  For any $n\geq4$ even, we establish a complete scattering theory for the
linear wave equation on the $(n+1)$-dimensional de Sitter space. We prove the
existence and uniqueness of scattering states, and asymptotic completeness.
Moreover, we construct the scattering map taking asymptotic data at past
infinity $\mathscr{I}^-$ to asymptotic data at future infinity $\mathscr{I}^+$.
Identifying $\mathscr{I}^-$ and $\mathscr{I}^+$ with $S^n,$ we prove that the
scattering map is a Banach space isomorphism on $H^{s+n}(S^n)\times
H^{s}(S^n),$ for any $s\geq1.$
  The main analysis is carried out at the level of the model equation obtained
by differentiating the linear wave equation $\frac{n}{2}$ times in the time
variable. The main result of the paper follows from proving a scattering theory
for this equation. In particular, for the model equation we construct a
scattering isomorphism from asymptotic data in $H^{s+\frac{1}{2}}(S^n)\times
H^s(S^n)\times H^s(S^n)$ to Cauchy initial data in
$H^{s+\frac{1}{2}}(S^n)\times H^{s+\frac{1}{2}}(S^n)\times
H^{s-\frac{1}{2}}(S^n)$.
