  Combined experimental and modeling studies of the magnetocaloric effect,
ultrasound, and magnetostriction were performed on single-crystal samples of
the spin-dimer system Sr$_3$Cr$_2$O$_8$ in large magnetic fields, to probe the
spin-correlated regime in the proximity of the field-induced XY-type
antiferromagnetic order also referred to as a Bose-Einstein condensate of
magnons. The magnetocaloric effect, measured under adiabatic conditions,
reveals details of the field-temperature ($H,T$) phase diagram, a dome
characterized by critical magnetic fields $H_{c1}$ = 30.4 T, $H_{c2}$ = 62 T,
and a single maximum ordering temperature $T_{{\rm max}}(45~$T$)\simeq$8 K. The
sample temperature was observed to drop significantly as the magnetic field is
increased, even for initial temperatures above $T_{{\rm max}}$, indicating a
significant magnetic entropy associated to the field-induced closure of the
spin gap. The ultrasound and magnetostriction experiments probe the coupling
between the lattice degrees of freedom and the magnetism in Sr$_3$Cr$_2$O$_8$.
Our experimental results are qualitatively reproduced by a minimalistic
phenomenological model of the exchange-striction by which sound waves
renormalize the effective exchange couplings.
