Astroparticle Physics

8.9 Problems 189where E max is the maximum energy at which the field theoryis valid. One expects the Standard Model of particle physicsto be incomplete, for example, at energies higher than thePlanck energy, E Pl ≈ 10 19 GeV, where quantum-gravitationaleffects come into play. So the vacuum energy mightbe roughlyϱ v ≈ EPl 4 ≈ 1076 GeV 4 . (8.39)But from the observed present-day value Ω Λ,0 ≈ 0.7, thevacuum energy density is3H02 ϱ Λ,0 = Ω Λ,0 ϱ c,0 = Ω Λ,08πG ≈ 10−46 GeV 4 . (8.40)The discrepancy between the naïve prediction and the observedvalue is 122 orders of magnitude.So something has clearly gone wrong with the prediction.One could argue that not much is understood about thephysics at the Planck scale, which is surely true, and so alower energy cutoff should be tried. Suppose one takes themaximum energy at the electroweak scale, E EW ≈ 100 GeV,roughly equal to the masses of the W and Z bosons. At theseenergies the Standard Model has been tested to high accuracy.The prediction for the vacuum energy density becomesϱ v ≈ EEW 4 ≈ 108 GeV 4 . (8.41)Now the discrepancy with the observational limit is ‘only’53 orders of magnitude. Perhaps an improvement but clearlynot enough.Finally one might argue that the entire line of reasoningwhich predicts vacuum energy might be incorrect. Butphenomena such as the Casimir effect discussed in Sect. 8.4have been observed experimentally and provide an importantconfirmation of this picture. So one cannot dismiss thevacuum energy as fiction; there must be some other reasonwhy its contribution to Ω is much less than expected. This iscurrently one of the most gaping holes in our understandingof the universe.discrepancy betweenexpectation and observationCasimir effect8.9 Problems1. Derive the relativistic relation (8.2) between redshift andvelocity of a receding galaxy.2. A gas cloud gets unstable if the gravitational energy exceedsthe thermal energy of the molecules constitutingthe cloud, i.e.,