Louise Riofrio recently pointed out that the inflation is in a bit of trouble due to the fact that it predicts a different curve than the one seen for the angular correlation of anisotropies of the Cosmic Microwave Background (CMB). An easily understood review of the CMB is given by The Cosmic Microwave Background for Pedestrians: A Review for Particle and Nuclear Physicists, astro-ph/0803.0834. The data excludes the curve expected by inflation at well above the 99% level:

Previously, Louise had explained the anomaly in a manner that I was too obtuse to understand, for example:

Views of the Cosmic Microwave Background may also indicate a spherical Universe. By measuring distances between acoustic peaks, scientists hope to complete a triangle and determine curvature. When a changing speed of light is accounted for, the angles do not add up to 180 degrees and the triangle is not flat. Most telling, the scale of density fluctuations is nearly zero for angles greater than 60 degrees. Like a ship disappearing over Earth’s horizon, the lack of large-angle fluctuations is smoking-gun evidence that the Universe is curved. Both lines of CMB data indicate that the curvature has radius R = ct.

And, in the comments of this post:

Mendo, I dearly wish for the Space/Time to answer your questions properly. Both relate to density fluctuations grown large by expansion. These can be modelled as Fourier series. The old inflationary paradigm says that fluctuations should be the samne at all scales, leading to a prediction corved ruled out by WMAP and COBE.

Theory predicts that the largest possible fluctuation has half-wavelength of 180 degrees. On the chart, these fluctuations cancel each other out above 60 degrees.

I didn’t find these very convincing, or even understandable. So I sat down to work out the CMB fluctuations under her model myself, from first principles, and sure enough, it does work! Quite nicely! Here’s the calculations:

**The Riofrio Variable Speed of Light**

Given the speed of light c, and age of the universe t, the observable universe has size R = ct. The usual is to think of this as R(t) = c t, that is, the observable universe increases as the universe ages. Riofrio’s insight was to put the time dependence onto the speed of light instead: c(t) = R/t. There’s a bunch of highbrow general relativity calculations to support this which you can find on her blog, and do not concern us here.

The reason I find this version of cosmology attractive is that it means that spacetime is stable. This is compatible with the flat space version of GR that was found by the Cambridge Geometry Group, (which uses Clifford algebra) and about which I need to write a longer post, and to improve my website on the subject. I want to redo it in a faster loading, simpler, version like my density matrix website with lots of literature references.

A flat space version of GR avoids the science fiction problems of the standard GR, and it also makes it natural that current space appears to be very close to flat. The usual gravity theories have difficulty explaining how it comes to be that the universe hasn’t collapsed into a little ball or whatever. Similarly, Riofrio’s cosmology is clean and simple. Space is big, flat, and matter doesn’t mess it up much.

There’s a great deal of modern commentary to the effect that variable speed of light theories are pointless because the speed of light can always be assumed to be constant, so light itself defines length. They see the speed of light as simply being the conversion ratio between distance and time. But there is nothing unnatural in having a conversion ratio be non constant. And most significantly, c(t) = R/t is a very simple equation. **This is far far far more restrictive than what one can do while keeping the speed of light constant but allowing the metric of spacetime to vary**. So Louise’s idea has fewer arbitrary constants or assumptions.

And Louise’s cosmology **is a lot easier **to make calculations with. Most particularly, in the flat space version of Louise’s cosmology, angles are trivial to compute. This makes it particularly easy to compute the angular dependency of the correlations of the CMB anisotropies.

**World’s Easiest Cosmology Calculation**

Suppose we travel at speed v(t) = kc(t) = kR/t, where k is a constant, from time t = a to t=b, in the direction x. In Louise’s model, the calculation is easy to make:

The distance travelled depends only on the ratio of the beginning and ending time, and is logarithmic.

The CMB light comes to us from the time of “recombination,” by which word the cosmologists mean the time at which ionized matter combined to form hydrogen, helium, and “metals”. Before that time, the universe was largely opaque to light because it was scattered by all the free charges. After that time, the universe was largely transparent to light. This happened at around 380,000 years after the big bang, about 14,000,000,000 years ago.

Since the universe has been transparent since the time of recombination, we can see all the way back to that time, and the CMB is what we see. Furthermore, when we look in different directions, what we see is the recombination as it happened approximately 13,999,620,000 years ago [LOL]. That means that the recombinations we see at different directions now were once very widely separated. How far apart were they? Recombination light travels at speed c(t) so k=1. Using the above integral, the distance the recombination light has travelled is:

When we are looking at recombination light coming from two different directions with an angle between them of , this gives the distance between them as as can be seen from trigonometry:

**Acoustic Waves in Plasma**

In order for the two recombination regions to be correlated, they must have been in causal connection. When you read the cosmology literature, you find that before recombination, the universe was filled with ionized gas, a plasma. In this plasma, matter and energy moves around with “acoustic waves”. The speed of these waves is approximately as is explained in section 1.6 of The CMB for Pedestrians.

These acoustic waves are just like sound waves that we’re all familiar with. They tend to move stuff away from where it’s crowded, towards where it’s not crowded. Now suppose the universe was anisotropic at some early time s << 380,000 years. And suppose that there was a crowded spot over here, and a less crowded spot over there. If the distance between these two spots is short enough that the acoustic wave could get between them, then they can be correlated. If it’s longer than the distance sound can go between s and 380,000 years, then the two regions will be uncorrelated.

If the two spots are very close together, sound can pass back and forth between them multiple times. This will tend to even out there differences so we will see a positive correlation. But that correlation will turn negative if they are far enough apart that sound can only make one trip. This is all explained in section 1.6 of the above reference where they talk about multiple poles. The pole we’re concerned with here is the biggest one, the l=1 pole.

**Inflation**

Before the theory of inflation, it was assumed that the CMB would not be well correlated. When it was discovered that the CMB is very highly correlated in all directions, inflation was brought in as an explanation for the correlation. The inflation assumption was that distant parts of the universe were once in causal contact and this was when the regions became correlated. Later, as the universe expanded, the correlations remained. The inflation was supposed to have happened at some very very early time in the big bang, seconds, (which is conveniently before our physics theories would have to fall on their swords).

So far so good. But in requiring inflation to happen so early, they were able to calculate that the maximum angle in which the CMB would be correlated would be about 1 degree. You can move this number around by assuming various degrees of curvature in the universe. But the beauty of Riofrio’s theory on a flat space is that one does not need to do this. But to get the fact that the CMB is quite nearly isotropic, we do need to add another epoch.

**Corrleation Angles and 2nd Recombination**

So there are two problems in the CMB correlations. The first is that the data are close to isotropic. The second is that the very small anisotropies disappear beyond an angle of 60 degrees. To explain this data, we will suppose that the universe had three epochs. The first is a wide period where everything pre plasma happens. We’ll call it the “snuark epoch”. In this scheme, the overall high isotropies are caused by the very long duration snuark epoch, while the small anisotropies arise from the plasma epoch. The speeds of causality (i.e. how matter moves around in bulk) in the three eras are:

(1) The snuark epoch, from the Planck time, seconds to s years. Snuark acoustic waves move at speed the speed of light, or some high fraction of it, i.e. divided by the square root of 3.

(2) The plasma epoch, from s = 3 minutes to 380,000 years. Plasma acoustic waves carry information at speed .

(3) The dark epoch, from 380,000 years to present around 14 billion years. Light carries information at speed .

The ratio of the time durations of the snuark epoch will be sufficiently long, compared to the ratio of times since recombination, that this period would let its acoustic waves die down resulting in the very isotropic overall distribution of CMB radiation. Since this period is so isotropic, we can’t easily verify its duration. However, for the plasma epoch, we can verify s from the approximate maximum angle with nonzero correlation, around 60 degrees.

**The Photon Epoch**

When the temperature of the universe drops low enough that the snuarks recombine, the effective speed of causality drops, and this sudden decrease in the speed of sound allows the remaining failures of correlation to appear to us now in the above graph. According to the simplest cosmology trig calculation ever, the distance between two barely correlated spots in the recombination plasma is:

We make this equal to the distance that causality can travel during the plasma era, that is, and get:

or

years = 0.0048 years or about 1.75 days, which is kind of late, Wikipedia gives the photon epoch as lasting from 3 minutes to 380,000 years, so let’s reverse the calculation; we’ll use 3 minutes and compute the maximum correlation angle.

According to the Wikipedia timeline of the big bang, the photon epoch lasted from about 3 minutes to 380,000 years. Putting s at 3 minutes gives ln(380,000 years / 3 minutes ) = 25. Multiplying by sqrt(1/3) gives 14.4, and fitting that into the world’s simplest cosmology trig equation gives the maximum correlation angle as:

degrees

so theta should be about 86 degrees.

So really, for a back of the envelope calculation, this isn’t too far off. In fact, looking at the chart at the top of this post, 86 degrees looks about right. Victory!

**The Varying Speed of Light**

Given a flat background space, a natural interpretation of a varying speed of light is to suppose that the rate at which time passes is changing. Our clocks measure time based, in the final analysis, on the de Broglie frequencies of matter waves. This was discussed in the previous post, David Hestenes Electron Model, which suggested that the source of that frequency is the rate at which the left and right handed portions of particles oscillate back and forth.

In order to arrange for the speed of light to follow the pattern c(t) = R/t, we could assume that there is a particle which is present throughout the universe which stimulates the oscillation. As time goes on, the density of these particles steadily increases and this causes the oscillations to increase, and therefore our clocks to speed up, and therefore our measurement of the speed of light (relative to the scale of the universe as a whole) to increase.

On the quantum mechanical level, this sort of thing, a stimulated oscillation between left and right handed states, can be modeled by the same methods used to calculate the stimulated emission of radiation in lasers. I’ll eventually write a post showing how this can be used to calculate the Painleve gravitational field, at least to first order post Newtonian.

I have already earlier spoken about interpretation of variable c in TGD framework. About mass formula I cannot say anything.

The variable light velocity c_# can be defined in terms of M^4_+ proper time a to get from A to B along space-time sheet.

Since space-time sheets are in general bumpy, c_# < c holds true in general since it takes longer time to travel along lightlike geodesic of space-time sheet than that of imbedding space.

For Roberston Walker cosmologies one would have

a= Int c_#dt

rather than a=c_#t. a corresponds to the scale factor R.

Both imbeddable critical and over-critical cosmologies as well as string dominated cosmology with mass density going like 1/a^2 (M^4_+ proper time a corresponds to R) approach near initial singularity a situation in which and c_#= sqrt(g_aa} approaches to very small constant. One would thus have a= c_#t either exactly or in a good approximation for sufficiently early times.

For these cosmologies horizon radius is infinite which was one of the the original motivations for them. In TGD framework the finite size of space-time sheets is possible reason for the angle cutoff.

Hey Carl, this is brilliant! One question: do the parameters (14By, 380ky, 180s) depend on assuming c=c(t) in any way ? In other words, are they computed by assuming c is a constant ?

Cheers,

T.

Tommaso,

The whole calculation needs c(t) = 1/t, that is, the speed of light varies as 1/t, so c=1 at the present time.

One can attribute the big bang to three attributes of physics, the speed of light, the size of the universe, and the rate of passage of time.

Time and distance are changed by a gravitational potential so it is natural to let these vary. And in any coordinate system in GR, the speed of light must vary (defined in terms of dz/dt for geodesics).

So there are three basic ways of modeling cosmology. They correspond to leaving two of the above three variables constant and letting the third one take the punishment of creation.

The primary method is to leave the speed of light and the rate of passage of time constant. The result is that space explodes. This is the big bang.

If you let the speed of light vary while the passage of time are constant and space is fixed, then you get the calculation I did here.

And finally, if you let the speed of light be fixed and space constant, then you get a varying passage rate for time, i.e. a ds/dt kind of effect, and that I think is the correct way.

The advantage of these last two methods over the traditional big bang is that angles and geodesics are preserved in the geometry. That makes making angle correlation calculations easy.

Which gets back to the relationship between c(t) and t. If you assume a power law so c(t) = t^n, then the only power that gives a decent result is n=-1, the one I used. The reason is that the extremely large ratios of times (13 billion years to 380 thousand years) makes a logarithm necessary.

An alternative is to make the passage of time be exponential or logarithmic.

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So, is the curvature of the universe is different from the usual gravitational curvature? Because, if that was true, wouldn´t we expect that the dynamics of intense gravitational bodies be significantly different from what we observe, after all, they fit quite well with GR?

I’m sorry that I didn´t yet read the long article you posted… But I think, you could give a qualitative explanation :).

Sorry, I took that R as the curvature, because I read ” observable size of the universe”, instead of the correct way “size of the observable universe”.

Carl

you seem to have given Louise’s model some thought so I’m hoping you can help me with something. I’m trying to determine what the metric of Louise’s spacetime is (based on the article on her blog). I’ve asked her about it several times but she hasn’t responded to those queries. What do you believe her metric to be?

Yoyo, I don’t know what her “metric” is, I believe that the cosmos is flat and naturally prefer flat coordinates.

To see things her way, see her blog, probably this post, which details her solution to the Friedman equations (see this wikipedia article).

To get from the Friedman equations to a “metric”, you might find another wikipedia article, on the Friedman-Lematre-Robertson-Walker metric useful.

Hey, if you manage to slog through all that, do give us a report back.

Thanks Carl.

I’ve already gone through her derivation in detail. In itself it’s reasonably standard (it’s essentially the same as the presentation given by Einstein, just with different letters) but I have some problems with the interpretation of $R = ct$ as a solution of the Friedmann equations. For example, in the original Einstein derivation and most standard texts, the term corresponding to $R$ is a dimensionless scale factor in the case where the spatial curvature is flat ($z = 0$, Louise mistakenly calls this an “integration constant”). It can only be interpreted as having dimensions of length when $z \ne 0$, and even then, this is only achieved by rescaling the coordinates to absorb the spatial curvature.

The metric is simply the 2-tensor $g$ whose components are represented by the term $g_{ik}$, which appears in the left-hand side of the very first equation in Louise’s slides from above. Since she doesn’t state the original form of the metric, it’s not clear to me where $R(t)$ goes in the components $g_{ik}$.

I’m not sure how you reconcile Louise’s solution with a “flat” cosmos. She explicitly sets the spatial curvature (the curvature of the timelike hypersurfaces t = const.) to zero, but this is not the same as setting the curvature of the spacetime to zero. The Ricci scalar $R$ of the spacetime is non-vanishing even if the spatial curvature $z$ is zero.

In any case, it’s perfectly fine to have a metric on a flat space – that’s exactly what the Minkowski metric is. I’m not sure why you put it quotes.

I don’t believe in special relativity, why should I care about general relativity? I’m perfectly willing to take your word that this is not a solution, but it’s not something that bothers me in the least.

My post on the CMB is done using flat coordinates. I write “flat” because it’s really the coordinates that are flat, not necessarily the space. Certainly Minkowski coordinates don’t have c(t) =1/t so what I’m using are not that sort of flat.

When I rewrote the Schwarzschild and Painleve orbits into flat space differential equations (which was arduous) for the gravity simulation applet I got huge amounts of grief from folks who care about what “flat” means. But my whole purpose in doing this was to see what changes I could make to GR that would be consistent with observation. To do that, I wanted to have an exact version of GR on flat coordinates.

My post doesn’t depend on whether or not Louise has a solution to GR or not. If you assume a power law for the speed of light (in flat coordinates), and you look at the angular correlations in CMB, then you end up with the same equations, i.e. with the speed of light falling with time as c = 1/t.

From c(t) = 1/t it’s easy enough to write a metric. Just take the usual Minkowski metric and replace “c” with 1/t. I don’t claim that this is a solution to GR; frankly I doubt that it is, otherwise the cosmologists wouldn’t be chasing after dark energy (and dark matter, which is related).

Carl, you may not care about GR, but evidently Louise does, and uses perfectly standard GR to derive her result (incorrectly, I believe). You even stated in your post that “there’s a bunch of highbrow general relativity calculations to support this which you can find on [Louise’s] blog, and do not concern us here.” I find it very strange that you don’t believe in GR – except when Louise is the one using it! The nice thing about mathematics is that you don’t have to take my word for it, you can check it yourself.

[“Nice thing” is only true in some sort of philosophical ideal world. In fact, nobody has the time to check everything out. See “Gravity’s Rainbow,” which I should review, to see a sociologist’s view of theoretical and experimental physics.]But my point here is not to bash alternative theories to GR, or to bash your post on CMB. I don’t have a problem with the idea that GR might be wrong, or with mathematically consistent theories where $c$ varies, only that I don’t believe that it follows from Louise’s mathematical argument. In other words: it might be that Louise’s theory is correct, but nevertheless a varying $c$ does not follow from the argument she gives. You’re perfectly free to do calculations under the assumption that Louise’s theory is true, but I think it’s important to be clear that this is distinct from the issue of whether the theory can be derived from GR.

Ok, I’ll stop going on about that now, however there are a couple of statements you made above I’d like to comment on.

“The reason I find this version of cosmology attractive is that it means that spacetime is stable.”

I’m not sure what you mean by stable here, it’s still expanding and still came from an initial singularity. This is true even in Louise’s cosmology, as $c(t)$ blows up at $t = 0$.

[Read section 2.2 of this paper by Unzicker to see what I mean by “stable”. The assumption is that gravity turns on at time 0 and this effects the speed of light.]“A flat space version of GR avoids the science fiction problems of the standard GR, and it also makes it natural that current space appears to be very close to flat.”

When people talk about space being “flat” they

alwaysmeanspatially flatie. the spacelike 3-dimensional hypersurfaces are flat, but the 4-d manifold is still curved, as it must be in the presence of matter (obviously, working under the assumption that GR holds). As far as I understand it, this is also true in Louise’s derivation – if the curvature vanished then the entire RHS of her first equation would be zero.[For what I mean by “flat” see one of the excellent articles on the Cambridge geometry group’s “gauge gravity” by David Hestenes. To get you interested, here’s a quote: “A dramatic new twist on the physics-geometry connection has been introduced by Cambridge physicists Lasenby, Doran, and Gull with their flat space alternative to GR called gauge theory gravity.” In short, the flatness is in the coordinates, not in how light moves around.]“The usual gravity theories have difficulty explaining how it comes to be that the universe hasn’t collapsed into a little ball or whatever.”

GR has no difficulty explaining this. The universe will a) collapse if density ratio $\Omega > 1$, or b) expand forever if $\Omega <= 1$. The bit that GR doesn’t explain is the initial density – it can’t, because as formulated this is a boundary condition which (for GR at least) can only be inferred from observation. It would be nice if there was a theory that predicted that there can only be one possible initial density, and I suppose inflation is one candidate (requiring $\Omega$ to be exactly 1) but this is unsatisfying for many people.

[This is a well known problem called the “flatness problem.” The initial density has to be flat to within 1 part in 1-^59, which is similar to the string theory idiocy. If you haven’t come across this number yet, see the section on initial conditions in this paper, or just google flatness and big bang.]“Similarly, Riofrio’s cosmology is clean and simple. Space is big, flat, and matter doesn’t mess it up much.”

The “matter doesn’t mess it up much” part is not right – the very reason why space in this model is flat is because the spatial curvature has been set to zero, which forces $\Omega$ to be precisely 1.

[My comment is about the flatness problem. Let me spell it out clearly. If you assume the universe is flat and the speed of light changes, then you don’t have a flatness problem do you. Instead you have a speed of light problem, but that’s a heck of a lot easier to solve, and is the natural consequence of a universe where the force of gravity was initially turned on at t=0.]“From c(t) = 1/t it’s easy enough to write a metric. Just take the usual Minkowski metric and replace “c” with 1/t. I don’t claim that this is a solution to GR”

This is an interesting point. Actually, any old metric can be made a solution of Einstein’s equation as long as you don’t care about having a physically plausible stress-energy tensor. Just calculate the Einstein tensor from your favourite metric and declare that to be $T$ – instant solution. Usually this doesn’t work because you get an unphysical $T$ with (say) negative energies or other such unlikely features.

Ok, I’ve taken up enough of your time. To be clear, I don’t have any problem with alternative theories of gravity. My problem is that Louise appears to be using an incorrect mathematical argument to derive her theory.

[No non English comments without translation included. As far as incorrect mathematical arguments, I stated in the post that her methods are over my head. And the history of physics has a few examples of bad calculations that turned into big discoveries. Also, try to keep your comments shorter so that I don’t have to edit them like I did with this one.]Thanks for your patience then Carl. Funny you should mention “Gravity’s Rainbow”, I’m nearly finished reading it (for the second time).

I’ll close with one final observation. This comment caught my eye:

“This is a well known problem called the “flatness problem.” The initial density has to be flat to within 1 part in 1-^59, which is similar to the string theory idiocy.”

How come you call this “idiocy”, but give Louise a free pass when she does it? She uses a density $\rho$ which is completely uniform spatially for all time and states explicitly that $\Omega = 1$ is a consequence of her theory at the bottom of this page.

Oooops.

I meant “Gravity’s Shadow”. The book “Gravity’s Rainbow” is fiction.

Yes, I realised that shortly after replying – I was talking about the Pynchon novel. I should get around to reading Harry’s book sometime as well.

I notice you didn’t answer my question though…

[Carl writes: I won’t let my blog be used by one person to go on and on about the same issue, especially one that is peripheral to what I am doing here. For yoyo’s further commetns on Louise Riofrio’s mathematics, please see: yoyodyne-inc.blogspot.com.]Yoyo, I put 6 responses to your earlier comments directly into your comment above

[in italics]. You’ll have to reread your comment to see them.As far as the “idiocy” comment, I don’t see any reason to amplify what I said. There are occasions when physics fails. To continue to believe it is the action of an idiot.

Suppose you are in a bar and a guy gives you a deck of cards and asks you to shuffle them and then give them back. He then asks you to look at the cards in the deck one by one. Without being able to see them, he names them. The probability of pulling a stunt like that is 1/52! ~= 1.2 x 10^-68.

So, are you going to then confidently bet him that he can’t do it twice? After all, according to the laws of mathematics he was only very lucky the first time. Hey, people can disagree but it’s clear to me that physics doesn’t work with the big bang.

All one can say is thank you, Carl. (and Tommaso too)

Of course I’m a big proponent of geometric algebra, so it’s nice to see that the guys at the Cambridge geometry group have a paper with a flat space cosmology.

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Courtesy of Kea, a nice description of the angular correlation problem is given in a multimedia presentation at the Perimeter Institute by Glenn Starkman Case Western University, October 30, 2008

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