please help me with these cosmology question except those pl

please help me with these cosmology question except those plotting problem.thank you so much.

Our Universe: According to cosmological parameter values measured by Planck (combined with other methods), our Universe is flat (K = 0), at present is filled by 68.8% dark energy (Ohm_A0 = 0.688), 31.2% matter (Ohm_M0 = 0.312), and 0.00922% radiation (Ohm_R0 = 9.22 10^-5), and is currently expanding at a rate H_0 = 67.51 km/s/Mpc. At which redshift z density of matter and dark energy are the same? of matter and radiation? Plot evolution of density of all matter components ln rho_i as a function of ln a, on the same graph, to scale. Plot evolution of fraction Ohm_i rho_i/rho_c for all matter components as a function of ln a, on the same graph, to scale. How long would it take the Universe to expand from zero to present size at a constant rate H_0? Numerically integrate Friedmann equation to plot the evolution of scale factor a as a function of time H_0 t. (If you are not sure how to do this, sketch the log-log plot instead, showing relevant asymptotics analytically.) How old is the Universe in this model? Does your calculation agree with the value quoted by Planck folks? (If you are not sure how to do this, calculate the age of matter-dominated universe Ohm_M =1, Ohm_A = Ohm_R = 0 analytically instead.)

Solution

1. Roughly at redshift of about z=0.4, which corresponds to about 4 billion years ago, assuming the standard CDM cosmology.  

For the record, you get this from
DE(a)=const
and
DM(a)a^3
with everything else negligible. Plug in the ratios today (a=1), and you get that the ratio is 1 at
a1/(1+z)0.7

I hope this answers your doubt