  We revisit the topic of invisible neutrino decay in the precision
cosmological context, via a first-principles approach to understanding the
cosmic microwave background and large-scale structure phenomenology of such a
non-standard physics scenario. Assuming an effective Lagrangian in which a
heavier standard-model neutrino $\nu_H$ couples to a lighter one $\nu_l$ and a
massless scalar particle $\phi$ via a Yukawa interaction, we derive from first
principles the complete set of Boltzmann equations, at both the spatially
homogeneous and the first-order inhomogeneous levels, for the phase space
densities of $\nu_H$, $\nu_l$, and $\phi$ in the presence of the relevant decay
and inverse decay processes. With this set of equations in hand, we perform a
critical survey of recent works on cosmological invisible neutrino decay in
both limits of decay while $\nu_H$ is ultra-relativistic and non-relativistic.
Our two main findings are: (i) in the non-relativistic limit, the effective
equations of motion used to describe perturbations in the neutrino--scalar
system in the existing literature formally violate momentum conservation and
gauge invariance, and (ii) in the ultra-relativistic limit, exponential damping
of the anisotropic stress does not occur at the commonly-used rate $\Gamma_{\rm
T} =(1/\tau_0) (m_{\nu H}/E_{\nu H})^3$, but at a rate $\sim (1/\tau_0) (m_{\nu
H}/E_{\nu H})^5$. Both results are model-independent. The impact of the former
finding on the cosmology of invisible neutrino decay is likely small. The
latter, however, implies a significant revision of the cosmological limit on
the neutrino lifetime $\tau_0$ from $\tau_0^{\rm old} \gtrsim 1.2 \times 10^9\,
{\rm s}\, (m_{\nu H}/50\, {\rm meV})^3$ to $\tau_0 \gtrsim (4 \times 10^5 \to 4
\times 10^6)\, {\rm s}\, (m_{\nu H}/50 \, {\rm meV})^5$.
