They are noisier because the entrance pupils of their smaller lenses are generally smaller than on the lenses that provide the same angle of view for larger sensor formats. There are physical limitations in making large entrance pupils for smaller focal lengths, as pupils bigger than f/0.5 are impossible even theoretically.You still get more light from a smaller focal length (at the same pupil size) . That's why we describe the amount of light coming through the lens as a function of the focal length and pupil size, or in other words we use F-stop notation.
a 10mm aperture on an 18mm lens that gives a certain fov on APS-C lets in more light than a 10mm aperture on a 28mm lens that would produce the same fov on FF.
You don't get get more light from the subject
at the same entrance pupil size. What you get is higher illuminated sensor
, but you get it only because the same amount of light is concentrated on smaller sensor area. However, the shot noise on the image of an area of the subject is exactly the same, because the amount of photons originated from this area and reaching the sensor is exactly the same.
The amount of light the lens gathers through the same entrance pupil from the same solid angle of the scene is exactly
the same no matter if the lens is going to focus this light on a 5 sq. micron pixel or on a 50 sq. micron pixel. As long as it's the same pixel on the output device
, it will carry the same amount of shot noise.
Think about it.
Thats why zoom lenses that aren't constant aperture get darker when you zoom in. The pupil isnt getting smaller, you just have less light entering the lens because you are constricting the FOV.
But they are shooting different subjects!
There is even a more "ideal" object for comparison than zoom lenses. It's teleconverters. They increase the focal length keeping the entrance pupil constant. If you shot the same sporting event (i.e. the same shutter speed) with the wide open lens with and without a teleconverter ("push" the darker lens shots with higher sensor ISO) you will find the noise less apparent on a full frame of the shot without the teleconverter. But that's because you are shooting different subjects. Crop your non-TC shot to cover the same subject as your TC shot, convert both images to the same pixel count and see if the difference in the noise still exists.