Five parameters are all you need (in ΛCDM)

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.