Regarding the Deceleration of the Universe

Draft: February 1, 2008 In the standard big bang model, the expansion rate of the Universe is predicted to slow down as the Universe evolves. ,2 The deceleration parameter, q0, measures the present deceleration rate. The deceleration causes a deviation from the linear relation between distance and red shift (the Hubble law) at large red shifts. A proposed approach for determining q0 is to measure the red shifts and distances of Type Ia supernovae, whose luminosities may be calibrated so that the inverse-square-law can be used to judge their distance. To date, the strategy has been to use measurements at low red shift z ≤ 0.1 to precisely determine the linear distance-red shift relation and then focus on measuring the deviation from the linear law using supernovae at high red shifts z = 0.35− 0.6 to determine q0. This note makes a simple point that is important in interpreting the supernovae results: At red shift z = 0.35− 0.6, the deviation from the linear Hubble relation does not depend on q0 alone, but also on the matter-energy content of the universe. It is well-known that this dependence on matter-energy content is significant at sufficiently high red shifts, but it has not been generally appreciated that the dependence is non-negligible at red shifts as modest as z = 0.35. The effect is easy to understand. Observations at non-zero red shift measure properties of the universe at earlier times when the light was emitted. In some models, such as open models with low matter density, the deceleration rate at z ≈ 0.5 is comparable to the deceleration rate today (q0). In other models, such as those with non-zero cosmological constant, the universe is undergoing a fairly rapid transition from a decelerating, matterdominated universe at z = 0.35 towards an accelerated expansion in a universe dominated by a non-zero cosmological constant. The two models predict a different deviation from the linear Hubble law at z = 0.35 even if the present deceleration rate q0 is the same. The “luminosity distance” between a given source and us is defined as d2L ≡ L/4πF where L is the emitted energy per unit time and F is the energy received per unit time. An