  Spin correlations of the frustrated pyrochlore oxide
Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been investigated by using inelastic neutron
scattering on single crystalline samples ($x=-0.007, 0.000,$ and $0.003$),
which have the putative quantum-spin-liquid (QSL) or electric-quadrupolar
ground states. Spin correlations, which are notably observed in nominally
elastic scattering, show short-ranged correlations around $L$ points [$q =
(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$], tiny antiferromagnetic Bragg
scattering at $L$ and $\Gamma$ points, and pinch-point type structures around
$\Gamma$ points. The short-ranged spin correlations were analyzed using a
random phase approximation (RPA) assuming the paramagnetic state and two-spin
interactions among Ising spins. These analyses have shown that the RPA
scattering intensity well reproduces the experimental data using temperature
and $x$ dependent coupling constants of up to 10-th neighbor site pairs. This
suggests that no symmetry breaking occurs in the QSL sample, and that a quantum
treatment beyond the semi-classical RPA approach is required. Implications of
the experimental data and the RPA analyses are discussed.
