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# Astrophysics/Equation of motion for the radius of the universe

Question
I found that the formula of equation of motion for the radius of the universe is :
R" = -4/3*(π3,14...)*G*(%matter density)
Questions :
1)Why is writed R" (second derivative) ?
2)Just as example is possible to put some numbers in that formula to clarify better to me the question ?
R" = -4/3*3,14...*6,67*10^-11*(R which value to insert?)*(%matter density ,which value to insert?).

Hello,

In answer to your question (1), the reason we have the second derivative R" is because of the analog to Newton's gravitational acceleration equation, e.g.

R" = g = - GMs/R^2

And as the sphere for the universe expands outward, gravity reduces the velocity. This is why we call R" the "deceleration parameter".

As for your other inquiry, putting numbers in the formula is not that much help because you are working with an incomplete metric. What I mean is that the presence of the cosmological constant term (which you omitted and is equal to (lambda R/3) can no longer be omitted since the evidence shows lambda is not equal to zero)

This means you have to return to the Friedmann equation and rec-compute the values of the Hubble constant H) , the critical density (rho) and the deceleration parameter in the context of this term.

Remember the scale factor (R) is not directly observable in any case, we only have the actual observables derived from it like the Hubble constant, H. So in a way asking for numbers to insert is like putting the cart before the horse. You need to know the Hubble constant and other observables and then work within the Friedmann template to arrive at the model of the universe you want.

My point is that all of these templates work in tandem. You have to be aware of exactly what your objective is before you can make effective use of the particular equations (and also be familiar with the differences between the Freidmann, Robertson-Walker, de Sitter templates).

A good book that I'd recommend if you can get it is:

'Foundations of Modern Cosmology' by John F. Hawley and Katherine A. Holcomb.

They lead you through the models and observables step by step and show what is feasible and what isn't.  The text is also an excellent introduction for anyone embarking on a study of general relativity, assorted self-consistent models and basic cosmological theory in general.

Astrophysics

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Papers appearing in Solar Physics, Journal of the Royal Astronomical Society of Canada, Journal of the Barbados Astronomical Society, Meudon Solar Flare Proceedings (Meudon, France). Books: 'Fundamentals of Solar Physics', 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level', 'Astronomy & Astrophysics: Notes, Problems and Solutions', 'Modern Physics: Notes, Problems and Solutions'

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