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I have been reading about weak gravitational lensing and I am trying to calculate the dispersion ##\langle M_{ap}^2\rangle## of the aperture mass for a singular isothermal sphere acting as a lens for distant objects.
I need some guidance on how to actually carry out the calculation of the power spectrum. It is defined as the Fourier transform of the two point correlation function, but how does one actually compute that function? All the information I have found just defines each quantity in terms of the other, which is not very helpful to me. I feel like I'm going around in circles.
To provide some context:
In The book "Gravitational Lensing: Strong, Weak and Micro", Peter Schneider includes a discussion on the 2-point correlation function for a homogeneous and isotropic random field ##g(\vec x)##. The main points being that (in 2d):
##\langle g(\vec x) g^*(\vec y) \rangle = C_{gg}(|\vec x - \vec y|)## is the two point correlation function.
By going to Fourier space we can show that ##\langle g(\vec k) g^*(\vec k') \rangle = (2\pi)^2 \delta(\vec k - \vec k') \int \mathrm{d^2}y \quad e^{i \vec y . \vec k} C_{gg}(|\vec y|)##
The power spectrum is then defined by the Fourier transform of the two point correlation function.
##P_g(|\vec k|) = \int \mathrm{d^2}y e^{i \vec y . \vec k} C_{gg}(|\vec y|)##
The aperture mass is defined as being ##M_{ap} (\vec \theta) = \int d^2 \theta' \kappa(\vec \theta')U(|\vec \theta - \vec \theta'|)##, where ##\kappa## is the convergence and ##U## a suitable filter function.
By following a similar approach as for the random field, expressing the two point correlation function of the convergence in Fourier space, I have been able to arrive at the desired expression for ##\langle M_{ap}^2\rangle##.
##\langle M_{ap}^2\rangle = \frac{1}{(2\pi)^2}\int d^2 \vec l \mathrm{P_{\kappa}}(\vec l) |U(\vec l)|^2 ##
But I am no closer to actually being able to compute what the power spectrum is.
Thank you if you're still reading!
I need some guidance on how to actually carry out the calculation of the power spectrum. It is defined as the Fourier transform of the two point correlation function, but how does one actually compute that function? All the information I have found just defines each quantity in terms of the other, which is not very helpful to me. I feel like I'm going around in circles.
To provide some context:
In The book "Gravitational Lensing: Strong, Weak and Micro", Peter Schneider includes a discussion on the 2-point correlation function for a homogeneous and isotropic random field ##g(\vec x)##. The main points being that (in 2d):
##\langle g(\vec x) g^*(\vec y) \rangle = C_{gg}(|\vec x - \vec y|)## is the two point correlation function.
By going to Fourier space we can show that ##\langle g(\vec k) g^*(\vec k') \rangle = (2\pi)^2 \delta(\vec k - \vec k') \int \mathrm{d^2}y \quad e^{i \vec y . \vec k} C_{gg}(|\vec y|)##
The power spectrum is then defined by the Fourier transform of the two point correlation function.
##P_g(|\vec k|) = \int \mathrm{d^2}y e^{i \vec y . \vec k} C_{gg}(|\vec y|)##
The aperture mass is defined as being ##M_{ap} (\vec \theta) = \int d^2 \theta' \kappa(\vec \theta')U(|\vec \theta - \vec \theta'|)##, where ##\kappa## is the convergence and ##U## a suitable filter function.
By following a similar approach as for the random field, expressing the two point correlation function of the convergence in Fourier space, I have been able to arrive at the desired expression for ##\langle M_{ap}^2\rangle##.
##\langle M_{ap}^2\rangle = \frac{1}{(2\pi)^2}\int d^2 \vec l \mathrm{P_{\kappa}}(\vec l) |U(\vec l)|^2 ##
But I am no closer to actually being able to compute what the power spectrum is.
Thank you if you're still reading!