How Much Early Dark Energy Would the Tension Actually Take? The Required Fraction, Window by Window

Early dark energy — a component contributing roughly a tenth of the cosmic density near recombination and then diluting away — is the most studied early-universe response to the Hubble tension. This paper computes, by direct integration of the sound-horizon integral, the density fraction such a component must actually reach to shrink the sound horizon by the full 7.78 percent the SH0ES ladder requires, as a function of the redshift above which it acts. Acting above redshift 1200, the required fraction is 16.2 percent of the total density; above 2000, 22.8 percent; above 3500 — the canonical epoch at which early-dark-energy models are designed to peak — 34.0 percent; and above 5000, 44.2 percent. The canonical ten-percent fraction, evaluated in the same windows, delivers only 61 percent of the required shrinkage in the most favourable window and only 27 percent at its own design epoch. A six-percent fraction delivers between 12 and 36 percent of the target depending on window. The arithmetic therefore separates two claims that are often run together: early dark energy at its canonical size is a partial remedy that moves the inferred Hubble constant partway toward the ladder, and a full remedy requires a component two to four times larger than the canonical one, sustained over the whole window above its switching epoch — a demand that the microwave damping tail and polarisation data then grade severely. The computation is simulated from stated assumptions; it is not a measurement, and it models the component as a step in redshift, which flatters it relative to any smooth realisation.