Varying couplings in the early universe: correlated variations of α and G C.J.A.P. Martins $^{a,b}$, Eloisa Menegoni$^{c}$, Silvia Galli$^{d,e}$, Gianpiero Mangano$^{f}$, and Alessandro Melchiorrid a Centro de Astrof´ısica, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal b CTC, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom c ICRA, International Center for Relativistic Astrophysics, University of Rome, “La Sapienza,” P.le Aldo Moro 2, 00185 Rome, Italy d Physics Department and INFN, Universit` a di Roma “La Sapienza”, Ple Aldo Moro 2, 00185 Rome, Italy e Laboratoire Astroparticule et Cosmologie (APC), Universit´ e Paris Diderot, 75205 Paris cedex 13 and f INFN and Physics Department, University of Naples “Federico II”, Via Cintia, 80126 Naples, Italy The Cosmic Microwave Background anisotropies provide a unique opportunity to constrain simul- taneous variations of the fine-structure constant α and Newton’s gravitational constant G . Those correlated variations are possible in a wide class of theoretical models. In this brief paper we show that the current data, assuming that particle masses are constant, gives no clear indication for such variations, but already prefers that any relative variations in α should be of the same sign of those of G for variations of ∼ 1%. We also show that a cosmic complementarity is present with Big Bang Nucleosynthesis and that a combination of current CMB and BBN data strongly constraints simultaneous variations in α and G . We finally discuss the future bounds achievable by the Planck satellite mission. PACS numbers: 98.80.Cq, 04.50.+h, 95.35.+d, 98.70.Vc The behavior of nature’s fundamental couplings is a subject of much recent interest. There is ample ex- perimental evidence showing that they run with energy, and many particle physics and cosmology models sug- gest that they should also roll with time. This explains why the European Space Agency (ESA) and the Euro- pean Southern Observatory (ESO) now list varying fun- damental constants among their key science drivers for the next generation of facilities. Recent technological de- velopments have for the first time provided us with tools to accurately test this hypothesis. Two recent reviews [1, 2] discuss most of these developments. In particular, there are controversial claims for time variations of the fine-structure constant α [3] and the proton-to-electron mass ratio µ [4] at redshifts z ∼ 1 − 3, as well as of possible spatial variations of the latter in the Galaxy [5]. While these have so far not been confirmed by independent analyses [6–9], their potential implications certainly warrant further study. In the present work we focus on the cosmic microwave background (CMB) as a means to probe the behavior of these couplings at red- sifts z ∼ 10 $^{3}$, but we will also discuss the importance of constraints at much higher redshifts ( z ∼ 10 $^{10}$), coming from Big Bang Nucleosynthesis. Typically, in any sensible theory where a coupling is rolling, one generically expects the others to do so as well, though possibly at fairly different rates. Since the rolling is expected to be due to the same underlying mechanism (the most natural of which will be a dynamical, funda- mental scalar field), the rates of change of the various couplings will in fact be related in any given theory. From an experimental point of view, one can either take the simplifying assumption that only the coupling Speaking of variations of dimensional constants obvi- ously has no physical significance: one can design any variation by defining appropriate units of length, time and energy. However, one is free to choose an arbitrary dimensionful unit as a standard and compare it with other quantities. If one assumes particle masses to be constant (as we will do in this paper), constraints on the gravitational constant G are in fact constraining the di- mensionless product of G and the nucleon mass squared. With this caveat, constraints on a rolling G provide key information on the gravitational sector. Somewhat para- doxically, G was the first constant to be measured but is now the least well known, a result of the weakness of The importance of α as a fundamental physics probe stems from the fact that it is ubiquitous in electromag- netic processes. In the past this has been extensively used to constrain the fine-structure constant [13–20]. These studies yielded results consistent with no variation, but due to degeneracies with other cosmological parameters the accuracy is lower than that of low-redshift measure- ments: it is only with the latest available data that con- straints stronger than the percent level have been ob- tained [21]. one wants to constrain is varying while the others are con- stant, or try to constrain joint variations at the expense of a more complicated analysis. In the latter case one can ei- ther choose a particular set of such relations (thereby con- straining only a particular theory, though usually quite tightly) or phenomenologically treat the different varia- tions as independent (thus obtaining model-independent but usually weaker constraints). In this work we will use the CMB to constrain possible variations of α and Newton’s constant G , assuming them to be related by a phenomenological parameter whose value will be differ- ent in various fundamental physics scenarios. I. INTRODUCTION arXiv:1001.3418v3 [astro-ph.CO] 26 Jan 2010