Questions about SHO Equations in paper about celerite HOT 3 CLOSED

Hi @victoriafoing,

Thanks for the questions!

Equation (20) can be derived by computing the absolute value squared of equation (4):

Then the power spectrum (their equation 6) is given as:

With the substitutions:

and

this yields an expression which is proportional to our equation (20). The \sqrt{2/\pi} results from the way we defined the Fourier transform (see footnote on page 3 of our paper), and the P_X(\nu) is a constant as we assume that \epsilon(t) = \omega_0^2 x(t) is white noise, and thus the Fourier transform is a constant value at all frequencies.

Then, S₀ is the normalization constant of the power spectrum, which is the Fourier transform of the kernel.

I find it easiest to interpret S₀ in terms of the kernel definition in our equation (23). When
, then
, which is the limiting amplitude of the kernel at zero time lag (which is in addition to a white-noise amplitude which is only present at zero time lag). The kernel has units of the y² (let's say this is [y]²), and
has units of 2\pi/[t] (using whatever "time" unit you are using, [t], or other independent vector for a general GP),
which is in units of the quantity you have a vector of measurements of, and so S₀ should equal
and have units of [y]²[t]. The quantities Q and
can be fit to your dataset.

from celerite.

"i" is the imaginary unit and when you take the absolute value squared of the numerator, it is equal to the denominator, so it cancels.

from celerite.

Hi @ericagol!

Thank you for your answers. What does i represent in equation 4? How do we get rid of the numerator in equation 4 after taking the absolute value squared?

from celerite.