  Amorphous packings prepared in the vicinity of the jamming transition play a
central role in theoretical studies of the vibrational spectrum of glasses. Two
mean-field theories predict that the vibrational density of states $g(\omega)$
obeys a characteristic power law, $g(\omega)\sim\omega^2$, called the non-Debye
scaling in the low-frequency region. Numerical studies have however reported
that this scaling breaks down at low frequencies, due to finite dimensional
effects. In this study, we prepare amorphous packings of up to $128000$
particles in spatial dimensions from $d=3$ to $d=9$ to characterise the range
of validity of the non-Debye scaling. Our numerical results suggest that the
non-Debye scaling is obeyed down to a frequency that gradually decreases as $d$
increases, and possibly vanishes for large $d$, in agreement with mean-field
predictions. We also show that the prestress is an efficient control parameter
to quantitatively compare packings across different spatial dimensions.
