# Dark Energy Hubble Equation

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1. Dec 11, 2013

### TimeFall

Hello! This is my first post, so go easy on me! I'm working through Scott Dodelson's book Modern Cosmology http://books.google.com/books?id=3oPRxdXJexcC&pg=PA23&source=gbs_toc_r&cad=4#v=onepage&q&f=false and I am a bit confused about equation 2.85:
$$\rho_{DE} \propto e^{-3 \int^a \frac{da'}{a'} [ 1+ w(a')]}$$

where $\rho$ is the energy density for dark energy, a is the scale factor, and w is the equation of state parameter defined as $w = \frac{P}{\rho}$.

My problem is that when I derive this equation from the dark energy continuity equation (which is eq. 2.55 in Dodelson, and I am able to derive with no problems):
$$\dot{\rho} + 3H\rho(1+w(a)) = 0$$

I get:
$$\rho_{DE} \propto e^{-3 \int_{1}^{a} \frac{[1+w(a')]da'}{a'}}$$

Which I'm assuming is what Dodelson's is (he doesn't have a lower limit on his integral, thus implying it is an integral over all a). However, this paper : http://www.aanda.org/articles/aa/pdf/2004/12/aah4738.pdf
in equation 7, has the limits flipped, but still has the negative sign on the 3. I'm inclined to believe both, since one is a textbook and the other is a peer-reviewed paper, but they cannot both be correct. I cannot, for the life of me, derive the version in the paper, but I have been able to get Dodelson's version (where the limits go from 1 to a). I just wanted to see if I was doing something stupid in deriving this equation, or if I am, in fact, correct.

I've also looked through several other books and papers, but they all seem to give it as a function of redshift:
$$\rho_{DE} \propto e^{3 \int_{0}^{z} \frac{[1 + w(z)]dz}{1 + z}}$$
Which I am also able to get from the continuity equation by just changing variables from a to z using:
$a = \frac{1}{1 + z}$.

If you change variables from z to a in equation 2.85, you get exactly what I got. The trouble is that I need this equation as a function of a because I need to put it into a larger code that already uses a instead of z.
Thank you very much for any help, and sorry for the ramble!

Last edited: Dec 11, 2013
2. Dec 12, 2013

### Chalnoth

Sounds like the main concern here is a sign error. The difference in sign out front will determine whether the energy density grows or shrinks over time for different values of w(a). So one easy to resolve this and make certain you have the sign in front correct would be to set w(a) = -2 and verify that the energy density of dark energy is lower in the past, and set w(a) = 0 and verify that it scales as 1/a^3 (as normal matter).

3. Dec 12, 2013

### George Jones

Staff Emeritus
Yesterday, I used (2) to try and derive (7), and I seemed to find a sign mistake in (7). I just used Chalnoth's suggestion of setting $w=0$, and, again, this indicates a sign mistake in (7).

Note also another mistake in (7), i.e., the missing exponent of 1/2.

4. Dec 12, 2013

### Chalnoth

What missing exponent of 1/2?

5. Dec 12, 2013

### George Jones

Staff Emeritus
(7) should end with $]^{1/2}$, as (6) does; either that, or both H and H_0 should be squared.

6. Dec 12, 2013

### Chalnoth

Oh, I see what you're saying. I was going by what you wrote here on PF, not the text of the paper. Yes, that is a clear typo in the paper.

7. Jan 27, 2014

Thank you!