How Robust is the Cosmic Distance with Tip of Red Giant Branch against Stellar Population Variations?
The tip of the red giant branch (TRGB) provides a key standard candle for extragalactic distance measurements and for refining the Hubble constant. We test its robustness by quantifying how metallicity, $\alpha$-element enhancement, age, and initial helium abundance modulate the TRGB luminosity, using synthetic composite color--magnitude diagrams in the $I$ and $F814W$ bands. We find that metallicity and $\alpha$-element enhancement are the primary drivers of TRGB variation, while age introduces only a modest effect and helium abundance is negligible. At fixed age and helium content, increasing the mean metallicity by 0.5 dex or the $\alpha$-element enhancement by 0.3 dex produces the well-known systematic dimming of 0.046 and 0.050 mag, respectively, in $M_I^{\rm TRGB}$, and of 0.093 and 0.044 mag, respectively, in $M_{F814W}^{\rm TRGB}$. By comparison, changes in age of 3~Gyr and in initial helium abundance of 0.10 yield minor luminosity shifts, with average changes of 0.031 and 0.009~mag, respectively, in $M_I^{\rm TRGB}$, and of 0.035 and 0.027 mag, respectively, in $M_{F814W}^{\rm TRGB}$, substantially smaller than those caused by variations in metallicity or $\alpha$-element enhancement. For mixed stellar populations under typical stellar-halo metallicity conditions, the net variation in $M_I^{\rm TRGB}$ arising from each combination of the $\alpha$-element enhancement, age, and initial helium abundance remains below 0.028~mag, well within reported systematic uncertainties. Together, these results reaffirm the TRGB as a highly robust distance indicator and support its continued use as an independent anchor for precision cosmology in the era of the Hubble-tension debate.