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Primordial magnetic fields generated in the very early universe are one of the candidates for the origin of magnetic fields observed in galaxy clusters. After recombination, the Lorentz force acts on the residual ions and electrons to generate density fluctuations of baryons. Accordingly these fluctuations induce the early formation of dark haloes which cause the Sunyaev–Zel'dovich (S–Z) effect in cosmic microwave background radiation. This additional S–Z effect due to primordial magnetic fields amplifies the angular power spectrum of cosmic microwave temperature anisotropies on small scales. This amplification depends on the comoving amplitude and the power‐law index of the primordial magnetic fields spectrum. Comparing with the small scale CMB observations, we obtained the constraints on the primordial magnetic fields, i.e.   B ≲ 2.0  nG for   n =−2.9  or   B ≲ 1.0  nG for   n =−2.6  where  B  is the comoving amplitude of magnetic fields at   h  − 1   Mpc and  n  is the power‐law index. Future S–Z measurements have the potential to give constraints tighter than those from temperature anisotropies and polarization of cosmic microwave background induced by the magnetic fields at the recombination epoch.

Sunyaev–Zel'dovich power spectrum produced by primordial magnetic fields