How Does the Dirac Delta Function Identity Apply in Equation (27) Derivation?

[Safinaz]

Homework Statement

May you please let me know if there an identity of a Dirac Delta function in momentum space that tells if:

Relevant Equations

$F(k_1) \delta^3 (k_1) \times F(k_2) \delta^3 (k_2) = \frac{2 \pi^3}{k^2} \delta(k_1-k_2) P(k)$
Then :
$P(k) = - 4 ( F(k_1) + F(k_2) )$

I need help to understand how equation (27) in this paper has been derived.

The definition of P(k) (I discarded in the question $\eta$ or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively.

In my question I defined:
$F(k_1) = \frac{1}{\sqrt{H^2-k_1^2}} sinh (\sqrt{H^2-k_1^2} (\eta-\tilde{\eta}_1)) \frac{1}{H\tilde{\eta}_1} [m^2 Y_i Y_j-\frac{1}{H\tilde{\eta}_1} Y'_i Y'_j ]$

and
$F(k_2) = \frac{1}{\sqrt{H^2-k_2^2}} sinh (\sqrt{H^2-k_2^2} (\eta-\tilde{\eta}_2)) \frac{1}{H\tilde{\eta}_2} [m^2 Y_i Y_j-\frac{1}{H\tilde{\eta}_2} Y'_i Y'_j ]$

So in (27) $F(k_1)$ and $F(k_2)$ are added while according to (26) they are multiplied , so what is the identity of $\delta^3(k)$ and $\delta(k_1-k_2)$ which lead to Eq.(27) ?

Any help is appreciated!

 

[anuttarasammyak]

Relevant Equations: $F(k_1) \delta^3 (k_1) \times F(k_2) \delta^3 (k_2) = \frac{2 \pi^3}{k^2} \delta(k_1-k_2) P(k)$
Then :
$P(k) = - 4 ( F(k_1) + F(k_2) )$

k appears only in RHS. What is the definition of k?

LHS is not zero only when k_1=k_2=0. RHS is not zero only when k_1=k_2. They do not seem compatible.

 

[Safinaz]

k appears only in RHS. What is the definition of k?

In a previous step in the paper, a transformation has been done from $x$ space to $k$ space by fourier-transforming

 

[Safinaz]

k appears only in RHS. What is the definition of k?

LHS is not zero only when k_1=k_2=0. RHS is not zero only when k_1=k_2. They do not seem compatible.

I try to figure out the identity that leads to equation (27) in the menstioned paper. and writting $F(k_1)$ or $F(k_2)$ just to simplify. So that I'm asking about the correct identity of Dirac Delta

 

[vela]

In a previous step in the paper, a transformation has been done from $x$ space to $k$ space by fourier-transforming

That explains what $\mathbf{k}_1$ and $\mathbf{k}_2$ are. It doesn't explain what $\mathbf{k}$ is in the expression for the power spectrum.

I try to figure out the identity that leads to equation (27) in the menstioned paper. and writting $F(k_1)$ or $F(k_2)$ just to simplify. So that I'm asking about the correct identity of Dirac Delta

I have the same concern as @anuttarasammyak. There's no point trying to find some identity you think you need if what you're starting with is wrong.

I moved this thread to the advanced physics forum. Perhaps someone with expertise in gravitational waves can shed some light on your question.