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

  • #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!
 
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  • #2
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.
 
  • #3
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
 
  • #4
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
 
  • #5
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.
 

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

What is the Dirac Delta Function?

The Dirac Delta Function, denoted as δ(x), is a mathematical function that is zero everywhere except at x = 0, where it is infinitely high such that the integral over its entire domain is equal to one. It is often used to model an idealized point source or impulse in mathematical physics and engineering.

Why is the Dirac Delta Function important in Equation (27) derivation?

The Dirac Delta Function is important in the derivation of Equation (27) because it allows for the simplification of integrals involving point sources or impulses. It effectively "picks out" specific values from a continuous function, which is essential for solving differential equations or performing Fourier transforms.

How does the sifting property of the Dirac Delta Function apply in the derivation?

The sifting property of the Dirac Delta Function states that for any continuous function f(x), the integral of f(x) multiplied by δ(x - a) over all space is equal to f(a). This property is used in the derivation of Equation (27) to isolate specific terms or values within an integral, simplifying the overall expression.

Can you provide an example of using the Dirac Delta Function in Equation (27)?

Consider an integral involving the Dirac Delta Function, such as ∫ f(x) δ(x - a) dx. Using the sifting property, this integral simplifies to f(a). In the context of Equation (27), similar integrals can be simplified by applying this property, which helps in reducing complex expressions to more manageable forms.

What are the common pitfalls when applying the Dirac Delta Function in derivations?

Common pitfalls include misunderstanding the nature of the Dirac Delta Function as a distribution rather than a traditional function, misapplying the sifting property, and neglecting the proper handling of limits and boundaries in integrals. It is crucial to carefully apply the properties of the Dirac Delta Function to ensure accurate results in derivations.

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