The Memory of Primordial Gravitational Waves

Primordial gravitational waves, after they enter the horizon and decay away, leave a residual displacement in test particles: a memory, in analogy with gravitational waves generated by astrophysical sources. The late-time distance between test particles is related to the one at early times by $\xi^i_{\rm late} = \frac{a_{\rm late}}{a_{\rm early}} (\delta^i_j -\frac12 \bar h^i_j)\xi^j_{\rm early}$. Therefore, the deformation of an initial spherical shell does not depend on the cosmological evolution, but only on the primordial value $\bar h^i_{j}$ of the gravitational wave. The memory is thus related to the adiabatic tensor mode that maps the unperturbed FLRW geometries at early and late times; this is analogous to the relation between memory in Minkowski spacetime and the BMS group. The primordial memory is also connected to the consistency relations of cosmological correlators, as the flat-space memory is related to the soft theorems for gravitational wave emission. We comment on the signature of the effect on the CMB $B$-modes and on the large-scale structure. There is also a primordial memory effect that is subleading in the spatial gradients of the wave: it is encoded in the rotation of free-falling gyroscopes.