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marcus

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What is your preferred way of remembering and writing down the Friedmann equations and in what form do you include dark energy in them?

Heres the way that seems best to me. It is a balance between making them uncluttered but not so simplified as to lose intelligibility. For one bit of notation, I refer to the article "Making Sense of the New Cosmology"

http://xxx.lanl.gov/PS_cache/astro-ph/pdf/0202/0202008.pdf

by Michael Turner (U Chicago and Fermilab). It is a well-written

recent survey article.

He uses the energy density

ρ_X to stand for the dark energy.

Some others say rho sub Λ

because its associated with the "Cosmological

Constant" but I like rho sub X because

nobody really knows what it is---it's just some

energy that behaves a certain

way, in particular the pressure is at least

roughly equal to minus the energy density

(exactly, if it turns out to be in fact the

cosmological constant.)

Anyway the density and pressure arguments in

the Friedmann equations are

ρ = ρ_m + ρ_X

p = p_m + p_X

Intuitively Fr. equations are about a_t and a_t,t ____often written a-dot and a-doubledot: the first and second time-derivatives of the scale factor a(t). This is a dimensionless factor usually adjusted to equal one at the present moment. To keep time and space commensurable I am going to work with a_ct and a_ct,ct.

It simplifies the equations a bit to use a force constant F = c^4/G.

a_ct,ct/a = -(4π/3)(ρ + 3p)/F

(a_ct/a)^2 = (8π/3)ρ/F - k/a^2

k is the sign of curvature factor and in the most interesting (flat) case is equal to zero. This makes the equations still simpler. And we can take ordinary time derivative of a if we divide the lefthand side by c^2. Doing that, in the k = 0 (flat) case results in these two Friedmann equations:

(1/c^2) a_t,t/a = -(4π/3)(ρ + 3p)/F

(1/c^2)(a_t/a)^2 = (8π/3)ρ/F

This form of the equations is intuitive to me. Pressure and energy density are dimensionally the same type quantity and dividing either one by a force gives one over an area---curvature.

Lots of times people write the equations and set basic constant in them (like c and G) equal to one. So the constants disappear and clutter is reduced. But then, with so much set equal to one, it is harder to parse---at least I find the physics harder to picture. So I wanted to reduce clutter some but not set any basic constants equal to one. This notation F (for the Planck force actually---that's what c^4/G is) provides for an uncluttered intelligible version, I think.

BTW F = 12x10^43 newtons----very roughly 10^40 metric tonsforce. The same constant c^4/G, or its reciprocal G/c^4,

plays a central role in the Einstein equation, the main equation in GR. So it makes conceptual sense to have it stand out visibly.

Any comment?

Heres the way that seems best to me. It is a balance between making them uncluttered but not so simplified as to lose intelligibility. For one bit of notation, I refer to the article "Making Sense of the New Cosmology"

http://xxx.lanl.gov/PS_cache/astro-ph/pdf/0202/0202008.pdf

by Michael Turner (U Chicago and Fermilab). It is a well-written

recent survey article.

He uses the energy density

ρ_X to stand for the dark energy.

Some others say rho sub Λ

because its associated with the "Cosmological

Constant" but I like rho sub X because

nobody really knows what it is---it's just some

energy that behaves a certain

way, in particular the pressure is at least

roughly equal to minus the energy density

(exactly, if it turns out to be in fact the

cosmological constant.)

Anyway the density and pressure arguments in

the Friedmann equations are

ρ = ρ_m + ρ_X

p = p_m + p_X

Intuitively Fr. equations are about a_t and a_t,t ____often written a-dot and a-doubledot: the first and second time-derivatives of the scale factor a(t). This is a dimensionless factor usually adjusted to equal one at the present moment. To keep time and space commensurable I am going to work with a_ct and a_ct,ct.

It simplifies the equations a bit to use a force constant F = c^4/G.

a_ct,ct/a = -(4π/3)(ρ + 3p)/F

(a_ct/a)^2 = (8π/3)ρ/F - k/a^2

k is the sign of curvature factor and in the most interesting (flat) case is equal to zero. This makes the equations still simpler. And we can take ordinary time derivative of a if we divide the lefthand side by c^2. Doing that, in the k = 0 (flat) case results in these two Friedmann equations:

(1/c^2) a_t,t/a = -(4π/3)(ρ + 3p)/F

(1/c^2)(a_t/a)^2 = (8π/3)ρ/F

This form of the equations is intuitive to me. Pressure and energy density are dimensionally the same type quantity and dividing either one by a force gives one over an area---curvature.

Lots of times people write the equations and set basic constant in them (like c and G) equal to one. So the constants disappear and clutter is reduced. But then, with so much set equal to one, it is harder to parse---at least I find the physics harder to picture. So I wanted to reduce clutter some but not set any basic constants equal to one. This notation F (for the Planck force actually---that's what c^4/G is) provides for an uncluttered intelligible version, I think.

BTW F = 12x10^43 newtons----very roughly 10^40 metric tonsforce. The same constant c^4/G, or its reciprocal G/c^4,

plays a central role in the Einstein equation, the main equation in GR. So it makes conceptual sense to have it stand out visibly.

Any comment?

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