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CMB power spectra induced by primordial cross-bispectra between metric perturbations and vector fields

We study temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation sourced from primordial cross-bispectra between metric perturbations and vector fields, which are generated from the inflation model where an inflaton and a vector field are coupled. In case the vector field survives after the reheating, both the primordial scalar and tensor fluctuations can be enhanced by the anisotropic stress composed of the vector fields during radiation dominated era. We show that through this enhancement the primordial cross-bispectra generate not only CMB bispectra but also CMB power spectra. In general, we can expect such cross-bispectra produce the non-trivial mode-coupling signals between the scalar and tensor fluctuations. However, we explicitly show that such mode-coupling signals do not appear in CMB power spectra. Through the numerical analysis of the CMB scalar-mode power spectra, we find that although signals from these cross-bispectra are smaller than primary non-electromagnetic ones, these have some characteristic features such as negative auto-correlations of the temperature and polarization modes, respectively. On the other hand, signals from tensor modes are almost comparable to primary non-electromagnetic ones and hence the shape of observed $B$-mode spectrum may deviate from the prediction in the non-electromagnetic case. The above imprints may help us to judge the existence of the coupling between the scalar and vector fields in the early Universe.