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

Safinaz
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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!
 
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Safinaz said:
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.
 
anuttarasammyak said:
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
 
anuttarasammyak said:
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
 
Safinaz said:
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.

Safinaz said:
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.
 
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