In a big bang cosmology there is a parameter called spectral index that measures deviation of scalar curvature perturbation from scale invariance. I want to know what is this invariance?
The idea is simple. In order to get a not-so-homogeneous universe today, the assumption goes, the primordial universe had to have small random density (scalar) fluctuations which grew into the stars and galaxies of today. Some of these initial density fluctuations would be small in size. Others would be large in size. Are small and large fluctuations equally likely to occur? If so, then if you took two pictures of the early universe and magnified one a thousand times, it would still look exactly the same as the other. This is scale invariance. On the other hand if, say, the early universe favored tiny fluctuations over large fluctuations, then the two pictures would not look the same; the original picture would contain predominantly small fluctuations, the magnified one, predominantly large fluctuations.
If you plot the amount of fluctuations against size (in other words, wavelength), you get a power spectrum. If the amount of fluctuations on all scales is the same, the plot would be a horizontal line. However, if we have, say, more fluctuations at small scales and less at larger scales, the plot will have a slope. This slope can be approximated by assuming a simple exponential relationship. The exponent is usually expressed as n - 1, where n is the spectral index. So n = 1 means an exponent of 0 and a horizontal line (scale invariance); any other value means some deviation from scale invariance.
The idea is simple. In order to get a not-so-homogeneous universe today, the assumption goes, the primordial universe had to have small random density (scalar) fluctuations which grew into the stars and galaxies of today. Some of these initial density fluctuations would be small in size. Others would be large in size. Are small and large fluctuations equally likely to occur? If so, then if you took two pictures of the early universe and magnified one a thousand times, it would still look exactly the same as the other. This is scale invariance. On the other hand if, say, the early universe favored tiny fluctuations over large fluctuations, then the two pictures would not look the same; the original picture would contain predominantly small fluctuations, the magnified one, predominantly large fluctuations.
If you plot the amount of fluctuations against size (in other words, wavelength), you get a power spectrum. If the amount of fluctuations on all scales is the same, the plot would be a horizontal line. However, if we have, say, more fluctuations at small scales and less at larger scales, the plot will have a slope. This slope can be approximated by assuming a simple exponential relationship. The exponent is usually expressed as n - 1, where n is the spectral index. So n = 1 means an exponent of 0 and a horizontal line (scale invariance); any other value means some deviation from scale invariance.
Power spectrum can be written as some factors times a power
of k. Here k is the wave number, which is (2 pi/lambda) lambda
is the wavelength, so it carries a length scale. If power spectrum
has k^0, then essentially k drops out as k^0 is a constant. Then
power spectrum becomes scale independent.
A perfect scale invariance corresponds to idealized de Sitter
universe. A slight deviation from absolute scale invariance
is needed. A Harrison-Zel'dovich spectrum has two parameters
to implement that. One is amplitude and another spectral
index.
For a different approach about the expansion of the Universe, which is based on basic physics, please see the article at my Research-Gate site. The model predicts the expansion, the meaning of the Hubble constant, and the acceleration of the expansion. It also predicts that the present mass is only ten percent of the original mass. It replaces negative energy and dark matter. It solves the gravity ‘problem’ which led to the proposal of dark matter.
Reaction-Kinetic Expansion of the Universe and the Hubble Constant, 5-7-2014
Ingo H Leubner
Please, let me know about anything that may be my editorial error, and what you think about it. Thank you. Ingo
I did model the process of the Big Bang as a crystallization process. The transition from an energy singularity to mass particles forces the appearance of anti-energy, and defines it as gravity. I wrote the manuscript over ten years ago and presented it at many national and international conferences. I am in the process of reviewing the manuscript and publishing it at my Research-Gate Site. If you are interested, please, email me, and I will send you the file.
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