# Expansion of the universe

1. Feb 28, 2015

### Suraj M

So I know that the universe expanded faster than the speed of light. Then it still should be expanding faster than the speed of light,right?
Then, how is the light produced during the big bang catching up to us now?? I mean the radio signals and the image of the early universe?

2. Mar 1, 2015

### Staff: Mentor

No, this is not correct. There is no single "speed" at which the universe expands. It is true that, in a certain system of coordinates, some objects are moving away from other objects at a coordinate speed faster than $c$. But this does not mean quite what you think it means. See below.

When people talk about objects in the universe moving apart "faster than light", that doesn't really mean what it says. Nothing ever moves faster than light beams that are at the same spatial location. And in a curved spacetime, there is no unique way to compare speeds of objects at spatially separated locations (like us here on Earth and a galaxy a billion light-years away). You have to pick a particular set of coordinates.

In the coordinates that are usually used in cosmology (called "comoving" coordinates), galaxies more than a certain distance away from us, called the "Hubble radius", are moving away from us at a coordinate speed faster than $c$. But light emitted by those galaxies in the direction away from us is moving away from us even faster than the galaxies themselves; i.e., in these coordinates, light itself can move "faster than light".

But what about light emitted towards us by those galaxies? It is true that, in "comoving" coordinates, this light will not get closer to us; for a galaxy right at the Hubble radius, light that it emits towards us will stay at that distance (the Hubble radius). However, that won't necessarily continue to be the case. The Hubble radius itself changes with time. If the Hubble radius is increasing with time (which it was up until a few billion years ago), then light emitted by a galaxy that was at the Hubble radius when it was emitted (or even outside it), will end up inside the Hubble radius and will start getting closer to us, and will eventually reach us. That's how we can see things from the early universe that may have been receding "faster than light" when they emitted the light we see now (i.e., were outside the Hubble radius at the time of emission).

Since a few billion years ago (as best we can tell), the expansion of the universe has been accelerating, which means the Hubble radius is now decreasing with time instead of increasing. If this continues (which as far as we can tell it will), objects that we can see now will eventually be lost to view as they move outside the Hubble radius.

3. Mar 1, 2015

### Suraj M

I knew the part about the hubble radius increasing,
then why aren't received as light are they? they are received with a low frequency how do they loose their energy?
Ok so, if we consider the original singularity as the reference, then the universe wasn't expanding faster than the speed of light from that point??only with respect to other particles?

4. Mar 1, 2015

### marcus

I think "accelerating" means something slightly different. Hubble radius continuing to increase but at a slower rate. The scale factor a(t) second derivative a''(t) > 0.
The "lost to view" is more a matter of redshifting to the point of being undetectable. Some of the language we use is confusing.

I think this is what Peter meant by "lost to view": it's like the fact that we can't see something falling in through the horizon of a black hole. We continue to see the object at the horizon (its light growing redder, and coming to us in longer and longer wavelengths), until our eyes and instruments can not detect the light. Meanwhile the object has long since fallen through the horizon.

Last edited: Mar 1, 2015
5. Mar 1, 2015

### Staff: Mentor

Light from distant objects is redshifted because the universe expanded between the time it was emitted and the time we see it. Yes, this means the light loses energy. Energy is not conserved in an expanding universe.

The original singularity is not a point in space; there is no one place in the universe now where the universe is "expanding from".

6. Mar 1, 2015

### Suraj M

Oh yes that theory, every particle in the universe is the centre of the universe!!
Could you please tell me then what the statement 'Expanding faster than the speed of light' actually means.

7. Mar 1, 2015

### Staff: Mentor

I can't find a handy quick reference for this, so let's work it out by hand.

We'll use the flat FRW solution where the metric is

$$ds^2 = - dt^2 + \left( a(t) \right)^2 \left( dr^2 + r^2 d\Omega^2 \right)$$

The Hubble radius is the $r$ coordinate at which the coordinate velocity of a "comoving" object is $1$ (we're using units in which $c = G = 1$). Put another way, at the Hubble radius, the proper distance $R = a r$ between "comoving" observers at $r = 0$ (presumed to be us here on Earth) and $r = r$ is increasing at rate $1$. So we have

$$1 = \frac{dR}{dt} = \frac{da}{dt} r = \frac{1}{a} \frac{da}{dt} a r = H R$$

where $H$ is the usual "Hubble parameter" $\dot{a} / a$ relating "recession speed" to distance. So we see that the proper distance $R$ to the Hubble radius is just the reciprocal of the Hubble parameter $H$.

Now we can just use the Friedmann equation for $H$:

$$H^2 = \frac{1}{R^2} = \frac{1}{3} \left( 8 \pi \rho + \Lambda \right)$$

which becomes

$$R^2 = \frac{3}{8 \pi \rho + \Lambda}$$

We have $\rho$ decreasing with time and $\Lambda$ constant. So it looks like you are right; the proper distance to the Hubble radius does not start decreasing if the expansion of the universe is accelerating; it just asymptotes to a constant limiting value.

However, I was also right . See follow-up post.

Last edited: Mar 1, 2015
8. Mar 1, 2015

### Staff: Mentor

Can you give a reference? This is not my understanding of what the standard theory in cosmology says.

See posts #2 and #7.

9. Mar 1, 2015

### marcus

that's correct! The present Hubble radius is 14.4 billion LY and its asymptotic value is 17.3 billion LY. We can denote that by c/H

The present value of H is 1/144 % per million years and H is expected to continue decreasing to an asymptotic value of H = 1/173% per million years.

I think the condition that we get accelerated a(t), that a''(t) > 0 is something like
H(t)2 < 3H2 (assuming spatial flatness and matter dominant over radiation.)

Last edited: Mar 1, 2015
10. Mar 1, 2015

### Suraj M

the cosmological principle, i don't know much about it, i just saw this

11. Mar 1, 2015

### Staff: Mentor

That's a side effect of what I meant, yes. The key point is that light emitted by the object will never reach us. See below.

Here's why. The proper distance to the Hubble radius continues to increase. But the coordinate distance to the Hubble radius decreases when the expansion is accelerating (while it increases if the expansion is decelerating). And if we're trying to figure out whether light emitted by a "comoving"object will ever reach another "comoving" object, it's the coordinate distance (in "comoving" coordinates) that counts.

Let's work this out by hand too. The coordinate distance $r$ is the proper distance $R$ divided by the scale factor $a$. If we plug in our equation for $R$ at the Hubble radius from post #7, we get

$$r^2 = \frac{3}{a^2 \left(8 \pi \rho + \Lambda \right)}$$

So the time dependence of $r$ at the Hubble radius depends on the time dependence of $a$.

For a matter-dominated universe (that is spatially flat), we have $a \propto t^{2/3}$ and $\rho \propto a^{-3}$. If we neglect $\Lambda$, that gives $r \propto t^{2/3}$, which is increasing with time.

For a dark energy-dominated universe (that is spatially flat), we have $a \propto e^t$ and $\Lambda$ constant. If we neglect $\rho$, that gives $r \propto e^{-2t}$, which is decreasing with time.

The reason it's the coordinate distance that matters is simple: a given "comoving" object is always at the same $r$ coordinate. So whether or not that object can send us light signals depends on the $r$ coordinate of the Hubble radius relative to the object's $r$ coordinate. If the $r$ coordinate of the Hubble radius is increasing with time, then "comoving" objects will gradually "come into view" as their $r$ coordinate comes inside the Hubble radius. But if the $r$ coordinate of the Hubble radius is decreasing with time, then "comoving" objects will gradually go out of view as their $r$ coordinate goes outside the Hubble radius.

(If we want to think of things in terms of proper distance, what is happening is that, during the matter-dominated phase, the proper distance to the Hubble radius was increasing faster than the proper distance between "comoving" observers. But during accelerated expansion, the proper distance to the Hubble radius increases slower than the proper distance between "comoving" observers. Using coordinate distance just makes this easier to see because "comoving" observers are always at the same $r$ coordinate.)

12. Mar 1, 2015

### Staff: Mentor

13. Mar 1, 2015

### Staff: Mentor

Every particle is the center of their own observable universe, but not the center of the entire universe, as there is no such thing to the best of our knowledge.

14. Mar 1, 2015

### wabbit

Analogies only go so far, but you may find the ants on a balloon analogy helpful to get an intuition about this.

15. Mar 3, 2015

### Quds Akbar

The universe's expansion has slowed down, the Universe expanded much more rapidly than it would now in its first moments.

16. Mar 3, 2015

### Staff: Mentor

More precisely: the universe expanded very fast during the initial inflation phase; then, up until a few billion years ago, the expansion was slowing down; now it is speeding up again.

17. Mar 3, 2015

### Suraj M

Why is that? where is all that variation coming from?

18. Mar 3, 2015

### Staff: Mentor

During the inflation phase, the universe was in a "false vacuum" state, equivalent to a very large cosmological constant, which caused exponentially accelerating expansion. At the end of inflation, there was a phase transition in which the energy in the false vacuum was transformed into ordinary matter and radiation, expanding very fast, plus a small remaining positive cosmological constant (dark energy). That ordinary matter and radiation dominated the dynamics, causing decelerating expansion, until a few billion years ago, when its density became small enough that the density of dark energy began to dominate the dynamics; that's when the expansion began to accelerate again.

19. Mar 3, 2015

### Suraj M

So what can actually happen in the future, can the density of dark energy become small enough so that there is again decelerating expansion ?

20. Mar 3, 2015

### Staff: Mentor

The density of dark energy does not change with time. The density of matter and radiation decreases with time (at least, it does if the universe is expanding); that's why they are now smaller than the density of dark energy, even though they were larger in the past.