Lights dimming after I installed subwoofer and amp in 2007 Hyundai Sonata 3.3L V6

First off, I upgraded my wiring with 4/0 welding cable as so;  engine block to battery
neg terminal,  battery neg terminal to frame, battery neg terminal to strut tower driver
side, strut tower passenger side to engine block, alternator positive to battery terminal
pos. I installed 0awg power wire (with fuse) to a 1000 watt RMS amp
(http://www.jlaudio.ca/jx1000-1d-car-audio-jx-amplifiers-99397)  and have it grounded to
the frame (easily made via fuel sender access panel behind seats). I have the amp's
gain sensitivity set to 47.4 volts and am running a singIe sub
(http://www.jlaudio.ca/12w7-3-car-audio-w7-subwoofer-drivers-92112). My alternator is
stock 130amp, which i have been told by many people involved in high end car audio,
should be able to handle the load of this amp, yet my lights are dimming when I have
volume high and the subs hit. My battery was tested and it's at 700amps. Is my
alternator failing perhaps? Do I need a bigger battery?  A secondary agw deep cycle
battery wired in parallel?  A higher output Alternator?  plz help and thanks in advance.