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Higher Curvature Supergravity, Supersymmetry Breaking and Inflation

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Higher Curvature Supergravity, Supersymmetry Breaking and Inflation CERN-PH-TH/2014-134 Higher Curvature Supergravity, Supersymmetry Breaking and Inflation∗ Sergio Ferraraa;b;c and Alex Kehagiasd;e a Physics Department, Theory Unit, CERN, CH 1211, Geneva 23, Switzerland b INFN - Laboratori Nazionali di Frascati Via Enrico Fermi 40, I-00044 Frascati, Italy c Department of Physics and Astronomy, University of California Los Angeles, CA 90095-1547, USA d Physics Division, National Technical University of Athens, 15780 Zografou, Athens, Greece e Department of Theoretical Physics 24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland dedicated to the memory of Bruno Zumino Abstract In these lectures, after a short introduction to cosmology, we discuss the supergravity embbeding of higher curvature models of inflation. The supergravity description of such models is presented for the two different formulations of minimal supergravity. arXiv:1407.5187v2 [hep-th] 23 Jul 2014 ∗ Lectures given by S.Ferrara at the Erice ISSSP 2014, 52th Course, 24 June-3 July, 2014 1 Introduction These lectures are devoted to the application of higher curvature supergravity to a particular class of cosmological models for inflation in which the “inflaton” field is identified with the \scalaron". The latter is a purely gravitational mode which arises when we add to the Einstein-Hilbert action a term quadratic in the (scalar) curvature 2 2 Lmodified = LEH + αR = R + αR : (1) This theory is \dual" to standard Einstein gravity coupled to a scalar field. The revival of these models was motivated by the fact that the recent experiments Planck [1] and BICEP2 [2] seem to favor simple one-field cosmological models for inflation even if there is a tension between the two experiments. In fact, while for the slow-roll parameter 2 n = 1 − 6 + 2η ≈ 1 − ≈ 0:96 (2) S N i.e., the spectral index of scalar perturbations, the same formula agrees, for the other slow-roll parameter r, the tensor-to-scalar ratio, a value of 12 r = (3) N 2 as in Starobinsky inflation [3] and Higgs Inflation [4], seems to be favorite by the Planck collabo- ration, which reports r < 0:08; (4) whereas chaotic models [5] like the quadratic one, predicts typically 8 r = ≈ 0:2; (5) N and are favored by BICEP2. The slow-roll parameters , η and the number of e-folds N(∼ 50−60) during inflation [6] are defined in terms of the scalar potential V (φ) of a canonically normalized inflaton field with Lagrangian 1 L = L − gµν@ φ∂ φ − V (φ) (6) E 2 µ ν as 1 V 0 2 V 00 Z φinit V = ; η = ;N = 0 dφ : (7) 2 V V φend V In the next section, we will recall some well known matter from Cosmology, and, in particular, the description of the inflaton as a component of a cosmological perfect fluid. In section 3. we 2 describe the Starobinsky model. In section 4. we present the supergravity embedding of the Starobinsky and chaotic inflation in the two different formulations of the N = 1 supergravity. In addition, we show how integrating out the sgoldstino multiplet, the Volkov-Akulov-Starobinsky supergravity emerges. Finally, section 5 contains our conclusions. 2 A (short) Introduction to Cosmology The Standard Model for Cosmology describes the Universe as made of different forms of energy densities acting as sources of the gravitational field. The latter in turn is described by a Friedmann- Lema^ıtre-Robertson-Walker geometry (FLRW) with metric dr2 ds2 = −dt2 + a(t)2 + r2dΩ2 ; (8) 1 − kr2 where the three-dimensional constant-time slice is a maximally symmetric space, the Riemann tensor of which is therefore satisfies k R = (g g − g g ) ; k = 0; ±1 : (9) ijkl a2 ik jl il jk For k > 0; k < 0 and k = 0 we refer to closed, open or flat Universe, respectively. a(t) is the \scale factor" which tells us how big is the 3D slice at (comoving) time t. The above assumptions are motivated by the \Copernican Principle", namely that our Universe looks isotropic and homoge- neous. Isotropy says that space looks the same in any direction and homogeneity that the metric looks the same everywhere. If a space is isotropic everywhere, then it is homogeneous as well. If we have isotropy and homogeneity, then the 3D slice is a maximally symmetric space, i.e., SO(4) SO(3; 1) a) (k > 0); b) (k < 0); c) E (k = 0) : (10) SO(3) SO(3) 3 The Einstein equations are written as 1 G = R − g R = κ2T ; (R = R µ ; κ2 = 8πG = M −2) ; (11) µν µν 2 µν µν µ P where Tµν is the energy-momentum (stress energy) tensor, or, equivalently, 1 R = κ2 T − g T ; (12) µν µν 2 µν which implies 1 T = T µ = − R: (13) µ κ2 3 Note that if the energy-momentum tensor is that of a vacuum energy, Tµν = −Λgµν (14) where Λ = const:; by the energy-momentum conservation µν rµT = 0; (15) we get that R = 4κ2Λ = const: (16) and we have a maximally symmetric space-time, which is de Sitter (DS) (Λ > 0), anti-de Sitter (AdS) (Λ < 0) or Minkowski (Λ = 0). But what is the general form of the energy-momentum tensor Tµν? We can answer to this question if we make the picture of the Universe as being made by perfect fluids, described by an energy density ρ and a pressure p. In this case, the energy-momentum tensor can be expressed as 1 1 T = (ρ + p)U U + pg = (ρ + p) U U + g + (3p − ρ)g ; (17) µν µ ν µν µ ν 4 µν 4 µν where U µ is the relativistic four-velocity vector. We may impose an equation of state p = w ρ ; (18) where p = p(t), ρ = ρ(t) and w = const:, with jwj ≤ 1 by the dominant energy condition. Moreover, the stress tensor conservation (15) implies ρ_ a_ = −3(1 + w) ; (19) ρ a which gives that ρ / a−3(1+w) : (20) Note in particular that a vacuum energy Tµν / gµν as in Eq. (14) implies that ρ + p = 0; (w = −1) ; (21) µ while a traceless stress tensor Tµ = 0 implies 1 3p − ρ = 0; (w = ; radiation) : (22) 3 4 The value of the scalar curvature is 2 µ 2 R = −κ Tµ = κ (1 − 3w)ρ (23) for a single component perfect fluid, or 2 X R = κ (1 − 3wi)ρi (24) i in the case of a multi-component perfect fluid. For w = −1, we have de Sitter or anti-de Sitter space-time depending on the value of ρ (DS for ρ > 0 and AdS for ρ < 0) whereas, R ≥ 0 for 1 w ≤ 3 and ρi ≥ 0. Because of the symmetries of the FLRW geometry, the Einstein equations now read 2 G00 = κ T00 ; (25) 2 Gij = κ Tij ; (26) which give the equations for the scale factor a_ 2 8πG k = ρ − ; (27) a 3 a2 a¨ 4πG = − (ρ + 3p) ; (k = ±1; 0) : (28) a 3 A dot and a double dot denote first and second derivatives with respect to comoving time t. By introducing the Hubble parameter a_ H = (29) a and noticing that a¨ = H_ + H2 ; (30) a we can rewrite the Einstein equations as (Friedmann equation) 8πG k H2 = ρ − ; (31) 3 a2 k H_ = −4πG(ρ + p) + : (32) a2 By defining 3k 1 1 ρ = − ; w = − ; (33) k 8πG a2 k 3 5 Friedmann equations (31,32) are written as 8πG H2 = (ρ + ρ ) ; (34) 3 k 8πG H_ = −4πG(1 + w)ρ − ρ ; (35) 3 k or, in the case of a multicomponent perfect fluid 8πG X H2 = ρ ; (36) 3 i i _ X H = −4πG (1 + wi)ρi ; (37) i 2 (where ρi includes ρk). By dividing Eq. (31) by H and defining the density parameter 8πG Ω = ρ ; (38) 3H2 we get k Ω − 1 = : (39) a2 In addition, if we also define the critical energy density 8πG ρ = ; (40) crit 3H2 we have that ρ Ω = : (41) ρcrit Similarly, Eq. (36) can be compactly written as X Ωi = 1 ; (42) i where ρi Ωi = : (43) ρcrit The second Friedmann equation (32) can be written in terms of the \deceleration parameter" a¨ 1 q = −a = − (H_ + H2) ; (44) a_ 2 H2 6 as 4πG 4πG q = (ρ + 3p) = (1 + 3w)ρ ; (45) 3H2 3H2 or for several components of energy densities 4πG X 1 X q = (1 + 3w )ρ = (1 + 3w )Ω : (46) 3H2 i i 2 i i i i Note that Ωk does not contribute to q (since wk = −1=3). As a result, Einstein equations for an FLRW geometry may be written as X 1 X Ω = 1; q = (1 + 3w )Ω : (47) i 2 i i i i Values for w include w = 0 (dust-baryonic matter) ; (48) 1 w = − (curvature) ; (49) 3 w = 1 (fast roll scalar field) : (50) Whenever a single term dominates, we say that the Universe is dominated by that component of (pi; ρi) or (wi = pi/ρi; ρi). For example, for w = 0 we have a matter dominated Universe, 1 1 for w = 3 a radiation dominated, for w = −1 a vacuum-energy dominated and for w = − 3 a curvature dominated Universe. When only one component dominates, the Friedmann equation a_ 2 8πG / a−3(1+w) (51) a 3 can be easily integrated to give ( 2 t 3(1+w) ; w 6= 1 a ∼ (52) eHt; w = −1; (H = const:) : A scalar field (inflaton) can be viewed as a perfect fluid component of the Universe with Lagrangian 1 L = − gµν@ φ∂ φ − V (φ) ; (53) φ 2 µ ν and stress energy tensor φ @Lφ Tµν = − @νφ + gµνLφ : (54) @@µφ 7 In components, we find for the latter 1 T φ = φ_2 + V (φ); (55) 00 2 1 T φ = φ_2 − V (φ) g ; (56) ij 2 ij which gives, after comparing with the standard form of the stress tensor of a perfect fluid T00 = ρ, Tij = pgij ; (57) that the energy density and the pressure of the inflaton is 1 1 ρ = φ_2 + V (φ); p = φ_2 − V (φ) : (58) 2 2 Therefore, we get that ρ + p = φ_2; ρ + 3p = 2(φ_2 − V ) ; (59) µ _2 ρ − p = 2V; Tµ = φ − 4V; (60) so that we have p = −ρ, (w = −1) for φ_2 V (φ) ; (61) p = ρ, (w = 1) for φ_2 V (φ) ; (62) wheras, in general 1 φ_2(1 − w) = (1 + w)V: (63) 2 The matter equations δLφ δLφ − @µ = 0 (64) δφ δ@µφ together with the Einstein equations are explicitly written as k 8πG 1 H2 + = φ_2 + V (φ) ; (65) a2 3 2 k H_ − = −4πGφ_2 ; (66) a2 ¨ _ φ + 3Hφ + Vφ = 0 : (67) 8 Inflation claims to solve the flatness and horizon problems.
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