# Why is the observable universe so big?

The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction?

• Do you have a reference for that 80 billion light year figure? I've personally never heard that before. – dagorym Jun 22 '11 at 13:10
• en.wikipedia.org/wiki/Observable_universe has 93 glyr for the diameter. Easy to forget whether you're speaking opf radius or diameter in this context. I nearly did, in looking up the number. Considering how fuzzy these kinds of measurements are, I think 80 glyr is credible for another, perhaps older, estimate. Professionals use redshift as a directly-measurable proxy for distance because it is so difficult to get a reliable translation of redshift to distance. – Andrew Jun 22 '11 at 13:58
• Well, yeah... Big Bang and all... – Andrew Jun 22 '11 at 23:36
• Note that your only referring to the observable universe, not the actual universe size. The actual universe size could be much much bigger than 80 billion light years across. The fraction that we can observe is smaller than the actual size. – Phil Feb 20 '13 at 4:32
• If I'm not mistaken, this question has been asked - and answered - before: physics.stackexchange.com/q/12049 Someone had linked the OP to: physics.stackexchange.com/q/11803 – mikhailcazi Jul 11 '13 at 13:43

This question implicitly refers to the visible universe, but we should state that explicitly, as otherwise the question doesn't make any sense.

It may seem like we shouldn't be able to see more than 13.7 billion light-years (13.7 giga-light-years, or glyrs) away, but that reasoning omits the expansion of spacetime according to General Relativity. A photon emitted from somewhere near the beginning of the Universe would have traveled nearly 13.7 glyrs if you had measured each light-year just as the photon crossed it, but since those light-years that you measured have expanded since the photon passed through, that distance now adds up to about 80 glyrs.

• Dear Andrew, which of the two passages below reflect what you mean to convey: (1) The visible universe is now 80 bly's across, or (2) the visible universe is 80 bly's across. – Lucy Meadow Dec 12 '16 at 18:47
• The first statement is modified with the inclusion of 'Now', and is equivalent to stating "the visible universe is 13.7 bly's across, but were it possible to measure the distance right now instantaneously (C*[infinity]) the actual displacement is 80 bly's. The second statement is equivalent to "leaving aside the redshift, the information payload of the photon contains two numbers, one representing the displacement at source, and the other representing the payload on arrival. – Lucy Meadow Dec 12 '16 at 18:48
• But this is not true - the photon carries only one quanta of visual information. Either the physical packet of visual information is consistent with a displacement at emission of 13.7 bly's or it is consistent with a displacement at emission of 50 bly's or whatever it's supposed to be. – Lucy Meadow Dec 12 '16 at 18:48
• The last sentence is wrong. The radius of the observable universe is about 45.7 billion light years. A photon emitted as cosmic microwave background radiation (the oldest light we can see) has therefore travelled 45.7b ly to reach us. No light has travelled more than this distance since the universe began. – Chappo May 2 '18 at 3:03
• So I may better understand, may you provide a citation/source where the latter number is derived? - the distance the primordial photon has traveled accounting for cosmological expansion. This number is oft quoted but I wanna see a derivation – N. Steinle Feb 14 at 19:15

The radius of the observable universe is about 46 billion light years, which is considerably greater than its age of about 14 billion years. Since the radius of the observable universe is defined by the greatest distance from which light would have had time to reach us since the Big Bang, you might think that it would lie at a distance of only 14 billion light years, since $x=ct$ for motion at a constant velocity $c$. However, a relation like $x=ct$ is only valid in special relativity. When we write down such a relation, we imagine a Cartesian coordinate system $(t,x,y,z)$, which in Newtonian mechanics would be associated with a particular observer's frame of reference. In general relativity, the counterpart of this would be a Minkowski coordinate frame, but such frames only exist locally. It is not possible to make a single frame of reference that encompasses both our galaxy and a cosmologically distant galaxy. General relativity is able to describe cosmology using cosmological models, and this description is successful in matching up with observations to a high level of precision. In particular, no objects are observed whose apparent ages are inconsistent with their distances from us.

One way of describing this difference between special relativity's $x=ct$ and the actual distance-time relationship is that we can think of the space between the galaxies as expanding. In this verbal description, we can imagine that as a ray of light travels from galaxy A to galaxy B, extra space is being created in between A and B, so that by the time the light arrives, the distance is greater than $ct$.

None of this has anything to do with inflation. Inflation makes certain testable predictions about cosmological observations (e.g., it predicts that the universe is spatially flat), but it's irrelevant for understanding why the radius of the observable universe has the size it does in comparison to the age of the universe. Inflation may not even be correct. If inflation turns out never to have happened, it will have no effect on this particular question.

It turns out that we can get a surprisingly good estimate of the size of the observable universe using a simplified FRW cosmological model consisting only of dust, i.e., nonrelativistic matter. The approximation is good because the universe has spent most of its history dominated by matter, with only a very short period early on that was radiation-dominated, and another fairly recent era that is dominated by the cosmological constant. In accord with the current observational data, we make a second approximation, which is that the universe is spatially flat. In a spatially flat FRW model, the $r-t$ part of the metric is of the form $ds^2=dt^2-a^2dr^2$, where the scale function $a$ depends on time. For a photon, $ds=0$, and we can then show that the proper distance traversed by a photon since shortly after the Big Bang is given by $L=a \int dt/a$. For a matter-dominated solution, $a$ is proportional to $t^{2/3}$, and we find $L=3t$. This is quite close to the $L/t$ ratio of about 3.3 given by the most realistic models. It also makes sense that the result is somewhat greater than 3, because the universe has now entered an era in which its expansion is accelerating. In the future, $L/t$ will become greater and greater.

• Ben, the sticking point (just personally for me) is referring to an observable universe. Observable means the physical packet of visual information in a quanta of incoming light contains two displacement values, one being the displacement at emission and the other being the displacement at absorption inclusive of the expansion of space during transit. Leaving the cosmological redishift to one side, that is clearly not the case. – Lucy Meadow Dec 12 '16 at 19:04
• What that means is that the physically observable universe is 13.7 bly's but that a real size of 80 bly's is inferable from theory. That is not the same as a direct measurement. The observable universe is therefore 13.7 blys – Lucy Meadow Dec 12 '16 at 19:05
• Ben, can you expand on your math above? How do we get from 13.7 to 80? Diameter = 13.7 radius * 2 = 27.4 27.4 * 3.3 = 90.42 90.42 is quite a lot more than 80. This is in line with the wikipedia estimate of 90.68 here en.wikipedia.org/wiki/… – Keith Knauber Dec 1 '17 at 21:15

The universe is commonly defined as the totality of everything that exists, including all physical matter and energy, the planets, stars, galaxies, and the contents of intergalactic space.

No one knows if the universe is infinitely large, or even if ours is the only universe there is.

Although our view of the universe is limited, our imaginations are not. Astronomers have indirect evidence that the universe of galaxies extends far beyond the region we can see. But no one knows if the whole universe is infinitely large - large beyond limit.

According to the leading theories, other parts of the universe may look very different from our own - and may even have different laws of nature. We may never be able to find out for sure. But it is possible that clues to the answer lie in plain view, just waiting to be discovered!

I should note also the "80 billion light years across" it doesn't count as a contradiction. I don't know what your reference is but I believe that this concerns the region that we can see yet of this Universe.

You can travel as close to the speed of light as you want, but (assuming you're made from matter) you can't travel at the speed of light or faster.

So there's no reason why galaxies can't be receding from each other at 99.9999% the speed of light. However, this isn't the whole story, as the expansion of spacetime can do funny things. The universe isn't actually expanding as you'd expect from everyday life, as a explosion would, and there wasn't a "single point" that you can point to and say "that's where the big bang happened", it happened where you're sitting, and on the dark side of the moon, and just around the corner from Alpha Centauri...everywhere in fact.

Picture a graph - the point on the graph aren't moving away from each other (ie point A goes from (1,1) to (2,2) and point B goes from (5,5 to 7,7)) - instead, the whole graph, piece of paper and all, is being stretched.

This stretching is allowed to happen faster than the speed of light, as no matter actually has to change coordinates for it to happen.

In relation to the title of your question, the best answer is to ask "compared to what?".

## Briefly...

Consider a photon emitted 13.8 Gly ago (Gly = billion light years). As it travels through space, not only its getting farther away from its source due to traveling at the speed of light $c$, but photons distance also increases as the universe grows.

## Technically...

To get technical, the expansion of the universe is captured by the scale factor $a(t)$ which is governed by the Friedmann's equation:

$$\frac{(a'(t)/a(t))^2}{H_ 0^2}=\frac{\Omega _{\text{R0}}}{a(t)^4}+\frac{\Omega _{\text{M0}}}{a(t)^3}+\frac{\Omega _{\text{\kappa 0}}}{a(t)^2}+\Omega _{\text{\Lambda 0}}$$

where the Hubble constant $H_0 = 67.8 \frac{\text{km}/\text{s}}{\text{Mpc}} = 0.0693 /\text{Gyr}$, the present value of the radiation density $\Omega _{\text{R0}} = 0.0000905$, the present value of the matter density $\Omega _{\text{M0}} = 0.308$ (which mainly consist of dark matter) , the present value of the curvature density $\Omega _{\text{$\kappa $0}} = 1 - (\Omega _{\text{R0}} + \Omega _{\text{M0}} + \Omega _{\text{$\Lambda $0}}) = 1$, and the present value of the cosmological constant $\Omega _{\text{$\Lambda $0}} = 0.692$. These values are from the Plank Collaboration 2015. This equation can only be solve numerically. The following figure shows scale factor as a function of time:

In cosmology, the amount of distance a photon can travel in a given time is known as the comoving distance or the comoving horizon denoted by $\eta$:

$$\eta = \int_{0}^{t}\frac{c dt}{a(t)}$$

which again has to be evaluated numerically. By numerically integrating from 13.799 Gyr ago until the present time, a particle horizon of 43.5 Glyr is obtained which is the radios of the observable universe. One should take this value with a grain of salt since it is highly sensitive to $H_0$.

## Biologically...

I would like to focus on "Why the observable universe is so big?" from different perspective. Alternatively we can ask why the observable universe is 13.8 Gyr old! This is not such an odd question to ask if we state it as the following: "Why are we, as observers, happen to look at the universe 13.8 Gyr after the big bang?" Given that intelligent life on earth took ~3.5 Gyr to rise and the conditions of the universe were too harsh to permit evolution of life during the first ~10 Gyrs, one could expect that most pockets of intelligent observers would form and study the universe ~15 Gyr after the big bang. However, it is very surprising to find ourselves at the dawn of formation of life given that life would flourish for many giga-years to come.

Yes this does sound like it might be a contradiction but in fact it isn't. It's because the universe has been expanding rapidly in every direction since the big bang and our observations are limited by the speed of light.

For example if we observe a distant quasar that appears to be 10 billion light years away, the light from the quasar is 10 billion years old (which is why quasars are known for being some of the most ancient phenomena in the universe). In the time it took for that light to reach us the universe has been expanding. In fact the expansion has been accelerating all that while so the distance between us and that quasar is at this present time considerably larger than 10 billion light years.

If we had a means to observe distant objects as they appear right at this instance, not only would we have a type of time machine but we might observe the universe to be 80 billion light years across, although I cannot vouch for the accuracy of the 80 billion light year figure given on the wikipedia page.

Astronomers and physicists also struggle with the question of "what is it that the universe expanding into". Is it expanding into empty space and if so is there anything that lies out there beyond our universe at some vast distance? If so, the universe may be infinite.

Or does the universe fold back in on itself on some higher dimensional plain i.e. if we had a hypothetical means to travel faster that the rate of the expansion of the universe and travelled in a straight line, would we eventually end up back in the same spot? In this scenario the universe would have a theoretical boundary at any one time.

• The two final paragraphs are both completely wrong. – Ben Crowell Jun 24 '13 at 14:20
• Also "the expansion has been accelerating all that while" is wrong. Acceleration only began about 5 billion years ago when dark energy began to dominate. Before that, gravity dominated and the rate of expansion had actually been slowing. – Chappo May 2 '18 at 5:07

This is not a contraction. Special Relativity does not say that nothing can travel faster than the speed of light. Rather, it says that no regular matter can travel faster than the speed of light. Therefore, instead of thinking galaxies move away from each other, think that the space between them is increasing. That would mean that the space itself can beat the speed of light.

• It seems information can't travel faster than the speed of light either. I think the distinction between traveling through space versus traveling because of space expansion is important for this. – Brandon Enright Feb 12 '14 at 22:55

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