Comoving Distance and Lookback Time

Solve Eq. (13) by integration to find $\eta$ as a function of $t$

$$\eta = c \int_0^t \frac{dt}{a(t)}$$ (40)

but change variables to integrate over $a$ rather than $t$

$$\eta = \int_0^a \frac{c}{a} \left(\frac{dt}{da}\right) da$$ (41)

The lower limit of the integration over $a$ is $0$ since at $t=0$ the scale of the universe is $0$. We substitute for $\dot{a}$ from the Friedmann Equation (Eq. 30) to obtain the integral for the conformal time [1]

$$\eta(a) = \int_0^a \left(\left(\frac{a}{a_0}\right) \Omega_... ...}{a_0}\right)^4 \Omega_{\Lambda}\right)^{-1/2} \frac{da}{a_0}$$ (42)

Since $a/a_0$ is related to the observable $z$, it is useful to change variables again and evaluate

$$\eta(z) = \int_0^{u_z}\left(u\Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (43)

with

$$u_z = \frac{1}{1+z}$$ (44)

from Eq. 19. Therefore

$$\eta(t_0) = \int_0^1\left(u\Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (45)

and

$$\eta(t) = \int_0^{1/1+z}\left(u\Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (46)

We can now evaluate the comoving distance from Eq. 20

$$\chi = \eta(t_0) - \eta(t)$$ (47)

$$\chi = \int_{1/1+z}^1\left(u\Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (48)

The dimensionless conformal $\chi$ is scaled by $a_0$ to give the comoving distance $d_{comoving}$ in physical units

$$d_{comoving} = a_0 \int_{1/1+z}^1\left(u\Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (49)

The lookback time is the difference between the age of the universe now and the age when the light we observe now was emitted [5]. Integrate Eq. 11 to obtain the age of the universe $t$ from the conformal time or distance $d\eta$

$$t = \int_0^\eta \frac{a}{c} d\eta$$ (50)

Change variables to integrate on $a$

$$t = \int_0^a \frac{a}{c} \frac{d\eta}{da} da$$ (51)

and substitute for $d\eta/da$ using the result in Eq. 42

$$t = \int_0^a \frac{a}{a_0} \left(\left(\frac{a}{a_0}\right) ... ...eft(\frac{a}{a_0}\right)^4 \Omega_{\Lambda}\right)^{-1/2} da/c$$ (52)

Notice the $da/c$ is $t_0 d(a/a_0)$ and that with $u=a/a_0$ we have

$$t = t_0 \int_0^{u_z} u \left(u \Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (53)

With

$$u_z = \frac{1}{1+z}$$ (54)

this becomes an equation for the age of the universe as a function of $z$

$$t(z) = t_0 \int_0^{1/1+z} u \left(u \Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (55)

The lookback time is the difference in ages at two epochs

$$t_{lookback} = t(0) - t(z)$$ (56)

$$t_{lookback} = t_0 \int_{1/1+z}^1 u \left(u \Omega_m + \Omega_r + u^4 \Omega_{\Lambda}\right)^{-1/2} du$$ (57)

As a practical matter, these equations cannot be integrated in closed form except in a limiting case where $\Omega_r$ and $\Omega_{\Lambda}$ are neglected. Programs that generate Partiview database files from the Sloan Sky Survey files would incorporate numerical integration routines to evaluate Eqs. 49 and 57 for each galaxy and quasar. The equations given here reproduce the values in Table 1 of Gott et al. [1], using the same WMAP parameters that he used. For selected entries from SDSS DR1, our results agree with the speck files that are now included in the extragalactic database that comes with Partiview.