Cosmological look-back time
= Cosmological look-back time
{title2=$t_L(z)=\int_0^z d\bar z/[(1+\bar z)H(\bar z)]$}
The look-back time is the cosmic-time interval between emission at <redshift> $z$ and reception today. Differentiating $1+z=R_0/R(t)$ gives $dt/dz=-1/[(1+z)H]$ and hence the displayed <integral>. The radial comoving-distance <integral> instead has denominator $H$ alone; elapsed time and present spatial separation answer different operational questions.