Spherical collapse in DHOST theories fails for the beyond-Horndeski parameter β1 above about 10^-7 because the scalar-field gradient becomes imaginary, and the halo mass function is suppressed relative to ΛCDM.
In order for the theory to propagate one scalar and two tensor degrees of freedom, the functions f andaI must obey degeneracy conditions
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Spherical collapse in DHOST theories and EFT of dark energy
Spherical collapse in DHOST theories fails for the beyond-Horndeski parameter β1 above about 10^-7 because the scalar-field gradient becomes imaginary, and the halo mass function is suppressed relative to ΛCDM.