- #1

Safinaz

- 260

- 8

- 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!

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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