We present semi-analytic models of the epoch of reionization focusing on the differences between continuous and bursty star formation (SF). Our model utilizes physically motivated analytic fits to 1D radiative transfer simulations of HII regions around dark matter halos in a representative cosmic volume. Constraining our simulations with observed and extrapolated UV luminosity functions of high redshift galaxies, we find that for a fixed halo mass, stellar populations forming in bursty models produce larger HII regions which leave behind long-lived relic HII regions which are able to maintain partial ionization in the intergalactic medium (IGM) in a manner similar to an early X-ray background. The overall effect is a significant increase in the optical depth of the IGM, $τ_e$, and a milder increase of the redshift of reionization. To produce $τ_e =0.066$ observed by Planck and complete reionization by redshift $z_{\rm re}\sim 6$, models with bursty SF require an escape fraction $f_{\rm esc}\sim 2\%-10\%$ that is 2-10 times lower than $f_{\rm esc}\sim 17\%$ found assuming continuous SF and is consistent with upper limits on $f_{\rm esc}$ from observations at $z=0$ and $z\sim 1.3-6$. The ionizing photon budget needed to reproduce the observed $τ_e$ and $z_{\rm re}$ depends on the period and duty cycle of the bursts of SF and the temperature of the neutral IGM. These results suggest that the tension between observed and predicted ionizing photon budget for reionization can be alleviated if reionization is driven by short bursts of SF, perhaps relating to the formation of Population~III stars and compact star clusters such as proto-globular clusters.

17 pages, 11 figures, 1 table. Submitted to MNRAS