============================================== Sloppiest way to get the sound horizon (a0*rs) ============================================== tL ~ 400,000 years zL ~ 1000 a0/aL = 1+zL ~ 1000 [Note: aL = a(tL)] cs = c/sqrt[3(1+R)] ~ c/2 1 pc ~ 3 light years The physical distance traveled by the light from t=0 to t=tL: ~3*c*tL (for the matter-dominated Universe) Thus, the Physical distance traveled by the sound wave from t=0 to t=tL in units of pc: 400,000 * 3 / 3 / 2 Extrapolate this to the present-time: 400,000 * 3 / 3 / 2 * 1000 ~ 200 Mpc!! ============================= Slightly better approximation ============================= tL ~ 380,000 years zL ~ 1090 cs ~ c/2 1 pc ~ 3.26 light years The physical distance traveled by the light from t=0 to t=tL: ~2.3*c*tL [Note: The physical distance during the radiation era: 2*c*tL. The Universe at t=tL was not completely dominated by the matter yet.] Thus, the Physical distance traveled by the sound wave from t=0 to t=tL in units of pc: 380,000 * 2.3 / 3.26 / 2 * 1090 ~ 146 Mpc!! =============================================================================== a0*rL ~ 14 Gpc, which looks awfully similar to the age of the Universe, 14 Gyr. Why? Is this a coincidence? =============================================================================== a0*rL ~ 14 Gpc * 3.26 ~ 46 billion light years The physical distance traveled by light for 14 Gyr: 14 Gyr * 3 * c ~ 42 billion light years (for the matter-dominated Universe) But the present-day Universe is accelerating due to dark energy, pushing the distance a bit further. Thus, 14 Gyr * 3.3 * c ~ 46 billion light years (for the LCDM Universe) Thus, the reason that a0*rL ~ 14 Gpc looks similar to 14 Gyr is because the conversion factor from pc to light years is approximately the same as the factor for the physical distance traveled by light in the LCDM Universe.