We investigate the generation of gravitational waves due to the gravitational
instability of primordial density perturbations in an early matter-dominated
era which could be detectable by experiments such as LIGO and LISA. We use
relativistic perturbation theory to give analytic estimates of the tensor
perturbations generated at second order by linear density perturbations. We
find that large enhancement factors with respect to the naive second-order
estimate are possible due to the growth of density perturbations on sub-Hubble
scales. However very large enhancement factors coincide with a breakdown of
linear theory for density perturbations on small scales. To produce a
primordial gravitational wave background that would be detectable with LIGO or
LISA from density perturbations in the linear regime requires primordial
comoving curvature perturbations on small scales of order 0.02 for Advanced
LIGO or 0.005 for LISA, otherwise numerical calculations of the non-linear
evolution on sub-Hubble scales are required.Comment: 23 pages, 2 figure
