Ratchet Baryogenesis during Reheating

Transcription

1 Ratchet Baryogenesis during Reheating Neil D. Barrie Kavli IPMU (WPI) February 17, 2018 K. Bamba, N. B., A. Sugamoto, T. Takeuchi and K. Yamashita, arxiv: , and upcoming work. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

2 Outline 1 Introduction 2 The Mechanism and Dynamics 3 Calculating the Generated η 4 Conclusion and Future Work N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

3 Matter-Antimatter Asymmetry The asymmetry is described quantitatively by, η= nb nb ' s The Sakharov Conditions 1 Baryon number violation 2 C and CP violation 3 Period of non-equilibrium N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

4 Ratchet Mechanism Inspired by molecular motors in biological systems, and their ability to generate directed motion. Consider an inflaton and complex scalar carrying X charge during reheating. A derivative coupling between a complex scalar and inflaton. Directed motion in the complex scalar phase gives a non-zero X number density. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

5 The Model Interplay between the inflaton and complex scalar during reheating. Introduce a derivative coupling term describing their interaction. S = dx 4 g [g µν µ φ ν φ V 0 (φ, φ ) g µν µ Φ ν Φ U(Φ) + i ( Λ g µν φ ) ] µ φ ν Φ, where V 0 (φ, φ ) = λφ φ(φ φ )(φ φ) +... Satisfying the Sakharov Conditions 1 The complex scalar potential. 2 Derivative coupling interaction. 3 Reheating epoch. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

6 U(1) Global Symmetry and X Charge If λ = 0, the action will be invariant under the global U(1) transformation (φ, φ ) (e iα φ, e iα φ ), with α a constant. We identify the U(1) symmetry with X charge, which may be B or L. Take φ to have a unit charge under this symmetry, and Φ uncharged. The associated current, j µ = iφ µ φ. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

7 Starobinsky Inflation and Reheating The Starobinsky inflation is consistent with current data, has the potential, U(Φ) = 3µ2 Mp 2 (1 e 2/3Φ/M p ) 2 4 The Starobinsky model approaches 1 2 µ2 Φ 2 during reheating. An approximate matter dominated epoch (a t 2/3 ). The potential will be expressed as, U(Φ) 1 2 µ2 Φ 2 du(φ) dφ µ2 Φ. where µ M p, H i GeV and Φ i 0.62M p. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

8 Reparametrising Taking φ in polar form φ = 1 2 φ r e iθ, we find V (φ r, θ) = λφ 4 r sin 2 θ + and n X = j 0 = φ 2 θ r. Therefore, a non-zero n X requires non-zero φ r and θ. The action is now, S = d 4 x [ φ 2 g r 2 g µν µ θ ν θ λφ 4 r sin 2 θ g µν µ Φ ν Φ U(Φ) φ2 r Λ g µν µ θ ν Φ ] N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

9 Analysis of the Equations of Motion N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

10 Equations of Motion for θ and Φ Equations of motion including the inflaton decay rate Γ, ( Φ + 3H Φ ) ( + Γ Φ + du(φ) dφ ) φ2 r ( θ + 3H θ ) = 0, Λ ( θ + 3H θ ) + λφ 2 r sin(2θ) 1 ( Φ + 3H Φ ) = 0. Λ Rescaling coefficients by ( 1 φ2 r Λ 2 ) and rearranging, in reheating epoch, ( Φ + 2 ) t Φ ) + ( Γ Φ + µ 2 Φ + λφ 4 r Λ sin(2θ) = 0, ( θ + 2 ) t θ + 1 ( Γ Φ + µ Φ) 2 + λφ 2 r sin(2θ) Λ = 0. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

11 Behaviour of the Inflaton Want Φ(t) to be unaffected by the dynamics of θ. Require that µ 2 Φ(t) λφ4 r Λ sin(2θ). ( Φ + 3H + Γ) Φ + µ 2 Φ = 0. When Γ µ, the approximate solution to this equation is Φ(t) Φ i ( ) ti e Γ(t t i )/2 cos [ µ(t t i ) ], t We now require that φ 2 r /Λ 2 1 λ λ, µ µ, and Γ Γ. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

12 Behaviour of the θ The EoM for θ is, θ + (Γ θ + 3H) θ + λφ 2 r sin(2θ) + 1 Λ ( ) Γ Φ(t) + µ 2 Φ(t) = 0. where we have added a decay term for θ, defined by Γ φ. Utilising the solution to the inflaton EoM, and neglecting Φ decay term, θ + (Γ θ + 3H) θ + p sin(2θ) + q(t) cos [ µ(t t i ) ] = 0, where p = λφ 2 r, q(t) = µ2 Φ i Λ ( ) ti e Γ(t t i )/2. t N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

13 Possible Cases p q(t), ( θ + 3H θ) = 1 d ( ) t 2 t 2 θ = q(t) cos [ µ(t t i ) ], dt which can be integrated to yield, ( µ θ(t) 2 ) Φ i ti [ ] = Λ t 2 e Γ(t t i )/2 {cos [µ(t t i )] + µt sin [µ(t t i )]} + θ simply oscillates around zero, because the X violation has vanished. p q(t), θ + 3H θ + p sin(2θ) = 0 Friction term damps θ until θ settles into a minima no persistent non-zero θ, as C and CP breaking term ignored. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

14 Sweet Spot Condition and Driven Motion We need p q(t) X, C and CP violating terms all contribute. SSC: λφ 2 r µ2 Φ i Λ H d H i We thus obtain the following simplified equation: θ + (Γ θ + 3H d ) θ + p sin(2θ) = q(t d ) cos [ µ(t t i ) ], This is analogous to a forced pendulum where LHS represents acceleration, damping, and gravitation, RHS is a sinusoidal driving torque with amplitude q and frequency µ. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

15 Nature of the Driven Motion N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

16 Phase Locked States Solutions increasing monotonously in time with small amplitude modulations. Known as phase-locked states in the study of chaotic behaviour of the forced pendulum. Studies of electric current passing through Josephson junctions. Reparameterise to: Θ 2θ, τ [ 2p (t t i ) π ], ω µ, µ 2p The EoM becomes, Q p 2 t d Θ + 1 Q Θ + sin Θ = γ cos(ωτ), where, γ q(t d) p. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

17 Solution and X Number Density The generic phase-locked state solution to the above equation has the form Θ(τ) = Θ 0 + nωτ α m sin(mωτ φ m ), m=1 where (n, m) are integers. For these solutions, we can calculate the X number density n X as p p n X = φ 2 r θ = 2 φ2 r Θ = 2 φ2 r nω ( ) n = µφ 2 r 2. where n/2 is the number of rotations of the phase θ per oscillation of Φ. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

18 Generated Asymmetry Parameter We assume that there is no additional production of entropy after reheating η reh = n ( ) X µφ 2 ( ) n r ad. s T 3 reh a rh Using numerical calculations it is possible to determine the generated baryon asymmetry. n depends on the validity of the phase-locked state and its stability, so can be determined by numerical simulations. Now to attempt to approximate n analytically. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

19 Estimation of n Starting with the EoM for θ in the simplified form, where the friction term has been dropped, and the SSC satisfied, θ + λφ 2 r (sin(2θ) + cos(µt)) = 0, Reparametrising using τ = µt and ξ = 2θ µ2, 2λφ 2 r ( ) 2λφ ξ 2 + sin r µ 2 ξ + cos(τ) = 0, Let n 0 = 2λφ2 r µ 2, ξ + sin (n 0 ξ) + cos(τ) = 0. Assume that directed motion is present and of the form ξ τ +... N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

20 Estimation of n Assume the oscillating terms of ξ are sub-dominant, ξ + sin(n 0 τ) + cos(τ) = 0, The above equation of motion is easily solved and has the general solution, ξ = a + bτ + sin(n 0τ) n 2 0 Consistent with our initial assumptions, + cos(τ), < ξ >= 1 < θ >= λφ2 r µ n = 2λφ2 r µ 2, which is consistent with numerical calculations. Now n > 1 φ r > 1 2λ µ, as a requirement for driven motion. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

21 Dynamics of θ after SSC Violated Driven motion ends friction term pushes θ towards the minima, SSC violated φ can t produce a non-zero n, θ oscillates around minima due to inflaton, If we assume sin θ θ after the driven motion has ended, Can approximate the amplitude of oscillations as, sin 2θ H, H d After the SSC is violated there is no net production or washout of the X asymmetry; cannot realise simultaneous violation of C, CP and X. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

22 The Generated Asymmetry N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

23 Allowed Parameter Ranges Maximum scalar baryon energy density, Utilising the SSC, this gives, Require Φ is unaffected by θ, Allowed range for φ r, Limits on H d, 3M 2 ph 2 λφ 4 r sin 2 θ. µ 2 Φ(t) λφ4 r Λ λ H H d φ2 r Λ GeV > φ r > µ/ 2λ λ > n > 1. H 0 > H d > GeV 310 > n > 1. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

24 Estimation of the Generated Asymmetry Generated asymmetry, η reh = n X s ( ) λφ 4 ( ) r ad. µt 3 reh a rh where the dilution factor is given by, ( ) 3 ad = a rh ( π 2 ) ( ) g T 4 rh 90 Hd 2M2 p. Using the SSC, H d = λφ2 r H 0Λ µ 2 Φ 0 2n GeV, η = ( ) T rh λ 10 8 GeV. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

25 Coupling φ to the SM Interpret the charge as lepton number, Introduce the lepton-number preserving dimension four interaction, L int = y L φ ν c Rν R + h.c. describing the decay of the complex scalar lepton into a ν R ν R pair. Can generate a large Majorana mass when φ = φ r / 2, depending on the value of y L. The leptonic scalar identified as part of the seesaw mechanism. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

26 Coupling φ to the SM Interpret as a baryonic scalar, Introduce the following interaction, which becomes, L int = i ( φ Λ 2 µ φ ) 1 (ūγ µ u + 3 dγ µ d ), L int = φ2 r Λ θ 1 ( u 2 u + d d ), 3 Analogous to a chemical potential coupling to the baryonic current, shifting in favour of matter or antimatter, where µ B = φ2 r Λ 2 θ. N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27

27 Conclusion and Future Work Interplay between inflaton and scalar baryon/lepton during reheating, Driven motion can be modelled as a forced pendulum, Presence of phase-locked states, Asymmetry linearly dependent on the reheating temperature. Future work Further exploration of the mechanism Investigation of efficiency required. Other cosmological implications N. D. Barrie (Kavli IPMU (WPI)) Ratchet Baryogenesis during Reheating February 17, / 27