  One of the key predictions of the standard inflationary paradigm is the
quantum mechanical generation of the transverse and traceless tensor
fluctuations due to the rapid accelerated expansion of space, which later
constitute a stochastic background of primordial gravitational waves (GWs). The
amplitude of the (nearly) scale-invariant inflationary tensor power spectrum at
large scales provides us with crucial information about the energy scale of
inflation in the case of the minimal inflaton coupling to gravity. Furthermore,
the spectral energy density, $\Omega_{_{\rm GW}}(f)$, of the GWs at
sufficiently small scales (or, large frequencies $f$) serves as an important
observational probe of post-inflationary primordial dynamics. In fact, the
small-scale spectral tilt, $n_{_{\rm GW}} = \frac{{\rm d}\log{\Omega_{_{\rm
GW}}}}{{\rm d}\log{f}}$, of the spectral energy density of GWs is sensitive to
the (unknown) post-inflationary equation of state (EoS), $w$, of the universe;
with a softer EoS ($w < 1/3$) leading to a red tilt: $n_{_{\rm GW}} < 0$, while
a stiffer EoS ($w > 1/3$) resulting in a blue tilt: $n_{_{\rm GW}} > 0$. The
post-inflationary dynamics, however, is generically expected to be quite
complex, potentially involving a number of distinct phases. Hence, in this
work, we discuss the possibility of multiple sharp transitions, namely $w_1 \to
w_2 \to w_3 \to ... \to w_n$, in the EoS of the post-inflationary universe and
compute the corresponding spectral energy density of the inflationary GWs. We
explicitly determine the region of the parameter space $\lbrace{ w_1, \, w_2,
\, w_3, ..., w_n\rbrace}$ which leads to a potentially detectable signal in the
upcoming GW detectors, without violating the current constraints.
