Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,851 answers , 20,616 comments
1,470 users with positive rep
501 active unimported users
More ...

Axion field energy density

+ 2 like - 0 dislike
176 views

The problem

Let's have axion lagrangian (without $a F \wedge F $ terms) in FRLW Universe ($g_{\mu \nu} = diag(1, -R^{2}(t), -R^{2}(t), -R^{2}(t))$):
$$
L = \frac{1}{2}g_{\mu \nu}\partial^{\mu}a\partial^{\nu}a - \frac{m^{2}_{a}(t)}{2}a^{2}
$$

Corresponding EOM for spatially homogeneous field reads
$$
\ddot{a} + 3H\dot{a} + m^{2}_{a}(t)a = 0, \quad H = \frac{\dot{R}}{R}
$$

Corresponding solution for large times and for adiabatically changing mass $|\dot{m}_{a}| < m_{a}^{2}$ reads
$$
\tag 0 a(t) \approx a_{0}\left(\frac{R(t_{\text{in}})}{R(t)}\right)^{\frac{3}{2}}cos(\int m_{a}(t)dt)
$$

Corresponding energy density reads (here I've inserted the solution $(0)$ and used relation $H << m_{a}(t)$, as typical $H \sim \frac{1}{t}$, while $m_{a} \sim t^{2}$)
$$
\tag 1 \rho_{a} = T_{00} = \frac{1}{2}\left((\partial_{0}a)^{2} + m_{a}^{2}a^{2} \right) \approx m_{a}^{2}(t)a_{0}^{2}\left( \frac{R(t_{\text{in}} )}{R(t)}\right)^{3} \approx C_{a}m_{a}^{2}(T)T^{3},
$$

where I've used relations $R(t) \sim \sqrt{t}$ and $t \sim T^{-2}$.

But recently I've found the true expression for axion energy density (for example, in Rubakov's "Introduction to the Theory of the Early Universe: Hot Big Bang Theory", chapter 9):
$$
\tag 2 \rho_{a} \approx \tilde{C}_{a}m_{a}(T)T^{3},
$$

It could be obtained from the following way: from EOM $\ddot{a} + 3H \dot{a} + m_{a}^{2}a = 0$ we can obtain EOM for $\rho = \frac{1}{2}\dot{a}^{2} + \frac{1}{2}m_{a}^{2}a^{2}$,
$$
\dot{\rho}_{a} = \dot{a}\ddot{a} + \dot{m}_{a}m_{a}a^{2} + \dot{a}am_{a}^{2} = -3\dot{a}^{2}H + \dot{m}_{a}m_{a}a^{2}
$$

Let's average over an oscillations, $\langle m_{a}^{2}a^{2}\rangle = \langle \dot{a}^{2}\rangle = \rho_{a}$ (this action implies conditions $H << m_{a}(t), \dot{m_{a}}{a} << m_{a}^{2}(t)$), and thus
$$
\dot{\rho}_{a} = -3H\rho_{a} + \frac{\dot{m}_{a}}{m_{a}}\rho_{a},
$$

and the solution reads
$$
\rho_{a} \approx \left(\frac{R(t_{\text{in}})}{R(t)}\right)^{3}m_{a}(t)\rho_{a}(t_{\text{in}}) = \tilde{C}_{a}T^{3}m_{a}(T)
$$

The question

Expressions $(1)$ and $(2)$ for energy density for large times don't coincide; in particular, they state different behaviours of energy density with time (due to time dependence of axion mass). But both expressions are derived by taking into account conditions $H, \frac{\dot{m_{a}}}{m_{a}} << m_{a}$, so it seems that they must coincide.

Can you explain me where is the mistake in derivation of expression $(1)$?

asked Sep 13, 2015 in Theoretical Physics by NAME_XXX (1,010 points) [ revision history ]
edited Sep 13, 2015 by NAME_XXX

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$y$\varnothing$icsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...