Can you downgrade PSP firmware version 3.80 without a Pandora battery

No, you must use a Pandora's Battery. Only PSP firmware 3.50 or below can have custom firmware flashed over it without using a Pandora's Battery and Magic Memory Stick. With 3.50 and below you can use a Lumines UMD. With 3.03 and below you can use an unpatched GTA: Liberty City UMD. With 2.81 and below you can use HEN. I wrote a guide here for downgrading without a Pandora Battery:
http://honorshard.com/blog/2009/03/01/downgrade-psp-with-official-firmware-without-a-pandora-battery/ Feel free to check it out and if you like it please share it.
I had OFW 4.05 and successfully downgraded to 4.01 M33-2 without a Pandora Battery!