Fundamental Constants of Nature and their Possible Time Variation Author: Guillermo Bern´ ardez Gil. Facultat de F´ ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Advisor: Joan Sol` a Peracaula. Abstract: The possibility of a slow cosmic evolution of fundamental “constants” of Nature has been discussed for a long time without being consolidated; recently, however, more and more experiments are arising whose results seem to support this idea. Apart from that, in an expanding universe the vacuum energy density ρΛ is expected to be a time-evolving parameter rather than the rigid one proposed in the standard ΛCDM model; in fact, quantum field theory in curved space time suggests a slow evolution determined by the expansion rate of the universe H. In this work, we will try to obtain and develop some cosmological models with such a a dynamical vacuum energy density, and at the same time we will check whether they can provide an explanation for the alleged slow time variation of the fundamental constants. I. INTRODUCTION Nowadays the time and space evolution of fundamen- tal constants of Nature is a very active field of theoretical and experimental research that could provide interesting results in the near future. The idea of a cosmic time evolution of these fundamental constants of physics be- gan with Dirac in the thirties, when he proposed a time- evolving gravitational constant G [1]; since then, the pos- sibility of different dynamical ”constants” has been con- tinuously discussed and investigated, but many techno- logical improvements have been needed to finally notice possible evidences of these variations; see the review [2]. However, it is important to note that we still deal with low significance levels due to technical limitations on the current measurements. For instance, there are many tests indicating the pos- sibility that the value of the fine structure constant α em has changed over the cosmic evolution. Constraints on ˙ α em /α em ≡ (1/α em )dα em /dt can be deduced from limits on the position of nuclear resonances in natural fission reactors that have been working for the last few billions years; one renowned example is the natural reactor lo- cated at the Oklo uranium mine (Gabon). Moreover, direct astrophysical observations are becoming very rel- evant in this course. Note that if α em does not remain constant, we could also expect a time evolution of the masses of all nucleons due to the fact that the interac- tion responsible for the variation of α em should couple radiatively to nucleons. Likewise, it is also considered the proton-electron mass ratio μ pe ≡ m p /m e , which is known with high accu- racy, testing a possible cosmic time evolution of its value. Such a time-evolving μ pe , thus, could be interpreted as a change of the fundamental QCD scale parameter Λ QCD of the strong interaction, in the sense that a dynami- cal Λ QCD would originate a variation of the proton mass whereas it would not affect the electron one; we refer to [5] or [6] to see how m p and Λ QCD can be related. Some of these experiments consist on astrophysical tests, com- paring interstellar and laboratory spectrums, but there are also laboratory experiments where a time variation of the nucleon mass is tried to be observed by monitoring molecular frequencies using atomic clocks. What if an alternative cosmological model, instead of the standard ΛCDM one, can give us an explanation about the possible time variations of the particle physics constants mentioned above? From the cosmological point of view, the idea that both the cosmological term Λ and the gravitational coupling G could also be dynamical pa- rameters is, intuitively, a reasonable assumption in an expanding universe. Thus, we will look for new cosmolog- ical models compatible with a time-evolving pair (ρ Λ ,G), then trying to explain the feasible evolution of particle masses and couplings through them. II. COSMOLOGICAL MODELS WITH TIME-EVOLVING PARAMETERS As we have stated in the Introduction, we propose the idea that cosmic time variations of constants of particle physics may be related with time-evolving parameters of cosmology. In order to develop this theory, the aim of this section is to obtain a set of cosmological equations compatible with time variations of the Newton (G) and the cosmological (Λ) constants. From this set of equa- tions, we will finally raise some cosmological models. We start from the General Relativity field equations G μν − g μν Λ=8πGT μν in the presence of the cosmologi- cal term, where G μν is the Einstein tensor, and T μν the energy-momentum one of the isotropic matter and radi- ation of the universe. Assuming the expanding universe as a perfect fluid, with matter-radiation density ρ m and vacuum energy density ρ Λ := Λ/(8πG), we can rewrite them as G μν =8πG ˜ T μν , (1) where now ˜ T μν := T m μν + ρ Λ g μν is a diagonal tensor called the full energy-momentum tensor of the cosmic fluid. We will focus on solving (1) in the FLRW metric, ds 2 = dt 2 − a(t)dx 2 , being a(t) the time-evolving scale factor.