How is the horizon length related to the power law spectrum

In summary, the conversation discusses the scaling of the horizon wavenumber ##\frac{2\pi}{\chi_H}## in a matter dominated universe with respect to the power law spectrum of density fluctuations, ##P(k) \propto k^n##. The evolving particle horizon, ##\chi_H(a)##, is defined in terms of the comoving radius and the matter-dominated universe is expressed in terms of the scaling of ##\rho(a')## with ##a##. The relationship between the power spectrum and ##\chi_H## is unclear, and further assistance is requested.
  • #1
thinkLamp
16
0

Homework Statement


For a power spectrum density fluctuations ##P(k) \propto k^n##, I need to find the scaling (with respect to ##a##) of the horizon wavenumber ##\frac{2\pi}{\chi_H}## in a matter dominated universe in terms of ##n##. ##\chi_H(a)## is the evolving particle horizon, in a flat universe.

Homework Equations


$$
\chi_H = \int_0^t \frac{c \; \mathrm{d}t'}{a(t')} \;,
$$

The Attempt at a Solution


I know that the horizon distance is the comoving radius of the particle horizon

$$
\chi_H = \int_0^t \frac{c \; \mathrm{d}t'}{a(t')} \;,
$$

which I can also write as

$$
\chi_H = \int_0^a \frac{\mathrm{d}a'}{a'} \left( \frac{8 \pi G \rho(a') a'^2}{3 c^2} - K \right)^{-1/2}\;,
$$
and that the matter-dominated universe part is expressed in terms of ##\rho(a')##'s scaling with ##a##. But I'm not sure how this relates to the power law spectrum of density fluctuations.

How is the power spectrum related to ##\chi_H##?
 
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  • #2
Is there a specific way to express the scaling of ##\chi_H## with respect to ##a## and ##n##? Any help is appreciated!
 

Related to How is the horizon length related to the power law spectrum

1. How is horizon length related to the power law spectrum?

The horizon length is a measure of the maximum distance that light can travel in the universe. The power law spectrum refers to the distribution of energy at different frequencies in a system. The two are related because the horizon length affects the wavelengths of light that can be observed, which in turn influences the power law spectrum of the universe.

2. What is the significance of the power law spectrum in understanding the universe?

The power law spectrum helps us understand the distribution of energy in the universe, which is essential in studying the formation and evolution of galaxies, stars, and other celestial bodies. It also provides insights into the fundamental laws of physics that govern the behavior of matter and energy in the universe.

3. How is the horizon length measured?

The horizon length is typically measured using cosmological observations, such as the redshift of galaxies and the cosmic microwave background radiation. These observations allow scientists to calculate the distance that light has traveled since the beginning of the universe, which gives us an estimate of the horizon length.

4. Can the power law spectrum change over time?

Yes, the power law spectrum can change over time as the universe evolves. The distribution of energy in the universe is affected by various factors, such as the expansion of the universe, the formation of new structures, and the interactions between different types of matter. These changes can alter the power law spectrum and provide us with valuable insights into the evolution of the universe.

5. How does the power law spectrum relate to the Big Bang theory?

The Big Bang theory is a scientific explanation for the origin and evolution of the universe. The power law spectrum is closely linked to this theory as it provides evidence for the initial conditions of the universe at the time of the Big Bang. By studying the power law spectrum, scientists can gain a better understanding of the early universe and its development over time.

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