# Newtonian derivation of Friedmann equation

• Mike2
In summary: Friedmann equation from Newtonian mechanics using the force produced from the mass density of the universe. The cosmological constant is expressed in terms of an energy density also as: $\Lambda = \frac{{8\pi G}}{{c^4 }}\rho vac$ from which we can just as easily derive the Friedmann equation which includes the cosmological constant. However, in order to do this, the author needs to understand the units of the various terms in the Einstein field equations.
Mike2
I'm trying to derive Friedmann's equation for cosmology using Newtonian physics. I've got the force equation F=ma for the case without a cosmological constant. But now I'm trying to incorporate the cosmological constant into this force equation.

But I'm having trouble seeing how the cosmological constant can appear to apply a force on some particle at the edge of a spherical universe of radius R. I thought that I might get somewhere through dimensional analysis if I could understant the units of the cosmological constant from its use in the Einstein Field Equation where it was initially introduced. But I find I don't know the units of any of the other entities in the EFE - the units for the metric gab or the Ricci tensor or the Ricci scalar or the energy-momentum tensor. Do the units change for different values of ab? It seems everyone likes using elaborate mathematics - and I don't see much practical use - not even dimensional units of the things they talk about. Any help out there? Thanks.

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It isn't obvious that what you want to do is even possible. Why don't you do a course on general relativity first (or even just say the track one exercises in Misner, Wheeler & Thorne's book)?

cesiumfrog said:
It isn't obvious that what you want to do is even possible. Why don't you do a course on general relativity first (or even just say the track one exercises in Misner, Wheeler & Thorne's book)?
In the pdf file, starting with page 9, at:

http://www.astro.caltech.edu/~george/ay21/Ay21_Lecture02.pdf

the author derives the Friedmann equation from Newtonian mechanics using the force produced from the mass density of the universe.

And at:

http://en.wikipedia.org/wiki/Cosmological_Constant

the cosmological constant is expressed in terms of an energy density also as:

$$$\Lambda = \frac{{8\pi G}}{{c^4 }}\rho vac$$$

from which we can just as easily derive the Friedmann equation which includes the cosmological constant.

But I don't know how they got the last equation above. I think it may have something to do with the dimensional analysis of the terms in the Einstein field equations shown below. And I'm hoping someone here knows. Thank you.

$$$Rab - \frac{1}{2}Rgab + \Lambda gab = 8\pi Tab$$$

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## 1. What is the Friedmann equation?

The Friedmann equation is a fundamental equation in cosmology that describes the dynamics of the universe. It relates the rate of expansion of the universe to its energy content and the curvature of space. It was derived by Alexander Friedmann in the 1920s and is one of the key equations in the standard model of cosmology.

## 2. How is the Friedmann equation derived from Newton's laws?

The Friedmann equation is derived from the equations of motion in Newtonian cosmology. These equations describe the motion of particles in a homogeneous and isotropic universe. By assuming a perfect fluid model for the universe and applying the conservation of energy and momentum, the Friedmann equation can be derived.

## 3. What does the Friedmann equation tell us about the universe?

The Friedmann equation tells us about the evolution and dynamics of the universe. It describes how the rate of expansion of the universe changes over time and how this is affected by the energy content and curvature of the universe. By solving the Friedmann equation, we can understand the past, present, and future of the universe.

## 4. How does the Friedmann equation relate to dark energy and dark matter?

The Friedmann equation includes terms for both dark energy and dark matter, which are two components of the universe that are not directly observed but are inferred from their effects on the expansion and structure of the universe. These terms account for the acceleration of the expansion of the universe and the gravitational pull that affects the motion of galaxies, respectively.

## 5. Can the Friedmann equation be applied to all cosmological models?

While the Friedmann equation is derived from Newtonian cosmology, it can also be applied to other cosmological models, such as general relativity. However, it may need to be modified to account for different assumptions or structures in these models. Additionally, the Friedmann equation is only valid for a homogeneous and isotropic universe, so it may not apply to more complex or non-uniform universes.

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