The exact relationship between the scale factor “a” and cosmic time “t” for our universe, considered flat, is given by the following expression:
$$\displaystyle t=\dfrac 1{H_0} \int_0^{a} \dfrac{dx}{\sqrt{\Omega_{\Lambda_0} x^2+\Omega_{M_0} x^{-1}+\Omega_{R_0} x^{-2}}}$$
The best values currently available (given for Planck Mission 2018), for the parameters are:
$H_0=67.66 \ (km/s)/Mpc$
$\Omega_{R_0}=9.15 \cdot 10^{-5}$
$\Omega_{M_0}=0.31105425$
$\Omega_{\Lambda_0}=0.68885425$
An approximation of the relationship between the scale factor and time for our universe, (considered flat), when the radiation density ratio is negligible, is given by the following expression:
$$\displaystyle a \approx \left (\frac{\Omega_{M_0}}{\Omega_{\Lambda_0}}\right )^{1/3} \sinh^{2/3}\left ( \frac{3 H_0 \sqrt{\Omega_{\Lambda_0}}}{2} \ t \right )$$
This expression is the same as in equation 63 page 26 of the document "The picture of our universe: A view from modern cosmology", taking into account that:
$\displaystyle t_{\Lambda}=\left (\dfrac 4{3\Lambda c^2} \right )^{1/2}=\dfrac{2}{3 H_0 \sqrt{\Omega_{\Lambda_0}}}$
For the density ratios in the approximate expression, we will use:
$\Omega_{M_0} \approx 0.3111$
$\Omega_{\Lambda_0} \approx 0.6889$
Now, we make a table in which for each cosmic time from the beginning until now, we tabulate the scale factor calculated with the exact expression (by numerical methods), the scale factor calculated with the approximate expression, and the relative error we make if we keep the approximate expression.
The value of the error we accept will determine from what cosmic time the approximate expression is acceptable
* If we do not accept an error greater than 20% the approximation is valid from t=180 thousand years until now.
* If we do not accept an error greater than 10% the approximation is valid from t=1 million years until now.
* If we do not accept an error greater than 2% the approximation is valid from t=22 million years until now.
* And if we only accept an error of less than 1% the approximation is valid from t=71 million years until now, ...
Best regards. 374