Axi-Higgs Cosmology
  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.
