A new parametrization of the reionization history is presented to facilitate robust comparisons between different observations and with theory .
The evolution of the ionization fraction with redshift can be effectively captured by specifying the midpoint , duration , and asymmetry parameters .
Lagrange interpolating functions are then used to construct analytical curves that exactly fit corresponding ionization points .
The shape parametrizations are excellent matches to theoretical results from radiation-hydrodynamic simulations .
The comparative differences for reionization observables are : ionization fraction | \Delta x _ { \text { i } } | \lesssim 0.03 , 21cm brightness temperature | \Delta T _ { \text { b } } | \lesssim 0.7 \text { mK } , Thomson optical depth | \Delta \tau| \lesssim 0.001 , and patchy kinetic Sunyaev-Zel ’ dovich angular power | \Delta D _ { \ell } | \lesssim 0.1 \mu \text { K } ^ { 2 } .
This accurate and flexible approach will allow parameter-space studies and self-consistent constraints on the reionization history from 21cm , CMB , and high-redshift galaxies and quasars .