Integrals of tensor perturbations (Weinberg, p. 315)

[jouvelot]

Hi,

I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that
$$
\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),
$$ where $e_{ij}(\hat{q})$ is any symmetric traceless matrix-valued function of $\hat{q}$ with $\hat{q}_i e_{ij}(\hat{q}) = 0$, $\hat{p}$ is the unit vector parallel to $p$ and $f$ any function of the scalar product $\hat{p}.\hat{q}$.

Is there a simpler way to get this formula than to proceed by using cumbersome spherical or euclidian coordinate expressions? I submitted this question on the Math forum on PF, but got no suggestions.

Thanks in advance.

Bye,

Pierre

 

[AndyW]

Dear Pierre,

Thank you for sharing your interesting formula and question. It seems that your formula is a result of using the orthogonality and normalization properties of the unit vector $\hat{p}$ and the symmetric traceless matrix-valued function $e_{ij}(\hat{q})$. To simplify the calculation, you can try using the properties of the spherical harmonics, which are commonly used in cosmology and have similar orthogonality and normalization properties. You can also try using tensor notation, which can help to simplify the expression and make it more compact. I hope this helps and good luck with your research!