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:(adsbygoogle = window.adsbygoogle || []).push({});

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

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!

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# Dark Energy Hubble Equation

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