Warm intermediate inflationary universe models with a generalized dissipative coefficient / 34

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1 Warm intermediate inflationary universe models with a generalized dissipative coefficient Departamento de Física Universidad de Chile In collaboration with: Ramón Herrera and Marco Olivares CosmoSur III August 2015, Córdoba, Argentina

2 Outline 1 Basics of the Big-Bang cosmological model 2 Cold Inflation 3 Intermediate inflationary model 4 Warm inflation 5 Warm intermediate inflationary models with a generalized dissipative coefficient General Relativity Brane-World Gravity 6 Conclusions

3 The Big-Bang cosmological model General Relativity (GR) G µν = κt µν, where κ = 8πG An isotropic and homogeneous expanding universe is described by the Friedmann-Robertson-Walker (FRW) metric [ ds 2 = dt 2 + a(t) 2 dr 2 ( ] 1 kr 2 + r 2 dθ 2 + sin 2 θdφ 2). Energy-momentum tensor for a perfect fluid T µν = (ρ + P) u µ u ν + Pg µν Friedmann and acceleration equations (ȧ ) 2 H 2 = κ a 3 ρ k a 2 ä a = κ (ρ + 3P) 6

4 Cosmic Inflation Some Big-Bang cosmological model shortcomings: The flatness problem The horizon problem The exotic relics problem The density fluctuations problem These problems are solved by a phase of acceleration during the very early universe: ä > 0 P < ρ/3 (Starobinsky (1980); Guth (1981); Albrecht, Steinhardt (1982); Linde (1982)) The standard theory of inflation predicts that the large scale structure of the universe can be traced back to quantum vacuum fluctuations of a weakly coupled field, called inflaton, during the inflationary expansion.

5 In cold inflaton, interactions with other d.o.f. are important only during (p)reheating. Interactions of the inflaton with other fields are considered negligible during inflation: ρ r + 4Hρ r = 0, (H = ȧ a const.) The radiation density during inflation redshifts away: ρ r a 4 Universe is supercooled during inflation A separate reheating phase after inflation brings the universe to a radiation dominated epoch.

6 Inflation driven by a scalar field Homogeneous scalar field in a FRW universe Equations of motion ρ φ = φ 2 H 2 = κ V (φ), P φ = φ 2 2 V (φ) ( ) φ2 2 + V (φ), φ + 3H φ + Vφ = 0 ä > 0 φ 2 < V (φ) Slow-roll conditions: φ2 /2 V (φ) and φ 3H φ Slow-roll parameters During inflation ɛ H, η H 1. Number of e-folds N ɛ H Ḣ(t) (ä > 0 ε < 1) H(t) 2 Ḧ(t) η H 2H(t)Ḣ(t) t2 N H dt t 1

7 Cosmological perturbations Inflationary observables Curvature perturbation: R = Hδφ/ φ, δφ = H 2π (Guth, Pi (1982); Hawking (1982); Bardeen et al. (1983)) k 0 = Mpc 1 Scalar spectral index: ( ) k ns 1 P R = P R (k 0 ), P R (k 0 ) = amplitude, k 0 n s 1 = d ln P R d ln k 2η 6ε, where ε = m2 p 2 ( Vφ ) 2 V and η = m 2 V φφ p V Tensor-to-scalar ratio: r = P T P R 16 ε

8 Inflation and CMB observations Inflationary observables NASA/JPL-Caltech/ESA

9 Inflation and CMB observations Inflationary observables Planck 2013 results (P. A. R. Ade et al., arxiv: ): ( P R (k 0 ) = n s = ± r < 0.12 ) 10 9 BICEP2 r = 0.2 (P. A. R. Ade et al., Phys. Rev. Lett. 112, (2014)) Planck 2015 results (P. A. R. Ade et al. (2015)): ( P R (k 0 ) = n s = ± r < ) 10 9

10 Intermediate inflationary model Intermediate inflation was introduced as an exact solution for a particular scalar field potential of the type ( Barrow (1990); Barrow et al. (2006)) V (φ) = 8A2 (2Aβ) β/2 κ(β + 4) 2 ( ) 6 β2 κφ where A > 0 and β 4(1 f ) f, with 0 < f < 1 The solution for the scale factor is given by a(t) = exp (A t f ) (κφ) β, At lowest order in the slow-roll approximation, this model predicts β(β 2) n s 1 κφ 2 r 8β(1 n s )/(β 2)

11 Intermediate inflation or late-time aceleration? The main motivation to study intermediate inflationary model comes from string/m theory It has been found that for a dark energy model, a Gauss-Bonnet interaction in four dimensions with a dynamical dilatonic [ scalar field ( ) ] coupling, leads to a solution of the form a(t) = exp 2 κn t1/2 (Sanyal (2007)), where n > 0 Recently, Cid et al. (2015) have found intermediate solutions of the form a(t) = exp (α t p ), where α > 0 and 0 < p < 1, as late-time solutions in a modified Jordan-Brans-Dicke theory

12 Intermediate inflationary model in light of recent data P. A. R. Ade et al. (2015) Barrow et al. (2006) This model lies outside the join 95 % CL contour region for any value of the parameter f, or equivalently β, being disproved by the observations (P. A. R. Ade et al. (2013), (2015) and del Campo (2014) )....but what if the universe was not supercooled? del Campo (2014)

13 Warm inflation Ian G. Moss PLB , Yokoyama and Maeda PLB , A. Berera and L. Z. Fang PRL , Berera PRL Interactions of the inflaton with other d.o.f. are important during inflation and generate dissipative terms small fraction of the inflaton energy density can be converted to radiation. Inflationary expansion occurs concurrently with particle production. Radiation can eventually dominate the energy density without a separate reheating phase.

14 Warm inflation at work The corresponding Friedmann equation reads H 2 = κ 3 (ρ φ + ρ r ), where ρ φ = φ V (φ). The dynamics for ρ φ and ρ r in the warm inflationary scenario is described by the equations and ρ φ + 3H(ρ φ + P φ ) = Γ φ 2, ρ r + 4Hρ r = Γ φ 2, where P φ = φ 2 2 V (φ) and Γ represents the dissipation coefficient responsible of the decay of the scalar field into radiation during inflation. In general Γ = Γ(T, φ).

15 Warm inflation at work During the inflationary epoch ρ φ ρ r. In this way, φ 2 = 2Ḣ κ(1 + R), where R Γ 3H. For the strong (weak) dissipation regime, we have R 1 (R 1). Cold or warm inflation: T < H or T > H During warm inflation the radiation production is quasi-stable, i.e., ρ r 4Hρ r and ρ r Γ φ 2 ρ r Γ φ 2 4H = ΓḢ 2κH(1+R). Radiation is not necessarily redshifted. Considering that for the radiation field ρ r = C r T 4, where T, C r = π 2 g /30 and g is the number of relativistic degrees of freedom, we obtain [ ] 1/4 ΓḢ T =. 2κC r H(1 + R)

16 Warm inflation model building Dissipative coefficient Γ Adiabatic, close-to-equilibrium approx. SUSY protects the potencial from large radiative corrections (M Bastero-Gil et al. (2011)) This can be realised with the superpotential (M Bastero-Gil et al. (2013)) W = g 2 ΦX 2 + h i 2 XY 2 i, i = 1,..., N Y The scalar component of the superfield Φ describes the inflaton field φ X is the superfield for the heavy catalyst fields χ The last term allows the heavy catalyst field to decay into light degrees of freedom in the supermultiplet Y Inflaton Catalyst fields Light fields φ χ, ψ χ y,ψ y Generalized dissipative coefficient (Zhang (2009); Bastero-Gil et al. (2011) T m Γ(T, φ) = C φ ; m = 3, 1, 0, 1 φm 1

17 Fluctuations in warm inflation In the presence of a thermal bath, when T > H, the quantum fluctuations of the inflaton are dominated by the thermal. The inflaton fluctuations during warm inflation are related to a Langevin equation for the inflaton field (Berera (1995)) δφ(k, t) + (3H + Γ) δφ(k, t) + φδγ + (k 2 a 2 + V φφ )δφ(k, t) = ξ(k, t), where the noise term ξ(k, t) drives scalar field fluctuations with amplitude δ(k, t) Curvature perturbation: Weak dissipative regime : Moss (1985); Berera (1995) Strong dissipative regime : R = Ḣ φ δφ, δφ2 = k F T 2π 2 Γ << 3H, T > H, k F = H Γ >> 3H, T > H, k F = ΓH Berera (2000)

18 Fluctuations in warm inflation A much more rigorous description of the fluctuation amplitude can be found in Hall et. al (2004) and I. G. Moss (2007), where π δφ 2 [ ] 1/2 (3H + Γ)H T, 2 Tensor perturbations are not affected by dissipation, however, the tensor-to-scalar-ratio will change ( ) H 16 ε r T (1 + R) 5/2 In the strong regime, the tensor-to-scalar ratio is suppressed by the factor T /H as in the weak regime, but also by the factor R 5/2. That is why in the strong regime the model always predicts a very low r.

19 General dissipative coefficient in warm intermediate and logamediate inflation R. Herrera, M. Olivares, and N. Videla, Phys. Rev. D 88, (2013). We consider a general form for the dissipative coefficient T m Γ(T, φ) = C φ, m = 3, 1, 0, 1 φm 1 We study how warm inflation works with this generalized coefficient when the universe evolves according to the intermediate inflationary model a(t) = exp (A t f ); 0 < f < 1, A > 0 We study our model in both weak and strong dissipative regimes We utilize data from WMAP 9 and Planck in order to constrain the parameters in our model according to the theory of cosmological perturbations.

20 Intermediate inflation: weak dissipative regime m = 1 Evolution of the ratio R = Γ/3H versus the primordial tilt n s (left panel) and the evolution of the tensor-to-scalar ratio r versus n s (right panel) for the case m = 1. In both panels we use three different values of the parameter C φ, κ = 1 and C r = 70. In the lower panel, we show the two-dimensional marginalized constraints (68 % and 95 % C.L.) on the inflationary parameters r and n s, derived with the nine-year WMAP data.

21 Intermediate inflation: weak dissipative regime Evolution of the tensor-to-scalar ratio r versus n s for the case m = 1. In both panels we use three different values of the parameter C φ, κ = 1 and C r = 70. In the left and right panels we show the two-dimensional marginalized constraints (68 % and 95 % C.L.) on the inflationary parameters r and n s, derived with the nine-year WMAP and Planck data, respectively

22 Intermediate inflation: strong dissipative regime Figura: Evolution of the ratio R = Γ/3H versus the primordial tilt n s (left panel) and the evolution of the tensor-to-scalar ratio r versus n s (right panel) for the case m = 1. In both panels we use three different values of the parameter C φ, κ = 1 and C r = 70

23 Summary of the constraints on C φ Dissipative regime Scale factor Γ = C φ T m φ m 1 Weak a(t) = e Atf m = 0 m = 1 m = 1 Strong a(t) = e Atf m = 0 m = 1 m = 1 Constraint on C φ 10 9 < C φ < < C φ < < C φ < < C φ < < C φ < < C φ < 1

24 Warm intermediate inflation in the Randall-Sundrum II model in the light of Planck and BICEP2 results R. Herrera, M. Olivares, and N. Videla, Eur. Phys. J. C 75, 205 (2015). Cavaglia (2003) We analyze the possibility that a higher dimensional scenario, in particular the RS II brane-world model (Randall and Sundrum, (1999)), can describe the dynamics of the Universe in its very early epochs. We propose this possibility in the context of warm inflation scenario for a Universe evolving according to the intermediate scale factor, and how a generalized form of dissipative coefficient Γ influences the dynamics of our model Using recent data from BICEP2 experiment and Planck satellite we constrain the parameters in our model, according to the theory of cosmological perturbations.

25 Warped DGP model K.-i. Maeda, S. Mizuno and T. Torii, PRD 68, (2003) where and S bulk = M S brane = d 5 X (5) g M S = S bulk + S brane, [ 1 2κ 2 5 κ 2 5 ( (5) R 2Λ 5 ) + (5) L m ], d 4 x [ ] 1 g K ± + L brane (g αβ, ψ), κ 2 5 = m 3 5 is the 5-dimensional gravitational constant. L brane = τ + L m

26 Friedmann equation for the RS II model Effective four-dimensional Friedmann equation for a flat FRW metric [ H 2 = κρ 1 + ρ ] + Λ 2τ 3 + E 0 a 4, ] where κ = 1 6 τκ4 5 and Λ = 1 2 [Λ κ4 5 τ 2 E 0 /a 4 : dark radiation term. Observational constraints on τ: τ > (1 MeV) 4 (Nucleosynthesis) τ (10 TeV) 4 (Deviation from Newton s law)

27 Warm inflation on the RS II model: basic equations We will ignore the dark radiation term E 0 and we shall restrict to the case Λ = 0. a 4 In this form [ H 2 = κ ρ 1 + ρ ]. 2 τ We will assume that the total energy density ρ is confined to the brane satisfying the continuity equation, so for ρ φ and ρ r we have the same conservation equations as before. During warm inflation ρ φ ρ r and ρ r Γ φ 2. In this way, we obtain and V = τ 4H [ ] φ 2 = 2 ( Ḣ) /2 H2, 3κ (1 + R) κ τ [ ] T = Γ( Ḣ) 1/4 [ ] /8 H2, 6 κc γ H(1 + R) κ τ ( H2 κ τ ) + Ḣ(2 + 3R) 6(1 + R) ( H2 κ τ ) 1/2

28 Cosmological perturbations: Strong dissipative regime We may parameterize the scalar power spectrum in terms of the number of e-folds ( 3(5f 6) P R = k(j[n]) 8 exp B[J[N]] ) [ ] 1 + 2(Af 3/16 )2 (J[N]) 2(1 f ) (m = 3), K κτ and [ ] (3m 6) 3[f (2+m) 2m] P R = γ m(j[n]) 8 ( B 3(1 m) m[j[n]]) 3 m 1 + 2(Af )2 (J[N]) 2(1 f ) 16 κτ (m 3), where B and B m represent incomplete Beta functions, and J(N) = [ ] 1 1+f (N 1) f Af Using the observational values for P R and n s at the pivot scale, we may find constraints on the parameters A and f for a given value of C φ

29 Cosmological perturbations: Strong dissipative regime The power spectrum of the tensor perturbations is more complicated in our model, because in brane-world gravitons propagate into the bulk. Tensor-to-scalar ratio (m = 3) [ r = 6κ π 2 k (Af )2 (J[N]) (f +2) 8 exp 3 B[J[N]] ] [ ] 1 + 2(Af )2 (J[N]) 2(1 f ) 3 16 F 2 (N) K κτ r = ( 6κ π 2 (Af ) 2 (J[N]) 1 [6m+f (10 3m) 16] 3 m 8 k m 2 B m[j[n]] K m ) 3(m 1) 3 m [ 1 + 2(Af )2 (J[N]) 2(1 f ) κτ ] (3m 6) 16 F 2 (N) (m 3), where F (x) = [ 1 + x 2 x 2 sinh 1 (1/x)] 1/2 and x = Hm p 3/(4πτ)

30 Weak dissipative regime In the left plots, the solid, dotted, and dashed lines correspond to the pairs (A = 0,19, f = 0,31), (A = 0,28, f = 0,32), and(a = 0,32, f = 0,30), respectively. In the right, we have (A = 0,47, f = 0,29), (A = 0,43, f = 0,28), and (A = 0,39, f = 0,29). For all plots, we have used the values C γ = 70, m p = 1, and τ = 10 14

31 Strong dissipative regime In both plots, the solid, dotted, and dashed lines correspond to the pairs (A = 1, , f = 0,75), (A = 7, , f = 0,87) and (A = 5, , f = 0,97). For all plots, we have used the values C γ = 70, m p = 1, and τ = 10 14

32 Summary of the constraints on C φ Dissipative regime Scale factor Γ = C φ T m φ m 1 Weak a(t) = e Atf m = 1 m = 0 m = 3 m = 1 Strong a(t) = e Atf m = 1 m = 0 m = 3 m = 1 Constraint on C φ < C φ < < C φ < < C φ < < C φ < < C φ < 10 9 C φ > 10 1 n s > 1 n s > 1

33 Conclusions Dissipation in warm inflation causes a friction term Γ in the inflaton s equation of motion, leading to particle and radiation production with inflationary expansion. We have studied how a generalized form of Γ works in warm intermediate inflationary model in the context of GR and RS II brane-world model. In the slow-roll approximation, we have found analytical expressions for the corresponding effective potential, power spectrum, scalar spectral index, and tensor-to-scalar ratio in both dissipative regimes We have obtained, in both dissipative regimes regimes, constraints on the parameters of the model from the BICEP2 and Planck dats, where we have considered the constraint on the n s-r plane, and from the essential condition for warm inflation and the weak (strong) dissipative regime Observational implications: Blue (n s < 1) and red spectra (n s > 1) are possible Low tensor-to-scalar ratio r 0.12 in the strong dissipative regime Running of the spectral index? Non-gaussianity f NL at strong dissipation?

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