Cosmological equivalence principle and the weak-field limit

The strong equivalence principle is extended in application to averaged dynamical fields in cosmology to include the role of the average density in the determination of inertial frames. The resulting cosmological equivalence principle is applied to the problem of synchronization of clocks in the observed universe. Once density perturbations grow to give density contrasts of order 1 on scales of tens of megaparsecs, the integrated deceleration of the local background regions of voids relative to galaxies must be accounted for in the relative synchronization of clocks of ideal observers who measure an isotropic cosmic microwave background. The relative deceleration of the background can be expected to represent a scale in which weak-field Newtonian dynamics should be modified to account for dynamical gradients in the Ricci scalar curvature of space. This acceleration scale is estimated using the best-fit nonlinear bubble model of the universe with backreaction. At redshifts $z\lesssim 0.25$ the scale is found to coincide with the empirical acceleration scale of modified Newtonian dynamics. At larger redshifts the scale varies in a manner which is likely to be important for understanding dynamics of galaxy clusters, and structure formation. Although the relative deceleration, typically of order ${10}^{-10}{\mathrm{ms}}^{-2}$, is small, when integrated over the lifetime of the universe it amounts to an accumulated relative difference of 38% in the rate of average clocks in galaxies as compared to volume-average clocks in the emptiness of voids. A number of foundational aspects of the cosmological equivalence principle are also discussed, including its relation to Mach’s principle, the Weyl curvature hypothesis, and the initial conditions of the universe.

Figure 1

A set of particles undergoes an isotropic spatial 3-volume expansion in a spatially flat local region. For the same initial conditions, provided that we consider time intervals over which deceleration of the expansion is negligible, it is impossible to locally distinguish the case of particles at rest in a dynamically expanding cosmological space from particles moving isotropically in a static Minkowski space. One spatial dimension is suppressed.

Figure 2

The semi-tethered lattice. Point particle observers in a homogeneous spatial lattice, initially expanding with uniform velocity, are attached to nearest neighbors by strings. Each string is fixed at one end. The arrowheads denote the free end of each string wound on a spool at a neighboring particle, to which the observers apply brakes in a synchronized fashion according to a predetermined plan. The time evolution of the lattice follows a course similar to that of the spatial grid in Fig. 1, but with deceleration.

Figure 3

Two equivalent situations: (a) in Minkowski space observers in separate semi-tethered lattices, initially expanding at the same rate, apply brakes homogeneously and isotropically within their respective regions but at different rates; (b) in the universe which is close to homogeneous and isotropic at last scattering, observers in separated regions initially recede from each other at the same rate, but experience different locally homogeneous isotropic decelerations as local density contrasts grow. In both cases there is a relative deceleration of the observer congruences, and those in the region which has decelerated more will age less.

Figure 4

The magnitude of the relative deceleration scale, $\alpha$, as a function of redshift: (a) for redshifts $z<0.25$; (b) for redshifts $z<2$. The central curve shows the value for the best-fit parameters $f_{\mathrm{v}0}=0.76$, $H_0=61.7\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$ (${\overline{H}}_0=48.2\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$) from Ref. 23. The dashed curves show the corresponding results with $1\sigma$ uncertainties, which are largely due to the large uncertainty on $f_{\mathrm{v}0}$, which is not tightly constrained by supernovae data. The upper dashed curve corresponds to $f_{\mathrm{v}0}=0.67$ and the lower dashed curve to $f_{\mathrm{v}0}=0.88$. In panel (a) the horizontal dotted lines indicate the bounds of the empirical acceleration scale of MOND when normalized for $H_0={61.7}_{-1.1}^{+1.2}\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$.

Figure 5

The magnitude of the dimensionless ratios $\alpha /(c\overline{H})$ (solid curve) and $\alpha /(cH)$ (dashed curve), where $\alpha (z)$, $\overline{H}(z)$, and $H(z)$ are varied with redshift, for best-fit values of $H_0$, $f_{\mathrm{v}0}$.