Why does Costa's loop PLL bandwidth need to be narrow? and why can't it track a certain frequency offset? and does that hold even if it has a filter with a pole at DC?
Costa's loop is related to traditional PLLs. One way of characterizing a PLL is by its loop bandwidth and damping factor. The damping factor determines the loop response (under-damped, over-damped or critically damped). The loop bandwidth determines how fast a PLL/Costas will achieve a "lock". The higher the loop bandwidth, the shorter the acquisition time. However, higher loop bandwidths result in higher tracking error (and hence higher locking threshold SNR - See graph below). Therefore, the loop bandwidth is a tradeoff between acquisition time and tracking error. In some systems, the PLL/Costas start with a higher loop bandwidth before acquisition. After acquisition, the loop bandwidth is reduced to minimize tracking error.
The following terms associated with general PLL can answer your second question
- Pull-in range: is the maximum initial frequency difference between the input and VCO center frequencies both in positive and negative directions, for which the PLL eventually achieves the phase-locked condition. The pull-in range is related to the dynamics of the PLL
- Lock-in range: is the frequency range over which the PLL achieves the phase-locked condition without cycle slips
A little Googling can help you to find the specific formulas of the two terms above for the Costas Loop.