The CALYMHA survey: Ly$\alpha$ escape fraction and its dependence on
  galaxy properties at $z=2.23$
  We present the first results from our CAlibrating LYMan-$\alpha$ with
H$\alpha$ (CALYMHA) pilot survey at the Isaac Newton Telescope. We measure
Ly$\alpha$ emission for 488 H$\alpha$ selected galaxies at $z=2.23$ from HiZELS
in the COSMOS and UDS fields with a specially designed narrow-band filter
($\lambda_c$ = 3918 {\AA}, $\Delta\lambda$= 52 {\AA}). We find 17 dual
H$\alpha$-Ly$\alpha$ emitters ($f_{\rm Ly\alpha} >5\times10^{-17}$ erg s$^{-1}$
cm$^{-2}$, of which 5 are X-ray AGN). For star-forming galaxies, we find a
range of Ly$\alpha$ escape fractions (f$_{\rm esc}$, measured with 3$"$
apertures) from $2$\%$-30$\%. These galaxies have masses from $3\times10^8$
M$_{\odot}$ to 10$^{11}$ M$_{\odot}$ and dust attenuations E$(B-V)=0-0.5$.
Using stacking, we measure a median escape fraction of $1.6\pm0.5$\%
($4.0\pm1.0$\% without correcting H$\alpha$ for dust), but show that this
depends on galaxy properties. The stacked f$_{\rm esc}$ tends to decrease with
increasing SFR and dust attenuation. However, at the highest masses and dust
attenuations, we detect individual galaxies with f$_{\rm esc}$ much higher than
the typical values from stacking, indicating significant scatter in the values
of f$_{\rm esc}$. Relations between f$_{\rm esc}$ and UV slope are bimodal,
with high f$_{\rm esc}$ for either the bluest or reddest galaxies. We speculate
that this bimodality and large scatter in the values of f$_{\rm esc}$ is due to
additional physical mechanisms such as outflows facilitating f$_{\rm esc}$ for
dusty/massive systems. Ly$\alpha$ is significantly more extended than H$\alpha$
and the UV. f$_{\rm esc}$ continues to increase up to at least 20 kpc
(3$\sigma$, 40 kpc [2$\sigma$]) for typical SFGs and thus the aperture is the
most important predictor of f$_{\rm esc}$.
