universe Article Inflationary Implications of the Covariant Entropy Bound and the Swampland de Sitter Conjectures Dibya Chakraborty 1 , Cesar Damian 2, * , Alberto González Bernal 1 and Oscar Loaiza-Brito 1   Citation: Chakraborty, D.; Damian, C.; González Bernal, A.; Loaiza-Brito O. Inflationary Implications of the Covariant Entropy Bound and the Swampland de Sitter Conjectures. Universe 2021, 7, 423. https:// doi.org/10.3390/universe7110423 Academic Editors: Giuseppe Dibitetto and Ivonne Zavala Received: 25 September 2021 Accepted: 3 November 2021 Published: 6 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Departamento de Física, Universidad de Guanajuato, Loma del Bosque No. 103 Col. Lomas del Campestre C.P 37150 Leon, Guanajuato, GTO 36050, Mexico; dibyac@fisica.ugto.mx (D.C.); jgbernal.physics@gmail.com (A.G.B.); oloaiza@fisica.ugto.mx (O.L.-B.) 2 Departamento de Ingeniería Mecánica, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago Km 3.5 + 1.8 Comunidad de Palo Blanco, Salamanca, GTO 36885, Mexico * Correspondence: cesar.damian@ugto.mx Abstract: We present a proposal to relate the de Sitter conjecture (dSC) with the time dependence of fluxes via the covariant entropy bound (CEB). By assuming an early phase of accelerated expansion where the CEB is satisfied, we take into account a contribution from time-dependent flux compactifi- cation to the four-dimensional entropy which establishes a bound on the usual slow-roll parameters η H and ǫ H . We also show an explicit calculation of entropy from a toroidal flux compactification, from a transition amplitude of time-dependent fluxes which allows us to determine the conditions on which the bounds on the slow-roll parameters are in agreement to the dSC. Keywords: fluxes; entropy; swampland; multi-field inflation 1. Introduction The Swampland program has reached great advances in the last few years in its pursuit of characterize those features distinguishing effective theories that can be consis- tently completed into a quantum gravity theory (Landscape) from those which do not (Swampland) [1–4]. The boundary between the Landscape and Swampland is usually defined by a series of bounds on quantities of the proper effective theory which, in turn, are expressed in terms of the Planck mass M p . In the limit where gravity decouples, i.e., M p → ∞, the Swampland constraints vanish. A question to be answered is whether these boundaries can be traced back to some essential microscopic physics in the Quantum Gravity Theory, or accordingly, to some fundamental constraints in the 10-dimensional String Theory formulation. Many different bounds have been proposed as Swampland conjectures, although it is expected that all of them are related in some way through more fundamental principles (see review [3] and lectures [4] for references). Some of them are based on solid grounds, such as the distance conjecture and the weak gravity conjecture, while others, as the de Sitter conjectures, were initially motivated by some empirical evidence based on string models [5,6], and appropriately refined into its final form in order to be compatible with some well known effective scenarios such as the Higgs potential [7] among others. The refined version of the conjecture was motivated by stability of de Sitter extrema [8], by cosmological arguments through the slow-roll parameter η V [9], as well as by entropy arguments together with the distance conjecture on the dS space [10]. Many logic and solid arguments have been followed in the literature [11–20] so far with the purpose to give stronger ground basis to this conjecture. A matter of particular interest to us is the series of arguments supporting the de Sitter conjecture (dSC) based on the laws of thermodynamics [21–24]. Relating the dSC to thermodynamic principles, such as a positive temperature phase and the concavity of the entropy functional, respectively, have also been considered in the literature. Furthermore, as stochastic effects become relevant, it is found Universe 2021, 7, 423. https://doi.org/10.3390/universe7110423 https://www.mdpi.com/journal/universe