Take alooke at http://sigitharyadi.net/multidicipline/non-intercept-linear-regression-id/ and my other calculator which use the " Haryadi Index" in various problems.

My method will be accurate not depending on the sample population. This is because my formula " Haryadi Index" was combined the "harmony" equation ( which was contained in Herhindahl-Hirschman Index, fractal equatiom, Keppler equation etc) with the "gradation" equation ( which was contained in Gini Index, Gaussion distribution, Pearson correlation, lorenz gamma equation etc).

Thanks for the reply Sigit. I don't see how I should use your software, we can only have one actual measurement since it is the mean of the whole of the observable universe. My question relates to the distribution obtained by extrapolating what readings beyond observation would be based on the law that describes a Fourier transform for wavelengths much longer than the size of our observable region.

I'll add another reply giving the background information, how would you enter that into your method?

Here are a few resources that cover this topic. I won't go into details as there doesn't seem to be much interest in the question but if anyone has any questions, I will be delighted to explain as much as I understand.

http://www.astro.caltech.edu/~george/ay21/Cosmo_8.3_DensFluct.pdf

http://www.astro.caltech.edu/~george/ay21/Cosmo_8.4_FluctGrowth.pdf

The above are from the Caltech undergraduate course on cosmology which is available here, the section headed "Lecture 8" is the part of direct interest:

http://www.astro.caltech.edu/~george/ay21/

https://ned.ipac.caltech.edu/level5/March01/Coles/Coles2.html

I limit my comment on your power spectrum image, where I see you have a number of data populations of the galaxy clustering that are not entirely on any of the standard curves (CMB normalization, Barryons, primordial HZ spectrum and CDM), which only close to one of the curves when your independent variable has an value of - 2 to -0.75. My suggestion is to limit the analysis to that area only, while in other areas you have to re-measure, but if you believe that the measurement data is correct, then you should create a new hypothesis or theory that has not been thought of by experts in your field right now.

Dear Sigit,

Thanks for your answer. The image above does contain more information than is needed, it was one I found which puts the question into context. Having done some further research, I think it also suggests an answer. The H-Z spectrum is the dashed line which has a known slope. The thin region marked "CMB normalisation" through which it passes gives a datum point on the line and the lack of "running" (i.e. non-linearity) therefore defines it at all points to the left.

Based on a paper by Max Tegmark, the distribution for a flat universe is ergodic so the distribution of multiple samples should be the same as that within a sample. Near the end of page 3, he writes: ".. and cosmic microwave background measurements have established that the trend towards uniformity continues all the way out to the edge of our observable universe (R∼1027m), where ∆M/M∼10−5.". One page 4 he writes "This quantum mechanism generates initial conditions that are for all practical purposes random, producing density fluctuations described by what mathematicians call an ergodic random field. Ergodic means that if you imagine generating an ensemble of universes, each with its own random initial conditions, then the probability distribution of outcomes in a given volume is identical to the distribution that you get by sampling different volumes in a single universe."

Article Parallel Universes

Dear George,

Back to my first suggestion, please try using my method, which I have created the calculator, i.e. http://sigitharyadi.net/multidicipline/non-intercept-linear-regression-id/Procedure: (1) From the data you have, specify the dependent and independent variables, where you are sure that there is a linear relationship that does not have an intercept (i.e. when the independent variable is 0, the independent variable must be zero), (2) Use my website, then you will get a regression equation accompanied by the level of correlation between two variables you have set, and the confidence level of the regression equation, (3) After that, you can use the regression equation to extrapolate to the area outside the range of your measurement data (and possibly for an interpolation within the measurement area), where the standard deviation or variance, or the possibility of miscalculation is equal to one minus the correlation index of the calculation you get from the my calculator.