You can fill the bottles to about 1/2" below the rim. That reduces headspace. Some leave even less space, but I don't want to risk breaking bottles. If you leave no headspace or too little, the bottle could break when it gets warmer.
I compared the color of two beers from the same batch, poured into glasses - one with 2" headspace and one with 1/2". The one with 1/2" was a little lighter. Not a huge difference, but lighter.
I compared the color of two beers from the same batch, poured into glasses - one with 2" headspace and one with 1/2". The one with 1/2" was a little lighter. Not a huge difference, but lighter.