Evidence for Expansion Velocity change

[yogi]

I came across the following quote on the home page of a cosmology professor and I have seen similar statements in other contexts:

"Although the rate of expansion of the observable Universe is constant at one light year per year, and the local rate of expansion is more or less constant at one light year expansion per light year of distance per 10 to 15 billion years, the rate at which individual objects move away from us is accelerating, because they are getting further away from us. An object now 200 million light years away is moving away from us at about 200 million light years per 10 to 15 billion years. But in 10 to 15 billion years, it will be 400 million light years away, and since it will be twice as far away, it will be moving away from us twice as fast. In other words, the local rate of expansion is constant, but distant objects move away from us faster and faster, over (very) long periods of time"

My question - since redshift is determined only by the ratio of distances - what is the
bases for the conclusion that the relative velocity between two objects will change with time - in other words, the velocity-distance law can be satisfied by assuming the relative recessional velocities remain constant and the Hubble factor H decreases linearly as the universe expands (assuming q = 0)

 

[Chalnoth]

I came across the following quote on the home page of a cosmology professor and I have seen similar statements in other contexts:

"Although the rate of expansion of the observable Universe is constant at one light year per year, and the local rate of expansion is more or less constant at one light year expansion per light year of distance per 10 to 15 billion years, the rate at which individual objects move away from us is accelerating, because they are getting further away from us. An object now 200 million light years away is moving away from us at about 200 million light years per 10 to 15 billion years. But in 10 to 15 billion years, it will be 400 million light years away, and since it will be twice as far away, it will be moving away from us twice as fast. In other words, the local rate of expansion is constant, but distant objects move away from us faster and faster, over (very) long periods of time"

My question - since redshift is determined only by the ratio of distances - what is the
bases for the conclusion that the relative velocity between two objects will change with time - in other words, the velocity-distance law can be satisfied by assuming the relative recessional velocities remain constant and the Hubble factor H decreases linearly as the universe expands (assuming q = 0)

Sort of a simple way of explaining it is that as we look far away, we look back in time. Thus looking at the expansion rate as a function of distance tells us how the expansion rate changes in time. And that tells us whether or not there is acceleration.

 

[mikeph]

He is saying that as the separation of two objects increases, a simple Hubble law expansion (ie. a'(t)/a(t) nonzero) implies acceleration of the two objects away from each other. Nothing to do with cosmological expansion that we know about, where a''(t)a(t)/[a'(t)]^2 is nonzero (positive).

 

[Chalnoth]

He is saying that as the separation of two objects increases, a simple Hubble law expansion (ie. a'(t)/a(t) nonzero) implies acceleration of the two objects away from each other. Nothing to do with cosmological expansion that we know about, where a''(t)a(t)/[a'(t)]^2 is nonzero (positive).

H = const, however, is an exponential expansion, which is accelerating.

 

[Skolon]

If I understood right, acceleration mean that if now the value of H is 71 (km/s)/Megaparsec in future H will have an other value. True?

A value of this acceleration was calculated? What kind of acceleration is that: constant acceleration, exponential or else?

 

[marcus]

If I understood right, acceleration mean that if now the value of H is 71 (km/s)/Megaparsec in future H will have an other value. True?
...

No actually that is not true. When astronomers talk about "accelerated expansion" they are talking about a number a(t) called the scale factor. Intuitively a(t) gives a handle on the average separation between galaxies.
a(t) increasing (the first derivative a'(t) > 0) means univ. expanding.
a'(t) increasing (the second derivative a"(t) > 0) means expansion accelerating.

That has no very simple or direct connection with the Hubble ratio H(t). So you can get confused if you try to think of accelerated expansion directly in terms of H(t).

In fact in the standard cosmo model that essentially everybody uses, we have acceleration, which means a"(t) is positive and a'(t) is increasing. And also it is true that H(t) has been decreasing, and is currently decreasing, and is predicted by the model to continue decreasing indefinitely but at a slower and slower rate. H(t) is around 71 now (you can put in the units) and it is predicted H(t) -> roughly 60.

According to the latest numbers I've seen H(t) asymptotically levels out around 64 km/s per Mpc. But for several years the best estimate has been around 61. In any case it keeps decreasing forever but more and more slowly so it has that estimated limiting value.

I guess the moral is, if you want to understand the expansion process, and get a quantitative sense of what acceleration means, then get to know the scale factor a(t).

A convention that is often used in defining a(t) is to "normalize" it by setting a(present) = 1.

A value of this acceleration was calculated? What kind of acceleration is that: constant acceleration, exponential or else?

Well before we talk about the second time-derivative a"(t) let's make sure you understand the scale factor a(t) itself. Here's a little exercise to check your understanding.

Let's call the present time t₀, the time corresponding to redshift z = 0.
So t₀ is the present age of the universe.

Let's say that t₁ is the time corresponding to redshift z = 1. Suppose we point a telescope at a galaxy and measure the wavelengths and discover that they are twice as long as when they were emitted. That means z+1 = 2. The factor the wavelengths are stretched is always one more than the redshift, z+1.

So t₁ is the age of the universe when that galaxy emitted the light which is now arriving to us with a redshift z = 1, which means wavelengths twice as long.

What was the scale factor when that galaxy emitted the light?

You tell me what is a(t₁).

 

[Skolon]

I just read your answer (and question).
Because I don't use to think in terms of z (I'm not an expert in cosmology) I need more time to answer.

 

[Wallace]

Excellent summary Marcus, that should be stickied or libraried.

 

[Skolon]

Last night I was very engaged, so, just now I can come here again.

If wavelength of light is double when arrive on Earth compared with emitting wavelength, then we can say that the scale of Universe is (roughly) also double.
That mean: a(t)=2*a'(t) or a'(t)=a(t)/2. Right?

Or we can (I suppose) use z to have: a'(t)=a(t)/z.

 

[Chalnoth]

Last night I was very engaged, so, just now I can come here again.

If wavelength of light is double when arrive on Earth compared with emitting wavelength, then we can say that the scale of Universe is (roughly) also double.
That mean: a(t)=2*a'(t) or a'(t)=a(t)/2. Right?

Or we can (I suppose) use z to have: a'(t)=a(t)/z.

Almost. Rather, it would mean:

$$a(t_0) = 2a(t_1)$$

Where $$t_0$$ is the time now, and $$t_1$$ is the time when the light was emitted.

 

[Skolon]

Oh, sorry. Of course, a(t) is a function, what is changing is t (t₀ is now and t₁ was the moment of light emitting). a'(t) is the first derivative of a(t), my mistake.

Then the question is how can we calculate t₁. I suppose we must use the speed of light (c) and the proper distance traveled by light. It doesn't look to be so easy.

 

[Wallace]

You're exactly right Skolon. Measuring 'z' tells you 'a' in a straightforward way, but we cannot 'measure' the time part, at least not directly. That part depends on the whole cosmology model, which requires an interplay between the theory (based on General Relativity and other known physics) and a host of observations.

 

[Chalnoth]

Oh, sorry. Of course, a(t) is a function, what is changing is t (t₀ is now and t₁ was the moment of light emitting). a'(t) is the first derivative of a(t), my mistake.

Then the question is how can we calculate t₁. I suppose we must use the speed of light (c) and the proper distance traveled by light. It doesn't look to be so easy.

Precisely :) As Wallace pointed out, this depends upon the entire cosmological model.

So what we do instead is compare the redshift (which almost directly measures a) to some measure of distance. With supernovae, for instance, we compare the redshift of the supernova to its brightness. We then make use of the relationship between redshift and brightness for many supernovae to determine what cosmology best fits the data (as well as error bars on the cosmological parameters).

 

[Skolon]

I realize that is a very hard task.

How far are we in developing an accurate answer? I mean, we know the value of t₁ with a sensibility of billions, millions or thousands years? This is complicated I suppose by the fact that expansion is not ... I don't know how to say ... perfectly isotropic? Starting with a specific resolution of measuring, the distribution of matter (in fact of dark matter) must be take into account when we measure the expansion.

 

[Chalnoth]

I realize that is a very hard task.

How far are we in developing an accurate answer? I mean, we know the value of t₁ with a sensibility of billions, millions or thousands years? This is complicated I suppose by the fact that expansion is not ... I don't know how to say ... perfectly isotropic? Starting with a specific resolution of measuring, the distribution of matter (in fact of dark matter) must be take into account when we measure the expansion.

Well, this isn't something that is usually calculated for most cosmological observations. Usually we just stick to the cosmological parameters (such as matter density, curvature, the current expansion rate, etc.). You'd have to do some extra work in determining from the errors on these cosmological parameters what the error in light travel time is. Usually this just isn't done.

But the WMAP team was a bit more thorough, and did these calculations for the CMB, at least:
http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm_sz_lens_wmap5.cfm
(the above link is for the simplest cosmological model)

They find:
$$t_0 = 13.69 \pm 0.13 Gyr$$
$$t_1 = 380,081 \pm 5,840 yr$$
That's a measurement accuracy of about a percent. But the CMB provides exceptionally good estimates of such things, so other measures like supernovae might not be quite so good.

 

[Wallace]

Over the enormous scales we have measured the expansion, we can actually say that it isotropic. There are local fluctuations, but there is no large scale deviations from anisotropy observered. In fact, the perturbations actually help a lot in terms of tying down the cosmological model. We can work out from theory what kind of perturbations we'd expect given a set of model parameters and the observations of the flucations we see, as well as the average expansion, tell us the more likely values for the parameters.

As for the precisions, normally this is quoted in terms of the model parameters, things like the Hubble paramter today, matter density paramter today, $$\sigma_8$$ which is a way of describing the 'strength' of perturbations at present and so on. From these you could derive a value and uncertainty for the 't' corresponding to any particular 'a'. The uncertainty would be bigger the smaller the value of a, maxing out for very small a which gives you the age of the Universe. The uncertainty for this is +/- a couple of hundred million years I think on current data.

 

[Skolon]

They find:
$$t_0 = 13.69 \pm 0.13 Gyr$$
$$t_1 = 380,081 \pm 5,840 yr$$
That's a measurement accuracy of about a percent. But the CMB provides exceptionally good estimates of such things, so other measures like supernovae might not be quite so good.

Interesting. That mean that the Universe doubled its size from the (almost) its beginning to now. If t₂ is the moment when the Universe will have a double sized relative to present, what is the value of t₂?
So, I will make a small image of expansion acceleration.

 

[Chalnoth]

Interesting. That mean that the Universe doubled its size from the (almost) its beginning to now. If t₂ is the moment when the Universe will have a double sized relative to present, what is the value of t₂?
So, I will make a small image of expansion acceleration.

Much, much more.

Sorry for the confusion. The $$t_1$$ above was the time from the emission of the CMB. The CMB is at a redshift of 1089, which means our universe has expanded by a factor of 1090 since the CMB was emitted. The CMB itself also has imprints of the physics that went on before, so it most definitely expanded by many times that amount.

 

[Wallace]

Hang on, something has been waylaid. The Universe is many many times the size it was originally today. Remember those values of t tell you nothing about what 'a' was at those time, to get the connection between a and t ( i.e. find the function a(t) ) you need to solve the full cosmological model.

For instance, the t_1 mentioned, the time when the CMB photons were last scattered, occurred at around z=1110, which means the universe has gotten roughly 1100 times bigger since then.

Edit: Okay I'm out :) looks like the cross talk between Chalnoth and I posting while the other is typing is confusing things! I'll leave this to your capable hands Chalnoth...

 

[Skolon]

Expressing the cosmological distances with conventional terms is very confusing. For instance, if we say that "a galaxy is a 1 billion light years away" what this is really mean?
1. If all moves stopped (moves in space and expansion) the light will travel from that galaxy (center of it) and us for 1 billion years.
2. The light traveled 1 billion years before we observed it.
3. When light was emitted from galaxy, the distance was "1 billion light years", but because expansion the light traveled more.
4. None of above.

So, I understand why cosmologist are using z. But that doesn't say anything about topology of Universe and his dynamics. What are cosmologist using to describe that aspects of Universe?