  In this work we shall study the implications of a subclass of $E$-models
cosmological attractors, namely of $a$-attractors, on hydrodynamically stable
slowly rotating neutron stars. Specifically, we shall present the Jordan frame
theory of the $a$-attractors, and by using a conformal transformation we shall
derive the Einstein frame theory. We discuss the inflationary context of
$a$-attractors in order to specify the allowed range of values for the free
parameters of the model based on the latest cosmic-microwave-background-based
Planck 2018 data. Accordingly, using the notation and physical units frequently
used in theoretical astrophysics contexts, we shall derive the
Tolman-Oppenheimer-Volkoff equations in the Einstein frame. Assuming a
piecewise polytropic equation of state, the lowest density part of which shall
be chosen to be the WFF1, or APR or the SLy EoS, we numerically solve the
Tolman-Oppenheimer-Volkoff equations using a double shooting python-based
"LSODA" numerical code. The resulting picture depends on the value of the
parameter $a$ characterizing the $a$-attractors. As we show, for large values
of $a$, which do not produce a viable inflationary era, the $M-R$ graphs are
nearly identical to the general relativistic result, and these two are
discriminated at large central densities values. Also, for large $a$-values,
the WFF1 equation of state is excluded, due to the GW170817 constraints. In
addition, the small $a$ cases produce larger masses and radii compared to the
general relativistic case and are compatible with the GW170817 constraints on
the radii of neutron stars. Our results indicate deep and not yet completely
understood connections between non-minimal inflationary attractors and neutron
stars phenomenology in scalar-tensor theory.
