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.
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:
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:
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.