  Energy Momentum Squared Gravity (EMSG) is a cosmological model where the
scale factor is non vanishing at all times and hence does not favor big bang
cosmology. However, the profile of density in the radiation dominated universe
shows that EMSG supports inflationary cosmology. Inflationary cosmological
models are successful in providing convincing answers to major cosmological
issues like horizon problem, flatness problem and small value of cosmological
constant but hitherto no model of inflation has been observationally confirmed.
Owing to this, Varying Speed of Light (VSL) were introduced which are a class
of cosmological models which disfavor inflation and propose an alternative
route to solve these cosmological issues by just allowing the speed of light
(and Newtonian Gravitational constant) to vary. VSL theories were motivated to
address the shortcomings of inflation but do not address the shortcomings
related to the initial big bang singularity. In this spirit, we present here a
novel cosmological model which is free from both the "initial big bang
singularity" and "inflation" by incorporating a mutually varying speed of light
$c(t)$ and Newtonian gravitational constant $G(t)$ in the framework of EMSG. We
report that in EMSG, for a dust universe ($\omega=0$), cosmological models for
a time varying $c(t)$ and $G(t)$ and constant $c$ and $G$ are
indistinguishable, whereas for a radiation dominated universe ($\omega = 1/3$),
a mutually varying $c(t)$ and $G(t)$ provides an exiting alternative to
inflationary cosmology which is also free from initial big bang singularity. We
further report that for an ansatz of scale factor representing a bouncing
cosmological model, the VSL theory can be applied to a quadratic $T$ gravity
model to get rid of "inflation" and "big bang singularity" and concurrently
solve the above mentioned cosmological enigmas.
