A few cosmological implications of tensor nonlocalities

The possibility that general relativity (GR) may be modified on different length scales is an area of active research [1]. The overwhelming majority of theories proposed to replace GR involve local functions of the degrees of freedom. Some of the most popular proposals work purely with the metric tensor, constructing functions of the curvature tensors and scalars [2]. Others couple the metric with other new gravitational degrees of freedom such as scalars [3,4], vectors [5,6], and tensors [7–10], which can arise in a variety of settings. Very few proposals consider the possibility that the gravitational action might be fundamentally nonlocal. In some sense, the minimalist approach to modifying GR is to look at the effective action for gravity that arises from one-loop correction from gauge and matter fields [11–14]. While, at first sight, nonlocal interactions may seem to be a step too far, they seem to consistently crop up, and when they do, they can have far-reaching consequences. So, for example, the feted trace anomaly which was uncovered in the 1970s [15] played an instrumental role in the first model of inflation constructed by Starobinsky [16]. Surprisingly, it is this model that seems best to fit the current measurements of large scale structure coming out of the analysis of the Planck satellite data [17]. The role of the trace anomaly in the early Universe was also highlighted in Ref. [18], where it was shown that it could be used to generate large scale seed magnetic fields. While in detail, the proposal is problematic (it needs many fermion families and large gauge groups), it is still one of the frontrunners in explaining how these fields formed in the early Universe. Over the years, a number of proposals for the origin, and effect of, nonlocal terms in the gravitational action have been put forward [19–23]. More recently, Deser and Woodard [24] have embraced the possibility of nonlocal terms and proposed that the gravitational action could take the form