Could dark energy be heat? Could dark matter be stuff that is not seen because of its refraction limits?

Heat has to go somewhere. Would the light and heat of the stars in the universe amount up to being dark matter ...and dark energy ... Dark matter would just be small pieces of stuff. Dark energy would just be heat that has gone into space.

If dark energy was just heat then a warp-drive could be as simple as a sphere with one very hot side.

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Heat can't just live in the vacuum. Because heat is a form of energy and energy is equivalent to mass, heat has to have a material carrier. The most "neutral" type of heat that you probably meant is the electromagnetic radiation. Dark matter can't be made out of electromagnetic radiation because it would escape, in 100,000 years, away from the galaxy. Instead, dark matter must be made of something that survives in a halo around the galaxy and permanently modifies its gravitational field. In fact, we know that most of the dark matter must be cold dark matter which moves very slowly – much slower than the speed of light. So light is the maximally wrong candidate for dark matter.

Dark energy is the "opposite" of electromagnetic radiation, too. Dark energy, by definition, has a negative pressure close to $p=-\rho$, minus energy density. On the other hand, radiation has $p = +\rho/3$, a very different equation of state.

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By "heat" I'm going to assume you mean electromagnetic radiation, because outside of the interiors of stars and other similarly dense regions, heat cannot be conducted or convected.

The answer is simply : No. Dark matter cannot be tiny bits of matter that we cannot see. To help you understand it better, let me explain a bit of history first - Dark matter was originally hypothesised because when people measured this thing called a "galaxy rotation curve" which measures how fast something is moving around the centre of the galaxy.

They expected a fall in the velocity that matched newton's laws of gravity (which still work extremely well in systems like this). Except instead of the velocity of bodies dropping off as $1/r^{2}$ after a point, they found it to be roughly constant.

This was essentially the motivation for hypothesising dark matter. Now to why it isn't possible to be 'small bits of matter' - Matter has a tendency to clump because of gravity. This is how stars form, and basically any of the astrophysical bodies and structures exist. The individual Hydrogen atoms and molecules clump together (gravitationally bound) until they get massive enough (and hot enough) to trigger fusion. In the same way any tiny bits of matter would just clump together until it got bigger and bigger, till where we could see it. There are other reasons as well - any 'regular matter' emits spectral lines, which aren't seen for dark matter.

The 'heat that has gone into space', is basically all the radiation that we receive from stars, in terms of light that we can see (optical), or x-rays, or radio waves, or microwaves (traditionally what we refer to as heat radiation). All this radiation does exert pressure, but not nearly enough to cause the accelerating expansion of the universe.

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