Can you install custom firmware on a psp running on 4.05 official firmware without using a Pandoras battery

definitely no the best u can downgrade without a Pandora battery is 3.03.... what you can do is use the Pandora battery to downgrade and the results will be great I've done it on my psp and when i downgraded my psp i got to download as many games that can fit into my psp(full games of course) so downgrade your psp using a Pandora battery :)