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### March 29, 2010

#### Big Bang for beginners-12: Measuring the rate of expansion of the universe

(My latest book *God vs. Darwin: The War Between Evolution and Creationism in the Classroom* has just been released and is now available through the usual outlets. You can order it from Amazon, Barnes and Noble, the publishers Rowman & Littlefield, and also through your local bookstores. For more on the book, see here. You can also listen to the podcast of the interview on WCPN 90.3 about the book.)

For previous posts in this series, see here.

We seem to be living in a runaway expanding universe. Given that we are confined to such a tiny region of what seems like an infinite space, how can we know so much about it? It is indeed a tribute to the doggedness of the scientific endeavor that we can investigate the universe so methodically and tease out answers to questions that at first glance might seem hopelessly out of reach. In this post, I want to give some further background about how we have figured out some of this information.

For example, how do we know the speeds of distant galaxies? The speed with which a distant galaxy is receding from us can be obtained from something called the 'red-shift' of the light emitted by it.

To understand how that is done, we first need to know that each element (hydrogen, oxygen, or whatever) emits a characteristic *pattern* of wavelengths of light (denoted by the symbol λ) that is unique to it and can be measured in the laboratory. So by observing the pattern of wavelengths emitted by a star we can tell what elements that star contains. If the universe is expanding (or contracting), then between the time that the light was emitted by that distant star and the time it reaches us, space would have expanded (or contracted) and the wavelength of the light would have also increased (or decreased) because of the expansion of the space. This difference Δλ=λ(received)-λ(emitted) tells us, if it is a negative number, that the star is 'moving' towards us (i.e., space is contracting) or, if it is a positive number, that it is 'moving' away from us (i.e., space is expanding). In the former case, the light is said to be 'blue-shifted' and in the latter case, it is 'red-shifted'. The size of Δλ tells us the rate at which the space is changing.

The shift is usually measured by the quantity z, obtained by dividing the change in the wavelength of the light by the wavelength of that same line as measured in the laboratory. i.e., z=Δλ/λ. So for example if we measure a spectral line for a given element in the laboratory to be 630 nm (λ) and we measure the same line from a distant star and find it to be red-shifted to triple its value (1890 nm), then Δλ=1890-630=1260 nm and hence z=1260/630=2.0.

If space is stretched in a short interval of time, then the *increase* in separation distance of two objects embedded in space will be proportional to the distance separating them, as can be seen by our old raisin bread analogy. So the speed of separation v (obtained by dividing the increase in separation distance by the time taken) will also be proportional to the separation distance d for the two objects. This gives Hubble's law, that the speed v of a receding galaxy is related to its distance from us by v=Hd, where H is the constant of proportionality and is called the Hubble constant. (See this paper titled *The redshift-distance and velocity-distance laws*, Edward Harrison, *The Astrophysical Journal*, 403:28-31,1993 January 20.)

If the rate of expansion of the universe is constant in time (i.e., H does not change with time), it can be shown that v/c=z (where c is the speed of light), so measuring z gives us the value of the recessional speed v. Note that z can be greater than 1, so we can have speeds that are greater than the speed of light. This is not a violation of the laws of relativity because the speeds we are talking about are the speeds due to the expansion of *space* and there is no limit to that. It is the *local motion of objects relative to space* that cannot exceed the speed of light. (Note: There are different ways of defining time and distance (and hence velocity) for the expanding universe. But while these may give different values of each quantity, the basic idea holds that recessional speeds due to the expansion of space can exceed the speed of light.)

Measuring the distance to distant galaxies is much more difficult (which I will not go into) but it can be done, though it has higher uncertainties associated with it, By obtaining the values of z (and hence deducing v) and d for a large number of distant galaxies and plotting the straight-line graph with v on the vertical axis and d on the horizontal axis, we can obtain the value H from the slope of the graph.

Note that although we refer to H as the Hubble 'constant', what that means is that we use the same value for all the observable objects *at one particular time*. It is possible that the value of H is changing with time. If so, at a different age of the universe, the speeds of separation may be more or less, and for each of those times we would have to (in theory) calculate the value of the Hubble constant from the slope of the graph, though we cannot do so directly in practice because the only time we have is now, so we have to infer its variation from theory. But since the value of H can vary with time, the value for the present time is customarily written as H_{o}.

If the recessional speed v of any given galaxy has been constant over the age of the universe (i.e., the space of the universe has been expanding at a steady rate), and if all the galaxies started out together at one point in space, then v=d/T, where d is the current separation distance and T is the age of the universe. Hence by combining this with v=Hd we get the simple relationship that T=1/H. *So measuring the Hubble constant as the slope of the v-d graph immediately enables us to obtain an estimate for the age of the universe*. The current value of H is 2.37x10^{-18}s^{-1}, which gives an age of the universe that is 4.22x10^{17}seconds or 13.4 billion years.

Of course, this result depends on the assumption that the speeds of all the galaxies have been constant over the age of the universe. If the rate of expansion has been slowing down so that the speeds in the past were greater than they are now, the actual age will be less than 13.4 billion years. If the expansion has been speeding up, then the age will be greater. The current best estimates for the age of the universe place it as 13.73 (+/- 0.15) billion years.

The measured value of the red-shift z also tells us when the light was emitted by the distant galaxy, as a fraction of the time that has elapsed since the Big Bang. i.e, as a fraction of the age of the universe. The relationship is a complicated one that depends on the relative domination of matter versus the cosmological constant in the universe. As a rough approximation for a flat universe, this fraction is given by 1/(1+z)^{3/2}. So in the case of a star or galaxy that has the value of z=2.0, this fraction works out to 0.192. If we take the age of the universe as 13.7 billion years, the star must have emitted its light 2.6 billion years after the Big Bang, or 11.1 billion years ago.

The current record for the highest observed red-shift is z=8.2 from an object known as GRB 090423, where GRB stands for 'gamma ray burst' and is believed to be emitted by a dying star. A value of z=8.2 corresponds to a source that emitted its light at about 1/28 the age of the universe or about 490 million years after the Big Bang. More precise calculations place the figure at 630 million years, so we are seeing something that happened almost at the beginning of our universe.

That's all for the mathematical background (except for the post-script below). In the last few posts in this series, I will get back to the verbal descriptions.

Next: Where did all the stuff in the universe come from?

**POST SCRIPT: The Doppler shift**

At the risk of getting too much into the weeds of theory, I want to deal with an issue that is confusing about the cause of the galactic red-shifts.

The shift in wavelengths above was described as being due to the expansion of *space* itself. But the shifting of light wavelengths is normally associated with something called the Doppler effect that says that if a source of light and the detector of light are moving relative to each other *in a fixed space*, the wavelength of light measured by the detector will also be different from the wavelength of light emitted by the source. The main point to bear in mind with wavelength shifts due to the Doppler effect (when compared to the expansion of space itself) is that in this view, speeds can never exceed the speed of light.

If the source and detector are moving towards each other, the detected wavelength is shorter than the emitted wavelength (this is called a 'blue shift') while if they are moving away from each other, the wavelength gets longer (called a red-shift), which is similar to the effects due to the expansion of space.

In the case of Doppler shifts, the relationship of z=Δλ/λ to the speed v of the moving objects is given by

z=√[(1+v/c)/(1-v/c)] -1.

We can turn this around to get

v/c=(z^{2}+2z)/(z^{2}+2z+2).

So knowing the speed v, we can get z and vice versa. So for the above case of z=2.0, the speed of the galaxy is given by v/c=0.8 and thus the galaxy is moving at four-fifths the speed of light.

In the early days of cosmology, space was assumed to be fixed and the red-shift of distant galaxies was thought to be caused by the Doppler shift as they moved away in space. But now it is more common to say that the red shift is caused by the expansion of space, not the motion of objects in space, so the interpretation of z is different and its relationship to the recessional speed is different such that there is no restriction that the recessional speed be less than the speed of light.

So how do we reconcile these two views? If we want to think of the positions of galaxies changing with time, rather than space itself expanding and the galaxies fixed in space, then we can use the Doppler shift but we have to add to that the additional shift due to the photon traveling through a gravitational field on its way to us. If we do that, then the end result is the same in both cases. As cosmologist Edward Wright says:

This depends on how you measure things, or your choice of coordinates. In one view, the spatial positions of galaxies are changing, and this causes the redshift. In another view, the galaxies are at fixed coordinates, but the distance between fixed points increases with time, and this causes the redshift. General relativity explains how to transform from one view to the other, and the observable effects like the redshift are the same in both views.

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## Comments

Mano,

These posts have been very interesting. Have you ever read 'Starts With a Bang' on Scienceblogs? He has recently been writing a similar series on dark matter which I find fascinating.

Mano,

> It is the local motion of objects relative to space that cannot exceed the speed of light.

If objects can move faster than light relative to each other (through expansion of space), then how do we know objects can't also exceed the speed of light relative to space?

How do you measure your speed relative to space? (I think I asked this before, and now I'm confused again.)

Thanks,

Robert

Robert,

Imagine a ruler that is lying on your desk. You can measure the speed of an ant walking on the ruler by taking the difference in its position (as measured by the markings of the ruler at two different times) and dividing by the time difference. This speed cannot exceed the speed of light, and is what I refer to as the local motion of objects in space.

But something else could be happening, and that is that the ruler itself could be getting stretched. In that case, even a stationary ant (i.e., one who always stays at the same ruler marking) would acquire a speed simply due to the expanding ruler. It is this speed that we refer to as the speed of expansion of space itself. This can exceed the speed of light.

Of course, we cannot measure this latter speed directly with a ruler because that would require us to have another ruler that was somehow outside of our expanding space. So we measure this speed indirectly, using the red-shift.

Mano, the "porno" commenter looks like a spam bot. It's just quoting random fragments of previous comments.

Paul,

You are right. I thought it was a little weird but gave it the benefit of the doubt. I have erased it and my response.

Thanks!

Mano,

So the expansion of space is a theory to explain the observed expansion. I think I have a way to distinguish our universe from one that "exploded". A repulsive force, like an explosion, would necessarily produce a distribution of velocities with a measurable "center." You can find the center of an explosion, such as an exploding star, by plotting the gradient of the divergence of the velocities of all the particles. In our universe, by contrast, the divergence is constant at all points, the gradient is zero, and you have no distinguishable center. This is certainly an unexpected observation!

Robert,

Another way is with the cosmic microwave background radiation. If there was a center, then there would be one region in the sky that was hotter than the rest. But the radiation is remarkably uniform, whichever way we look, up to 1 part in 100,000. Hence we can infer that the explosion had no center.

Mano,

I thought of something. An image of a distant galaxy would be an image of a time when that galaxy was much closer to us and to everything else. An image of a nearby galaxy would be relatively recent, of a time when that galaxy was still far away. So, if you want to see what things looked like when they were very close to us, you have to look very far away?

I have a blog against the expansion of the universe, with arguments that show this is impossible.

How the people consider there are evidences for the Big Bang I study these evidences.

I have arguments and Hypotheses in: http://bigbangno.wordpress.com

Thanks.