  We present new Big Bang Nucleosynthesis (BBN) limits on the cosmic expansion
rate or relativistic energy density, quantified via the number $N_\nu$ of
equivalent neutrino species. We use the latest light element observations,
neutron mean lifetime, and update our evaluation for the nuclear rates $d+d
\rightarrow He3 + n$ and $d+d \rightarrow H3 + p$. Combining this result with
the independent constraints from the cosmic microwave background (CMB) yields
tight limits on new physics that perturbs $N_\nu$ and $\eta$ prior to cosmic
nucleosynthesis: a joint BBN+CMB analysis gives $N_\nu = 2.898 \pm 0.141$,
resulting in $N_\nu < 3.180$ at $2\sigma$. We apply these limits to a wide
variety of new physics scenarios including right-handed neutrinos, dark
radiation, and a stochastic gravitational wave background. We also search for
limits on potential {\em changes} in $N_\nu$ and/or the baryon-to-photon ratio
$\eta$ between the two epochs. The present data place strong constraints on the
allowed changes in $N_\nu$ between BBN and CMB decoupling; for example, we find
$-0.708 < N_\nu^{\rm CMB}-N_\nu^{\rm BBN} < 0.328$ in the case where $\eta$ and
the primordial helium mass fraction $Y_p$ are unchanged between the two epochs;
we also give limits on the allowed variations in $\eta$ or in $(\eta,N_\nu)$
jointly. Looking to the future, we forecast the tightened precision for $N_\nu$
arising from both CMB Stage 4 measurements as well as improvements in
astronomical \he4 measurements. We find that CMB-S4 combined with present BBN
and light element observation precision can give $\sigma(N_\nu) \simeq 0.03$.
Such future precision would reveal the expected effect of neutrino heating
($N_{\rm eff}-3=0.044$) of the CMB during BBN, and would be near the level to
reveal any particle species ever in thermal equilibrium with the standard
model.
