ\ll M_p$, where $M_p$ is the reduced Planck mass. We will provi…
We investigate the possibility of a trans-Planckian duality, which exchanges a manifold of events (space-time), with a manifold of momenta (energy-momentum). Gravity has a dual counter-part, that is, a geometric theory defined on the manifold of momenta. We provide a mathematical framework that can possibly realize this idea, and analyze its classical behaviour.
We investigate a model based on a generalised version of the Fourier transform for curved space-time manifolds. This model is possible if the metric has an asymptotic flat region which allows a duality to be implement between coordinates and momenta, hence, the models's name, trans-Planckian. The theory and the action are based on the postulate of the absolute egalitarian relation between coordinates $x$ and momenta $p$. We show how to implement this construction in a cosmological setting, on a Friedman-Robertson-Walker metric background, where the asymptotic time infinity plays the role of th…
It was recently proposed that a field theory cannot be consistent with quantum gravity if it allows a mode shorter than the Planck length to exit the Hubble horizon. This is called the Trans-Planckian Censorship Conjecture (TCC). We discuss the implications of the TCC on the possible shape of the inflaton potential in single-field slow-roll inflation. We point out that (1) there is generically an initial condition in which the total e-folding number $N_\text{total}$ is doubled or more compared to the e-folds necessary for the cosmic microwave background fluctuations, and (2) a sizable negative…
The standard, scale-invariant, inflationary perturbation spectrum will be modified if the effects of trans-Planckian physics are incorporated into the dynamics of the matter field in a phenomenological manner, say, by the modification of the dispersion relation. The spectrum also changes if we retain the standard dynamics but modify the initial quantum state of the matter field. We show that, given {\it any} spectrum of perturbations, it is possible to choose a class of initial quantum states which can exactly reproduce this spectrum with the standard dynamics. We provide an explicit construct…
The ``trans-Planckian'' challenge in cosmology appears when we trace the present physical wavelengths of fluctuations backwards in time. They become smaller and smaller until crossing the Planck scale where conventional QFT is challenged, so that unknown ultraviolet physics may be traced in the observable cosmological fluctuations. Usually this issue is addressed in the inflationary context, but trans-Planckian reasoning is much broader. We examine this logic in a simple example of scalar quantum field theory in the expanding and contracting Milne universes, where wavelengths of the eigenmodes…
Motivated by the old trans-Planckian (TP) problem of inflationary cosmology, it has been conjectured that any consistent effective field theory should keep TP modes `hidden' behind the Hubble horizon, so as to prevent them from turning classical and thereby affecting macroscopic observations. In this paper we present two arguments against the Hubble horizon being a scale of singular significance as has been put forward in the TP Censorship Conjecture (TCC). First, refinements of TCC are presented that allow for the TP modes to grow beyond the horizon while still keeping the de-Sitter conjectur…
In this paper we explored in detail a phenomenological model of modified single field natural inflation in light of recent cosmological experiments, BICEP2. Our main goal is to construct an inflationary model which not only predicts the important cosmological quantities such as $(n_s, r)$ compatible with experimental observation, but also is consistent with the low energy effective theory framework. Therefore, all the fundamental scale apart from $M_p$ and quantities of our interest should be within the sub-Planckian region. In order to achieve our goal we modify the usual single field natural…
One of the firm predictions of inflationary cosmology is the consistency relation between scalar and tensor spectra. It has been argued that such a relation -if experimentally confirmed- would offer strong support for the idea of inflation. We examine the possibility that trans-Planckian physics violates the consistency relation in the framework of inflation with a cut-off proposed in astro-ph/0009209. We find that despite the ambiguity that exists in choosing the action, Planck scale physics modifies the consistency relation considerably. It also leads to the running of the spectral index. Fo…
We examine the constraints cosmological observations can place on any trans-Planckian corrections to the primordial spectrum of perturbations underlying the anisotropies in the Cosmic Microwave Background. We focus on models of trans-Planckian physics which lead to a modulated primordial spectrum. Rather than looking at a generic modulated spectrum, our calculations are based on a specific model, and are intended as a case study for the sort of constraints one could hope to apply on a well-motivated model of trans-Planckian physics. We present results for three different approaches -- a grid s…
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 co…
The recently proposed trans-Planckian censorship conjecture (TCC) amounts to the claim that inflation models with an inflationary energy scale larger than Lambda_inf^max ~ 10^9 GeV belong to the swampland, i.e., cannot be embedded into a consistent theory of quantum gravity. In this paper, we point out that this constraint can be readily satisfied in D-term hybrid inflation (DHI), which is a well-motivated inflation scenario in the context of supersymmetric grand unification. In DHI, the amplitude of the primordial scalar power spectrum originates from a Fayet-Iliopoulos term of the order of t…
We study the effects of the trans-Planckian dispersion relation on the spectrum of the primordial density perturbations during inflation. In contrast to the earlier analyses, we do not assume any specific form of the dispersion relation and allow the initial state of the field to be arbitrary. We obtain the spectrum of vacuum fluctuations of the quantum field by considering a scalar field satisfying the linear wave equation with higher spatial derivative terms propagating in the de Sitter space-time. We show that the power spectrum does not strongly depend on the dispersion relation and that t…
Following a self-contained review of the basics of the theory of cosmological perturbations, we discuss why the conclusions reached in the recent paper by Kaloper et al are too pessimistic estimates of the amplitude of possible imprints of trans-Planckian (string) physics on the spectrum of cosmic microwave anisotropies in an inflationary Universe. It is shown that the likely origin of large trans-Planckian effects on late time cosmological fluctuations comes from nonadiabatic evolution of the state of fluctuations while the wavelength is smaller than the Planck (string) scale, resulting in an…
