Dynamical curvature in a nonstandard cosmological model

We consider a nonrelativistic cosmological model introduced in [1] and derived as the nonrelativistic limit (or approximation at sub-Hubble scales) of a general relativistic model in [3, 4]. The latter is defined by an energy-momentum tensor containing only dust and a nontrivial energy flow. The nonrelativistic limit contains in leading order a 1st-order relativistic contribution to the spatial curvature whose time-dependence drives the accelerated expansion of the Universe (we do not need any kind of dark energy). Analytic solutions of the model are fixed by three constants (initial conditions). In the present paper we use our model as a toy model by adjusting the three constants in two different ways to a second order polynomial fit by Montenari and R\"as\"anen [5] to the observed expansion rate $H(z)$ for $z \lesssim 2$ (mainly cosmic chronometer data). In scenario 1 we adjust our model to this fit and its derivative at the self-consistently determined transition redshift $z_t$. In scenario 2 we use the same fit at $z_t$ and in addition $H(z)$ at decoupling $(z = 1089)$. The Hubble parameter $H_0$ is taken from the polynomial fit in [5]: $H_0 = 64.2 km/s/Mpc$. For both scenarios we obtain a satisfactory agreement between predicted and observed $H(z)$ values. But the outcomes for the curvature function $k(z)$ are completely different: In scenario 1 we obtain a strong variation of $k(z)$ ranging from $k(0) = - 1.216$ up to $k(2.33) = 0.718$. On the other hand scenario 2 shows an almost constant value for $k(z) \sim - 1$ for all $z \lesssim 2$ in agreement with the polynomial fit to one of the FRW consistency conditions performed in [5].