Clarifying transfer function approximations for the large-scale
  gravitational wave background in $\Lambda$CDM
  The primordial gravitational wave background (GWB) offers an exciting future
avenue of discovery for new physics. Its information content encodes multiple
eras in the early Universe's history, corresponding to many orders of magnitude
in frequency and physical scale to be measured today. By numerically solving
for the GW transfer functions we provide simple yet accurate formulas
describing the average power of the large-scale energy spectrum of the GWB for
arbitrary primordial tensor power spectra. In doing so we can pedagogically
explain and clarify previous GWB literature, highlight the important
cosmological parameters of various GWB features, and reveal multiple ways in
which cancelling conceptual errors can give deceptively accurate results. The
scales considered here are particularly important for CMB probes of the GWB,
via $B$-modes and spectral distortions. In particular, we carefully study the
effects of both neutrino damping, and the precise nature of the transition
between the radiation-dominated (RD) and matter-dominated (MD) eras. A
byproduct of numerically solving the problem is the ability to study the
robustness of common approximations in the literature. Specifically, we show
that a numerical treatment is especially important around the RD--MD
transition, and for a brief moment of history where neutrino damping occurs
during MD. In passing we also discuss the effects of late acceleration caused
by dark energy -- showing that this can be neglected in most practical GWB
applications -- and the effects of changing relativistic degrees of freedom on
the GWB at very small-scales.
