  This work studies the cosmology of $\chi^{3/2}$-MOND gravity by Bernal et.
al. (2011). This theory is a modification to Einstein's General Relativity (GR)
that uses a dimensionless curvature scalar $\chi$ by rescaling the Ricci scalar
$R$ by some characteristic length scale $L_M$, as well as a set of modified
field equations that follows from a $3/2$-power Lagrangian. The characteristic
length scale is assumed to be built from the universal constants of the theory
and the parameters of the system in question. In the weak field limit, this
theory recovers Milgrom's (1983) Modified Newtonian Dynamics (MOND). MOND is a
proposal that corrects Newtonian gravitational laws below an acceleration
threshold $a_0\approx1.2\times{10}^{-10}m/s^2$ to explain the anomalous
flattening of galactic rotation curves without imposing any dark matter
components. In the cosmological case, this work asserts that the characteristic
length scale is of the order $c^2/a_0$. This specific value is motivated in two
ways: (1) it is shown that this scale defines a convergence of GR and MOND at
some critical mass (with this as the corresponding length); (2) this length
scale is shown to be an extremal value of $L_M$ independent of the mass
parameter. The established length scale is then used in the case of cosmology;
the FLRW metric is plugged in into the modified field equations and the two
modified Friedmann equations are derived incorporating the MOND effects by a
manifest appearance of the constant $a_0$.
