# Intuition Behind Scale Invariance Power Spectrum

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In summary, the book "Statistical physics for cosmic structures" defines scale invariance as the requirement that the normalised mass variance over a sphere of radius equal to the horizon distance should be constant, regardless of the time at which it is computed. This means that the mass variance at any given time should be the same, regardless of the size of the horizon at that time. This concept is believed to be true in de Sitter expansion, where there are no time-dependent dynamics that would cause a scale dependence on the perturbations.
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In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that light could have traveled in cosmological time ##t##.

In other words, the normalised mass variance over a sphere of radius the horizon distance should be independent of time. So if we computed the mass variance at some time ##t_0## when the horizon was ##R_H(t_0)## this should be the same as if we computed it at any other ##t_1## when the horizon was ##R_H(t_1)## even though ##R_H(t_0)## might be much smaller than ##R_H(t_1)##.

I am trying to get intuition for why one would believe such a requirement to be true. Does anyone have some enlightening explanations/insights?

Scale invariance is only true for de Sitter expansion in which $\dot{\rho} = 0$. The background is steady state, so there are no time-dependent dynamics that would impart a scale dependence on the perturbations.

## 1. What is scale invariance in the context of power spectrum?

Scale invariance refers to the property of a system or signal where the overall shape or structure remains the same regardless of the scale at which it is observed. In the context of power spectrum, it means that the power spectrum of a signal remains the same when the scale of the signal is changed, such as by zooming in or out.

## 2. Why is scale invariance important in the study of power spectrum?

Scale invariance is important in the study of power spectrum because it allows us to analyze signals at different scales without losing important information about the underlying structure of the signal. It also allows us to compare and combine power spectra from different sources, making it a useful tool in various fields such as astrophysics, engineering, and biology.

## 3. How is scale invariance related to fractals?

Scale invariance and fractals are closely related concepts. Fractals are objects or patterns that exhibit self-similarity at different scales, meaning that the same structure is repeated at different levels of magnification. This is similar to the property of scale invariance, where the overall shape remains the same regardless of the scale. Many natural phenomena, such as coastlines, mountain ranges, and trees, exhibit both scale invariance and fractal properties.

## 4. What is the role of logarithms in understanding scale invariance in power spectrum?

Logarithms play a crucial role in understanding scale invariance in power spectrum. By taking the logarithm of the power spectrum, we can transform it into a scale-invariant form, making it easier to analyze and compare across different scales. This is because logarithms "flatten out" exponential growth, allowing us to see patterns and relationships that would be obscured in the original power spectrum.

## 5. How is scale invariance used in practical applications?

Scale invariance has various practical applications, such as in image and signal processing, where it is used to analyze and compare images or signals at different resolutions. It is also used in the analysis of financial markets, where it helps to identify long-term trends and patterns. Additionally, scale invariance is used in computer simulations and modeling, as well as in the study of natural phenomena, such as earthquakes and turbulence.

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