According to the "PLANCK" data the Universe's composition is the following: $4,89 \%$ of usual (baryon) matter (the previous estimate according to WMAP data was $4,6 \%$), $26.9 \%$ of dark matter (instead of previous $22,7 \%$) and $68.25 \%$ (instead of $73\%$) of dark energy. The Hubble constant was also corrected; the new value is $ H_0 = 67.11 km\ s^{-1}\ Mpc^{-1}$ (the previous estimate was $70 km\ s^{-1}\ Mpc^{-1}$).
1
Problem 1
problem id:
Compare estimates of the age of Universe according to the "PLANCK" data and that of WMAP.
solution
\[a(t)=a_0\left(\frac{\Omega_{m0}}{\Omega_{\Lambda0}}\right)^{1/3} \left[sh\left(\frac32\sqrt{\Omega_{\Lambda0}}H_0t\right)\right]^{2/3};\]
\[t_0=\frac23H_0^{-1}\Omega_{\Lambda0}^{-1/2}arsh\sqrt{\frac{\Omega_{\Lambda0}}{\Omega_{m0}}} =t_{\Lambda}arsh\sqrt{\frac{\Omega_{\Lambda0}}{\Omega_{m0}}} =t_{\Lambda}arcth\sqrt{\Omega_{\Lambda0}}.\]
According to the WMAP data
\[t_{\Lambda}\equiv\frac23H_0^{-1}(\Omega_{\Lambda0})^{-1/2}\simeq10.768\times10^9\ years\]
and \[t_0=13.7\ Gyr.\]
According to the "PLANCK" data
\[t_{\Lambda}=11.74\ Gyr\]
and \[t_0=13.8\ Gyr.\]
2
Problem 2
problem id:
Estimate the age of Universe corresponding to termination of the radiation dominated epoch.
solution
\[\rho_m=\frac{a_0^3}{a^3}\rho_{m0}=\rho_r;\]
\[\frac a{a_0}=\frac{\rho_{r0}}{\rho_{m0}}=\frac{\Omega_{r0}}{\Omega_{m0}}=2.9088\times10^{-4};\]
\[1+z=\frac{a_0}a\to z^*=3436.9;\]
\[t(z^*)=\frac1{H_0}\int\limits_0^{a/a_0}\frac{dx}x \frac1{\sqrt{\Omega_{\Lambda0}+\Omega_{curv}x^{-2}+\Omega_{m0}x^{-3}+\Omega_{r0}x^{-4}}};\]
\[t(z^*=3436.9)=50.152\ years.\]
3
Problem 3
problem id:
Estimate the age of Universe corresponding to termination of the matter dominated epoch.
solution
\[\rho_m=\frac{a_0^3}{a^3}\rho_{m0}=\rho_\Lambda;\]
\[\frac a{a_0}=\left(\frac{\rho_{m0}}{\rho_{\Lambda}}\right)^{1/3}= \left(\frac{\Omega_{m0}}{\Omega_{\Lambda}}\right)^{1/3}=0.7719\times10^{-4};\]
\[1+z=\frac{a_0}a\to z^*=0.2956;\]
\[t(z^*)=\frac1{H_0}\int\limits_0^{a/a_0}\frac{dx}x \frac1{\sqrt{\Omega_{\Lambda0}+\Omega_{curv}x^{-2}+\Omega_{m0}x^{-3}+\Omega_{r0}x^{-4}}};\]
\[t(z^*=0.2956)=10.309\ Gyr\to 3.508\ Gyr \ ago.\]