I would like to get a tight-ish estimate for the expectation of $e^{ax_1^2}$ where $x$ is drawn randomly from the unit sphere in $n$ dimensions and $x_1$ is the first coordinate.
More specifically I want to compare the expectation with $e^a$ and want to know whether the ratio
$\frac{e^a}{E[e^{ax_1^2}]}$
is bounded by an exponential in $n$ or is polynomial in $n$? (It might of course depend on $a$ and I am willing to have be logarithmic in $a$)