Induced gravitational waves provide a powerful probe of primordial perturbations in the early universe through their distinctive spectral properties. We analyze the spectral energy density ${\mathrm{\Omega }}_{\mathrm{GW}}$ of gravitational waves induced by isocurvature scalar perturbations. In the infrared regime, we find that the spectral slope $n_{\mathrm{GW}}\equiv \mathrm{d}\ln {\mathrm{\Omega }}_{\mathrm{GW}}/\mathrm{d}\ln k$ takes the log-dependent form $3-4/\ln ({\tilde{k}}_{*}^2/6k^2)$, where ${\tilde{k}}_{*}$ represents the effective peak scale of the primordial scalar power spectrum. This characteristic behavior differs markedly from that of adiabatic-induced gravitational waves, establishing a robust observational discriminant between isocurvature and adiabatic primordial perturbation modes.

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• Cosmology
• Gravitational waves