  Modifications to the Mukhanov-Sasaki equation in loop quantum cosmology (LQC)
have been phenomenologically explored using polymerization of the connection
and related variables in the classical expressions in order to capture the
quantum gravity effects in cosmological perturbations which replace the
classical big bang by a big bounce. Examples of this strategy include the
dressed metric and the hybrid approaches whose inter-relationship at an
effective level was demonstrated by the authors recently. In this manuscript,
we propose a new family of the effective mass functions in the modified
Mukhanov-Sasaki equation of LQC by investigating the polymerization of a
particular form of the classical mass function in terms of variable
$z_s$($=a\dot \phi/H$) which relates the Mukhanov-Sasaki variable with the
comoving curvature perturbation. Using a generalized ansatz motivated by
quantum gravity effects in the background dynamics we find alternative
effective mass functions which are distinct from those used in the dressed
metric and the hybrid approaches with differences originating from the
non-commutativity of the evaluation of the Poisson brackets and the
polymerization procedures. The new effective mass functions acquire four
correction terms in the effective potential whose exact forms are closely tied
up with the ansatz used for polymerizing the inverse Hubble rate. In contrast
to earlier works, one of these correction terms can in principle produce
sizable effects even when the bounce is kinetic dominated. Our investigation
opens a new window to explore the phenomenological implications of a large
family of effective mass functions in LQC which can potentially lead to
significant departures from the dressed metric and the hybrid approaches in the
bounce regime.
