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We study gravitational waves induced from the primordial scalar perturbations
at second order around the reheating of the Universe. We consider reheating
scenarios in which a transition from an early matter dominated era to the
radiation dominated era completes within a timescale much shorter than the
Hubble time at that time. We find that an enhanced production of induced
gravitational waves occurs just after the reheating transition because of fast
oscillations of scalar modes well inside the Hubble horizon. This enhancement
mechanism just after an early matter-dominated era is much more efficient than
a previously known enhancement mechanism during an early matter era, and we
show that the induced gravitational waves could be detectable by future
observations if the reheating temperature $T_{\text{R}}$ is in the range
$T_\text{R} \lesssim 7\times 10^{-2}$GeV or $20 \, \text{GeV} \lesssim T_\text{R} \lesssim 2 \times 10^7 \, \text{GeV}$. This is the case even if the
scalar perturbations on small scales are not enhanced relative to those on
large scales, probed by the observations of the cosmic microwave background.

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