If the electroweak Higgs vacuum expectation value $v$ in early universe is $sim 1 %$ higher than its present value $v_0=246$ GeV, the $^7$Li puzzle in BBN and the CMB/$Lambda$CDM tension with late-universe measurements on Hubble parameter are mitigated. We propose a model of an axion coupled to the Higgs field, named ``axi-Higgs, with its mass $m_a sim 10^{-30} - 10^{-29},{rm eV}$ and decay constant $f_a sim 10^{17} - 10^{18},{rm GeV}$, to achieve this goal. The axion initial value $a_{rm ini}$ yields an initial $Delta v_{rm ini}/v_0 sim 0.01$ throughout the BBN-recombination epoch and a percent level contribution to the total matter density today. Because of its very large de Broglie wavelength, this axion matter density $omega_a$ suppresses the matter power spectrum, alleviating the CMB/$Lambda$CDM $S_8/sigma_8$ tension with the weak-lensing data. It also explains the recently reported isotropic cosmic birefringence by its coupling with photons. Adding the axion ($m sim 10^{-22},$eV) in the fuzzy dark matter model to the axi-Higgs model allows bigger $Delta v_{rm rec}$ and $omega_a$ to address the Hubble and $S_8/sigma_8$ tensions simultaneously. The model predicts that $Delta v$ may be detected by the spectral measurements of quasars, while its oscillation may be observed in the atomic clock measurements.
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