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 similar to 10(7)/(1 + z)(2) M-circle dot, where z is the redshift. Binaries heavier than similar to 10(8)/(1 + z) M-circle dot 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 greater than or similar to 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.