Armstrong et al. have recently presented new ways of combining signals to precisely cancel laser frequency noise in spaceborne interferometric gravitational wave detectors such as LISA. One of these combinations, which we will call the “symmetrized Sagnac observable,” is much less sensitive to external signals at low frequencies than other combinations, and thus can be used to determine the instrumental noise level. We note here that this calibration of the instrumental noise permits smoothed versions of the power spectral density of stochastic gravitational wave backgrounds to be determined with considerably higher accuracy than earlier estimates, at frequencies where one type of noise strongly dominates and is not substantially correlated between the six main signals generated by the antenna. We illustrate this technique by analyzing simple estimators of gravitational wave background power, and show that the instrumental sensitivity to broad-band backgrounds at some frequencies can be improved by a significant factor of as much as $(f\tau {/2)}^{1/2}$ in spectral density $h_{\mathrm{rms}}^2$ over the standard method, where f denotes frequency and $\tau$ denotes integration time, comparable to that which would be achieved by cross-correlating two separate antennas. The applications of this approach to studies of astrophysical gravitational wave backgrounds generated after recombination and to searches for a possible primordial background are discussed. With appropriate mission design, this technique allows an estimate of the cosmological background from extragalactic white dwarf binaries and will enable LISA to reach the astrophysical confusion noise of compact binaries from about 0.1 mHz to about 20 mHz. In a smaller-baseline follow-on mission, the technique allows several orders of magnitude improvement in sensitivity to primordial backgrounds up to about 1 Hz.

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