definitely, the installer in sault ontario has another BATTERY to supply the peak power. this is not a crazy guy who does something for nothing. putting in the capacitor does the same job, but in a smaller cheaper package. 500 microfarad is 1/2 of a farad, which is a nice hefty number of electrons. the capacitor acts as a reservoir - at all times, the reservoir is filling, until it is full. when a peak comes along, the electrons are released to the amplifier, draining the capacitor, but not emptying it completely. after the peak is over, the capacitor fills again. it acts just the same as a reservoir of water before a hydroelectric dam. bass uses a lot of current, and you can supply this current with lighter guage of wire if you have a capacitor right next to the amp. this is very simple.
also, we are dealing with DC, not AC, so there is no phase shift.