Asymptotically safe inflation from quadratic gravity

Transcription

1 Asymptotically safe inflation from quadratic gravity Alessia Platania In collaboration with Alfio Bonanno University of Catania Department of Physics and Astronomy - Astrophysics Section INAF - Catania Section, INFN - Catania Section 5 October 2015 A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

2 Introduction Power spectrum and primordial perturbations Quantum generation of initial spectrum Quantum fluctuations (a δφ k ) + (k 2 + z z Two-point correlations function: ) (a δφ k ) = 0 z = a ( tφ) H 0 R k R p 0 = (2π)3 2 k 3 P R(k) δ 3 (k + p) R = H δφ t φ Spectral index and tensor-to-scalar-ratio ( ) ns 1+ log(k/k 0) P R (k) = A k 2 s k 0 dns d log(k) P t (k) = A t ( k k 0 ) nt r At A s k Mpc 1 pivot scale A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

3 Introduction Slow-roll inflation Slow-roll inflation and power spectrum Scalar field - inflaton - us a perfect fluid with: ρ φ 1 2 ( tφ) 2 + V (φ) p φ 1 2 ( tφ) 2 V (φ) Definitions of slow-roll parameters: ɛ(φ) = 1 ( V (φ) 2 κ 2 V (φ) ) 2 η(φ) = 1 κ 2 ( V ) (φ) V (φ) Slow-roll inflation ɛ(φ) 1 η(φ) 1 The values of the slow-roll parameters identify: Spectral index n s, tensor-to-scalar-ratio r n s = η(φ i ) 6 ɛ(φ i ) r = 16 ɛ(φ i ) A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

4 Introduction Slow-roll inflation Inflationary models and cosmic data A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

5 Introduction Super-Planckian initial conditions Problem: Super-Planckian initial conditions L, l H N e l Pl L(t) l H (t) l Pl t t t Pl as t tr t N H(t) a/t a > 1 A. Bonanno, M. Reuter, JCAP 08 (2007) 024 A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

6 Beyond the Einsten-Hilbert truncation AS inflation Weinberg Phys. Rev. D 81, (2010) Consider a general truncation to obtain a de Sitter solution which is unstable but lasts N > 60 e-folds. Let us consider the UV Lagrangian: L k = k 2 16πg(k) [R 2 k2 λ(k)] β(k)r 2 (1) Where g k, λ k, β k are dimensionless running coupling constants, such that: lim {g k, λ k, β k } = {g, λ, β } k A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

7 Beyond the Einsten-Hilbert truncation By solving the linearized flow equations for g(k), λ(k) and β(k), we obtain: g k = 6πc 1 k 2 6πµ c 1 (k 2 µ 2 ) (2) ( ) θ k 2 32 β k = β + b 0 λ k c 0 k 2 (3) Where: µ is an infrared renormalization scale θ 3 is the critical exponent relative to β(k) b 0, c 0 and c 1 are free parameters µ 2 b 0, c 0, c 1 relevant directions of the UV critical surface A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

8 Beyond the Einsten-Hilbert truncation The effective action can be obtained by substituting into the UV action: λ uv λ(k) g uv g(k) β uv β(k) Cutoff scale identification? k as function of the cosmological time t: k(t) = ξ/t, ξ > 0 (4) However this identification breaks diffeomorphism invariance. By noting that: a(t) t α R 1/t 2 k 2 We can identify: k 2 ξr (5) A. Bonanno, Phys. Rev. D 85, (2012) E. J. Copeland, C. Rahmede, I. D. Saltas, Phys. Rev. D 91, (2015) A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

9 Beyond the Einsten-Hilbert truncation We thus obtain the effective action at the inflationary era: S = 1 g [R 2κ 2 + αr 2 θ R 2 ] m 2 Λ d 4 x (6) Where: κ 2 = 48 π 2 c 1 6πµ 2 23(µ 2 +2 ξ c 0 )c 1 8πG N Λ = µ2 c 0 (6π 23c 1 ) 6πµ 2 23(µ 2 +2ξc 0 )c 1 α = 2 µ θ 3 b 0 M 2 pl A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

10 Conformal transformation Inflation in f(r) gravity model Let us consider the following general action: S[g µν ] = 1 g {R + F (R)} d 4 2κ 2 x If F (R) 0, we can do a conformal transformation: So that: S[g E µν] g µν g E µν = ϕg µν { ge ϕre 2κ 2 1 } 2 g µν E µφ ν φ V (φ) d 4 x V (φ) = 1 2κ 2 {(ϕ 1) χ(ϕ) F (χ(ϕ))} ϕ2 ϕ = e 2 3 κφ A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

11 Conformal transformation In our case: ( ϕ(χ) 1 + α 2 θ ) 3 χ 1 θ 3 χ m 2 (7) This relation, due to its non-linearity, cannot be inverted. In our case, as θ 3 is rather close to unity, and in order to have an analytical expression for V (φ), we set θ 3 = 1. In this way we explicitly obtain the two branches: χ ± = 3 ( 27α 2 m 4 + 8m 2 ϕ 8m 2 ± 3 ) 3 27α 8 4 m α 2 m 6 (ϕ 1) With the reality condition χ 1 27m 2 α 2 /16. A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

12 Conformal transformation The scalar inflationary potential is the following: V ± (φ) = m2 e 2 2 { 3 κφ ( κ e 3 κφ 1) + 3 α Λ + ) ( ) +3 α (α e 3 κφ 16 ± 6 α α α e 3 κφ ( ( ) ( ) 32α α e 3 κφ 8 α ) α e 3 κφ 16 We can study the inflationary scenario coming from the potential V (φ). A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

13 Constraints on (α, Λ) The shape of the potential depends on the values (α, Λ). Examples: V Φ 5 Α 20 1, Α 0 V Φ 2 4 Α Φ 3 Α 10 2 Α Α Α 4.5 Α Α Φ 10 We choose ranges for α and Λ such that: V (φ) has a minimum (oscillatory phase); We can have a graceful exit from inflation V (φ min ) 0. A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

14 Constraints on (α, Λ) These features are verified for V (φ) = V + (φ) if α [1, 3] and Λ [0, 1.5] V Φ 1 2 Α 1.3 Α 1.6 Α Α 2.2 Α 2.5 Φ 3 A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

15 Risults: Spectral index and tensor-to-scalar ratio Planck 2015: n s = ± r < 0.11 AS inflation: n s [0.968, 0.970] r [0.005, 0.006] A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

16 Oscillatory phase after inflation Oscillatory phase after inflation After the end of inflation, the inflaton field φ begins to oscillate around the minimum φ min of V (φ). To study this phase, we can approximate: Where: φ min = φ min (α, Λ) a(α, Λ) = V (φ min ) b(α, Λ) = 2 V (φ min) V (φ min ) V (φ) a 2 [ (φ φmin ) 2 b ] A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

17 Oscillatory phase after inflation In particular: φ min (α, Λ) = ( 3 2 3α 3 ( α 2 4 ) 32αΛ + 4 ( α 2 6 ) α 3) 6α (α 2 8) (α 2 + 2) 64αΛ + 8 (α 2 9) α 3 a(α, Λ) = α2 3 α Λ 4 α 3 α + 36 α α 24 b(α, Λ) = 8α ( 15α 4 3α 6 96Λ + 8α 2 (15 + 4Λ) ) α 8 3 ( α2 3α Λ 4α (α 2 9) α ) 2 25α α 6 384α 2 Λ ( α2 3α Λ 4α (α 2 9) α ) 2 48α 4 (21 + 4Λ) 1024Λ(3 + Λ) ( α2 3α Λ 4α (α 2 9) α ) 2 A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

18 Oscillatory phase after inflation The time evolution of the field φ(t) is given by φ(t) + 3 H(t) φ(t) + V (φ(t)) = 0 (8) Where: Putting: H(t) = [ ( )] 1 1 1/2 3 2 φ(t) 2 + V (φ(t)) x(t) = a (φ(t) φ min ) y(t) = φ(t) The original equation is equivalent to the following dynamical system: ẏ = [ ( 3 2 y 2 + x 2 ab )] 1 2 y a x ẋ = (9) a y A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

19 Oscillatory phase after inflation The long time behavior is determined by the sign of ab = 2 V (φ min ) V (φ min ) < 0 Limit cycle behavior (our case) V (φ min ) > 0 (φ min, V (φ min )) is an attractive node V (φ min ) = 0 is an Hopf bifurcation point A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

20 Oscillatory phase after inflation This analysis can be useful to determine the scale factor a(t). We obtain: [ ( 2/3 3 sin 4 V (φ min) t)] V (φ min ) < 0 a(t) = t 2/3 V (φ min ) = 0 (10) [ ( 2/3 3 sinh 4 V (φ min) t)] V (φ min ) > 0 a t A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21

21 Conclusions Conclusions Conclusions Our model is significatly different from the Starobinsky model because it predicts a tensor-to-scalar ratio which is significantly higher, and a dynamics characterized by a limit-cycle behavior at the inflation exit; It is in agreement with Planck 2015 data; Present CMB data can put important constraints on the structure of the effective Lagrangian at the Planck scale; Limitation: simple tensorial structure of the effective Lagrangian which assumes a functional dependence of the f (R) type. A. Bonanno, A. Platania AS inflation from quadratic gravity 5 October / 21