φ 2 or not φ 2 : Checking the Simplest Universe Paolo Creminelli, 1, 2 Diana L´ opez Nacir, 1, 3 Marko Simonovi´ c, 4, 5 Gabriele Trevisan, 4, 5 and Matias Zaldarriaga 2 1 Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy 2 Institute for Advanced Study, Princeton, NJ 08540, USA 3 Departamento de F´ ısica and IFIBA, FCEyN UBA, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabell´ on I, 1428 Buenos Aires, Argentina 4 SISSA, via Bonomea 265, 34136, Trieste, Italy 5 Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34136, Trieste, Italy The simplest inflationary model V = 1 2 m 2 φ 2 represents the benchmark for future constraints. For a quadratic potential the quantity (ns − 1) + r/4 + 11(ns − 1) 2 /24 vanishes (up to corrections which are cubic in slow-roll) and can be used to parametrize small deviations from the minimal scenario. Future constraints on this quantity will be able to distinguish a quadratic potential from a PNGB with f  30M pl , set limits on the deviation from unity of the speed of sound |cs − 1|  3 · 10 −2 (corresponding to an energy scale Λ  2 · 10 16 GeV), and on the contribution of a second field to perturbations ( 6 · 10 −2 ). The limiting factor for these bounds will be the uncertainty on the spectral index. The error on the number of e-folds will be ΔN ≃ 0.4, corresponding to an error on the reheating temperature ΔT rh /T rh ≃ 1.2. We comment on the relevance of non-Gaussianity after BICEP2 results. Motivations. The recent detection of B-modes in the polarization of the CMB by BICEP2 [1] indicates a high level of primordial tensor modes. This requires [2]a large excursion of the inflaton during inflation, Δφ  M pl , which challenges the naive expectation that higher- dimension operators suppressed by powers of M pl spoil the slow-roll conditions. While, before BICEP2, the cru- cial question for inflation was “large or small r?”, we are now facing a new dichotomy: “φ 2 or not φ 2 ?” The two possibilities are qualitatively different. A large field model which is not quadratic, say V ∝ φ 2/3 , suggests an interesting UV mechanism, like monodromy inflation [3] for instance. If data will, on the other hand, favour a quadratic potential, the simplest explanation will be that inflation occurs at a generic minimum of a poten- tial whose typical scale of variation f is much larger than the Planck scale. Indeed an approximate shift-symmetry gives rise to potentials that are periodic in φ/f , like for instance V =Λ 4 (1 − cos(φ/f )) [4]. For f ≫ M pl infla- tion occurs near a minimum of the potential, where one can approximate V ∝ φ 2 . In string theory it seems dif- ficult to obtain a parametric separation between f and M pl , although there is no issue at the level of field theory [5]. Therefore, if quadratic inflation will remain compat- ible with the data, it will be important to study small deviations from it, to understand to which extent the quadratic approximation holds and to limit other possi- ble deviations from the simplest scenario of inflation. Inflationary predictions must face our ignorance about the reheating process and the subsequent evolution of the Universe. All this is encoded in the number of e-folds N between when the relevant modes exit the horizon and the end of inflation. The dependence on N is rather strong (see Fig. 1) and it will become larger than the experimental sensitivity on n s and r. To study small deviations from V = 1 2 m 2 φ 2 we have to concentrate on a combination of observables which does not depend on N ( 1 ). At linear order in 1/N , given that for a quadratic potential n s − 1= −2/N and r =8/N , a prediction which is independent of N is obviously (n s − 1)+ r/4 = 0. Since corrections at second order in slow-roll will not be completely negligible in the future, it is worthwhile to go to order 1/N 2 . Using the explicit formulas at second order in slow-roll [6] it is straightforward to verify 2 that for a quadratic potential (n s − 1) + r 4 + 11 24 (n s − 1) 2 =0 , (1) up to corrections of order N −3 ( 3 ), which we can safely ignore. Assuming that data will favor a φ 2 potential we can use the equation above to study how sensitive we will be to small departures from the simplest scenario. The best measurement of the tilt comes from Planck [7] n s −1= −0.0397±0.0073, while the recent value of r mea- sured by BICEP2 is r =0.20 +0.07 −0.05 (without foreground 1 The power spectrum normalisation fixes m ≃ 1.3 · 10 13 GeV, V ≃ (1.9 · 10 16 GeV) 4 and Δφ ≃ 15 M pl , assuming N = 60. 2 Up to second order in slow-roll we have ns − 1=2η − 6ǫ − 2C(12ǫ 2 + ξ)+ 2 3 (η 2 − 5ǫ 2 + ξ) + (16C − 2)ηǫ , r = 16ǫ  1 − 4ǫ 3 + 2η 3 +2C(2ǫ − η)  , where C ≡ −2 + ln2 + γ, with γ = 0.57721 ... the Euler- Mascheroni constant, and the slow-roll parameters are defined as ǫ ≡ M 2 pl 2  V ′ V  2 , η ≡ M 2 pl V ′′ V , ξ ≡ M 4 pl V ′′′ V ′ V 2 . 3 Up to 1/N 3 corrections we can equivalently write (ns − 1) + r/4 + 11/384 · r 2 = 0. This form can be useful in future given that the error on (r/4) 2 is expected to be smaller than the one on (ns − 1) 2 . arXiv:1404.1065v1 [astro-ph.CO] 3 Apr 2014