Higher Harmonics of Double White Dwarfs in the Centihertz Band: Linking LISA and DECIGO

Technical Summary: Higher Harmonics of Double White Dwarfs in the Centihertz Band

Problem Statement
Short-period double white dwarfs (DWDs) are primary sources of low-frequency gravitational waves (GWs) for the Laser Interferometer Space Antenna (LISA). While LISA is expected to detect the fundamental quadrupole mode ($k=2$) from thousands of Galactic systems, this mode alone does not provide direct information on the binary mass ratio. Higher harmonics (specifically the first $k=1$ and third $k=3$ harmonics) arise at 0.5 post-Newtonian (PN) order and scale with the mass asymmetry parameter $\Delta$. Detecting these harmonics is crucial for constraining mass ratios, which in turn informs the stability of Roche-lobe overflow and the evolutionary pathways of DWDs (e.g., merger vs. AM CVn formation). Previous studies suggested that while most Galactic DWDs are too weak for higher harmonic detection with LISA, rare nearby systems might be observable. However, a systematic statistical assessment of the detectability of these harmonics across the Galactic population, particularly in the context of future mid-band observatories, was lacking.

Methodology
The authors constructed a synthetic population of 4,000 inspiraling DWDs to evaluate the detectability of PN harmonics in a realistic setting. The methodology involved:

• Population Synthesis: A steady-state population was generated by injecting binaries at a minimum quadrupole frequency ($f_2 = 4$ mHz) and evolving them under gravitational radiation reaction. Component masses were drawn from a two-component Gaussian mixture (representing He and CO white dwarfs), and frequencies were assigned based on a steady-state inspiral distribution ($dN/df_2 \propto f_2^{-11/3}$). The sample was restricted to $f_2 \geq 5$ mHz to ensure individual resolvability.
• Signal Modeling: The waveform model included the fundamental quadrupole mode and the 0.5PN order odd harmonics ($k=1, 3$). The amplitude of the odd harmonics scales as $A \beta \Delta$, where $\beta$ is the PN velocity parameter and $\Delta$ is the fractional mass difference. The analysis accounted for inclination dependence and the specific noise spectral densities of LISA, Taiji, TianQin, DECIGO, and BBO.
• Noise and Detection Criteria: The study utilized angle-averaged noise power spectral densities, including instrumental noise and the extra-galactic white-dwarf confusion foreground. For DECIGO and BBO, the effective sensitivity was scaled by a factor of 3 to account for three independent detector units. A conservative signal-to-noise ratio (SNR) threshold of $\rho_k = 5$ was adopted for the odd harmonics, treating the search as a targeted analysis following the identification of the dominant quadrupole mode.

Key Contributions and Results

• LISA Limitations: The study confirms that for the vast majority of Galactic DWDs, the higher harmonics remain undetectable with LISA. In a sample of 1,858 binaries with $f_2 \geq 5$ mHz, zero systems exceeded the SNR threshold of 5 for the first or third harmonics using LISA alone. Only a single case was detected in the hypothetical LISA–Taiji–TianQin (LTT) network configuration. This indicates that LISA will provide a nearly complete census of the population via the quadrupole mode but will not yield systematic constraints on mass ratios via higher harmonics.
• Decihertz Observatory Capabilities: In contrast, planned mid-band observatories (DECIGO and BBO) operating in the $\sim 0.01–0.1$ Hz band show significant potential. For the same sample of 1,858 binaries, DECIGO (with a 10-year mission) is predicted to detect the third harmonic ($k=3$) in approximately 135 systems (roughly 7.3% of the sample), with BBO detecting 170 systems.
• Frequency and Distance Dependence: The detectability of the third harmonic is strongly concentrated at higher frequencies within the band. Of the 135 systems detectable by DECIGO, 106 have $f_2 \geq 7$ mHz and 43 have $f_2 \geq 10$ mHz. This is driven by the $f_2^{1/3}$ scaling of the PN parameter $\beta$ and the rapid decline of the extra-galactic confusion noise at higher frequencies. Detection fractions decrease with distance but remain non-negligible even at 7.5–10 kpc.
• First Harmonic: The first harmonic ($k=1$) is significantly harder to detect than the third, with only 1 system expected to be detectable by DECIGO in the sample.

Significance and Claims
The paper argues that these results establish a "sequential observational strategy" for space-based gravitational-wave astronomy. LISA serves as the foundation, providing a secure census of Galactic DWDs through quadrupole detections. Future decihertz missions (DECIGO/BBO) build upon this foundation by extracting mass-ratio information from the higher harmonics of a significant subset of these systems.

The authors emphasize that while the absolute detection counts depend on the specific population model (varying by roughly 10% with changes in mass distribution assumptions), the qualitative conclusion remains robust: roughly 10% of inspiral binaries with detectable quadrupole emission above 5 mHz will have measurable third harmonics with decihertz observatories. This capability enables statistical constraints on the mass ratio distribution, offering direct observational access to DWD formation channels and binary configurations that are currently inaccessible. The study highlights the critical importance of the $\sim 10$ mHz regime as a bridge between the millihertz and decihertz domains, providing unique astrophysical insights into compact binary evolution.