# How do we work out the light travel time on a cosmic scale?

I just read this article in the AUSTRALIAN SKY & TELESCOPE magazine, Nov/Dec 2022 Issue 140, on P16, KEEP YOUR DISTANCE: How far away are the objects we see in the universe? And on P23:

"And here is where our story takes a mind-bending turn:"

What does distance even mean in the expanding universe? On scales of the Solar System, we can understand it fairly easily. But for really remote galaxies, cosmic expansion makes the concept of distance quite tricky. In fact, many cosmologists protest that giving distances for anything farther than a couple of billion light-years should be avoided.

Suppose you measure a galaxy's redshift to be z=1.5, meaning that visible light emitted with a wavelength of 500 nm by the galaxy has been shifted by 1.5 * 500 = 750 nm to an observed infrared wavelength of 1250 nm. The Hubble-Lemaitre Law tells you that the light has been travelling through expanding space for some 9.5 billion years. Intuitively, you'd conclude that the galaxy is 9.5 billion light-years away.

However, you can't simply convert the light-travel time into a distance. When the light was emitted 9.5 billion years ago, the universe was smaller and the galaxy was a "mere" 5.8 billion light-years away. Because space is expanding, it took the light 9.5 billion years to reach the Earth. But by the time the light finally arrives here, the galaxy's "true" (or proper) distance has increased to 14.6 billion light-years.

How do they know that the light has travelled for 9.5 billion years? How do you work/figure it out? And, is it now really 14.6 B light-years to the other galaxy or the original/starting point? The other galaxy has also moved away!

• Does this answer your question? Is the distance covered always equal to the time taken? Feb 1 at 8:37
• @ProfRob: The stuff in [square brackets] is what I put in, for my own benefit, and as a matter of fact, 21 B LYs sounds to me to be more like what the proper distance should be now! The other galaxy has also moved away. Are my questions, at the end, really that unclear!? Feb 1 at 15:40

Let's first be clear that there is no unique way to identify the time or distance between two events. This is true in every relativistic context; just think about relativistic time dilation and length contraction. In fact, the proper time along light's path through spacetime (in vacuum) is always zero. So what does light travel time actually mean?

Fortunately, there is stuff everywhere in the universe, and that stuff is easy to track, in the sense that you can uniquely associate a flow velocity with each point, at least at scales much larger than galaxies (such that the matter is effectively a fluid, in the technical sense). This means you can consider times and distances as measured by the stuff.

The idea is that you can define the time to be that measured by clocks that were synchronized at the beginning of the universe, when all the stuff would have been in the same place, and those clocks simply move along with the rest of the stuff. This is likely how the article is defining the light travel time: it's the time difference between the source's clock at the emission time and the target's clock at the receiving time.

With this array of clocks, you can also define distances to be those measured "at constant time". Here's a conceptual illustration of the relevant distances. The important feature of this illustration is that I am depicting the source ("them") and the target ("us") in a symmetrical way, moving apart at the same speeds. This is demanded by how I defined the clocks, above. We need the speeds to be the same so that the clocks of the source and the target run at the same rate.

There are nevertheless ways in which the picture is inexact. It's on a flat screen, while the Universe's spacetime is curved. Also, even in flat spacetime, the constant-clock-time surfaces would be hyperbolas (due to time dilation), not the horizontal lines that the picture suggests.

• This doesn't tell the OP how the 9.5 billion years is calculated. I believe the issue is not with the concept you have drawn but in the actual calculation. The quoted article says Hubble's law is used in the form $cz/H_0$, but this in fact gives 21 billion light years. Feb 1 at 8:11
• @ProfRob: The stuff in [square brackets] is what I put in, for my own benefit, and as a matter of fact, 21 B LYs sounds to me to be more like what the proper distance should be now! The other galaxy has also moved away. Feb 1 at 15:38
• I second @ProfRob . This answer does not fully answers the question. A more detailed explanation is required, especially for users that can't see the image for any reason. Without the visual aid, the worth of this question drops to zero. I downvoted it. Feb 1 at 16:11
• Updated with more detail. Still a conceptual answer; thanks @ProfRob for the numerical answer.
– Sten
Feb 2 at 19:30

The light travel time is calculated as the cosmic time interval between when the light was emitted and when it was received at the Earth.

Calculation of this quantity depends on the cosmological parameters that describe the expansion of the universe because, for a given initial separation between the distant galaxy and the Earth (or where the Earth would eventually form), a faster-expanding universe would mean the light took longer to reach us and vice-versa.

There is also a connection between redshift and the cosmic time at which the light was emitted and between the cosmological parameters and the age of the universe now. As a result, there is a non-straightforward relationship between the light travel distance and the redshift, $$z$$ of a galaxy.

$$d_{LT} = \frac{c}{H_0}\int^{z}_{0} \frac{dz}{(1+z)E(z)^{1/2}}\ ,$$ where $$E(z) = \Omega_m (1+z)^3 + \Omega_\Lambda +\Omega_k (1+z)^2 + \Omega_r (1+z)^4$$ with $$H_0$$ equal to the present Hubble constant (about 70 km/s/Mpc), $$\Omega_m$$ the current ratio of gravitating matter density to the critical density (about 0.3), $$\Omega_\Lambda$$ the current ratio of dark energy density to the critical density (about 0.7) and $$\Omega_k$$ and $$\Omega_r$$ are the equivalent density ratios for "curvature" and radiation, which are small in the current universe.

The integral needs to be (in general) calculated numerically. The derivation of this formula can be found in most cosmology textbooks or here. A convenient "cosmology calculator" that implements these equations can be found here. For $$z=1.5$$ in a flat universe with the parameters estimated from the Planck data, then I get $$d_{LT} \simeq 9.5$$ billion light years and the light travel time is thus 9.5 billion years.

The distance to the galaxy was smaller than this when the light was emitted and, because the expansion has continued since then, the galaxy is further away than this now. The figure of 14.6 billion light years is the approximate proper distance to the emitting galaxy (if it still exists?) now.

The proper distance now (also called the co-moving distance) is calculated as $$d_{\rm prop} = \frac{c}{H_0}\int^{z}_{0} \frac{dz}{E(z)^{1/2}}\ .$$

The proper distance when the light was emitted (9.6 billion years ago) is equal to the proper distance now multiplied by the scale factor then (the scale factor now being defined as unity), which was $$(1+z)^{-1}$$. We thus get a proper distance when the light was emitted of $$14.6/2.5 = 5.8$$ billion light years.

• Minor comment: the integral can be done analytically for the flat matter+$\Lambda$ case.
– Sten
Feb 2 at 19:33

Bright objects have emission spectra, within them are very narrow clearly defined frequencies that are due to the arrangements of electrons in that type of element. By looking at these frequencies they give a "fingerprint" that is unique for each element, complex objects like stars and clouds of dust are made up of multiple different elements but each element's frequencies are still visible.

Because the universe is growing the travelling light has its wavelength stretched out, longer wavelength light is towards the red end of the spectrum which is why the phenomenon is called "redshift". These fingerprints are redder than they would be for a nearby object. The amount of this shift tells us about how long the light has been travelling and how much the universe has grown since it was emitted.

Now that sounds a little circular in that we use redshift to work out the travel time, as the redshift amount depends on the travel time... However redshift isn't the only way to measure distance there are ways of measuring distance against types of objects with a known brightness and nearer objects with parallax, by using these different techniques astronomers have built up a catalogue of how redshift varies with distance/time.

TLDR: We know the light is that old because we knew what colour it was when it started and light doesn't just change colour by itself, but from being stretched which takes time.

• I don;t think the OP is confused about how redshift is measured or that higher redshift means the light is "older". The questions asks about the origin of the numbers in the quoted article and how they are calculated. Feb 1 at 17:43