Priorities in gravitational waveforms for future space-borne detectors: vacuum accuracy or environment?

In preparation for future space-borne gravitational-wave (GW) detectors, should the modelling effort focus on high-precision vacuum templates or on the astrophysical environment of the sources? We perform a systematic comparison of the phase contributions caused by (1) known environmental effects in both gaseous and stellar matter backgrounds, or (2) high-order post-Newtonian (PN) terms in the evolution of mHz GW sources during the inspiral stage of massive binaries. We use the accuracy of currently available analytical waveform models as a benchmark value, finding the following trends: the largest unmodelled phase contributions are likely environmental rather than PN for binaries lighter than ~10⁷/(1 + z)² M_⊙, where z is the redshift. Binaries heavier than ~10⁸/(1 + z) M_⊙ do not require more accurate inspiral waveforms due to low signal-to-noise ratios (SNRs). For high-SNR sources, environmental phase contributions are relevant at low redshift, while high-order vacuum templates are required at z ≳ 4. Led by these findings, we argue that including environmental effects in waveform models should be prioritized in order to maximize the science yield of future mHz detectors.