Deep space communication systems operate on very long distance and the on-board energy generator capacity is very limited which results a very low signal to noise ratio (SNR) at the reception. This is the reason why near Shannon limit error correcting codes are used. Nevertheless, to take advantage of this gain, the coherent demodulation is mandatory and the carrier phase synchronization must be provided at a more restrictive SNR. Spectrum resources allocated for deep space missions are limited (X-band at 8 GHz), and in order to optimize the spectral efficiency, the Consultative Committee for Space Data System (CCSDS) recommended pre-coded GMSK modulation (Gaussian Minimum Shift Keying) with Gaussian filter bandwidth-bit period product BTb=0.5 for Category B missions (deep space missions) and GMSK BTb=0.25 for category A missions (low altitude missions). This paper discusses a blind phase detector [2] derived from maximum a posteriori (MAP) criterion and Laurent expansion for GMSK modulation. In order to evaluate the performance of this phase detector in a closed loop structure at very low SNR, we consider two others simplified versions. We develop a comprehensive study of these three different structures in both linear and nonlinear domains. We also present the results obtained by computer simulation using low rate error-correction code (Turbo 1/6). The aim of this work is to compare the performance of these three phase detectors and estimate the impact of the simplification that have been made to obtain the two simplified versions.