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We present results from a quantitative spectroscopic analysis conducted onarchival Keck/HIRES high-resolution spectra from the California-$Kepler$ Survey(CKS) sample of transiting planetary host stars identified from the $Kepler$mission. The spectroscopic analysis was based on a carefully selected set of FeI and Fe II lines, resulting in precise values for the stellar parameters ofeffective temperature (T$_{\rm eff}$) and surface gravity (log $g$). Combiningthe stellar parameters with $Gaia$ DR2 parallaxes and precise distances, wederived both stellar and planetary radii for our sample, with a median internaluncertainty of 2.8$\%$ in the stellar radii and 3.7$\%$ in the planetary radii.An investigation into the distribution of planetary radii confirmed the bimodalnature of this distribution for the small radius planets found in previousstudies, with peaks at: $\sim$1.47 $\pm$ 0.05 R$_{\oplus}$ and $\sim$2.72 $\pm$0.10 R$_{\oplus}$, with a gap at $\sim$ 1.9R$_{\oplus}$. Previous studies thatmodeled planetary formation that is dominated by photo-evaporation predictedthis bimodal radii distribution and the presence of a radius gap, orphoto-evaporation valley. Our results are in overall agreement with thesemodels. The high internal precision achieved here in the derived planetaryradii clearly reveal the presence of a slope in the photo-evaporation valleyfor the CKS sample, indicating that the position of the radius gap decreaseswith orbital period; this decrease was fit by a power law of the form R$_{pl}$$\propto$ P$^{-0.11}$, which is consistent with photo-evaporation andEarth-like core composition models of planet formation.

A Spectroscopic Analysis of the California-Kepler Survey Sample: I. Stellar Parameters, Planetary Radii and a Slope in the Radius Gap