  Given a simple graph $G$ and an integer $k$, the goal of $k$-Clique problem
is to decide if $G$ contains a complete subgraph of size $k$. We say an
algorithm approximates $k$-Clique within a factor $g(k)$ if it can find a
clique of size at least $k / g(k)$ when $G$ is guaranteed to have a $k$-clique.
Recently, it was shown that approximating $k$-Clique within a constant factor
is W[1]-hard [Lin21].
  We study the approximation of $k$-Clique under the Exponential Time
Hypothesis (ETH). The reduction of [Lin21] already implies an
$n^{\Omega(\sqrt[6]{\log k})}$-time lower bound under ETH. We improve this
lower bound to $n^{\Omega(\log k)}$. Using the gap-amplification technique by
expander graphs, we also prove that there is no $k^{o(1)}$ factor
FPT-approximation algorithm for $k$-Clique under ETH.
  We also suggest a new way to prove the Parameterized Inapproximability
Hypothesis (PIH) under ETH. We show that if there is no $n^{O(\frac{k}{\log
k})}$ algorithm to approximate $k$-Clique within a constant factor, then PIH is
true.
