Speed of light during inflation

[Gerinski]

It is often said that although the expansion of the universe may create the illusion of objects in very far regions of the universe to recede from each other at superluminal speed, that does not violate c because locally they never move faster than c. This I can understand.

How would things look like during the inflationary period is more confusing to me though. The rate at which space expanded was huge even in a local context. Without caring to do any math, the space separation which at a certain moment was, say 1,000 km, after a few seconds would have become hundreds of thousands or billions of kilometers, as 'new kilometers got pumped in between'. Even locally, if you tried to measure the speed at which objects receded from each other (ok there were no 'objects' yet but go along with me), you would get apparent superluminal speeds, wouldn't you? (or more precisely, you would not be able to measure it because all the neighbour 'objects' would just disappear from sight with the space between you both stretching faster than the distance the emitted photon traveling at c towards you could travel). Although this would not really violate c either, the effect if it could have been measured would be that even locally things appeared to be getting apart from each other 'faster than c'. Or not?

Thanks!

 

[phinds]

Yes, things would then, as now, have been receding from each other faster than c, it's just that as you point out it would have been happening at smaller scales. Right now things at the edge of our observable universe are receding from us at about 3c but they are about 47BILLION light years apart.

During inflation, things very close together would have been receding from each other at that speed.

Still, as you say, no speeding tickets would be issued in either case.

 

[Gerinski]

Thanks,

So this means that (in the hypothetical case that the universe was transparent and photons could have traveled freely at c, and that there could have been any observer) any observer during the inflationary period would have just seen complete blackness around him, and effectively would have perceived that his own point was the complete existing universe, right?

 

[Mordred]

Thanks,

So this means that (in the hypothetical case that the universe was transparent and photons could have traveled freely at c, and that there could have been any observer) any observer during the inflationary period would have just seen complete blackness around him, and effectively would have perceived that his own point was the complete existing universe, right?

not necessarily we do not know if the beginning of the universe was finite or infinite. The observable portion in your example would be finite. However that does not necessarily mean the entirety of the universe

 

[Gerinski]

not necessarily we do not know if the beginning of the universe was finite or infinite. The observable portion in your example would be finite. However that does not necessarily mean the entirety of the universe

I mean, 'he' would not have been able to observe anything at all, so from his point of view there would be nothing else than his own point (even if with hindsight we know that there were other 'things', just that he could not observe them as they were becoming causally unrelated).

'we do not know if the beginning of the universe was finite or infinite'

What do you mean? we know that (shortly after the BB) it was spatially finite, don't we? anything that grows in size is by definition spatially finite at any given time (a different thing being that it may be infinite in time forwards, i.e. it may expand forever).

 

[bapowell]

I mean, 'he' would not have been able to observe anything at all, so from his point of view there would be nothing else than his own point (even if with hindsight we know that there were other 'things', just that he could not observe them as they were becoming causally unrelated).

No, he would see things that were within his causal horizon (which includes more than just his point, but all points within a radius $\propto H^{-1}$, where H is the Hubble parameter). Photons emitted from sources within his horizon would be visible to him as long as the source remained inside his horizon. This is possible as along as the source is not "comoving" with the expansion and moving at a speed sufficient to overcome the recession velocity imparted to it by the expansion.

 

[Gerinski]

Well, in a quick google search I could not find a precise figure for the rate of space expansion during inflation in order to calculate which sphere radius would have remained within his causal horizon but I guess that it would have been limited to a really tiny 'bubble' of space around 'him'.

The only quote I could quickly find is this (rather simplistic and possibly unprofessional one) which states that 'space itself expanded faster than the speed of light'

http://www.space.com/52-the-expanding-universe-from-the-big-bang-to-today.html

 

[bapowell]

The only quote I could quickly find is this (rather simplistic and possibly unprofessional one) which states that 'space itself expanded faster than the speed of light'

Yeah, that's a common and frustrating misinterpretation of inflation. It's true -- space does expand at a rate during inflation such that objects at a proper distance $\propto H^{-1}$ have superluminal recession velocities. But that's true of any expansion -- accelerated or otherwise! The key difference is that the Hubble radius expands faster than the expansion in standard, non-accelerated cosmologies, so that objects that might be receding superluminally today will not be in the future. During inflation, in contrast, the space expands more quickly than the Hubble radius, and so once objects are outside, they don't return. In the limit that the inflation is driven by a pure cosmological constant (so-called de Sitter expansion), the Hubble radius is coincident with the event horizon of the spacetime.

 

[Gerinski]

Yeah, that's a common and frustrating misinterpretation of inflation. It's true -- space does expand at a rate during inflation such that objects at a proper distance $\propto H^{-1}$ have superluminal recession velocities. But that's true of any expansion -- accelerated or otherwise! The key difference is that the Hubble radius expands faster than the expansion in standard, non-accelerated cosmologies, so that objects that might be receding superluminally today will not be in the future. During inflation, in contrast, the space expands more quickly than the Hubble radius, and so once objects are outside, they don't return. In the limit that the inflation is driven by a pure cosmological constant (so-called de Sitter expansion), the Hubble radius is coincident with the event horizon of the spacetime.

Thanks, so, if I understand this well, it is true that during inflation space even in your close vicinity expanded at such a rate that any 'objects' in it would recede from you at superluminal velocity, right?
Is there any calculation of what would have been the maximum radius around you which remained within your causal horizon during the fastest inflation period?

 

[Mordred]

Thanks, so, if I understand this well, it is true that during inflation space even in your close vicinity expanded at such a rate that any 'objects' in it would recede from you at superluminal velocity, right?
Is there any calculation of what would have been the maximum radius around you which remained within your causal horizon during the fastest inflation period?

during the inflationary epoch the universe was too hot to have matter, so your left with energy/radiation. Matter formed later so you wouldn't have objects as per se. Unfortunately one could calculate the rate of inflation, however their is no agreement on which inflationary paradigm is correct. Each inflationary model has differences in the number of e-foldings some may coincide but that's more coincidence than design. Some models say 60 e-folds other models say higher or lower.

One of the reasons we do not know which is correct is that we cannot observe this period of time, we simply cannot see that far back due to the lack of transparency of the plasma state. So any values we have are based on calculations of what we do know. Related calculations is the number of e-folds to solve the flatness problem and the horizon problem.

 

[bapowell]

Thanks, so, if I understand this well, it is true that during inflation space even in your close vicinity expanded at such a rate that any 'objects' in it would recede from you at superluminal velocity, right?

No, only objects outside the event horizon, approximately $d \sim H^{-1}$ away, must move superluminally. You can figure out exactly what this distance is if you know the energy scale of inflation, which has an upper limit around $H^2 = 10^{16}$ GeV or so and a lower limit set by about the scale of nucleosynthesis.

 

[bapowell]

Each inflationary model has differences in the number of e-foldings some may coincide but that's more coincidence than design. Some models say 60 e-folds other models say higher or lower.

This is the amount of inflation -- not the rate. The rate is set by the energy scale, which is also not known. But, it's possible to get a sense of the size of the horizon by calculating what it should be for reasonable energy scales.

 

[Mordred]

This is the amount of inflation -- not the rate. The rate is set by the energy scale, which is also not known. But, it's possible to get a sense of the size of the horizon by calculating what it should be for reasonable energy scales.

good point not sure why I stated rate wasn't thinking clearly.

 

[phinds]

Thanks,

So this means that (in the hypothetical case that the universe was transparent and photons could have traveled freely at c, and that there could have been any observer) any observer during the inflationary period would have just seen complete blackness around him, and effectively would have perceived that his own point was the complete existing universe, right?

My understanding is that the plasma was still dense enough during inflation that any hypothetical observer would have had his hypothetical *** fried off. It wasn't until 400,000 year AFTER inflation that the plasma cooled enough to allow light to disperse instead of constantly banging into things. Google "surface of last scattering"

 

[bapowell]

Inflation is essentially a supercooled phase transition -- the temperature is that of the vacuum during inflation.

 

[phinds]

Inflation is essentially a supercooled phase transition -- the temperature is essentially that of the vacuum during inflation.

Oh. Boy, did I have THAT wrong. Thanks.

 

[Gerinski]

My understanding is that the plasma was still dense enough during inflation that any hypothetical observer would have had his hypothetical *** fried off. It wasn't until 400,000 year AFTER inflation that the plasma cooled enough to allow light to disperse instead of constantly banging into things. Google "surface of last scattering"

Well, that's why I said 'hypothetical', meaning 'as a thought exercise even if historically not applicable'. This does not invalidate the question as a matter of principle. The question refers to the expansion of spacetime during the inflationary period, irrelevant of what was occupying it at that particular moment.

 

[marcus]

... the energy scale of inflation, which has an upper limit around $H^2 = 10^{16}$ GeV or so...

I glanced at a paper of Liddle where he was estimating the energy scale of inflation and saw and estimate of H ≈ 10^-6 m_planck

The basic unit of H is frequency (i.e. reciprocal time) so it makes sense to state H in either energy terms or mass terms. I think Liddle's figure was not an upper limit, but rather was what he judged sufficient to get 60 efolds.

Since Planck mass is about 10¹⁹ GeV, that would make Liddle's estimate around
H ≈ 10¹³ GeV

It would be nice to have a source for that upper limit figure that you give, if there's an arxiv link and it's not too hard to understand. As I recall you've published some about early universe/inflation yourself, but it would save fumbling around to have a link. The Liddle paper I looked at is over ten years old.

 

[Mordred]

this paper provides one estimate

http://arxiv.org/pdf/1008.5258v3.pdf "Observational constraints on the energy scale of inflation"

In this paper we have placed observational constrains
on the potential energy scale, the first and second deriva-
tive of the potential by using the 7-year WMAP data,
combined with the latest distance measurements from
the baryon acoustic oscillations in the distribution of
galaxies and measurement of the present-day Hubble con-
stant from supernova data. A previous upper limit from
the first WMAP data release, combined with large scale
structure data from the 2dF galaxy redshift survey found
V₀^1/4<∼2.7 × 10¹⁶ GeV at 90% C.L. [7, 8]. Our new up-
per limit on the energy scale of inflation is only slightly
stronger V₀ ^1/4<∼2.3 × 10¹⁶ GeV at 95% C.L., and shows
a degeneracy with the upper limit on the first derivative of the inflaton potential,
V ₁³¹<∼2.7 × 10¹⁵ GeV at 95%

this article covers this as well but is not easy to find the info nor understand partly due to its extreme length.
http://arxiv.org/abs/1303.3787 "Encyclopædia Inflationaris" it utilizes the slow roll approximation to compare alternate inflationary models and either contraints them or invalidates (in rare cases).
The article states that over 64 models are still viable.

 

[Mordred]

Found this paper covering constraints on inflation according to the latest Planck results

http://arxiv.org/abs/1303.5082

in the conclusions it places the upper energy-scale at 1.9*10¹⁶ Gev Planck+WMAP

 

[bahamagreen]

I don't know how to estimate it by energy, but just for fun I'll attempt a back 'o the napkin approach (and I'm sure it's wrong, too).

per Wiki,

"In physical cosmology, cosmic inflation, cosmological inflation, or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least (10^78) in volume, driven by a negative-pressure vacuum energy density.[1] The inflationary epoch comprises the first part of the electroweak epoch following the grand unification epoch. It lasted from (10^−36) seconds after the Big Bang to sometime between (10^−33 and 10^−32) seconds. Following the inflationary period, the universe continued to expand, but at a slower rate."

So a volume expansion factor of ^78 in ^-34 seconds...

So how small a bubble of space will continue to have any possible exterior interactions during this epoch?

Ri = Initial bubble radius before epoch
Rf = Final bubble radius after epoch (Ri^26)
(I'm using a radial expansion factor of ^26 as an approximation of the cube root of ^78...)
Tp = Period of epoch (^-34 s)
Tc = Time for light to travel Rf-Ri

Let's start by looking for Ri such that Tc=Tp
This is the initial bubble radius for which light speed and bubble radial speed are matched. For any interactions to occur during the epoch, the bubble will need to be equal to or smaller than this.

Let Rf-Ri = Rf since Ri is really small
So Rf is distance at c after Tc, which took Tp s.
Rf = 3^-26 m
So Ri = 4^-39 m This is the "borderline" bubble size

Initially, this suggests to me that unless the starting bubble radius is smaller than 4^-39 m prior to the expansion epoch, there will be no possible interaction or exchange with the exterior of the bubble - the rest of the universe seems "gone" during this epoch, at the end of which the universe appears locally and begins to build out radially.

Also, for smaller initial bubble radii, there will be a point within the epoch after which the outside universe appears locally and builds out radially prior to the end of the epoch, and for larger initial bubble radii there is a refractory period after the epoch where the exterior universe does not seem to exist for a while, then begins to appear locally and builds out radially.

So it appears that with small enough initial bubble radii, the point in time at which the emergence of interactions and exchanges may be allowed can be pushed back indefinitely approaching the onset of the epoch... in thinking about this, there are problems.

The rate of expansion is so great I'm pretty sure local frame dragging or something like that is going to be confounding the simple calculations of local light speed.

The calculation implies the emergence of interaction and exchange possibilities but the scale of inter-actors and exchangers themselves is not considered. The borderline initial bubble of 4^-39 m radius does not reach the Planck length until well into its expansion, so the availability of the universe with which to interact at scales smaller than this begs the question of what is available to mediate these interactions, and among what would these interactions be mediating?

 

[Gerinski]

That was very interesting thanks!
It changed my intuitive thought that inflation occurred when the universe was bigger than it actually was. The size and time scales are so mind-bogglingly small that as you say it's hard to comprehend whether even a concept such as the speed of light could have any meaning at all.

 

[bapowell]

The upper bound on the energy scale of inflation, $V$, is established empirically by the presence of a tensor perturbation in the CMB. These perturbations are the result of gravitational waves generated during the inflationary expansion. In short, the energy needed to be sufficiently high to create them. The expression, which makes use of the slow roll approximation, is:
$$V^{1/4} \approx r^{1/4}(4\times 10^{16})\, {\rm GeV}$$

The tensor-to-scalar ratio, $r=A_T/A_S$, is the conventional way to report the amount of tensor perturbation seen in the CMB: it's given in proportion to the larger scalar (or density) perturbation amplitude, $A_S$. The constant value $4\times 10^{16}$ is set by the scalar amplitude.

The latest Planck results give $r < 0.1$ for power law spectra at 95% CL.

I'd be happy to further discuss the origin of the upper bound expression if anyone is interested.

 

[marcus]

I'd be happy to further discuss the origin of the upper bound expression if anyone is interested.

What I'm wondering about is probably much less technical, more straightforward. What I saw in that paper by Liddle was an estimate of H at the start of inflation: the Hubble rate. I liked how he expressed it, in terms of Planck mass---which amounts to giving it in terms of Planck frequency.

Or, taking reciprocals, he was telling me the Hubble time in terms of the Planck time unit.

That paper by Liddle was from the 1990s. What I'm wondering is what would an estimate of Hubble rate would look like now?

The specifics don't matter so much, as it happened Liddle used symbols H₆₀ and H_end for H at start and H at the end of inflation, assumed adequate to get 60 efolds. As I recall he said H₆₀ ≈ 10^-6 m_Planck and H_end ≈ 10^-7 m_Planck. But I'm just looking for a rough handle on the inflation era Hubble rate that is more up-to-date than something from the 1990s.

 

[eloheim]

Resonaances: Planck About Inflation (04/20/13)

The above blog post at Resonaances discusses constraints for models of inflation from the new Planck data.

Okay, made me laugh:

The potential should be almost but not exactly flat, so that the scalar field slowly creeps down the potential slope; once it falls into the minimum inflation ends and the modern history begins. Clearly, that sounds like a spherical cow model rather than a fundamental picture. However, the single-field slow-roll inflation works surprisingly well at the quantitative level...{snip}...One can say that the slow-roll inflation is like a spherical cow model that correctly predicts only the milk yield, but also the density, hue, creaminess, and even the timbre of moo the cow makes when it's being milked.

The conclusion really caught my eye though, especially as applicable to the discussion at hand:

So, the current situation is interesting but unsettled. However, the limit r≲0.11 may not be the last word, if the Planck collaboration manages to fix their polarization data. The tensor fluctuations can be better probed via the B-mode of the CMB polarization spectrum, with the sensitivity of Planck often quoted around r∼0.05. If indeed the parameter ε is not much smaller than 0.01, as hinted by the spectral index, Planck may be able to pinpoint the B-mode and measure a non-zero tensor-to-scalar ratio. That would be a huge achievement because we would learn the absolute scale of inflation, and get a glimpse into fundamental physics at 10^16 GeV!. Observing no signal and setting stronger limits would also be interesting, as it would completely exclude power-law potentials. We'll see in 1 year.

Sounds exciting, no?!

 

[bapowell]

But I'm just looking for a rough handle on the inflation era Hubble rate that is more up-to-date than something from the 1990s.

In order to determine what the Hubble rate was at a particular time during inflation, we need to know the tensor amplitude on the corresponding scale in the CMB. The tensor amplitude is known to be, to lowest order, $P_T(k_0) = 16H(\phi_0)^2/\pi$. So, until we have a better understanding of the tensor contribution, we will always have only an upper bound on the energy scale.

 

[marcus]

..., until we have a better..., we will always have only an upper bound ...

I'm good with upper bounds. A reasonable upper bound for H (say around the start of inflation) would be nice to have.

Liddle in 1990s gave a figure for H which was about 10^-6 m_Planck.

What I'm wondering is, is that still an acceptable figure, say as upper bound? One problem I have is that I'm unsure how to translate from a bound on V to a bound on H. So quotes about V don't cure my curiosity. I want to know what the experts think a reasonable upper bound on H is (at start or some other benchmark time they use).

For definiteness let's write H_max for the upper bound.

If it actually turned out that current experts reckon H_max = 10^-6 m_Planck,

then I could say that the reciprocal, the Hubble time, had a LOWER bound of 10⁶ t_Planck.

 

[bapowell]

What I'm wondering is, is that still an acceptable figure, say as upper bound? One problem I have is that I'm unsure how to translate from a bound on V to a bound on H. So quotes about V don't cure my curiosity. I want to know what the experts think a reasonable upper bound on H is (at start or some other benchmark time they use).

I'll have a look again at Liddle's paper, but my strong feeling is that this estimate has not changed all that much -- it's been tracking our experimental constraints on tensors. I'll get back to you on this though (too nice of a day outside to read a physics paper ;))

As for the translation from $V$ to $H$, during slow roll they are directly related through the Friedmann equation:
$$H^2 = \frac{8\pi}{3 m_{\rm Pl}} V.$$

Slow roll is likely to be a very good approximation of inflationary physics when observable scales are leaving the horizon.

 

[marcus]

I'll have a look again at Liddle's paper, but my strong feeling is that this estimate has not changed all that much -- it's been tracking our experimental constraints on tensors. I'll get back to you on this though (too nice of a day outside to read a physics paper ;))

As for the translation from $V$ to $H$, during slow roll they are directly related through the Friedmann equation:
$$H^2 = \frac{8\pi}{3 m_{\rm Pl}} V.$$

Slow roll is likely to be a very good approximation of inflationary physics when observable scales are leaving the horizon.

Great! I think I can save you the trouble. I just checked Figure 2 on page 4 of
http://arxiv.org/pdf/1209.1609v2.pdf

And it gave Ashtekar's idea of the energy density at onset of inflation in terms of
ρ_Planck.

If I have an energy density I can just multiply by 8piG/3 to get H², or use the Friedman eqn proportionality directly.

According to Ashtekar energy density at onset of inflation is about 10^-11 ρ_Planck.

In the LQC picture the Hubble time shortly after bounce is ≈ t_Planck, so that would make the Hubble time at slowroll onset 10^11/2 t_Planck.

Between bounce and onset inflation, the Hubble rate would have declined by roughly 5.5 orders of magnitude.

This agrees with what Liddle said. His Hubble rate at onset inflation is around 10^-6 times the Planck frequency. Not essentially different from what Ashtekar said in 2012.

You were right to suspect that the estimate had not much changed.

 

[Mordred]

I'll have a look again at Liddle's paper, but my strong feeling is that this estimate has not changed all that much -- it's been tracking our experimental constraints on tensors. I'll get back to you on this though (too nice of a day outside to read a physics paper ;))

As for the translation from $V$ to $H$, during slow roll they are directly related through the Friedmann equation:
$$H^2 = \frac{8\pi}{3 m_{\rm Pl}} V.$$

Slow roll is likely to be a very good approximation of inflationary physics when observable scales are leaving the
horizon.

Ah that's what that equations for I've cone across it a few times but until now never understood it lol

Every paper I usually come across still places slow roll as a good fit.
Anyways thanks for the useful information your supplying on this thread.