Bounds for moments of symmetric square $L$-functions

We study the $2k$-th moment at the central point of the family of symmetric square $L$-functions attached to holomorphic Hecke cusp forms of level one, weight $\kappa$. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally. Assuming the truth of the generalized Riemann hypothesis, we also obtain sharp lower bounds for all real $0 \leq k < 1/2$ and sharp upper bounds for all real $k \geq 0$.