2 00 9 Modifying Gravity in the Infra - Red by imposing an “ Ultra - Strong ” Equivalence Principle ∗

The equivalence principle suggests to consider gravity as an infra-red phenomenon, whose effects are visible only outside Einstein’s free-falling elevator. By curving spacetime, General Relativity leaves the smallest systems free of classical gravitational effects. However, according to the standard semi-classical treatment, indirect effects of gravity can be experienced inside the elevator through the well-known mechanism of quantum particle production. Here we try a different path than the one historically followed: rather than imposing field quantization on top of a curved manifold, we attempt to upgrade the equivalence principle and extend it to the quantum phenomena. Therefore, we consider, and try to realize in a theoretical framework, a stronger version of the equivalence principle, in which all the effects of gravity are definitely banned from the elevator and confined to the infra-red. For this purpose, we introduce infra-red modified commutation relations for the global field operators (Fourier modes) that allow to reabsorb the time-dependent quadratic divergence of the vacuum expectation value of the stress-energy tensor. The proposed modification is effective on length scales comparable to the inverse curvature and, therefore, does no add any dimensional parameter to the theory. Based on the essay written for the Gravity Research Foundation 2009 Awards. [email protected] Modifications of General Relativity (GR) on the largest scales have been advocated in order to give account for the present acceleration of the Universe. These alternatives to GR look now particularly appealing in view of some emerging tensions between standard ΛCDM cosmology and large-scale observations [1]. Infra-red (IR) modifications of gravity typically involve large extra-dimensions and are effectively equivalent to giving a small mass to the graviton . In this note we explore a modification of GR of quite a different nature and contemplate the possibility that the very geometrical description of space-time as a metric manifold may break down on the largest scales. This point of view is provocative only in appearance; it aims in fact, rather conservatively, to recover the most genuine and intuitive physical content of the Equivalence Principle (EP), namely, the absence of any gravitational effect within each sufficiently small freefalling system. 1 Invitation: Gravity as an Infra-Red Effect. The equivalence principle (EP) can be formulated simply as follows: inside a sufficiently small free-falling elevator you do not see the (classical) effects of gravity. Amusingly, such a cornerstone of modern physics is actually stating what (where) gravity is not, rather than what (where) gravity is! Among the many celebrated implications of general relativistic physics, the view that we aim to stress here is that EP forces us to consider and describe gravity as an IR phenomenon, whose effects are visible only outside the free-falling elevator. How EP turned into a consistent theory is well known: gravity is beautifully encoded in GR as the geometry of the physical space-time and therefore its effects are automatically suppressed within those systems that are much smaller than the inverse curvature. By changing (curving) the large-scale structure of spacetime, GR makes the smallest systems free of classical gravitational effects. Notably, the IR scale where non-gravitational physics breaks down is not a parameter of the theory, but is set by the local curvature R. Schematically, in three dimensions, the area of a two-sphere of radius l and volume V receives corrections from flat-space expectation of the type A(l) = 4πl (1 +O(lR)) = (36πV ) (1 +O(RV )). (1.1) The effects of gravity, originally banned from the free-falling elevator, reappeared, after the developments of quantum theory, through what one might call the back door. The fields quantized on a curved manifold are sensitive to the global properties of spacetime because their modes are defined on the whole of it. As a result, inside the elevator, you will generically experience, and possibly detect with your local instruments, particle creation because of non-local gravitational effects. Clearly, the process of quantum particle creation does not contradict EP, which was formulated within the framework of classical physics. Nevertheless, it is tempting to try a different path than the one historically followed: rather than imposing field quantization on top of a curved 1 manifold, here we attempt to upgrade the equivalence principle and extend it to the quantum phenomena. Thus, we will consider a stronger version of EP, in which all the effects of gravity are definitely forbidden inside the elevator, including the quantum effects that in the standard semi-classical treatment lead to particle creation. More precisely, Equivalence Principle, “Ultra-Strong” Version: For each matter field or sector sufficiently decoupled from all other matter fields, there exists a state, the “vacuum”, that is experienced as empty of particles in the vicinity of each free-falling observer. Since the effect that we want to cancel is sensitive to the global structure of spacetime, we argue that applying the above more severe version of EP forces a further substantial change at scales comparable to the curvature. Accordingly, we attempt to set up a theoretical framework for semi-classical gravity where the metric manifold structure of GR is systematically modified in the infra-red but holds in the vicinity of each point/event. An obvious warning is that, while EP is extremely well tested (see e.g. [2]), the proposed “ultra-strong” version is not. If any, experimental hints might actually be arguing against it, since, according to the current paradigm, cosmological fluctuations are generated during inflation precisely with the mechanism of quantum particle creation. On the other hand, the appeal of the model that we are going to propose (see also [3]) is that it contains no more parameters than GR itself, and therefore it is in principle very well testable.