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We revisit the method of using the photon-graviton conversion mechanism in the presence of the external magnetic field to probe small-scale primordial magnetic fields that may exist between the last scattering surface and the present. Specifically, we investigate impacts on the conversion efficiency due to the presence of matter, including the plasma collective effect and the atomic polarizability. In general, these effects tend to reduce the conversion probability. Under this more realistic picture and based on the precision of Cosmic Background Explorer's measurement of the cosmic microwave background (CMB) blackbody spectrum, we find an upper bound for the primordial magnetic field strength, B30 G, at the time of recombination. Although at present the bound based on the photon-graviton conversion mechanism is not as tight as that obtained by the direct use of CMB temperature anisotropy, it nevertheless provides an important independent constraint on primordial magnetic fields and at epochs in addition to the recombination. The bound can be significantly improved if the CMB blackbody spectrum measurement becomes more precise in future experiments such as Primordial Inflation Explorer.
Chen et al. (Thu,) studied this question.