Christ - colder air - as you say are more dense - which means that at a given RPM you are able to administer more fulel, thus developing more hp, because the colder and more dense air contains more oxygen...
As a rule of thumb:
- For every 3,3 degrees centigrade temperature drop the air density rises 1%
- If you for instance can lower the intake temp with 40 deg C you have a 12% greater air mass. This is equvivalent of a 1,78 psi "over pressure" which in theory shold mean that you have 12% more hp availeble, but in daylilife this is more likely to be around 8%.
In our modern engines you have theese censors:
- Airmass meeter
- temp censor
- oxygen censor
The ecu recives information on air temp, oxygen level, and administer feul accordingly, thus meaning, that at a given opening of the throttle/ RPM, you can administer more fule and generate more hp.
Cold air intake is a way of finding more torque/hp.
On a standard street car you often see the exaust manifold located to the front just behind the radiator, thus throwing lot of very hot air in to the engine room. Often the airbox is located on the hottest spot in the engine romm - on top of the engine.
In such a case you will often se air intake temps in the rage from 70 - 100 deg C.
If ambient temp is for instance 25 deg. C, you can lower the intake temp by isolating the intakesystem and heatshilding it with reflctive materials.
Often this is further improved by ruting a duct from to front of the car to just before the intake "mouth" / bell opening.
This will mean that you can generate more hp at a given RPM.
Now it's up to you (and your right foot) if you want to USE them
(Changing the lenght of the intake itselves will change the RPM where max torque occur. Either up or down depending on the specific car)
What I have stated here are fact of physics - colder air => more oxygen => burn more fuel => develop more hp at a given RPM.
Now - as I said it's up to the driver how to use the extra hp. And if you only use the forst ½ centimenter travel on the accelerator, this might indeed be very difficult to administer i a good way.
But dont deny the facts.