What sub under a grand?

Jughead, subs do a lot more than just go boom. Really good subs can be crossed over very high as long as they are integrated well. They can add unbelievable stage depth and realism. Gun shots in movies can have a startling effect and really fill the room. The timing of the high passed sub relative to the front mains is critical. So moving subs around the room,setting delays, eq,s, room correction all have a huge impact on frequency in the room which will in effect have a big impact on imaging. Bone crushing bass can start as high as 200hz and it's obvious Bo has played with these higher frequencies and room correction.
The other thing a good sub can do which is very important is take the load off your amp. 80hz is still a good load for most amps and speakers. IMO, sealed subs are the way to go.

The psa is a sealed down fire sub, and I understand a sub does more than go boom. imaging and thunderous bass is what I said I was looking for all along,and so far the psa has delivered. I'm plugged into my p5's low pass at 80hz and the explosions and gunfire from the ps4 while playing cod advanced warfare is unreal! I have 29 more day's in home trial to really get to know it and hear what music is like.so far I listened to 1 cd avenged sevenfolds heil to the king and the sub blended in well with my main speakers.

all still sounds good with the psa music has depth and nice punch and ht duty is where it really shines, it's driving my wife nuts she say's it sounds like a war zone upstairs! I don't think I have it placed in the right spot yet still trying to find that sweet spot. But for 750 delivered and being brand new I don't think you can beat it svs sb 2000 maybe? The only think I don't like about the psa is it's size it's a beast! but that's the only problem I find can find with it.

200hz is bone crushing lower midrange.

Good powered subs do bass best. Whether its needed or not depends on all the rest.

Lots of good subs for use with monitors and other speakers with limited bass extension available for under $1000.

If speakers are more full range to start with, then you need a bigger sub that is also good quality and can cover from your crossover point down to 20hz or so, in order to be considered "full range" in regards to what most people can clearly hear. There are fewer to choose from here for under $1000, but still many good choices I think.

Where to crossover will depend case by case on speakers ability to cover higher bass frequencies versus the subs. A good sub that can do it all well from 20 hz up past the range that most good speakers can cover well will be rarer.

And yes, adding a sub and lessening the bass demand for the amps driving the mains is pretty much always a good thing and benefits everything, not just the bass. If done right of course.

If you can find two good subs to fit the bill within budget, then two subs are always better than one for delivering balanced bass response throughout the room.

And 4 are better than two for that.

It never ends for an audiophile, but probably significant diminishing returns after that.