Is the Universe Expanding at the Speed of Light?

The short answer to the question “Is the universe expanding at the speed of light?” is “Yes, it is!”  Or, rather, I think it does and I will, shortly, explain why.  But first have I have to give you the slightly more involved answer.

The slightly more involved answer to the question “Is the universe expanding at the speed of light?” is “No, of course it isn’t!” 

If the universe were uniformly expanding at the speed of light, we could not exist – everything would be zooming away from everything at the speed of light and that would make things difficult.  Clearly what we see in our vicinity is not receding at the speed of light – not in the least because we can see it!  We are actually moving towards our nearest neighbours.  The Canis Major Dwarf galaxy, is 25,000 light years from us (actually closer to us than the centre of our own galaxy) – and would be moving away from us at about 500 m/s – if it wasn’t being “eaten” by our galaxy, which has a notional speed of between 130 and 600 km/s in the direction of the Hydra constellation.  Andromeda, the nearest “proper” galaxy (being a spiral galaxy rather than a dwarf galaxy or a cloud), is 2,540,000 light years away and in the direction that we are moving.  If it weren’t for the fact that we are due to collide in about 3.75 billion years, Andromeda and the Milky Way would be moving apart at about 50,000 m/s – but this is not enough to overcome whatever is putting us on the collision course.

However, if we look at more distant galaxies, what we see is that the further away they are, the faster they are moving away from us.  The relationship between the distance and the speed of recession is given by the Hubble Constant.  Over the past four years there have been at least four measurements:

• 2011 (Hubble) ~71.5 to ~76 km/s/Mpc
• 2012 (Spitzer) ~72 to ~76.5 km/s/Mpc
• 2012 (WMAP – after 9 years) 68.52-70.12 km/s/Mpc
• 2013 (Planck – after four years) 67.03 to 68.57 km/s/Mpc

We can be reasonably confident, therefore, that the value of the Hubble Constant lies somewhere between 67.03 and 76.5 km/s/Mpc.  (These are the figures I used for working out how fast our neighbours “should” be moving away from us.)

The bottom line is that, if something is sufficiently far away from us, the speed of recession could be the speed of light – we can’t see things that were receding at the speed of light (relatively to us) at the time that light was emitted from it, because that light will never reach us, but we can see the light from distant galaxies that was emitted billions of years ago and it has been calculated that these galaxies are currently receding at greater than the speed of light (noting that there is a simultaneity problem associated with distant moving objects, the concept of “now” or “currently” gets a little vague when there is no causal chain to keep us on track).

This is not, however, what I mean when I say that I think that the universe is expanding at the speed of light, because we could in one sense be saying that the universe is expanding faster than the speed of light.  I don’t think that that is the case.

To explain as simply as possible, I have to work through a hypothetical and hope that the reader realises at the end that this hypothetical might not actually be that hypothetical after all.  First though, I do have to briefly explain about Planck units.

Planck units are natural units based on the properties of free space alone.  Their relationship to each other is such that, in terms of Planck units, the speed of light in a vacuum is 1, the gravitational constant is 1 and so on (see here and here for more detail).  These units are, however, awkward to use on an everyday basis.  One unit of Planck time is equal to about 5.39106×10⁻⁴⁴ seconds, while one unit of Planck length equals about 1.616199×10⁻³⁵ metres.  Planck energy units, on the other hand, are relatively huge: 1.956x10⁹J.

Let’s say, that the Planck length and the Planck time represent the smallest possible division of space and time (see Return to Constants that Resolve to Unity).  Let’s further say, hypothetically, that the universe is expanding at precisely the speed of light … in other words, that for every unit of Planck time, the universe gets one unit of Planck length larger (in terms of radius).  Putting this in a table:

Age of the Universe in Planck units	Radius   of the Universe in Planck Units
1	1
2	2
3	3
4	4
…	…
1,000,000	1,000,000
…	…
8.08x1060	8.08x1060

This isn’t terribly complicated, I’m just adding one unit to time and one unit to the radius.  However, we can look at this in a slightly more complicated a way.

For every unit of Planck time, we add 1/n units of Planck time for each unit of Planck length that exists.  I’ll put this in a table to explain:

# of Planck Time units	Increment/Planck Length	# of Planck Length units	Increment
1	1	1	1
2	1/2	2	1
3	1/3	3	1
4	1/4	4	1
…	…	…	…
1,000,000	1/1,000,000	1,000,000	1
…	…	…	…
8.08x1060	1/8.08x1060	8.08x1060	1

This means, that for every unit of Planck length (1), we add one unit of Planck length (1), divided by the number of units of Planck time (N_pl).  This means that the rate of expansion would be, at any given time:

H_o     = 1 / N_pl (in units of Planck length, per unit of Planck time, per unit of Planck length)

Let’s say, however, that we wanted to know how many units of Planck length would be added for another unit of length.  How about if we used a megaparsec (Mpc)?  There are 1.91x10⁵⁷ units of Planck length in a megaparsec.  This would result in:

H_o     = 1.91x10⁵⁷ / N_pl (in units of Planck length, per unit of Planck time, per megaparsec)

Let’s say that we wanted to know the rate of expansion at 8.08x10⁶⁰ units of Planck time into the expansion of the universe:

H_o     = 1.91x10⁵⁷ / 8.08x10⁶⁰ (in units of Planck length, per unit of Planck time, per megaparsec)

       = 2.36x10^-4 units of Planck length, per unit of Planck time, per megaparsec

This is interesting, but the units aren’t particularly accessible, so let’s convert the Planck units to kilometres and seconds:

H_o     = 2.36x10^-4 * (metres per unit of Planck length) * (kilometres per metre) / (seconds per unit of Planck time)

       = 2.36x10^-4 * 1.616199×10⁻³⁵ * 0.001 / 5.39106×10⁻⁴⁴

       = 70.75 km/s/Mpc

This might be a familiar number.  This is partly because 8.08x10⁶⁰ is the current age of the universe (13.8 billion years).  In other words, if the universe is expanding at a rate of one unit of Planck length per unit of Planck time, we would expect to see a Hubble Constant of … pretty much exactly what we measure it to be.

Now, I’m not merely saying that some distant edge of the universe is moving away from us at a rate of one unit of Planck length for each unit of Planck time, because no matter what direction we look, we see the same rate of expansion for objects at the same distance.  The implication is that the added space appears to be evenly distributed throughout the universe, very much like the whole “fabric of space-time” were being stretched which is consistent with the concept that space-time is expanding as would the rubber of an inflating balloon.

If my “hypothetical” is right, then we should be able to determine the age of the universe from the Hubble constant and/or the Hubble constant from the age of the universe because they are the reciprocal of each other.  Of course, neither are particularly easy to tie down, so we might just find that the values are close but not identical.  If we find that the Hubble Constant and the reciprocal of the age of the universe  truly diverge (as in not just the measurements of the related values), then naturally my “hypothetical” won’t hold any water.  It would, however, be interesting to see if it is consistent with other observations.