# How do you calculate the lookback time distance to a given galaxy?

By lookback time distance I mean the same as Light travel distance. Can you calculate it by just knowing its redshift?

No you can't. Other information is required.

For low redshifts - let's say smaller than 0.1 - and by that I mean the wavelength increases by 10 percent, you might get away with using Hubble's law to estimate the distance and then get the look back time by dividing by the speed of light. $$t \simeq \frac{\lambda - \lambda_0}{H_0 \lambda_0},$$ where $H_0$ is the present day Hubble parameter of about 70 km/s per Mpc, $\lambda$ is the measured wavelength and $\lambda_0$ is the rest wavelength.

So far so good, you just need to know $H_0$. However for larger redshifts it gets horribly complicated because the Hubble parameter changes with time in a way that depends on the curvature of the Universe and hence on the cosmological parameters defining the matter density and dark energy density.

There is indeed a highly complicated, non- linear formula involving integrals that I will look up a reference for. But possibly the better way to proceed is to use a simple look-up table produced from such calculations with certain assumed values for ratios of matter and dark energy densities with respect to the critical density; a.k.a. $\Omega_M$ and $\Omega_{\Lambda}$.

The plot below is an example taken from http://www.astro.caltech.edu/~eran/MATLAB/Cosmology.html which shows look back time versus redshift for two different cosmologial models (but with the same value of $H_0$). The curves are very different at high redshifts, but converge at small redshifts.

Here is a cosmological calculator that can do the job for you. Enter the redshift and your assumptions about the Hubble parameter and other cosmological parameter and it will tell you the age of the universe at that redshift as well as the lookback time.

@RobJeffries has covered most points, but just to add a little. The lookback distance is directly proportional to the cosmological time elapsed since emission, whereas the cosmological red-shift depends on [basically] how much scale factor, which is a function of cosmological time, has changed since emission. If the Universe has been expanding for its history up to the present then the scale factor to present will be a strictly increasing function of cosmological time, so there will be a one-to-one correspondence between red-shift and lookback distance. NB this will not be true if, for example, the Universe was initially expanding but is currently contracting.

However how the scale factor changes is governed by the stress-energy of the contents of the Universe so the relationship between red-shift and lookback distance will not generally be simple.

Lookback time, T, and distance, D, can be calculated from the object’s redshift, z, and cosmological parameters.

Lookback time, or (less appropriately) light travel time, answers the following similar questions:

-How old are the photons presently recorded in the object's spectrum?

-How long did it take the light to traverse the distance between the object and the observer?

-What distance did the light have to travel through an expanding universe in order to arrive at the present time?

-When we observe a remote object, how far back in time do we see?

The relation between redshift and lookback time is quite complex, and is presented here with simple equations derived by regression analysis from actual results. Correlation coefficients are better than 0.999. The equations are based on the following cosmological parameters:

Omega Matter = 0.272

Omega Lambda = 0.728

Hubble Constant now = 70.4

The result is given in billions of years (By)

T = ( 19292 x Z ) / ( 1.3878 + Z ), for Z between 0 and 0.2

T = ( 0.589 + 13.87 x Z^1.25 ) / ( 0.852 + Z^1.25), for Z between 0.2 and 20

Solving the second equation for quasar APM 8279, with Z = 3.911, T = 12.10 By

Lookback time also describes the distance the quasar's light travelled through an expanding universe before reaching the observer. This lookback time distance, D, determines by how much the object's light intensity falls off due to distance and due to intergalactic medium extinction.

D = C x T, where C = 1, describes D in light years,

D= ( C x T ) / 3.26, where C = 1, describes D in parsecs

• The lookback time can't be calculated knowing only the redshift. You also need the cosmological parameters, as you say yourself (and as stated in the answer by Rob Jeffries, four years ago). – pela May 16 '19 at 19:06
• You are absolutely correct. I edited my post accordingly. – Rudy E. Kokich May 18 '19 at 2:23