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The specificity of the spectrum of relic gravitational waves formed at the inflationary and post-inflationary stages of the evolution of the early universe is considered for cosmological models based on modified theories of gravity in relation to the case of standard inflationary models based on Einstein's gravity. The possibility of detecting high-frequency relic gravitational waves by using the process of converting gravitons into photons in a constant and alternating magnetic field is considered. The sensitivity of detectors of this type is compared with the sensitivity of other existing and prospective detectors of high-frequency gravitational waves. Based on the analysis of the sensitivity assessment of various types of high-frequency gravitational wave detectors, a conclusion is made about the prospects for direct verification of cosmological inflation models using gravitational-wave detectors.