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.
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