Critical evaluation of Power Spectrum calculation in the book "The Inflationary Universe"

Cosmology is generally speaking based on two concepts: Physics and mathematics.
When you study the evolution of the universe than what you are studing is the evolution of different physical processes (i.e. chemical reactions) which happened with a "Big Bang" and which finally created or builded the present day universe consisting of galaxies, stars and planets.
The chalenge of course is to unravel and to describe those physical processes (these change of events) at most accurate and detailed as possible during the periods applicable. This is not easy.

A case in point is the interpretation of the Cosmic Micro Wave Background radiation versus the Inflation Theory. This subject is discussed in the book "The Inflationary Universe" by Alan H. Guth. For a book review select: The Inflationary Universe - Book review
Figure 14.1 at page 242 in the book by Alan H.Guth shows the following picture:

. . . . ^ . . . . . | . | . | . | Prediction . |______ of . Inflation . . --------------------------------------- 0 45 90 135 180

The text below this Figure reads:

The data from the COBE satelite gives the temperature T of the background radiation for any direction in the sky.

To understand the above figure select two points separated by some specific angle. Assume that the temperature of both points are T1 and T2. The temperature difference between those two points is defined as: (T1-T2)^2.
The horizontal axis shows the angle of separation between the two points
The vertical axis shows the average value of the summation of the temperature difference (T1-T2)^2 of all the points n considerd at a certain angle divided by n.

The calculation starts by selecting a random point (the central position). This point has temperature T1.
Next a circle, say 45 degrees, is drawn around this central position. The average temperature difference is calculated for all the points on this circle with temperature T2. In the program 16 points are considerd.
Next a different central position T1 is selected, the average temperature difference for that position is calculated and the running average temperature difference for 45 degrees is updated. etc etc.
Finally the result of 45 degrees is drawn in the figure.
Next a different angle is selected, say 50 degrees, and the whole process is repeated until all the angle are calculated.

What Figure 14.1 shows is that the line after 45 degrees is almost horizontal. This means that the difference in temperature between two points with a larger difference than 45 represents the average difference in temperatures between the whole universe. There is no correlation.
What Figure 14.1 also shows is that values below 45 degrees slowly drops to zero. This means that there are regios where the temperatures are more or less the same. There is a certain level of correlation. The smaller the angle the smaller this temperature difference.

The question is what is the relation between this Power Spectrum and the concept of inflation i.e. a rappid grow in the size of the universe in the first second after the Big Bang.
To test this I wrote a program in Visual Basic 2010.

Calculation Power Spectrum using Visual Basic

The Visual Basic Program to calculate the actual Power Spectrum is slightly different as described above.
Starting point is an array of 360 values which contains the average temperature value for all the angles. In the first cycle this whole array is updated for one specific central position point with temperature T1. Next a following central position point is selected. etc etc.
The central position for the different CMB radiation observations are identical.

The following two pictures are the result of the calculations using the real observed CMB radiation data.
The horizontal axis is 90 degrees. The number of calculated points is 360. The angle between the points is 0.5 degrees, that means all the angels between 0 and 180 degrees are calculated.
Picture 1 represents the result of 30 central positions
Picture 2 represents the result of 37510 central positions

Power Spectrum 30 * 1 central points Picture 1

Power Spectrum 310*121 positions Picture 2

The green curve shows the Power Spectrum of the Planck 21 March 2013.
The temperature of the CMB radiation is between 516.86 and -513.73. The average value is -44.1.
The average temperature difference of the calculated Power Spectrum is 193.
The red curve shows the Power Spectrum of a random simulation of Planck, the so called Bubble Universe. The temperature of the CMB radiation is between 600 and -599. The average value is -42.11
The average temperature difference of the calculated Power Spectrum is 202
The blue curve shows the Power Spectrum of WMAP 2009.
The temperature of the CMB radiation is between 206.15 and -214.61. The average value is -44.1.
The average temperature difference of the calculated Power Spectrum is 97.

For an image of the CMB radiation used to calculate the curves 1 and 2 (Planck based) select: Planck 21 March 2013 based CMB radiation simulation

Reflection part 1

When you compare Picture 1 and Picture 2, specific the green curve and the blue curve you will observe a big difference:

Each of the curves in Picture 1 have an almost identical shape as Figure 14.1. The blue curve (WMAP data) the most.
When you draw two lines though each curve, one roughly around the maximum values and one roughly around the minimum values the distance between those two lines is rather large. This is the so called "gray band"
Picture 2 shows three curves. When you draw the same two lines around each curve the distance of the "gray band" is almost zero. In fact you almost can draw a very smooth continuous line through each of the points.

The reason for this difference is the number of calculations performed. The more calculations the smoother the line through the points.

What is also important in Picture 2 is the difference in height between the Planck curve (Green) and the WMAP curve (Blue). The magnitude of the Planck curve is higher which is explained by the difference in magnitude of temperature of the Planck based CMB radiation data.

What is also important that the Planck curve in Picture 2 is flat starting from almost 20 degrees. The curve in Figure 14.1 is "flat" after 40 degrees. The WMAP curve is flat after roughly 40 degrees.
The fact that part of the curve is flat implies that the average temperature difference between far away points is the same. i.e. no correlation.
For short distance the curve implies that the temperatures are more or less the same. This implies a certain level of correlation.

Since inflation determines the shape of the spectrum but not the magnitude of the density perturbations, the magnitude of the predicted gray band was adjusted to fit the data.

Reflection part 2

What is also interesting, is to compare the Power Spectrum from Figure 14.1 calculated by Alan H Guth with the Power Spectrum calculated by the Planck team. The Planck team works with the parameter l. Small l values mean large degrees. Large l values mean small degrees. The value of l=100 is approximate equal to 1 degrees.
The importance of the Planck team Power Spectrum is that most information about the cosmological parameters comes from the large l values or small degrees. This means from the points in your own neighbourhood.
For more information See: Question 13: Is it possible to calculate the paramaters H0, omega(Lambda), Omega(M) alone using the CMB radiation

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Created: 19 June 2014

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