y}}dx \,x^3 e^{-b(x-1)}=e^b \f{6}{b^4}(f(b) -f(b e^{\m y} ))
,$$ where $b=HB/\m$ and $$f(b) =e^{-b}(1+b +b^2/2! +b^3/3! )
.$$ For $b<1$ approximately $f(b)\simeq 1-b^4 /12$, and for $y<\b{y}$ the integral is estimated as $$I\simeq \f{e^b}{2}\left(e^{4\m y} -1\right) 
< 2\f{\r}{\m}\exp\left\{9.1 \left(\f{\r}{\m}\right)^{1/2}\right\}
.$$ For small enough $\r (t_l )/\m$ the expression in the exponent in (\[f10\]) ł[f3a]{} K()\^[5/2]{}{9.1 ()\^[1/2]{}} can be made arbitrary small.

To conclude, contribution to the energy-momentum tensor due to graviton emission modifies solutions of the Einstein equations, however, in the period of late cosmology corrections to the leading-order solutions are numerically small as compared to the leading-order terms.

Let us consider the Friedmann equation (\[14a\]) with the terms $\c{T}_{ij}$ included. Calculating the first integral in (\[14a\]), we obtain ł[f12a]{} &&I\_[04]{} =\_[t\_l]{}\^t dt’Ţ\_[04]{}(0,t’ )(+(t’ ) ) a\^[-4]{} (0,t’)=\
&&(t) A\_[04]{}(( -) +(-)) ,where we have substituted (\[i6\]) ł[f14a]{} \^2 Ţ\_[04]{}= A\_[04]{}\^2 =-()\^2\^2 .The integral (\[f12a\]) has a strong dependence on the value of the lower limit. Therefore, in the integrand the slowly varying functions can be taken at the times when all the Standard model degrees of freedom are relativistic. In this period $A=166.2, \quad g_* (T)=106.7$ and $ A_{04}\simeq- 0.126 $. In the leading approximation $\r (t) \simeq 1/8\m 