Inflation and superluminal expansion

[phsopher]

Inflation is often referred to as a period of 'superluminal' or 'faster-than-light' expansion (e.g. see article on Wikipedia and hundreds of research papers on the subject). This has always bugged me. What exactly is superluminal about an inflating universe that does not apply to a non-inflating one? I mean in a dust/radiation dominated universe you also get regions which move away faster than light even though expansion is decelerating. If the universe expands at 70 km/s/Mpc then regions farther than 45 Gpc away are moving faster than light with respect to us.

So what is it that 'superluminal' is referring to? Is it that de Sitter space has an event horizon (i.e. light will never reach some regions of the universe) in contrast to dust/radiation dominated universe? I haven't really come across any explanation for the use of the term anywhere, despite its ubiquity.

 

[Mordred]

Inflation is an extremely rapid expansion that multiplied the universe roughly 60 e-folds in many orders less than 1 second. The rate of expansion is far far greater than what it is today. An e-fold is 2.71828 times the original size. Each e-fold is a separate multiplier of the last e-fold.

This is different than expansion today, in that the superluminal velocities we see are, as you say at great distance and depends on the separation distance. In inflation the volume of space is literally expanding faster than light, however as the volume is changing equally in all locations. energy-mass is distributed evenly and effectively dilutes evenly in distribution. However this is still not a violation of GR and SR for the same reasons that recessive velocity isn't.

 

[bapowell]

There is no difference between inflation and decelerated expansion when it comes to objects receding at superluminal speeds. This is a simple consequence of Hubble's Law: $v_{\rm rec} = Hr$, where $v_{\rm rec}$ is the recession velocity, $H$ is the expansion rate (Hubble parameter) and $r$ is the separation between the objects. Notice that there is a distance, $d_H = c/H$ at which objects begin to recede at light speed. This is the Hubble scale (or radius, or distance). Beyond this point, objects recede with superluminal recession velocities. All of that is true regardless of the nature of $H$ -- the rate of expansion. So superluminal recession happens in all expanding cosmologies.

So what's so special about inflation? Take a look at how the recession velocities change in time. If you know some calculus, take the time derivative of Hubble's Law, and you will find an expression for $\dot{v}_{\rm rec}$ (the recession acceleration, if you will), that is sort of like Hubble's Law:
$$\dot{v}_{\rm rec} = -H^2qr$$
Here $q$ is the deceleration parameter: it is positive when the universe is decelerating and negative when it is accelerating. When the universe decelerates, we see that $\dot{v}_{\rm rec} < 0$ -- all objects recede at a decelerated rate. This means that even though there might be objects receding at superluminal velocities today, they will eventually recede subluminally (this is why we can still receive light from objects that are receding superluminally -- because the photons they emit towards us eventually fall within the Hubble sphere and acquire a net velocity towards, rather than away from us). Now, in an accelerating universe, $\dot{v}_{\rm rec} > 0$ -- objects accelerate. Meanwhile, the Hubble radius itself only increases with a constant speed, and so this means that all objects will eventually recede with superluminal velocities in an accelerated cosmology. This implies the existence of an event horizon during inflation (in fact, the Hubble scale is the event horizon for pure de Sitter inflation), in which objects accelerate away from Earth and lose causal contact with it.

So, it's not about superluminal recession per se. It's about the long term fate of these objects: in a decelerating universe all objects will eventually become visible; whereas in an accelerating spacetime, once superluminal, always superluminal.

Sadly, the ubiquity of the term is a result of both cosmologists being sloppy with the language and bad science reporting. It sounds cool, I guess.

EDIT: I am preparing some notes on this subject specifically that I should complete in a few days. I'd be happy to share with you if interested.

 

[Mordred]

I am preparing some notes on this subject specifically that I should complete in a few days. I'd be happy to share with you if interested.

Even if the OP isn't I would be, I've always enjoyed your articles.

 

[phsopher]

There is no difference between inflation and decelerated expansion when it comes to objects receding at superluminal speeds. This is a simple consequence of Hubble's Law: $v_{\rm rec} = Hr$, where $v_{\rm rec}$ is the recession velocity, $H$ is the expansion rate (Hubble parameter) and $r$ is the separation between the objects. Notice that there is a distance, $d_H = c/H$ at which objects begin to recede at light speed. This is the Hubble scale (or radius, or distance). Beyond this point, objects recede with superluminal recession velocities. All of that is true regardless of the nature of $H$ -- the rate of expansion. So superluminal recession happens in all expanding cosmologies.

So what's so special about inflation? Take a look at how the recession velocities change in time. If you know some calculus, take the time derivative of Hubble's Law, and you will find an expression for $\dot{v}_{\rm rec}$ (the recession acceleration, if you will), that is sort of like Hubble's Law:
$$\dot{v}_{\rm rec} = -H^2qr$$
Here $q$ is the deceleration parameter: it is positive when the universe is decelerating and negative when it is accelerating. When the universe decelerates, we see that $\dot{v}_{\rm rec} < 0$ -- all objects recede at a decelerated rate. This means that even though there might be objects receding at superluminal velocities today, they will eventually recede subluminally (this is why we can still receive light from objects that are receding superluminally -- because the photons they emit towards us eventually fall within the Hubble sphere and acquire a net velocity towards, rather than away from us). Now, in an accelerating universe, $\dot{v}_{\rm rec} > 0$ -- objects accelerate. Meanwhile, the Hubble radius itself only increases with a constant speed, and so this means that all objects will eventually recede with superluminal velocities in an accelerated cosmology. This implies the existence of an event horizon during inflation (in fact, the Hubble scale is the event horizon for pure de Sitter inflation), in which objects accelerate away from Earth and lose causal contact with it.

So, it's not about superluminal recession per se. It's about the long term fate of these objects: in a decelerating universe all objects will eventually become visible; whereas in an accelerating spacetime, once superluminal, always superluminal.

Sadly, the ubiquity of the term is a result of both cosmologists being sloppy with the language and bad science reporting. It sounds cool, I guess.

EDIT: I am preparing some notes on this subject specifically that I should complete in a few days. I'd be happy to share with you if interested.

I'm aware of this. It's obvious that if the expansion is decelerating then stuff that moves away at some velocity will move away with a lesser velocity in the future, eventually transitioning from superluminal to subluminal. Conversely it is clear that if the expansion is accelerating then something moving away at a superluminal velocity will continue to move with a superluminal velocity. I don't see how this would warrant the term "superluminal expansion".

 

[phsopher]

This is different than expansion today, in that the superluminal velocities we see are, as you say at great distance and depends on the separation distance. In inflation the volume of space is literally expanding faster than light, however as the volume is changing equally in all locations. energy-mass is distributed evenly and effectively dilutes evenly in distribution.

I don't understand what you mean by this. Could you elaborate?

 

[bapowell]

I don't see how this would warrant the term "superluminal expansion".

It doesn't. It's a confused term. If anything, I would apply the word "subluminal" to inflation, as regards the growth rate of the Hubble scale.

 

[Mordred]

As Bapowell mentioned, the growth rate of expansion today is completely different, than inflation. Growth rate today is 67.3 km/s/Mpc. In inflation the exact rate and timing depends on which inflationary model so I can't give you an exact figure. However its roughly 60 E-Folds. So if the area your measuring is the same unit of scale as 1 Mpc, Multiply that number by 60 E-folds in far less than 1 second. These are not exact numbers but a very rough approximation. This google link shows it as being somewhere between -360 and minus -320 10*Log₁₀ of a second. Far far less than 1 pico second. The rough visual graph on the link should provide some clue as to how much faster.

http://en.wikipedia.org/wiki/Graphical_timeline_of_the_Big_Bang

Distances increasing by a factor of e⁶⁰ is the same as their increasing
by a factor of 10²⁶, so you can multiply our original 1Mpc by 10²⁶

 

[bapowell]

The inflationary expansion rate is given by the Hubble parameter during inflation, which, if the BICEP2 results hold, was around 10^16 GeV. I'll leave it as an exercise for the reader to convert that to km/s/Mpc.

 

[phsopher]

As Bapowell mentioned, the growth rate of expansion today is completely different, than inflation. Growth rate today is 67.3 km/s/Mpc. In inflation the exact rate and timing depends on which inflationary model so I can't give you an exact figure. However its roughly 60 E-Folds. So if the area your measuring is the same unit of scale as 1 Mpc, Multiply that number by 60 E-folds in far less than 1 second. These are not exact numbers but a very rough approximation. This google link shows it as being somewhere between -360 and minus -320 10*Log₁₀ of a second. Far far less than 1 pico second. The rough visual graph on the link should provide some clue as to how much faster.

http://en.wikipedia.org/wiki/Graphical_timeline_of_the_Big_Bang

Distances increasing by a factor of e⁶⁰ is the same as their increasing
by a factor of 10²⁶, so you can multiply our original 1Mpc by 10²⁶

I realize that the expansion rate was larger and that the universe was smaller. What I was confused by was the impression I got from your post that expansion during of inflation was somehow qualitatively different, i.e., "the volume of space is literally expanding faster than light" as opposed to "the superluminal velocities we see are, as you say at great distance and depends on the separation distance". This was the part I was having trouble with. Was your point simply that the distances at which you transition to a superluminal regime were much smaller than today?

 

[phsopher]

The inflationary expansion rate is given by the Hubble parameter during inflation, which, if the BICEP2 results hold, was around 10^16 GeV. I'll leave it as an exercise for the reader to convert that to km/s/Mpc.

Actually, I think that would be the energy density during inflation (or fourth root thereof). This would correspond to a Hubble rate of about H = 10¹⁴ GeV.

 

[bapowell]

Yep. Thanks for the correction.

 

[Mordred]

I realize that the expansion rate was larger and that the universe was smaller. What I was confused by was the impression I got from your post that expansion during of inflation was somehow qualitatively different, i.e., "the volume of space is literally expanding faster than light" as opposed to "the superluminal velocities we see are, as you say at great distance and depends on the separation distance". This was the part I was having trouble with. Was your point simply that the distances at which you transition to a superluminal regime were much smaller than today?

How do you define superluminal? If the rate of expansion between two measurement points is greater than the value for the speed of light its often termed superluminal. so say you compare expansion today, with 67.3 km/s/Mpc. set your initial distance between two particle at 100,000 metres. With the rate of change in expansion today, the change in one second will be smaller than 67.3 km. Light has no problem crossing that change in distance

Now do the same with inflation, set your initial distance apart at 100,000 metres. calculate the change in volume in one second. It will be a smaller fraction of change of 67.3 km/s. Light has no problem crossing that change in one second.

Now do the same for inflation. set your distance between two objects at 100,000 metres.

now calculate your change of distance by 60 E-folds, for simplicity sake we will simply multiply 100,000 metres and set the change at one second for a very long lasting inflation.

so the change in distance in one second is 100,000*10²⁶. Can light cross that distance in one second?

take it a step further, set your initial distance at 1 mm, then calculate the change of distance in one second for inflation

edit: please keep in mind 60 e-folds over 1 sec is a far slower rate of inflation, than 60 E-folds in less than 10^-32 seconds which is a figure from the BICEP paper I read though I don't recall if they had an e-fold value

 

[phsopher]

How do you define superluminal? If the rate of expansion between two measurement points is greater than the value for the speed of light its often termed superluminal. so say you compare expansion today, with 67.3 km/s/Mpc. set your initial distance between two particle at 100,000 metres. With the rate of change in expansion today, the change in one second will be smaller than 67.3 km. Light has no problem crossing that change in distance

Now do the same with inflation, set your initial distance apart at 100,000 metres. calculate the change in volume in one second. It will be a smaller fraction of change of 67.3 km/s. Light has no problem crossing that change in one second.

Now do the same for inflation. set your distance between two objects at 100,000 metres.

now calculate your change of distance by 60 E-folds, for simplicity sake we will simply multiply 100,000 metres and set the change at one second for a very long lasting inflation.

so the change in distance in one second is 100,000*10²⁶. Can light cross that distance in one second?

take it a step further, set your initial distance at 1 mm, then calculate the change of distance in one second for inflation

edit: please keep in mind 60 e-folds over 1 sec is a far slower rate of inflation, than 60 E-folds in less than 10^-32 seconds which is a figure from the BICEP paper I read though I don't recall if they had an e-fold value

Now I'm utterly confused. Those seem to me to be just arbitrary scales that you've chosen. I could play the same game with just the simple Einstein-de Sitter universe. The Hubble parameter is then given by H=2/3t where t is cosmic time.

So the Hubble rate today is about 70 km/s/Mpc. Consider a distance of 10^-3 parsec. In one second it will expand by 70 micrometers. Light has no trouble traveling this distance in one second.

Now consider this universe when it was 1 second old. Then the Hubble rate was about 10¹⁹ km/s/Mpc. Again take the distance of 10^-3 parsec which will now expand by about 10¹³ meters. Light has no hope of traversing this distance in one second.

Does this mean that the early stage of Einstein-de Sitter is somehow fundamentally different and deserves to be called "superluminal expansion" in contrast to the later stage? I don't see why, it's just a matter of choosing the scales appropriately.

And sure, with inflation the difference is even more dramatic, but it's not a qualitative difference. The point is that it is true of both an inflating universe and a decelerating one that there exist scales which expand superluminally (those outside the Hubble horizon) and scales which expand subluminally (those inside). I don't see why inflation is more superluminal so to speak.

I'm sorry if I'm missing your point or being otherwise thick, but I'm still confused as to what you mean.

 

[George Jones]

Inflation is often referred to as a period of 'superluminal' or 'faster-than-light' expansion (e.g. see article on Wikipedia and hundreds of research papers on the subject). This has always bugged me. What exactly is superluminal about an inflating universe that does not apply to a non-inflating one?

Many, many references, e.g., "Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe" by Davis and Lineweaver,

http://arxiv.org/abs/astro-ph/0310808

discuss superluminal expansion.

When the cosmological conventions for time and distance are used in special relativity, even special relativity has superluminal speeds.

 

[bapowell]

And sure, with inflation the difference is even more dramatic, but it's not a qualitative difference. The point is that it is true of both an inflating universe and a decelerating one that there exist scales which expand superluminally (those outside the Hubble horizon) and scales which expand subluminally (those inside). I don't see why inflation is more superluminal so to speak.

It isn't. You've answered your own question right here. People who say that inflation causes the universe to "expand at the speed of light" are in error or at least confused. That's all there is to it.

 

[phsopher]

OK, thank you. It's just that the practice of referring to inflation as "superluminal expansion" is so widespread in the literature that I wanted to make sure I wasn't missing some subtle (or even obvious) aspect of the issue that justified this.

 

[bapowell]

As long as you understand that there's an event horizon during inflation, you get the idea.

 

[Mordred]

no you have it the only difference is the scales. in the first example the distance scale is larger and rate of expansion much slower, so you need a larger separation to have a superluminal velocity. In the second case the rate is so much greater that superluminal velocity would have a much smaller distance scale. As pointed out super-luminal velocity is somewhat of a misnomer, its simply a consequence of rate of expansion and separation distance.

In your opening post you indicated you already understood this relation, so I didn't bother repeating the separation distance to recessive velocity relations.

"If the universe expands at 70 km/s/Mpc then regions farther than 45 Gpc away are moving faster than light with respect to us."

 

[phsopher]

no you have it the only difference is the scales. in the first example the distance scale is larger and rate of expansion much slower, so you need a larger separation to have a superluminal velocity. In the second case the rate is so much greater that superluminal velocity would have a much smaller distance scale. As pointed out super-luminal velocity is somewhat of a misnomer, its simply a consequence of rate of expansion and separation distance.

In your opening post you indicated you already understood this relation, so I didn't bother repeating the separation distance to recessive velocity relations.

"If the universe expands at 70 km/s/Mpc then regions farther than 45 Gpc away are moving faster than light with respect to us."

Ah ok. I interpreted your post as trying to explain why inflation should be called supeluminal. If the point is that scales where different back then then we are on the same page.

 

[abitslow]

Inflation is the fudge that Guth tossed at the problem we had back in the 60's & 70's in explaining WHY the CMB was so darned uniform (down to 1 part in 100,000). Ignoring H = (da/dt)/a that is, that expansion is time dependent, we (I should say they, I am too simple by far to contribute) needed to explain how it came to be that parts of the universe that are no longer causually connected (think thermodynamics) were able to reach (approximate) equilibrium. Guth said what if the universe "started out" near equilibrium and then disconnected (via superluminal expansion)? The rest is history. In the inflationary epoch, the distance between EVERY point in space-time is increasing at faster than c. Contrast this to the current situation where only sufficiently separated points have recessional velocities larger than c. Two totally different things. I typically call inflation "hyper-inflation" to better contrast it to the current 70 km/MPc/s 'expansion' we are in today. FWIW.

 

[bapowell]

In the inflationary epoch, the distance between EVERY point in space-time is increasing at faster than c.

No. Did you read any of the preceding discussion? This is easily seen to be false.

I typically call inflation "hyper-inflation" to better contrast it to the current 70 km/MPc/s 'expansion' we are in today. FWIW.

Why not just call it "inflation" like everyone else?

 

[bapowell]

As promised, I've prepared a note on some of the misconceptions surrounding inflation for anyone interested. I also elaborate a bit on the different kinds of cosmological horizon and their relevance to these misconceptions. pdf version is here: http://tangentspace.info/docs/horizon.pdf and html version is here: http://tangentspace.info/Articles/horizon.php. The notes are geared towards a general(ish) audience -- no knowledge of cosmology is assumed but some calculus is needed. Feel free to message me with comments/criticisms/corrections.

 

[Mordred]

Good article, one of the better ones I've read. One improvement you might consider. Your target audience is those that aren't familiar with cosmology. An old math professor used to always stress the importance in defining every variable and constant used. So any reader has the information on hand without needing to look them up. LOL I used to hate that math professor, now I can't thank him enough

Ie $\dot{v}$ , I know the answer but what does the dot mean to someone that doesn't ?

adding a lookup table would help also for the variables and constants.

other than that I don't see anything wrong in it and its nicely laid out and well explained

 

[bapowell]

Thanks Mordred! I'll add a lookup table.