Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 51of massless degrees of freedom, s = 1.8g ∗ n γ [186], and we have assumed an adiabatic post-reheating evolution. With theseassumptions we obtain the photon number densityn γ = 1.28g −3/4∗ (HM Pl ) 3/2 .Recalling that the ratio of the present baryon number to the lepton number originally generated in leptogenesis isapproximately n B /n L = 4/11 [181], in this model we obtainn B= 4.05 × 10 −5 g∗3/4n γ( ) 7/2 H( µ) 6Θ . (323)M Pl HA less naive approach could be to follow the dilution of n L and ρ with the expansion of the universe to the end of reheating.The final result is the same (see, however, [188] for a comment on this point). With the adiabatic expansion assumption, Eq.(323) can be compared directly to the present value of n B /n γ given in Eq. (307).8.3.5. Is Gravi-leptogenesis a viable model?In order to answer this question, one must numerically estimate n B /n γ . Substituting for Θ, Eq. (323) becomes( )n B√ −1/2 ( ) H µ 6≃ 2.9 × 10 −6 g∗3/4 ɛ N . (324)n γ M Pl M PlClearly, this ratio depends on five dimensionless parameters: g ∗ , H/M Pl , µ/M Pl and the slow-roll parameter ɛ and N .Some of these parameters are already constrained or given by theoretical considerations. For example, within the usualsupersymmetric particle physics models, g ∗ ∼ 100 is a reasonable choice. Moreover, WMAP data, through the densityperturbation ratio δρ/ρ (for a single field inflation), leads to an upper bound on H/M Pl ratio [1] of H/M Pl 10 −4 andɛ 0.01, or H 10 14 GeV . Finally, the factor N ≃ 10 2 − 10 10 is inferred from the string theory compactification and isproportional to the square of the four dimensional M Pl to the ten dimensional (fundamental) Planck mass [27] (see also theAppendix in [27]). If the gravi-leptogenesis model is to be viable, we then have thatΘ ≃ 10 −8 − 10 0 , (325)assuming that H saturates its current bound.The physically viable range for the parameter µ depends on the details of the underlying particle physics model onwhich our gravi-leptogenesis is based. For example, within the SM plus three heavy right-handed neutrinos (and the seesawmechanism), µ could be of the order of the right-handed neutrino mass, in which case µ 10 12 GeV. If we do not restrict theanalysis to the seesaw mechanism, µ could be larger, up to perhaps approximately the energy scale at which the effectivefield theory analysis breaks down. Therefore, depending on the details of the model10 −6 µ M Pl 10 −2 − 10 −1 . (326)Combining all these considerations, one finds that the gravi-leptogenesis model predictsn Bn γ( ) µ 6≃ 10 +5 Θ . (327)M PlGravi-leptogenesis thus possesses a large parameter space to explain the observed baryon asymmetry of the Universe, but itsfinal prediction strongly depends on the ultraviolet cutoff of the effective theory [189,190]. In turn, this cutoff dependencearises because lepton number is generated by graviton fluctuations, which we know are non-renormalizable in GR. Thisissue and that of initial conditions are currently being investigated.9. OutlookWe have provided a comprehensive review of CS modified gravity from a particle physics, astrophysical, and cosmologicalperspective. From the particle physics point of view, the presence of the CS term leads to a gravitational anomalycancellationmechanism in the standard model. In cosmology, the chiral anomaly works together with inflation to amplifythe production of leptons, leading to a viable model of leptogenesis. The parity violation present in the Pontryagin densityleads to direct modifications to GW generation and propagation.While all of these avenues are promising directions of research, there is still much to understand. One important issuethat needs to be addressed in more detail is that of the coupling strength and the potential of the CS field. In string theory,the perturbative evaluation of this coupling suggests a Planck suppression, while in LQG it is related to a quantum (Immirizi)ambiguity. The potential of the CS field, on the other hand, cannot be evaluated with current mathematical methods. Thus,new non-perturbative techniques need to be developed to fully evaluate these terms. In lack of a concrete, accurate estimatefor these terms, we have taken an agnostic view in this review: to study the observational consequences of the CS term