Trans-Planckian corrections to the primordial spectrum in the infra-red and the ultra-violet

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 indicate that local Lorentz invariance may be preserved to extremely high energies. In such a situation, to study the trans-Planckian effects, it becomes imperative to consider models that preserve local Lorentz invariance even as they contain a fundamental scale. In this work, we construct one such model and evaluate the resulting spectrum of density perturbations in the power-law inflationary scenario. While our model reproduces the standard spectrum on small scales, it naturally predicts a suppression of power on large scales. In fact, the spectrum we obtain has some features which are similar to the one that has recently been obtained from non-commutative inflation. However, we find that the amount of suppression predicted by our model is far less than that is required to fit the observations. We comment on the fact that, with a suitable choice of initial conditions, our approach can lead to corrections at the infra-red as well as at the ultra-violet ends of the spectrum.Comment: 11 pages, 3 figures, Revtex 4; References adde

Quantum gravitational corrections to the stress-energy tensor around the rotating BTZ black hole

Modes emerging out of a collapsing black hole are red-shifted to such an extent that Hawking radiation at future null infinity consists of modes that have energies beyond the Planck scale at past null infinity. This indicates that physics at the Planck scale may modify the spectrum of Hawking radiation and the associated stress-energy tensor of the quantum field. Recently, it has been shown that, the T-duality symmetry of string fluctuations along compact extra dimensions leads to a modification of the standard propagator of point particles in quantum field theory. At low energies (when compared to the string scale), the modified propagator is found to behave as though the spacetime possesses a minimal length, say, \lp, which we shall assume to be of the order of the Planck length. We utilize the duality approach to evaluate the modified propagator around the rotating Banados-Teitelboim-Zanelli black hole and show that the propagator is finite in the coincident limit. We compute the stress-energy tensor associated with the modified Green's function and illustrate graphically that the quantum gravitational corrections turn out to be negligibly small. We conclude by briefly commenting on the results we have obtained.Comment: v1. 7 pages, 2 figures; v2. 11 pages, 4 figures, discussion extended to the case of the rotating BTZ black hole, figures improve

Sub-leading contributions to the black hole entropy in the brick wall approach

[Abridged] We compute the canonical entropy of a quantum scalar field around static and spherically symmetric black holes through the brick wall approach at the higher orders (in fact, up to the sixth order in \hbar) in the WKB approximation. We explicitly show that the brick wall model generally predicts corrections to the Bekenstein-Hawking entropy in all spacetime dimensions. In four dimensions, we find that the corrections to the Bekenstein-Hawking entropy are of the form (A^n \log A), while, in six dimensions, the corrections behave as (A^m + A^n \log A), where A denotes the area of the black hole event horizon, and (m, n) < 1. We compare our results with the corrections to the Bekenstein-Hawking entropy that have been obtained through the other approaches in the literature, and discuss the implications.Comment: 21 pages, Revtex 4; Final verson - 22 pages, References added, Accepted in Phys. Rev.

Path integral duality modified propagators in spacetimes with constant curvature

The hypothesis of path integral duality provides a prescription to evaluate the propagator of a free, quantum scalar field in a given classical background, taking into account the existence of a fundamental length, say, the Planck length, \lp, in a {\it locally Lorentz invariant manner}. We use this prescription to evaluate the duality modified propagators in spacetimes with {\it constant curvature} (exactly in the case of one spacetime, and in the Gaussian approximation for another two), and show that: (i) the modified propagators are ultra violet finite, (ii) the modifications are {\it non-perturbative} in \lp, and (iii) \lp seems to behave like a `zero point length' of spacetime intervals such that \l = \l[\sigma^{2}(x,x')+ {\cal O}(1) \lp^2 \r], where $\sigma(x,x')$ is the geodesic distance between the two spacetime points $x$ and $x'$, and the angular brackets denote (a suitable) average over the quantum gravitational fluctuations. We briefly discuss the implications of our results.Comment: v1. 10 pages, no figures; v2. 11 pages, acknowledgments adde

Path integral duality and Planck scale corrections to the primordial spectrum in exponential inflation

The enormous red-shifting of the modes during the inflationary epoch suggests that physics at the Planck scale may modify the standard, nearly, scale-invariant, primordial, density perturbation spectrum. Under the principle of path-integral duality, the space-time behaves as though it has a minimal length $L_{_{\rm P}}$ (which we shall assume to be of the order of the Planck length), a feature that is expected to arise when the quantum gravitational effects on the matter fields have been taken into account. Using the method of path integral duality, in this work, we evaluate the Planck scale corrections to the spectrum of density perturbations in the case of exponential inflation. We find that the amplitude of the corrections is of the order of $({\cal H}/M_{_{\rm P}})$, where ${\cal H}$ and $M_{_{\rm P}}$ denote the inflationary and the Planck energy scales, respectively. We also find that the corrections turn out to be completely independent of scale. We briefly discuss the implications of our result, and also comment on how it compares with an earlier result.Comment: 12 pages, 1 figure, RevTex4 forma

Boundary Effective Field Theory and Trans-Planckian Perturbations: Astrophysical Implications

We contrast two approaches to calculating trans-Planckian corrections to the inflationary perturbation spectrum: the New Physics Hypersurface [NPH] model, in which modes are normalized when their physical wavelength first exceeds a critical value, and the Boundary Effective Field Theory [BEFT] approach, where the initial conditions for all modes are set at the same time, and modified by higher dimensional operators enumerated via an effective field theory calculation. We show that these two approaches -- as currently implemented -- lead to radically different expectations for the trans-Planckian corrections to the CMB and emphasize that in the BEFT formalism we expect the perturbation spectrum to be dominated by quantum gravity corrections for all scales shorter than some critical value. Conversely, in the NPH case the quantum effects only dominate the longest modes that are typically much larger than the present horizon size. Furthermore, the onset of the breakdown in the standard inflationary perturbation calculation predicted by the BEFT formalism is likely to be associated with a feature in the perturbation spectrum, and we discuss the observational signatures of this feature in both CMB and large scale structure observations. Finally, we discuss possible modifications to both calculational frameworks that would resolve the contradictions identified here.Comment: Reworded commentary, reference added (v2) References added (v3

Concept of temperature in multi-horizon spacetimes: Analysis of Schwarzschild-De Sitter metric

In case of spacetimes with single horizon, there exist several well-established procedures for relating the surface gravity of the horizon to a thermodynamic temperature. Such procedures, however, cannot be extended in a straightforward manner when a spacetime has multiple horizons. In particular, it is not clear whether there exists a notion of global temperature characterizing the multi-horizon spacetimes. We examine the conditions under which a global temperature can exist for a spacetime with two horizons using the example of Schwarzschild-De Sitter (SDS) spacetime. We systematically extend different procedures (like the expectation value of stress tensor, response of particle detectors, periodicity in the Euclidean time etc.) for identifying a temperature in the case of spacetimes with single horizon to the SDS spacetime. This analysis is facilitated by using a global coordinate chart which covers the entire SDS manifold. We find that all the procedures lead to a consistent picture characterized by the following features: (a) In general, SDS spacetime behaves like a non-equilibrium system characterized by two temperatures. (b) It is not possible to associate a global temperature with SDS spacetime except when the ratio of the two surface gravities is rational (c) Even when the ratio of the two surface gravities is rational, the thermal nature depends on the coordinate chart used. There exists a global coordinate chart in which there is global equilibrium temperature while there exist other charts in which SDS behaves as though it has two different temperatures. The coordinate dependence of the thermal nature is reminiscent of the flat spacetime in Minkowski and Rindler coordinate charts. The implications are discussed.Comment: 12 page

Minimal Length Scale Scenarios for Quantum Gravity

We review the question of whether the fundamental laws of nature limit our ability to probe arbitrarily short distances. First, we examine what insights can be gained from thought experiments for probes of shortest distances, and summarize what can be learned from different approaches to a theory of quantum gravity. Then we discuss some models that have been developed to implement a minimal length scale in quantum mechanics and quantum field theory. These models have entered the literature as the generalized uncertainty principle or the modified dispersion relation, and have allowed the study of the effects of a minimal length scale in quantum mechanics, quantum electrodynamics, thermodynamics, black-hole physics and cosmology. Finally, we touch upon the question of ways to circumvent the manifestation of a minimal length scale in short-distance physics.Comment: Published version available at http://www.livingreviews.org/lrr-2013-