Late-time MDO cosmology has exactly the ΛCDM Hubble function (Eqs. (22)–(26))

Late-time MDO cosmology reproduces the $\Lambda$CDM expansion law.

Consider the model with potential $V=V_1+V_2$, $V_1(\phi)=A^4(1-\phi^2/\phi_c^2)^2$, $V_2(\phi)=\tfrac12 m^2\phi^2+\tfrac\alpha4\phi^4$, where $\kappa, A,\phi_c>0$. Suppose that on a late-time interval $t>T$:

1. the scale factor $a$ is positive and differentiable, and the baryon density $\rho_b$ is differentiable;
2. the MDO field sits at the bottom of the symmetry-breaking potential, $\phi(t)=\phi_c$;
3. the only other matter is pressureless baryonic matter: $\dot\rho_b+3H\rho_b=0$ (Eq. (9) with $\rho_m=\rho_b$, $p_m=0$);
4. the first Friedmann equation (5) holds, $H^2=\frac{\kappa^2}{3}\left(1+\frac{\kappa^2\phi^2}{8}\right)\left[\frac{\dot\phi^2}{2}+V(\phi)+\rho_b\right]$.

Fix a reference time $t_0>T$ with $a(t_0)=1$ and $H_0=H(t_0)\neq0$. Set $\rho_{\mathrm{crit},0}=3H_0^2/\kappa^2$, $\Omega_b=\rho_b(t_0)/\rho_{\mathrm{crit},0}$, $\Omega_{DM,\phi}=\frac{\kappa^2\phi_c^2}{8}\Omega_b$, $\Lambda_{\mathrm{eff}}=\kappa^2\left(1+\frac{\kappa^2\phi_c^2}{8}\right)V_2(\phi_c)$ and $\Omega_{\Lambda,\phi}=\Lambda_{\mathrm{eff}}/(\kappa^2\rho_{\mathrm{crit},0})$. Then for every $t>T$

This is Eq. (25) of the paper: once the field is frozen at $\phi_c$, its gravitational (spinorial) coupling to baryons acts as a dark matter component scaling like $a^{-3}$, and the residual potential energy $V_2(\phi_c)$ acts as a cosmological constant.

Formalization Note The frozen-field regime is encoded exactly ($\phi\equiv\phi_c$, hence $\dot\phi=0$ and $V_1(\phi_c)=0$); the field equation (7) is deliberately not assumed, since the paper uses only Eqs. (12)/(15) in this regime. The hypotheses are stated on the open half-line $(T,\infty)$ rather than on all of $\mathbb R$, because a matter-filled expanding universe cannot have $a>0$ for all times. The positivity of $A$ is the paper's standing assumption.