Astroparticle Physics

8.8 Experimental Evidence for the Vacuum Energy 187mined by the Friedmann equation once the contributions tothe energy density are specified. Suppose the universe contains(non-relativistic) matter and vacuum energy; the lattercould be described by a cosmological constant Λ.Theircurrentenergy densities divided by the critical density ϱ c canbe written as Ω m,0 and Ω Λ,0 , respectively. Here as usual thesubscripts 0 denote present-day values. One should also considerthe energy density of radiation, Ω r,0 , from photons andneutrinos. Such a contribution is well determined from measurementsof the cosmic microwave background and is verysmall compared to the other terms for the time period relevantto the observations. So neglecting the relativistic particles,one can show that the luminosity distance, d L ,isgivenbyd L (z) = 1 + zH 0∫ z0[Ω Λ,0 + (1 + z ′ ) 3 Ω m,0+ (1 + z ′ ) 2 (1 − Ω 0 )] −1/2dz ′ . (8.36)It is convenient to carry out the integral numerically, yieldingd L (z) for any hypothesized values of Ω m,0 and Ω Λ,0 .Recall the luminosity distance is defined by F = L/4πdL 2 ,whereF is the flux one measures at Earth and L isthe intrinsic luminosity of the source. Astronomers usuallyreplace F and L by the apparent and absolute magnitudes, mand M, which are related to the base-10 logarithm of F andL, respectively (precise definitions can be found in standardastronomy texts such as [10] and in the glossary). The observationof a supernova allows one to determine its apparentmagnitude, m. The absolute magnitude, M, is unknown apriori, but is assumed to be the same (after some correctionsand adjustments) for all type-Ia supernovae. These quantitiesare related to the luminosity distance by( )dLm = 5log 10 + 25 + M. (8.37)1MpcNotice that a higher apparent magnitude m corresponds to afainter supernova, i.e., one further away.A plot of the apparent magnitude m of a sample of distantsupernovae versus the redshift is shown in Fig. 8.6 [11].The data points at low z determine the Hubble constant H 0 .The relation at higher z, however, depends on the matter anddark-energy content of the universe. The various curves onthe plot show the predictions from (8.36) and (8.37) for differentvalues of Ω m,0 and Ω Λ,0 . The curve with no darkcosmological constantluminosity distancemagnitudesHubble diagram