Redshift

How does the changing scale of the universe alter what we see? Since the universe is expanding while light is on its way to us, the galaxies which emitted photons in the past are no longer where they were, or even what they were. The difference in the age of the universe now and when light left the galaxy is known as the ``lookback'' time. In effect, it is the time it took light to reach us, but it is not the current light travel time to the object. The distance they are at ``now'' is called the ``comoving'' distance. It is the distance you would have to travel to make the trip between galaxies instantaneously.

The expansion of the universe alters the wavelength of the light that is emitted at one place in space and time, and then detected at another place and time. Photons on their way from a source to the telescope follow a radial path on a null ``geodesic'' at speed $c$. The angular terms in Eq. (3) are zero and the equation describing the path of the light in spacetime is

$$0 = - c^2 dt^2 + a^2(t)d\chi^2$$ (10)

$$c \frac{dt}{a(t)} = \pm d\chi$$ (11)

$$d\eta = d\chi$$ (12)

We have defined a dimensionless ``conformal'' time $\eta$ with

$$d\eta = c \frac{dt}{a(t)}$$ (13)

and selected the positive sign.

Suppose that a galaxy is at a conformal distance $\chi_e$ from us when it emits a maximum in the field of an electromagnetic wave at time $t_e$. The light follows a radial path to us and is observed at $t_{obs}$. The next maximum is emitted at $t_e + \Delta t_e$ and observed at $t_{obs} + \Delta t_{obs}$. According to Eq. 11 the conformal distance in either case is given by

$$\int_0^{\chi_e} d\chi = c \int_{t_e}^{t_{obs}}\frac{dt}{a(t)... ..._{t_e + \Delta t_e}^{t_{obs} + \Delta t_{obs}}\frac{dt}{a(t)}$$ (14)

The small time differences, $\Delta t_e = \lambda_e/c$ and $\Delta t_{obs} = \lambda_{obs}/c$, are related by the scale of the universe at the two epochs through

$$\frac{\Delta t_e}{a(t_e)} = \frac{\Delta t_{obs}}{a(t_{obs})}$$ (15)

$$\frac{\Delta t_{obs}}{\Delta t_e} = \frac{a(t_{obs})}{a(t_e)}$$ (16)

$$\frac{\lambda_{obs}}{\lambda_e} = \frac{a(t_{obs})}{a(t_e)}$$ (17)

The ratio of the observed wavelength to the emitted or ``rest'' wavelength defines the redshift $z$

$$1 + z = \frac{\lambda_{obs}}{\lambda_{e}}$$ (18)

In terms of the scale of the universe, the redshift observed now is given by the ratio of the scale of the universe now to what it was when the light was emitted, or

$$1 + z = a_0/a(t)$$ (19)

If we know the dependence of $a$ on $t$ we can connect $z$ to the comoving distance $a(t)\chi$. Suppose that at time $t$ a galaxy that is $\chi$ from us emits a photon that arrives and is detected at time $t_0$. Integrate Eq. (12) from $t$ to $t_0$ to obtain

$$\chi = \eta(t_0) - \eta(t)$$ (20)

This means that $\chi$, the dimensionless conformal distance, is the difference between the conformal time when we detect the photon and the conformal time when it was emitted. Values for $\eta$ can be found by explicitly integrating the defining Eq. 13 when $a(t)$ is known.