Small Scale Homogeneity and Expansion Rate Variance

In summary, the continuity equation describes the conservation of mass in a volume, and the variance of the expansion rate within that volume can be affected by various factors, including small scale homogeneity.
  • #1
Apashanka
429
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From the continuity equation ##\frac{\partial \rho}{\partial t}+\rho (\nabla • u)=0## where ##\rho## is the mass density and is homogeneous and ##u## is the velocity of expansion or contraction.
For an expanding volume this becomes ##\nabla•u=\frac{\dot v(t)}{v(t)}=\Theta## which gives the rate of expansion ,##v(t)## is the volume.
Now if we consider our volume ##V(t)## to be consist of small patches of volume ##v(t)## where ##\rho## is homogeneous in ##v(t)## ,local expansion rate of volume ##V(t)## is therefore ##\Theta## ,now if the variance of this ##\Theta## is taken over ##V(t)## it will be ##\sigma_\Theta=[\bar{\Theta^2}-(\bar \Theta)^2]##.
Now my question is it small scale homogeneity is sufficient for this non-zero variance of volume expansion rate??
 
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  • #2


I would like to clarify that the continuity equation is a fundamental equation in fluid dynamics that describes the conservation of mass in a volume. It is not specific to expansion or contraction, but rather applies to any changes in the volume.

In the case of an expanding volume, the equation can be simplified to ##\frac{\partial \rho}{\partial t}+\rho \frac{\partial v}{\partial t}=0##, where ##v## is the volume. This equation simply states that the change in mass within the volume is equal to the rate of change of the volume.

The concept of small scale homogeneity is not directly related to the variance of the expansion rate. The variance in the expansion rate, as described in the forum post, is a measure of the variability of the expansion rate within the volume. This can be affected by factors such as the distribution of matter and energy within the volume.

In order to accurately measure the variance of the expansion rate, it is important to consider the scale at which the measurements are being taken. If the volume is divided into smaller patches, as suggested in the forum post, the variance may change depending on the size and distribution of these patches.

In conclusion, while small scale homogeneity may play a role in the variance of the expansion rate, it is not the only factor to consider. Other factors, such as the distribution of matter and energy, can also impact the variability of the expansion rate within a given volume.
 

1. What is small scale homogeneity?

Small scale homogeneity refers to the uniformity of the distribution of matter on a small scale, typically on the order of a few hundred million light-years. It is a fundamental assumption in the standard cosmological model, and it means that the universe looks roughly the same at any given point in space.

2. How is small scale homogeneity related to the expansion rate variance?

The expansion rate variance is a measure of how much the expansion rate of the universe varies from one region to another. Small scale homogeneity implies that the expansion rate should be roughly the same in all regions, so a low variance would support this assumption.

3. What evidence do we have for small scale homogeneity?

One piece of evidence for small scale homogeneity is the cosmic microwave background (CMB) radiation. The CMB is a nearly uniform glow of radiation that fills the entire universe, and it is thought to be the leftover heat from the Big Bang. The fact that the CMB is nearly the same in all directions supports the idea of small scale homogeneity.

4. How does small scale homogeneity affect our understanding of the universe?

Small scale homogeneity is a crucial assumption in the standard cosmological model, as it allows us to make predictions about the large-scale structure of the universe. It also helps us understand the distribution of galaxies and the overall expansion of the universe.

5. Are there any challenges to the concept of small scale homogeneity?

There are some challenges to the concept of small scale homogeneity, as some observations have shown slight variations in the distribution of matter on small scales. However, these variations are still within the expected range and do not significantly affect our understanding of the universe as a whole.

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