Can Gibbons–Hawking radiation and inflation arise due to spacetime quanta?
Introduction
Black hole radiation and inflation are ones of the most intriguing aspects in cosmology. The first one being theorized by Stephen Hawking [1], has been attracted a considerable interest [2], [3], [4], [5], [6], [7]. Along with Unruh radiation that is “felt” by an accelerated observer[8], [9], [10], [11], Hawking radiation was originally derived based on Bogoliubov’s method [12], ever since many approaches [13], [14], [15], [16], [17], [18] including the complex path (or Hamilton–Jacobi) method [19], [20] have been developed to derive such a radiation. Hawking radiation is described as a tunnelling effect of particles across the black hole’s horizon [13]. In fact, such a thermal radiation is related to any geometrical background that possesses a horizon (i.e. the cosmological horizon of a black hole, de Sitter space and even the one of an accelerated observer).
The hypothesis of inflation, on the other hand, has been postulated by Alan Guth [21], [22]. His model relies on the assumption that the very early Universe has gone through a period of accelerated expansion that preceded the standard radiation-dominated era. Such a period of accelerated expansion offers a physical explanation of the cosmology’s biggest puzzles that the standard cosmological scenario cannot explain. Inflation drives any initially curved spacetime towards the spatial flatness observed today, hence answering the question: ”Why would the universe be perfectly spatially flat?”, it brings together all causally disconnected regions and extends the causal horizon beyond the present Hubble length, in such a away to answer the question: ”Why would the universe have the same temperature everywhere?” and it also brings a satisfactory solution to the magnetic monopoles problem, answering the question: ”Why are there no leftover high-energy relics?” To date, many simple as well as complex inflationary models have been proposed [23], [24], [25], [26], [27], [28], [29], [30]. In their simplest picture, inflationary models are based on a scalar field (inflaton field) with a fine-tuning potential, i.e., a flat potential. The latter is constrained by slow roll conditions, i.e., in order to trigger inflation the scalar field’s potential energy must dominate over its kinetic and gradient energy that can prevent its starting. This fine-tuning requirement puts the viability of such inflationary models into question.
To contribute to the ongoing literature on the two topics, firstly we address the Gibbons–Hawking radiation [31], [32] in anti-de Sitter space by incorporating a quantized spacetime. We want to stress, however, that our contribution is to show that this radiation has nothing to do with the tunnelling particle but it is due to the space curved geometry, precisely its quantum nature. Secondly we propose, inspired by the Tsallis q-formalism [33], [34], [35], a new inflationary model, with a minimum of fine-tuning, that we will call the q-inlfation. The latter, which is devoid of singularities, is triggered implicitly by the spacetime quanta but invokes the contribution of the “cosmological constant” carried by the quanta, explicitly. We believe in the importance of the present work because it kills two birds (Gibbons–Hawking radiation and inflation) with one stone (spacetime quantization). Throughout the paper, we consider the metric signature . Moreover, the units are chosen with .
Section snippets
The origin of Gibbons-Hawking radiation
In this section we derive the well-known Gibbons–Hawking temperature [31], [32] by means of a spacetime quantization framework firstly introduced by L.C. Céleri et al. [36], [37]. Their study, in which they show that Unruh effect can be obtained without changing the reference frame, focuses on deriving Unruh radiation in the accelerated fields scenario. For our purpose however, let us consider the two-sheet hyperboloid (see Fig. 1) governed by the parameterization
q-inflationary model
This section is dedicated to study the properties (i.e. power spectrum and spectral index) of large scale vacuum fluctuations in the q-inflationary scenario. The latter, which depends on the non extensive parameter measuring the deviation from the usual exponential expansion, is a strategy to (i) skirt scalar fields with fine-tuned potentials, (ii) skirt an exponential expansion that lasts forever and (iii) ensure a nearly but not perfectly scale-invariant spectrum, in agreement with
Observational constraints on the q-parameter
The value of the spectral index measured by the Planck collaboration including the results of WMAP and those based on the investigation of baryon acoustic oscillations (BAO) [41], [42], [43], is required to lie in the range [0.955–0.98] at CL, which rules out the scale invariant spectrum (i.e. ) at more than 5 confidence level. Based on this finding, we can safely discard inflationary models for which . Hence, we must pick out the range within which the value of must lie. For such
Data analysis
In order to impose constraints on the q-inflationary model’s parameters, we have used a modified version of the Boltzmann CAMB code [47], [48], [49] and the Monte Carlo Markov Chain (MCMC) analysis [50] provided by the publicly available CosmoMC package.2 Therefore, the inflationary sector has been modified by plugging in the following power spectrum parameterizations
Closing remarks
Despite the fact that the obtained results may allow the q-inflation to be a plausible candidate for describing the early universe, the current model is just a toy model and much enhancement has yet to be done. In particular, the reheating mechanism must be further developed. As one may notice, the current model, similarly to the power-law inflation, has an exit problem, i.e. inflation does not end via slow roll violation, i.e. . Nevertheless, as has been stated by Lucchin et al. [54]
CRediT authorship contribution statement
Naouel Boulkaboul: Conceptualization, Formal analysis, Investigation, Methodology, Validation, Visualization, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
I would like to thank Antony Lewis for kindly providing the numerical codes CosmoMC and CAMB. I am also grateful to Sukannya Bhattacharya and Savvas Nesseris for valuable instructions about numerical issues regarding CosmoMC, and I am particularly indebted to an anonymous reviewer for insightful comments.
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