So I have the equation (derived from the Friedman equations with $a(t_0)=1$ and the powers of $a(t)$ quoted)$$H^2 = H_0^2\left(\frac{\Omega_{R_0}}{a^4(t)}+\frac{\Omega_{M_0}}{a^3(t)}+\Omega_{\Lambda_0}+\frac{1-\Omega_0}{a^2(t)}\right)$$

Where the "$0$" subscript denotes the value of the quantities at the initial time $t_0$, the subscripts $R_0$, $M_0$, $\Lambda_0$ denote the density parameter of radiation, matter and dark energy respectively and $\Omega$ is the total density parameter.

It says that for $\Omega_{M_0}=1$, it follows that $\Omega_{R_0}=\Omega_{\Lambda_0}=0$.

How exactly does this follow? I think that I might be missing something here because I'm not entirely sure I understand how this is so straight forward.


They're imposing spatial flatness. $\Omega_0 = \Omega_{R_0} + \Omega_{M_0} + \Omega_{\Lambda_0}$. The curvature parameter $\Omega_k = 1 - \Omega_0$. Space is flat when $\Omega_k=0$, thus $\Omega_{R_0} + \Omega_{M_0} + \Omega_{\Lambda_0} = 1$. You can relax the flatness requirement, though, in the more general case, though all observations, so far, are consistent with $\Omega_k=0$.


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