Evolution of H$\alpha$ Equivalent Widths from $z \sim 0.4-2.2$: implications for star formation and legacy surveys with Roman and Euclid

We investigate the `intrinsic' H$\alpha$ EW distributions of $z \sim 0.4 - 2.2$ narrowband-selected H$\alpha$ samples from HiZELS and DAWN using a forward modeling approach. We find an EW - stellar mass anti-correlation with steepening slopes $-0.18\pm0.03$ to $-0.24^{+0.06}_{-0.08}$ at $z \sim 0.4$ and $z\sim 2.2$, respectively. Typical EW increases as $(1+z)^{1.78^{+0.22}_{-0.23}}$ for a $10^{10}$ M$_\odot$ emitter from $15^{+2.4}_{-2.3}$\r{A} ($z \sim 0.4$) to $67.7^{+10.4}_{-10.0}$\r{A} ($z \sim 2.2$) and is steeper with decreasing stellar mass highlighting the high EW nature of low-mass high-$z$ systems. We model this redshift evolving anti-correlation, $W_0(M,z)$, and find it produces H$\alpha$ luminosity and SFR functions strongly consistent with observations validating the model and allowing us to use $W_0(M,z)$ to investigate the relative contribution of H$\alpha$ emitters towards cosmic SF. We find EW$_0 > 200$ \r{A} emitters contribute significantly to cosmic SF activity at $z \sim 1.5 - 2$ making up $\sim 40$% of total SF consistent with sSFR $> 10^{-8.5}$ yr$^{-1}$ ($\sim 45 - 55$%). Overall, this highlights the importance of high EW systems at high-$z$. Our $W_0(M,z)$ model also reproduces the cosmic sSFR evolution found in simulations and observations and show that tension between the two can simply arise from selection effects in observations. Lastly, we forecast Roman and Euclid grism surveys using $W_0(M,z)$ including observational efficiency and limiting resolution effects where we predict $\sim 24000$ and $\sim 30000$ $0.5 < z < 1.9$ H$\alpha$ emitters per deg$^{-2}$, respectively, down to $>5\times10^{-17}$ erg s$^{-1}$ cm$^{-2}$ including $10^{7.2 - 8}$ M$_\odot$ galaxies at $z > 1$ with EW$_0 >1000$\r{A}. Both Roman and Euclid will enable us to observe with unprecedented detail some of the most bursty/high EW, low-mass star-forming galaxies near cosmic noon.