Quantum cosmology
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Problem 1
problem id: 150_dp24
Find the solutions corrected by LQC Friedmann equations for a matter dominated Universe.
solution
Solving the corrected first Friedmann equation and the conservation equation for a matter dominated Universe \[H^2=\rho3\left(1-\frac\rho{\rho_c}\right),\quad \dot\rho+3H\rho=0,\] one obtains the following quantities \[a(t)=\left(\frac34\rho_ct^2+1\right)^{1/3},\quad\rho(t)=\frac{\rho_c}{\frac34\rho_ct^2+1},\quad H(t)=\frac{\frac12\rho_ct}{\frac34\rho_ct^2+1}\] For small values of the energy density $\rho\ll\rho_c$ the solutions of standard Friedmann equations are recovered.