Scalar-Induced Gravitational Waves in a $\Lambda$CDM Cosmology

We reconsider the gravitational wave spectrum induced by scalar perturbations in spatially flat Friedmann-Lema\^itre-Robertson-Walker spacetimes, focusing on the matter- and $\Lambda$-dominated epochs. During matter domination, sub-horizon modes are not free and a commonly applied approximation for the derivative of the tensor perturbation is flawed. We show analytically that this leads to a significant overestimation of the energy density spectrum. In addition, we demonstrate that gauge-dependent non-oscillating tensor perturbations appear in the presence of a cosmological constant. Complementing the analytical calculations, we compute the according present-day spectrum numerically for a Planck-like $\Lambda$CDM cosmology, finding that non-oscillating growing modes appear during the transition between matter and $\Lambda$ domination in conformal Newtonian gauge.