The Gaia-Kepler Stellar Properties Catalog. II. Planet Radius
  Demographics as a Function of Stellar Mass and Age
  Studies of exoplanet demographics require large samples and precise
constraints on exoplanet host stars. Using the homogeneous Kepler stellar
properties derived using Gaia Data Release 2 by Berger et al. (2020), we
re-compute Kepler planet radii and incident fluxes and investigate their
distributions with stellar mass and age. We measure the stellar mass dependence
of the planet radius valley to be $d \log R_{\mathrm{p}}$/$d \log M_\star =
0.26^{+0.21}_{-0.16}$, consistent with the slope predicted by a planet mass
dependence on stellar mass ($0.24-0.35$) and core-powered mass-loss (0.33). We
also find first evidence of a stellar age dependence of the planet populations
straddling the radius valley. Specifically, we determine that the fraction of
super-Earths ($1-1.8 \mathrm{R_\oplus}$) to sub-Neptunes ($1.8-3.5
\mathrm{R_\oplus}$) increases from $0.61 \pm 0.09$ at young ages (< 1 Gyr) to
$1.00 \pm 0.10$ at old ages (> 1 Gyr), consistent with the prediction by
core-powered mass-loss that the mechanism shaping the radius valley operates
over Gyr timescales. Additionally, we find a tentative decrease in the radii of
relatively cool ($F_{\mathrm{p}} < 150 \mathrm{F_\oplus}$) sub-Neptunes over
Gyr timescales, which suggests that these planets may possess H/He envelopes
instead of higher mean molecular weight atmospheres. We confirm the existence
of planets within the hot sub-Neptunian "desert" ($2.2 < R_{\mathrm{p}} < 3.8
\mathrm{R_\oplus}$, $F_{\mathrm{p}} > 650 \mathrm{F_\oplus}$) and show that
these planets are preferentially orbiting more evolved stars compared to other
planets at similar incident fluxes. In addition, we identify candidates for
cool ($F_{\mathrm{p}} < 20 \mathrm{F_\oplus}$) inflated Jupiters, present a
revised list of habitable zone candidates, and find that the ages of single-
and multiple-transiting planet systems are statistically indistinguishable.
