Shortcomings of Shapiro delay-based tests of the equivalence principle on cosmological scales

Abstract

The "Shapiro delay" experienced by an astronomical messenger traveling through a gravitational field has been used to place constraints on possible deviations from the equivalence principle. The standard Shapiro delay used to obtain these constraints is not itself an observable in general relativity, but is rather obtained by comparing with a fiducial Euclidean distance. There is not a mapping between the constraints obtained in this manner and alternative theories that exhibit equivalence principle violations. However, even assuming that the comparison with the fiducial Euclidean distance is carried out in a way that is useful for some class of alternative theories, we show that the standard calculation of these constraints cannot be applied on cosmological scales, as is often done. Specifically, we find that the Shapiro delay computed in the standard way (taking the Newtonian potential to vanish at infinity) diverges as one includes many remote sources. We use an infinite homogeneous lattice model to illustrate this divergence, and also show how the divergence can be cured by using Fermi coordinates associated with an observer. With this correction, one finds that the Shapiro delay is no longer monotonic with the number of sources. Thus, one cannot compute a conservative lower bound on the Shapiro delay using a subset of the sources of the gravitational field without further assumptions and/or observational input. As an illustration, we compute the Shapiro delay by applying the Fermi coordinate expression to two catalogs of galaxy clusters, illustrating the dependence of the result on the completeness of the catalogue and the mass estimates.

Constraining violations of the Weak Equivalence Principle Using CHIME FRBs

“…Eq. 6 is consistent (Minazzoli et al 2019b) up to at least ∼ 400 Mpc (𝑧 ∼ 0.1) when Shapiro Time Delay is calculated from observed galaxy clusters. Now, in this paper, we focus on a time lag under gravity between two photons with different energies.…”

“…If both the violation of WEP and Δ𝑡 DM show an equivalent dependence on 𝜈 −2 obs , the two effects are not distinguishable (Ioka 2003;Macquart et al 2020;Inoue 2004). Until now, current studies of FRBs or any other sources have not found any such violation due to frequency (Einstein 1908;Tingay & Kaplan 2016;Nusser 2016;Minazzoli et al 2019a;Xing et al 2019).…”

“…However, for huge cosmological distances where 𝑧 1, it is not the proper calculation. In fact, for cosmological sources, 𝑡 𝑔𝑟 𝑎 and Δ𝑡 𝑔𝑟 𝑎 do not have a monotonic increment, as shown in Minazzoli et al (2019a). Thus, to calculate Δ𝑡 𝑔𝑟 𝑎 , we have to consider all the gravitational sources in the path of the radiation.…”

Neutron star mergers and how to study them

“…(36) uses an implicit coordinate system that is not gaugeinvariant, i.e., depending on the coordinate choice one can obtain positive or negative values (Gao and Wald 2000). In Minazzoli et al (2019) they consider Fermi coordinates associated to an observer, such that the delay is expressed in terms of an observed proper time, which results in a sum of terms with opposing sign. Using reasonable constructions for contributions to the total gravitational well experienced by propagating particles by considering the sum of catalogs they show the induced total (absolute) Shapiro delay is not monotonic, and indeed crosses zero in some cases.…”

“…Therefore, while we can use the small offsets for, e.g., GW-GRBs to state that we find no evidence for WEP violation, we cannot quantify robust limits on relative Shapiro delay. Minazzoli et al (2019) suggest this as motivation for the development of tests for specific alternative theories of gravity. We note that if the WEP is violated we will find evidence for it, we will just be unable to quantify the degree of violation with this test.…”