# Observing galaxies that recess faster than c

1. Apr 25, 2015

### virgil1612

Hello,

I've read the article at

http://pages.erau.edu/~reynodb2/LineweaverDavis_BigBang_SciAm_March05p36.pdf

that was recommended several times on this forum. At the question "Can we see galaxies receding faster than light?" they answer:
"Sure we can, because the expansion rate changes over time. The photon initially is unable to approach us. But the Hubble distance is not constant; it is increasing and can grow to encompass the photon. Once that happens, the photon approaches us and eventually reaches us"

I know that we can see these galaxies. I just don't understand their explanation. Could someone help me understand?

Thanks, Virgil.

2. Apr 25, 2015

### rootone

Due to expansion they may be moving away from you 'here and now' at a rate exceeding c.
However the light you see 'now' would have been emitted billions of years ago when they we not receding as fast, so those photons DO reach you eventually

3. Apr 25, 2015

### marcus

Their explanation is pretty good.
The galaxy is not moving in the usual sense. It is not approaching any destination, should not be thought of as going through space. The distance between stationary objects can increase (that's what curved geometry is about, distances can change, don't have to obey ordinary Euclidean expectations.

Most of the galaxies we can see with telescopes WERE ALREADY RECEDING FASTER THAN LIGHT WHEN THEY EMITTED THE LIGHT WHICH WE ARE NOW GETTING FROM THEM.

That is true for any galaxy with redshift greater than 1.6 and most visible galaxies have redshifts z > 1.6

Last edited: Apr 25, 2015
4. Apr 25, 2015

### virgil1612

When they emitted the light that we see NOW, shouldn't they have receded slower than c, so that THAT light eventually reaches us?
When talking about the moment the galaxies emit the light that we see now, are we doing some kind of velocity addition between velocity of light and the recessing velocity at that point?

5. Apr 25, 2015

### virgil1612

This makes sense. Thanks.

6. Apr 25, 2015

### marcus

What Rootone said actually does not make sense to me because it makes it seem that when the galaxy emitted the light we are getting the distance to it was actually increasing SLOWER than c. But the whole point is that for most galaxies we can see, we are getting light that was emitted by them when the distance to the source was increasing FASTER than c.

7. Apr 25, 2015

### virgil1612

I just begin reading cosmology (the descriptive way), but for me it's just the opposite. For a photon emitted by a distant galaxy to reach us, that galaxy, at the moment when it emitted the photon, should have moved, with respect to us, slower than light.

That's just the way I understand it and I could be wrong, but I'm sure that I've read things along these lines, I could try to find those references if it would be helpful.

8. Apr 25, 2015

### marcus

No, and it is very simple. There is no tricky "velocity addition" involved. Take a simple example of a galaxy with redshift z = 1.6 which emitted a photon we are now receiving. The distance to the galaxy and the space around the galaxy was increasing at c at the moment of emission.
So the photon made no progress. It was heading towards us but the distance between us did not decrease. It stayed essentially the same distance from us for a long time.

That distance, that is increasing exactly as fast as light travels is called the HUBBLE DISTANCE.

If a photon is trying to get to us and it is that distance from us, it makes no progress.
If it is WITHIN that distance it makes progress
If it is farther than that it gets swept back, loses ground, distance it has to go is getting bigger.

Lineweaver and Davis point out that the Hubble distance, or Hubble radius, INCREASES so it gradually reaches out and TAKES STRUGGLING PHOTONS IN.

If, today, a photon is emitted towards us at the Hubble radius (which is currently 14.4 billion LY) it will at first make zero progress, it will stay at that distance. But the Hubble radius is constantly increasing (because as a percentage rate, the expansion rate is decreasing) so after a while the Hubble radius will be 14.5 billion LY. thank goodness! says the photon, I am now within the Hubble radius! I am making progress towards my goal.

9. Apr 25, 2015

### virgil1612

Extremely interesting! Please give me some time to think about it, and I'll post back tomorrow.
Virgil.

10. Apr 25, 2015

### Chronos

11. Apr 25, 2015

### Staff: Mentor

Can you elaborate on this, Marcus? Is there a difference between the rate of expansion (which is increasing) and the rate of expansion as a percentage (which is decreasing)?

12. Apr 25, 2015

### marcus

There's a problem in the English language around the word "rate". People confuse it with "speed".

If a distance grows at a constant rate (say 1% per year) then it lengthens at an increasing speed. Of course! It is growing exponentially at 1% per year.
Exponential growth at a constant rate has this kind of "acceleration" because the principal is growing.

A distance can even grow at a gradually DECLINING rate and if the decline is slow enough it will still lengthen at an increasing speed. It can exhibit NEAR exponential growth, even if the rate slowly declines from 1% to .99% to .98% leveling out say at .97%.
It can "accelerate" even though the growth rate is declining.

In the standard cosmic model, based on the Friedmann equation, the distance growth rate has declined since very early times, and it is expected to continue declining.
What you hear a lot about is that people discovered in 1998 that the decline had slowed enough so that if you focus on a particular distance between two stationary objects that distance would grow at increasing speed. The transition from deceleration to acceleration we think happened around year 8 billion.

The distance growth RATE continues to decline, but gradually enough that this "near exponential growth" behavior has begun to be detectable.

If you calculate what the Hubble growth rate actually is (cut through the confusing units) it comes out to a percentage rate of 1/144 percent per million years and its decline seems destined to continue more and more gradually so that it levels off at a rate of 1/173 percent per million years.

this corresponds to a Hubble radius now of 14.4 billion LY and an eventual Hubble radius of 17.3 billion LY. You can see the past and future behavior of the Hubble radius (and thus its reciprocal, the hubble rate) by clicking on Lightcone---a table making calculator that embodies the standard cosmic model:
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
the R column gives the Hubble radius (aka the Hubble distance) in billions of LY.
It shows how it is currently 14.4 and expected to converge to 17.3 in future (as the Hubble rate declines)

Last edited: Apr 25, 2015
13. Apr 25, 2015

### Staff: Mentor

I'm having trouble converting what you've said into something I know about. I've rarely dealt with expansions and scale factors. Let me try to understand this using my new calculus 1 knowledge.

If I graph the distance to a galaxy undergoing this kind of motion over time, I get a function which is increasing (because it is moving away from us). Exponential growth is an exponential function, like F(t) = At, right?
The 1st derivative is the velocity, which is also increasing.
The 2nd derivative is the acceleration. In this type of motion is the acceleration increasing? I want to say yes but I can't work out the mathematical details at the moment. I just know that if the velocity is increasing, the acceleration is positive.

14. Apr 25, 2015

### marcus

That's right! the derivative of ex is ex
so the derivative of THAT is ex
all the derivatives are the same and they are all increasing.

If you look at a different timescale, or put some constant k in front then you have ekx and the derivatives involve powers of that constant term but that doesn't change the qualitative behavior . The derivatives all look like ex with different constant terms in front and they are all increasing.

That is growth at a constant rate . But you can have nearly the same shaped curves with growth at a
gradually declining rate if the decline is gradual enough. Think of it as ekx with k nearly constant but very slowly dwindling down to some k

Last edited: Apr 25, 2015
15. Apr 25, 2015

### Staff: Mentor

Thanks.

Is 'declining rate' talking about acceleration while 'lengthen at an increasing speed' talking about velocity?
That would mean velocity is increasing, and that the acceleration is positive, but that it is decreasing, right?

16. Apr 25, 2015

### Chalnoth

Another way to look at it is through the rate of expansion rather than distance. In the very early universe, the rate of expansion was much, much higher. But that rate has decreased over time as our universe has become less dense. Early-on, the high expansion rate caused many photons to fail to make progress towards us. But as that expansion rate dropped, a number of those photons were still close enough that they did start making progress.

To put some numbers on this, the photons from the cosmic microwave background that we see today were, at the time they were emitted, only about 46 million light years away. But that was in the very early universe, when the rate of expansion was much, much faster (as in something like 17,000 times faster). That really rapid rate of expansion easily increased the distance between us and those photons. That rapid expansion was such that it made it so that those photons actually had to traverse some 13.8 billion light years before they could actually reach us.

Because the rate of expansion isn't slowing down as much any longer (and the expansion rate will probably eventually become a constant), there are a great many photons that were emitted from further away that can never reach us.

17. Apr 25, 2015

### Monsterboy

Does this mean that the magnitude of acceleration of expansion of the universe is decreasing?

18. Apr 25, 2015

### Chalnoth

I'm not sure that's a good way to look at it.

First of all, "acceleration of expansion" is a confusing phrase. The expansion rate itself is going down and approaching a constant value. What an accelerated expansion means is that individual objects in the universe are accelerating away from one another.

I'm not so sure that the rate of acceleration between objects is decreasing, however. I'd have to do some fairly specific math to figure out whether the acceleration between objects is currently increasing or decreasing, however we do know that in the distant past, objects in our universe were decelerating with respect to one another, and quite rapidly. The fact that they're accelerating recently indicates that the rate of acceleration has increased a lot: it's gone from a negative value to a positive one. If the change in the rate of acceleration were simple enough, then that would imply that the current rate of acceleration between objects may still be increasing.

19. Apr 26, 2015

### Monsterboy

No , I didn't say objects , I meant accelerated expansion of space between the objects without the objects moving through space.

20. Apr 26, 2015

### Chalnoth

My point is that the distances between individual objects increase at an accelerating pace, but the rate of expansion is still dropping and approaching a constant value.

21. Apr 26, 2015

### marcus

Hi Virgil, glad you found that interesting, and I hope you will contribute some more to the discussion. Here is a simple graph of how that story. The red curve shows the past experience, collectively, of photons we are now receiving. The ones that were emitted recently had an easy straightforward job getting here. The ones emitted a long time ago, like in year 1 billion, at first got farther away because the space they were traveling thru got farther away (canceling their forward progress).

The blue curve is the Hubble distance which grows and in a sense reaches out to photons so that some of them eventually make it even though they may have, at first, gotten farther away (because the distance to the space they were traveling thru was increasing.)

You can see that some of the photons we are receiving today started out in year 1 billion at a distance of 4 billion LY (if you could have paused geometric expansion at that moment to have time to measure, with radar or string or however)
As long as they were outside the Hubble radius they made no progress, got farther way, their motion thru space canceled by the distance growth.
As long as the red curve is above the blue, things get worse for them.
But the blue curve is always rising, the Hubble radius is extending, and at a certain moment, in year 4 billion, the photon finds itself making ZERO progress but at least not being dragged back. It's forward motion thru space is exactly canceled by the space getting farther away.

The curves cross there, and after that, after year 4 billion, the photon makes progress and gradually reduces the distance to us.
It is always traveling thru space at the standard speed c. but now because it is within the Hubble radius the space it has to travel is not not growing so much.

You can see that the turnaround distance is about 6 billion LY, half way between the 4 and the 8. More precisely it is 5.8 billion LY.

So some of the photons that are coming to the HST (hubble space telescope) at this moment started out towards us in year 1 billion at a distance of 4 billion LY and got "dragged back" for the first 3 billion years so that they were actually 5.8 Gly away, and then were rescued by the slowing rate of distance growth (the Hubble distance is ONE OVER THE RATE, the reciprocal of the rate, so as the growth rate declines the Hubble distance increases, as the blue curve shows.)

Last edited: Apr 26, 2015
22. Apr 26, 2015

### marcus

Good point! Here is a plot of the distance growth rate (the socalled Hubble parameter) over time.
The present is at 0.8 on the x axis.
I don't happen to have a plot of it with the x axis labeled in billions of years and the present labeled "13.8"
The time scale here is 17.3 billion years = 1.
So 13.8 billion years = 0.8.
But aside from the time scale it gives a good idea of how the expansion rate has changed over time:

The growth rate is expressed as the fractional amount of growth per billion years if expansion would stay constant at that level for the whole billion year time interval.
Readers can see that at x = 0.1 the rate is 0.4 per billion years
The rate is such that if it stayed steady all that time then in a billion years the distance would be 0.4 or 40% longer.
Of course the rate does not stay steady, it declines. So maybe we should imagine a briefer time interval.
That same rate would be 0.0004 per million years. Or 0.04 percent per million years, if it's easier to think about it that way rather than as 0.0004.

Is it clear that at the present time (x = 0.8) the growth rate is about 0.07 per billion years? If you go out to x = 0.8 then the curve is a little more than one square up. One graph square is 0.05, two squares is 0.1. The curve is about 7/10 of two squares up, so it is at 0.07.

0.07 is about 1/14. So at the present rate, if it would remain steady that long, in a billion years a distance would increase by 1/14 of its size.

That is the same as saying the Hubble time is about 14 billion years, or expressing the same thing as a distance, the Hubble distance is 14 billion LY.
That is what is meant by saying the Hubble distance is the RECIPROCAL of the rate. It is a convenient handle on the growth rate which is easier to grasp than these very tiny numbers and which lengthens inversely as the rate declines.

As Chalnoth just said, the rate is dropping and approaching a constant value. You can see that in the plot.
The curve is leveling out around 0.06 or 1/17 per billion years.
By the same token, the Hubble time is INCREASING AND LEVELING OUT AT 17 BILLION YEARS or more precisely at 17.3 billion years. And another way of saying that is the Hubble distance is increasing towards a longterm level of 17.3 billion LY. Hubble time and distance are both convenient inverse handles on the growth rate.

That is what the longterm growth rate level of 0.06 shown by the curve means.

Last edited: Apr 26, 2015
23. Apr 26, 2015

### Chalnoth

By the way, I did some math, and as long as I calculated this correctly, the acceleration of the scale factor is:

$${d^2 a \over dt^2} = H^2\left(1 - {3 \over 2}{\rho_m \over \rho_c}\right)$$

This is assuming no spatial curvature and a cosmological constant. At late times, when this equation is valid, the matter density fraction drops monotonically, which means that the distance between objects accelerates at an increasing pace into the future (in the presence of a cosmological constant).

Last edited: Apr 26, 2015
24. Apr 26, 2015

### Jorrie

Is this equivalent to the classical deceleration parameter:

${d^2 a / dt^2} = aH_0^2 (\Omega_\Lambda - \Omega_m/(2a^3))$

as per prof. Peebles' 1992 (textbook)?

The latter is expressed in present values and I presume that your equation uses the values at time t.

25. Apr 27, 2015

### tkjtkj

You've presented for cosmology an extremely easy-to-imagine model ... But as we all know here, being imaginably logically clear does not mean it is correct.
It all goes back to the very logical (at the time!) voyage of the sun around our earth ;)

But i do like sitting here finally able to imagine !

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