FFT, How to decide if there is a signal among noise?
As a fellow astrophysicist, I suggest you use the standard 3 sigma above noise for detection, and 5 sigma above noise if you want to do analysis on what you find.

If your noise does not have a flat background, you need to subtract a "Flat", an estimation of that the background might be. If you know what the noise should look like, such as exponential, gaussian, etc., try subtracting such a curve. If not, try a first order polynomial. If not good enough, then go with a second order.

Beyond that, it becomes interpretation... What if that tendency you notice in the noise is actually a signal, spread over many frequencies?

Try computing the FFT of a pure white noise image, to get an idea of what your noise is supposed to look like. (Off course, remember that FFTs assumes your signal is periodic beyond its edges. Don't forget to zero-pad your data, and use a correct apodization, or limit your analysis to something smaller than half the size of your image.)