Inflation is today a part of the Standard Model of the Universe supported by the cosmic microwave background (CMB) and large scale structure (LSS) datasets. Inflation solves the horizon and flatness problems and naturally generates density fluctuations that seed LSS and CMB anisotropies, and tensor perturbations (primordial gravitational waves). Inflation theory is based on a scalar field $\varphi$ (the inflaton) whose potential is fairly flat leading to a slow-roll evolution. This review focuses on the following new aspects of inflation. We present the effective theory of inflation \`a la Ginsburg-Landau in which the inflaton potential is a polynomial in the field $ \varphi $ and has the universal form $V(\varphi) = N M^4 w(\varphi/[\sqrt{N}\; M_{Pl}])$, where $ w = {\cal O}(1) , \; M \ll M_{Pl} $ is the scale of inflation and $ N \sim 60 $ is the number of efolds since the cosmologically relevant modes exit the horizon till inflation ends. The slow-roll expansion becomes a systematic $ 1/N $ expansion and the inflaton couplings become {\bf naturally small} as powers of the ratio $ (M / M_{Pl})^2 $. The spectral index and the ratio of tensor/scalar fluctuations are $ n_s - 1 = {\cal O}(1/N), \; r = {\cal O}(1/N) $ while the running index turns to be $ d n_s/d \ln k = {\cal O}(1/N^2) $ and therefore can be neglected. The energy scale of inflation $ M \sim 0.7 \times 10^{16}$ GeV is completely determined by the amplitude of the scalar adiabatic fluctuations. A complete analytic study plus the Monte Carlo Markov Chains (MCMC) analysis of the available CMB+LSS data (including WMAP5) with fourth degree trinomial potentials showed: (a) the {\bf spontaneous breaking} of the $ \varphi \to - \varphi $ symmetry of the inflaton potential. (b) a {\bf lower bound} for $ r $ in new inflation:$ r > 0.023 \; (95\% \; {\rm CL}) $ and $ r > 0.046 \; (68\% \; {\rm CL}) $. (c) The preferred inflation potential is a {\bf double well}, even function of the field with a moderate quartic coupling yielding as most probable values: $ n_s \simeq 0.964 ,\; r\simeq 0.051 $. This value for $ r $ is within reach of forthcoming CMB observations. The present data in the effective theory of inflation clearly {\bf prefer new inflation}. Study of higher degree inflaton potentials show that terms of degree higher than four do not affect the fit in a significant way. In addition, horizon exit happens for $ \varphi/[\sqrt{N} \; M_{Pl}] \sim 0.9 $ making higher order terms in the potential $ w $ negligible. We summarize the physical effects of {\bf generic} initial conditions (different from Bunch-Davies) on the scalar and tensor perturbations during slow-roll and introduce the transfer function $ D(k) $ which encodes the observable initial conditions effects on the power spectra. These effects are more prominent in the \emph{low} CMB multipoles: a change in the initial conditions during slow roll can account for the observed CMB quadrupole suppression. Slow-roll inflation is generically preceded by a short {\bf fast-roll} stage. Bunch-Davies initial conditions are the natural initial conditions for the fast-roll perturbations. During fast-roll, the potential in the wave equations of curvature and tensor perturbations is purely attractive and leads to a suppression of the curvature and tensor CMB quadrupoles. A MCMC analysis of the WMAP+SDSS data including fast-roll shows that the quadrupole mode exits the horizon about 0.2 efold before fast-roll ends and its amplitude gets suppressed. In addition, fast-roll fixes the initial inflation redshift to be $ z_{init} = 0.9 \times 10^{56} $ and the {\bf total number} of efolds of inflation to be $ N_{tot} \simeq 64 $. Fast-roll fits the TT, the TE and the EE modes well reproducing the quadrupole supression. A thorough study of the quantum loop corrections reveals that they are very small and controlled by powers of $(H /M_{Pl})^2 \sim {10}^{-9} $, a conclusion that validates the reliability of the effective theory of inflation. The present review shows how powerful is the Ginsburg-Landau effective theory of inflation in predicting observables that are being or will soon be contrasted to observations.
Boyanovsky, D., Destri, C., de Vega, H., Sanchez, N. (2009). The effective theory of inflation in the standard model of the universe and the CMB+LSS data analysis. INTERNATIONAL JOURNAL OF MODERN PHYSICS A, 24(20/21), 3669-3864 [10.1142/S0217751X09044553].
The effective theory of inflation in the standard model of the universe and the CMB+LSS data analysis
DESTRI, CLAUDIO;
2009
Abstract
Inflation is today a part of the Standard Model of the Universe supported by the cosmic microwave background (CMB) and large scale structure (LSS) datasets. Inflation solves the horizon and flatness problems and naturally generates density fluctuations that seed LSS and CMB anisotropies, and tensor perturbations (primordial gravitational waves). Inflation theory is based on a scalar field $\varphi$ (the inflaton) whose potential is fairly flat leading to a slow-roll evolution. This review focuses on the following new aspects of inflation. We present the effective theory of inflation \`a la Ginsburg-Landau in which the inflaton potential is a polynomial in the field $ \varphi $ and has the universal form $V(\varphi) = N M^4 w(\varphi/[\sqrt{N}\; M_{Pl}])$, where $ w = {\cal O}(1) , \; M \ll M_{Pl} $ is the scale of inflation and $ N \sim 60 $ is the number of efolds since the cosmologically relevant modes exit the horizon till inflation ends. The slow-roll expansion becomes a systematic $ 1/N $ expansion and the inflaton couplings become {\bf naturally small} as powers of the ratio $ (M / M_{Pl})^2 $. The spectral index and the ratio of tensor/scalar fluctuations are $ n_s - 1 = {\cal O}(1/N), \; r = {\cal O}(1/N) $ while the running index turns to be $ d n_s/d \ln k = {\cal O}(1/N^2) $ and therefore can be neglected. The energy scale of inflation $ M \sim 0.7 \times 10^{16}$ GeV is completely determined by the amplitude of the scalar adiabatic fluctuations. A complete analytic study plus the Monte Carlo Markov Chains (MCMC) analysis of the available CMB+LSS data (including WMAP5) with fourth degree trinomial potentials showed: (a) the {\bf spontaneous breaking} of the $ \varphi \to - \varphi $ symmetry of the inflaton potential. (b) a {\bf lower bound} for $ r $ in new inflation:$ r > 0.023 \; (95\% \; {\rm CL}) $ and $ r > 0.046 \; (68\% \; {\rm CL}) $. (c) The preferred inflation potential is a {\bf double well}, even function of the field with a moderate quartic coupling yielding as most probable values: $ n_s \simeq 0.964 ,\; r\simeq 0.051 $. This value for $ r $ is within reach of forthcoming CMB observations. The present data in the effective theory of inflation clearly {\bf prefer new inflation}. Study of higher degree inflaton potentials show that terms of degree higher than four do not affect the fit in a significant way. In addition, horizon exit happens for $ \varphi/[\sqrt{N} \; M_{Pl}] \sim 0.9 $ making higher order terms in the potential $ w $ negligible. We summarize the physical effects of {\bf generic} initial conditions (different from Bunch-Davies) on the scalar and tensor perturbations during slow-roll and introduce the transfer function $ D(k) $ which encodes the observable initial conditions effects on the power spectra. These effects are more prominent in the \emph{low} CMB multipoles: a change in the initial conditions during slow roll can account for the observed CMB quadrupole suppression. Slow-roll inflation is generically preceded by a short {\bf fast-roll} stage. Bunch-Davies initial conditions are the natural initial conditions for the fast-roll perturbations. During fast-roll, the potential in the wave equations of curvature and tensor perturbations is purely attractive and leads to a suppression of the curvature and tensor CMB quadrupoles. A MCMC analysis of the WMAP+SDSS data including fast-roll shows that the quadrupole mode exits the horizon about 0.2 efold before fast-roll ends and its amplitude gets suppressed. In addition, fast-roll fixes the initial inflation redshift to be $ z_{init} = 0.9 \times 10^{56} $ and the {\bf total number} of efolds of inflation to be $ N_{tot} \simeq 64 $. Fast-roll fits the TT, the TE and the EE modes well reproducing the quadrupole supression. A thorough study of the quantum loop corrections reveals that they are very small and controlled by powers of $(H /M_{Pl})^2 \sim {10}^{-9} $, a conclusion that validates the reliability of the effective theory of inflation. The present review shows how powerful is the Ginsburg-Landau effective theory of inflation in predicting observables that are being or will soon be contrasted to observations.
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