Cosmological look-back time ($t_L(z)={\int }_0^{z}d\bar{z}/[(1+\bar{z})H(\bar{z})]$)

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.