How was democracy expanded during the Persian Wars?

Answer 1

It disappeared. Athens, at the forefront of introduction of democracy, rverted to oligarch in order to conduct the extended war effort and continued for another 20 years until it reemerged in the final stage of the 50 year war led by Ephialtes.

Answer 2

There was no democracy in Greece at the time of the Persian War.

It emerged in Athens when the city diverted the war funds of the anti-Persian alliance, which it continued to mulct from its allies after peace was made with Persia.

These funds were diverted to supporting Athens' own lifestyle, providing the resources to include all its people in employment, government and a degree of widespread prosperity not possible in such poor economies without the artificial largesse from the stolen funds.

Athens then encouraged other allied and subject cities towards democratic politics to complement its own.