Solve Buchert Equations for Dust Flow Averaging

[Apashanka]

Just trying to solve the buchert equations in this paper https://arxiv.org/abs/1112.5335 (eq 2.7,2.8 and 2.9) from eq.(2.3,2.4,2.5)
First the average is taken on the hypersurfaces orthogonal to the geodesic of the dust flow(along proper time $\tau$).
Taking the average of the first term of eq. 2.3 $<\frac{d\Theta}{d\tau}>=\frac{d}{d\tau}<\Theta>=3\frac{d}{d\tau}\frac{\dot a_d}{a_d}=3\frac{\ddot a_d}{a_d}-3(\frac{\dot a_d}{a_d})^2=3\frac{\ddot a_d}{a_d}-\frac{<\Theta>^2}{3}$
Averaging LHS of eq. 2.3 becomes
$3\frac{\ddot a_d}{a_d}-\frac{<\Theta>^2}{3}+\frac{<\Theta^2>}{3}$
(Given that $<\Theta>=3\frac{\dot a_d}{a_d}$ & $\Theta$ is the volume expansion rate.
Compairing with eq.2.7 rest are the same only fact is instead of $\frac{1}{3}$ there is $\frac{2}{3}$ in the $Q_d$??
Am I missing some calculation mistake??

 

[Ambitwistor]

Thank you for bringing this paper to our attention. I have taken a look at the equations you mentioned and I believe there may be a small mistake in your calculations. Let me explain:

In equation 2.3, the average of the first term $<\frac{d\Theta}{d\tau}>$ is taken on the hypersurfaces orthogonal to the geodesic of the dust flow. However, in your calculation, you have taken the average of the entire equation. This results in an extra term $<\Theta^2>$ which is not present in the original equation 2.3. This is why you are seeing a difference in the coefficient $\frac{1}{3}$ and $\frac{2}{3}$ in the $Q_d$ term.

To solve the Buchert equations, you need to take the average of each term separately, not the entire equation. This means that the LHS of equation 2.3 should be averaged as $3\frac{\ddot a_d}{a_d}$ and the RHS should be averaged as $\frac{<\Theta>^2}{3}$. This will result in the correct equation 2.7 with the coefficient $\frac{1}{3}$ in the $Q_d$ term.

I hope this helps clarify the issue. Please let me know if you have any further questions or concerns.