Shrinking the Sound Horizon in a Bounced Cosmology: How Much Early Expansion Boost the Hubble Tension Requires

If the local distance ladder is correct, the sound horizon at the drag epoch must be about 7.78 percent smaller than the standard history predicts. This paper computes, by direct integration of the sound-horizon integral, how large a pre-recombination expansion-rate enhancement a bounced cosmology must supply to achieve that reduction, as a function of the redshift at which the enhancement switches on. The standard integral reproduces the Planck sound horizon at 146.95 Mpc. An enhancement active above redshift 1200 must raise the expansion rate by 9.2 percent; above redshift 2000, by 13.8 percent; above redshift 5000, by 33.9 percent; and an enhancement confined above redshift 20000 cannot reach the target at any strength up to a factor of six, because only a small share of the sound horizon accumulates there. Only 12.7 percent of the sound horizon accumulates between the drag epoch and redshift 1300, which is why late, weak modifications fail. The density ceiling of the bounce frame, 9.72 × 10^57 kg per cubic metre on the neutron branch, corresponds to a maximum redshift of 1.05 × 10^22 and a maximum temperature of 2.87 × 10^22 kelvin, far above the epochs considered here, so the ceiling itself does not obstruct the required boost; the obstruction is the narrowness of the redshift window in which a boost is effective. The computation is simulated from stated assumptions, not measured.