# If an object 1 billion light years away emits light, does it take more than 1 billion years to reach us because of the expansion of the universe?

From page 7 of the recent (September 26, 2020) edition of Science News Magazine:

Detected May 21, 2019, the gravitational waves came from a source about 17 billion light-years from Earth, making this the most distant confirmed detection. Because of the universe's expansion, that distance corresponds to a travel time of about 7 billion years, meaning the gravitational waves were emitted when the universe was about half its current age.

But that means the object(s) were only about 4 billion light-years or so away when the merger of two black holes occurred, right? But it take 7 billion light-years, rather than approximately four, for the gravitational waves to reach us because of the expansion of the universe itself?

Is there an online calculator that can calculate this? Does anybody know the equation(s) interconverting current distance, original distance and travel time of the radiation?

• astro.ucla.edu/~wright/CosmoCalc.html – Alchimista Oct 15 '20 at 10:03
• Hi Kurt, did my post answer your question? It would be nice — both for me, but also for future readers — if you either accepted or asked for clarification :) – pela Nov 17 '20 at 16:15
• Sorry, Pela, I got distracted...... – Kurt Hikes Dec 3 '20 at 4:35
• No worries, thanks! :) – pela Dec 3 '20 at 9:51

Yes, in the time it takes light — or, in this case, gravitational waves (GWs) from the black hole merger event GW190521 — to travel from a source to an observer, the Universe expands, thus increasing the distance further.

### Various distance terms

In the following, "$$\mathrm{Glyr}$$" means a distance of a billion lightyears, while "$$\mathrm{Gyr}$$" means a time of a billion years.

There is a slight confusion, I think, in the quoted distance of $$17\,\mathrm{Glyr}$$ (Abbott et al. 2020): This is the so-called luminosity distance, which is the distance that satisfies the usual inverse-square law. This is not the same distance that you would measure if you were to freeze time and lay out meter sticks. This physical distance is smaller, only $$9.5\,\mathrm{Glyr}$$.

These values correspond to a redshift of $$z=0.82$$. That is, if GW190521 was luminous, its light would be redshifted by a factor $$(1+z)=1.82$$. In fact, in this case an electromagnetic counterpart was reported, albeit not with a measured redshift (Graham et al. 2020)

The time it took the GWs to travel to us is called the lookback time; it is the quoted $$7\,\mathrm{Gyr}$$. When GW190521 emitted the GWs we detect today, it was closer to us by a factor $$(1+z)$$. That is, its physical distance was only $$5\,\mathrm{Glyr}$$.

For a flat universe (which our Universe is to high precision) this is equal to the so-called angular diameter distance, called so because it is the distance that satisfies the usual relation between distance $$d$$, size $$D$$, and angle $$\theta$$, namely $$\theta = D/d$$.

### Relation between lookback time and distance

In every-day life, all these distance measures are the same, and in the Universe, for small distances they also coincide. But because of the expansion of the Universe, and because the Universe's components (matter, radiation, and dark energy) influence its geometry, as the distance of an object increases they become increasingly different.

You can find the equations here, or use a cosmological calculator such as Ned Wright's (as commented by Alchemista). Alternatively, you can calculate them in Python using the module astropy like this:

>>> from astropy.cosmology import Planck15
>>> from astropy import units as u
>>> from astropy.cosmology import z_at_value

>>> dL = 5.3 * u.Gpc                                 # Lum. dist. in giga-parsec quoted in Abbott+ 20
>>> z  = z_at_value(Planck15.luminosity_distance,dL) # Corresponding redshift
>>> print(z)
0.8174368585313242
>>> print(Planck15.lookback_time(z))
<Quantity 7.11401487 Gyr>
>>> print(dL.to(u.Glyr))                             # Convert parsec to lightyears
<Quantity 17.28628801 Glyr>
>>> print(Planck15.comoving_distance(z).to(u.Glyr))  # Comoving dist. is equal to phys. dist. today
<Quantity 9.53452323 Glyr>


I used this to plot the current distance to GW190521 and other objects as a function of lookback time:

• Light from an object which has a physical distance of $$1\,\mathrm{Glyr}$$ now, is redshifted by $$z = 0.070$$, its light has been traveling for $$0.97\,\mathrm{Gyr}$$, and it was $$0.93\,\mathrm{Glyr}$$ away from us when it emitted the light we see today.
• Light from an object that was $$1\,\mathrm{Glyr}$$ away when it was emitted, traveled $$1.03\,\mathrm{Gyr}$$ before reaching us with a redshift of $$z = 0.076$$, and the object is now $$1.076\,\mathrm{Glyr}$$ away.
As you can see, the difference is not very large, but as you go to higher redshifts, it increases. The hitherto most distant observed galaxy, GN-z11, has a redshift of $$z=11.09$$. It was only $$2.7\,\mathrm{Glyr}$$ from us when it emitted the light we see today, but in the $$13.4\,\mathrm{Gyr}$$ it took the light to reach us (most of the age of the Universe), GN-z11 moved out to a current distance of $$32.2\,\mathrm{Glyr}$$!
• @KurtHikes Don't worry, the various terms for cosmological distances continue to confuse even professional astronomers :) The "$d_\mathrm{L} = 17\,\mathrm{Glyr}$" isn't physical in the sense that this is the distance you would have to walk if you froze the Universe and set off toward GW190521. You would only have to walk 9.5 Glyr. As I wrote above, the "purpose" of the luminosity distance is that it satisfies the inverse-square law, and the reason that this is important is that we don't measure distance with meter stick, but rather with observations of light. – pela Dec 3 '20 at 10:16
• The reason that the physical distance doesn't satisfy the inverse-square law ($F_\mathrm{obs} = L_\mathrm{em}/4\pi d^2$) is not only a possible curvature, but also the expansion of the Universe. This has two effects; a "geometric", i.e. the "$d$" changes in between emission and detection, and an "energetic", i.e. the light also loses energy so that $F$ changes. – pela Dec 3 '20 at 10:19