Five parameters are all you need (in $\Lambda$CDM)

The standard cosmological model, with its six independent parameters, successfully describes our observable Universe. One of these parameters, the optical depth to reionization $\tau_\mathrm{reio}$, represents the scatterings that Cosmic Microwave Background (CMB) photons will experience after decoupling from the primordial plasma as the intergalactic medium transitions from neutral to ionized. $\tau_\mathrm{reio}$ depends on the neutral hydrogen fraction $x_\mathrm{HI}$, which, in turn, should theoretically depend on cosmology. We present a novel method to establish the missing link between cosmology and reionization timeline using symbolic regression. We discover the timeline has a universal shape well described by the Gompertz mortality law, applicable to any cosmology within our simulated data. Unlike the conventional tanh prescription, our model is asymmetric in time and a good fit to astrophysical constraints on $x_\mathrm{HI}$. By combining CMB with astrophysical data and marginalizing over astrophysics, we treat $\tau_\mathrm{reio}$ as a derived parameter, tightening its constraint to $<3\%$. This approach reduces the error on the amplitude of the primordial fluctuations by a factor of 2.3 compared to Planck's PR3 constraint and provides a commanding constraint on the ionization efficiency $\zeta_\mathrm{UV} = 26.9^{+2.1}_{-2.5}$. We expect further improvements in the near term as reionization constraints increase and our understanding of reionization advances.