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We employ the formalism developed in \cite{Mentasti:2023gmg} and\cite{Bartolo_2022} to study the prospect of detecting an anisotropicStochastic Gravitational Wave Background (SGWB) with the Laser InterferometerSpace Antenna (LISA) alone, and combined with the proposed space-basedinterferometer Taiji. Previous analyses have been performed in the frequencydomain only. Here, we study the detectability of the individual coefficients ofthe expansion of the SGWB in spherical harmonics, by taking into account thespecific motion of the satellites. This requires the use of time-dependentresponse functions, which we include in our analysis to obtain an optimalestimate of the anisotropic signal. We focus on two applications. Firstly, thereconstruction of the anisotropic galactic signal without assuming any priorknowledge of its spatial distribution. We find that both LISA and LISA withTaiji cannot put tight constraints on the harmonic coefficients for realisticmodels of the galactic SGWB. We then focus on the discrimination between agalactic signal of known morphology but unknown overall amplitude and anisotropic extragalactic SGWB component of astrophysical origin. In this case,we find that the two surveys can confirm, at a confidence level $\gtrsim3\sigma$, the existence of both the galactic and extragalactic background ifboth have amplitudes as predicted in standard models. We also find that, in theLISA-only case, the analysis in the frequency domain (under the assumption of atime average of data taken homogeneously across the year) provides a nearlyidentical determination of the two amplitudes as compared to the optimalanalysis.

Probing the galactic and extragalactic gravitational wave backgrounds with space-based interferometers