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tokens: one is a delta-function type in logarithmic scale given by where corresponds to the amplitude at the peak wavenumber k p , and another is a power-law type given by where is the amplitude at the normalization wavenumber k n . In this Letter, we set the normalization scale as k n = 1 Mpc −1 . First, we focus on the delta function type for the power spectrum. By substituting equation ( 8 ) into equation ( 3 ) with equation ( 7 ), we obtain the injected energy fraction to the CMB energy by the decaying PMFs as where a function of k p , is explicitly given by When k p is much larger than k D ( z i ) (> k D ( z μ )), the energy injection due to the decay of the PMFs becomes efficient before the BBN era. On the other hand, when k p is much smaller than k D ( z μ ), the energy injection from the PMFs occurs during the CMB distortion era. We focus on the case where the energy injection from the decaying of the PMFs is efficient after the BBN era and before the CMB distortion era, that is, the case where k p is set between the two decaying scales k D ( z i ) and k D ( z μ ). For such a case, we can approximately take (see Miyamoto et al. 2014 ). Combining equa-tion ( 2 ) and equation ( 10 ) yields a constraint on PMFs in the k p - plane. We present our new result obtained from magnetic reheating as a red line in Fig. 1 . Figure 1. The upper bound on the amplitude of the delta-function type for the power spectrum as a function of k p from the magnetic reheating. The limits from the BBN (Kawasaki Kusakabe 2012 ) and CMB distortions (Jedamzik et al. 2000 ) in blue and magenta lines, respectively, are shown in the same plot. For comparison, we plot limits from the BBN and CMB distortions in magenta and blue lines, respectively. We can see that our magnetic reheating constraint gives a tight limit on the PMFs on small sales from to 10 8 Mpc −1 . Next, we consider the power-law type of the power spectrum defined in equation ( 9 ). By using equation ( 3 ) with equation ( 7 ), we obtain Here, we focus on the magnetic reheating due to the fast-magnetosonic mode given in equation ( 5 ). In Fig. 2 , we plot a limit for the amplitude of the power-law type as a function of the spectral tilt n B in a red line. Figure 2. The upper bound on the amplitude of the power-law type for the power spectrum as the function of n B from the magnetic reheating of the fast-magnetosonic mode (red) and Alfvén-magnetosonic mode (blue). The constraint from the Planck (Planck Collaboration XIX 2016 ) is also shown. Note that n B = −3.0 corresponds to the scale-invariant power spectrum. We find that blue-tilted spectrum is strongly constrained compared with the scale-invariant spectrum ( n B = −3.0). It is shown that the causal mechanism of PMF generation predicts only a blue spectrum with n B ≥ 2 (Durrer Caprini 2003 ). Therefore, the constraint on the blue-tilted spectrum is suggestive for such a causal mechanism. So far we have considered the case that the Alfvén and