KK Towers in the Early Unvierse

Transcription

1 KK Towers in the Early Universe: Phase Transitions, Relic Abundances, and Applications to Axion Cosmology (IBS-CTPU) [arxiv:689] [arxiv:947] collaborators on this work: Keith Dienes (Arizona) Brooks Thomas (Lafayette) CosPA 7 Thursday, December 4 th, 7 / 4

2 Scalars in the Early Universe Impact of Mass-Generating Phase Transitions Additional scalar fields commonly appear in extensions of the SM, and tend to play an important role in early-universe cosmology These fields are often light due to shift symmetries at high scales, but are broken by some dynamics that enters in the effective lower-temperature theory ie, they undergo mass-generating phase transitions example: QCD Axion quintessence fields Q-balls chameleons DE motivated scalars in litte-higgs theories V (ϕ) massless degree of freedom at T Λ QCD dilatons axion-like particles candidate scalar fields Im{ϕ} branons majorons png bosons QCD axion familons string and geometric moduli scalar SUSY partners Re{ϕ} / 4

3 Scalars in the Early Universe Impact of Mass-Generating Phase Transitions Additional scalar fields commonly appear in extensions of the SM, and tend to play an important role in early-universe cosmology These fields are often light due to shift symmetries at high scales, but are broken by some dynamics that enters in the effective lower-temperature theory ie, they undergo mass-generating phase transitions example: QCD Axion quintessence fields Q-balls chameleons DE motivated scalars in litte-higgs theories V (ϕ) broken by instanton effects for T Λ QCD dilatons axion-like particles candidate scalar fields Im{ϕ} branons majorons png bosons QCD axion familons string and geometric moduli scalar SUSY partners Re{ϕ} / 4

4 Scalars in the Early Universe Impact of Mass-Generating Phase Transitions All of this can be important for model building: the energy density ρ carried by these scalar(s) at late times (used to compute abundances, overclosure bounds, etc) is generally sensitive to the timescale G over which such a phase transition unfolds With multiple fields {ϕ λ }, such transitions can generate off-diagonal elements in the mass matrix M, and thus mixing is also generated amongst the fields in a dynamical, time-dependent way M (t) = V eff (ϕ, ϕ, ) k,l ϕ k mass matrix M kl(t)ϕ l M m M m m,n m m m,n + MN m,n m,n m N,N constant masses M i m ij (t) generated during phase transition 3 / 4

5 A Two-Field Toy Model A Very Brief Review This has been found to have a surprising influence, even in the context of a simple but generic two-component toy model [arxiv:947]: [ ] ] M m (t) = +[ m M m m h (t) both enhancements and large suppressions to late-time energy density ρ _ (Δ G, ξ _ )/ ρ _ (,) - -3 ξ= ξ= ξ= ξ=9 ξ=99 ξ=999 m sum =4 Δm = θ max=4-4 Δ G constant term _λ ρ / _ ρ time-dependence generated terms distribution amongst fields extremely sensitive to phase transition timescale sequence of parametric resonances appear as mixing is saturated m sum =4 Δm = θ max=4 ξ= ξ= ξ= ξ=9 ξ=99 ξ=999 Δ G ϕ λ additional dynamical over/underdamping transitions: reoverdamping Δ G/τ G ϕ λ (Δ G) ϕ λ ( ) m sum = Δm = ϵ= -7 τ G=Δ G= -3 τ/τ G G phase transition timescale ξ mixing [, ) 4 / 4

6 A Two-Field Toy Model A Very Brief Review This has been found to have a surprising influence, even in the context of a simple but generic two-component toy model [arxiv:947]: [ ] ] M m (t) = +[ m M m m h (t) constant term time-dependence generated terms both enhancements and distribution amongst fields additional dynamical large suppressions to extremely sensitive to phase over/underdamping late-time energy This density is under the minimal transitionassumption timescale of only two transitions: components reoverdamping 3 m sum =4 m ϕ sum =4 Δ λ (Δ what happens in models with larger collections of G) G/τ fields, G Δm = Δm = ϕ λ ( ) such θ max=4 as those furnished 8 by models with extra dimensions? ρ _ (Δ G, ξ _ )/ ρ _ (,) - -3 ξ= ξ= ξ= ξ=9 ξ=99 ξ=999-4 Δ G _λ ρ / _ ρ 6 4 sequence of parametric resonances appear as mixing is saturated θ max=4 ξ= ξ= ξ= ξ=9 ξ=99 ξ=999 Δ G ϕ λ - - m sum = Δm = ϵ= -7 τ G=Δ G= -3 τ/τ G G phase transition timescale ξ mixing [, ) 4 / 4

7 Mass Generation in a KK Tower The Framework Consider a spacetime geometry M S /Z, ie an extra dimension compactified on a line segment, with a bulk scalar Φ(x µ, x ): S = d 4 xdx [ M Φ M Φ Φ shift symmetry forbids bulk mass ] +δ(x )L brane (ψ i, Φ) The 4D mass matrix then mixes the fields: M + c M = m m (t) (t) 4M + c m (t) M c /R 449 GeV m M c indicates highly mixed ensemble SM brane {ψ i (x µ )} V πr bulk Φ(x µ, x ) interactions with fields on brane can lead to an effective 4D mass m(t): V L brane(φ) = m (t) Φ + and we parameterize our ignorance: δ G m(t) m t G t / 4

8 Evolving the System In a flat FRW cosmology the KK modes {ϕ k } evolve as ϕ k + 3H(t) ϕ k + M kl(t)ϕ l =, l= which in general cannot be solved analytically due to the time-dependence in M kl near the phase transition perform numerics on truncated tower of N modes, and recover features through N limiting behavior 6 / 4

9 Survey of Four-Dimensional (N = ) Limit Standard Approximations Two approximations are commonly use in the literature to compute late-time abundances in single-field models that undergo such phase transitions: abrupt approximation ρ4d (where δg ) adiabatic approximation ρ4d ad (where m/m ) [exact ρ]/[abrupt solutions] ρ4d (δg ) 6 ρ4d () 3 [exact ρ]/[adiabatic solutions] ρ4d ρ4d ad 4 4 tζ m m / δg π /m G = δg = mtg N = N = 3 3 mtg 3 Even for N =, there are regions of parameter space that are inaccesible to the standard approximations, particularly in the m /tg regime 7 /4

10 Survey of Four-Dimensional (N = ) Limit Standard Approximations Two approximations are commonly use in the literature to compute late-time abundances in single-field models that undergo such phase transitions: abrupt approximation ρ4d (where δg ) adiabatic approximation ρ4d ad (where m/m ) [exact ρ]/[abrupt solutions] ρ4d (δg ) 6 ρ4d () 3 [exact ρ]/[adiabatic solutions] ρ4d ρ4d ad 4 4 tζ m m / δg can use numerical results to6 extract accurate analytical approximations 3 π /m G = δg = mtg N = N = 3 3 mtg 3 Even for N =, there are regions of parameter space that are inaccesible to the standard approximations, particularly in the m /tg regime 7 /4

11 Dynamics of the N > Tower A Qualitative Description suppresses/enhances modes by different amounts according to details of the phase transition energy density of modes in tower re-shuffling of energy densities t t t G t t G t t G Φ = implies displaced zero mode phase transition distribution frozen in / 4

12 Approaching Asymptotia: N Behavior of the Solutions It is instructive to examine the N asymptotic behavior of various late-time quantities while varying δ G (and taking m = M c ): different truncations [exact ρ]/[abrupt approx] [exact ρ]/[4d limit] the phase transition suppresses modes that exceed δ G t G π/λ, ie it accelerates the N convergence often leaving only a few modes that appreciably contribute to the total ρ 9 / 4

13 The KK Tower Limit: Extracting N Limit Suppressions, Tower Fractions, and Distributions Equipped with a method to efficiently compute asymptotia for large N, we now have the ability to compute results effectively for the full KK tower [exact ρ]/[abrupt approx] [exact ρ]/[4d limit] [tower fraction] ρ(δ G) ρ ρ() ρ 4D η 3 δg 3 t G = /M c mt G t G = /M c 3 4 mt G 3 3 tg = /Mc mt G η max λ { ρ λρ } fraction of abundance in subdominant modes / 4

14 The KK Tower Limit: Extracting N Limit Suppressions, Tower Fractions, and Distributions Equipped with a method to efficiently compute asymptotia for large N, we now have the ability to compute results effectively for the full KK tower [exact ρ]/[abrupt approx] [exact ρ]/[4d limit] [tower fraction] ρ(δ G) ρ ρ() ρ 4D η 3 δg 3 t G = /M c mt G t G = /M c again can extract general analytical approximations in different regions 3 4 mt G 3 3 tg = /Mc mt G η max λ { ρ λρ } fraction of abundance in subdominant modes / 4

15 The KK Tower Limit: Extracting N Limit Suppressions, Tower Fractions, and Distributions tower fraction η slices give ρ λ /ρ, the fractional abundance [blue slice lightest mode] δg heavier modes increasingly suppressed with larger δ G (as δ G t G π/λ) mt G / 4

16 The KK Tower Limit: Extracting N Limit Suppressions, Tower Fractions, and Distributions tower fraction η 3 area of pies total abundance of tower slices give ρ λ /ρ, the fractional abundance [blue slice lightest mode] δg heavier modes increasingly suppressed with larger δ G (as δ G t G π/λ) mt G 97 / 4

17 Example: Axion in the Bulk At this point we can drop the generality of Φ and apply our machinery to a specific model: for example a bulk axion-like particle (ALP) Our {t G, m X, M c } parameter space is mapped onto {Λ G, ˆf X, M c } associated confinement scale T t G = RH Mp 4g (T RH ) π g (Λ G )Λ 4 G effective 4D decay constant m X = C g Λ 4 3π G ˆf X ˆfX [GeV] 3 t G [s] Mct G= m X= Mc δg = m Xt G= 9 Λ G [GeV] t G [s] Mct G= m X= Mc δg = m Xt G= Λ G [GeV] t G [s] Mct G= m X= Mc δg = 3 m Xt G= Λ G [GeV] η [tower fraction] maximum tower fraction in M ct G m X t G transition suppresses heavier modes, confining maximum to M ct G m X t G / 4

18 Example: Axion in the Bulk [exact ρ]/[abrupt approx] ρ(δ G) ρ() 3 t G [s] 4 7 Mct G= m X= Mc δg = [exact ρ]/[4d limit] ρ ρ 4D t G [s] 4 7 Mct G= m X= Mc δg = ˆfX [GeV] 9 enhancements and large suppressions relative to abrupt approximation Mct G= m X= Mc m Xt G= δg = 3 Mct G= m X= Mc m Xt G= δg = 3 presence of extra dimension produces significant additional suppression of ρ for m M c mt G ˆfX [GeV] m Xt G= 9 Λ G [GeV] m Xt G= Λ G [GeV] 3 / 4

19 The Take-Away Message Models of non-minimal scalar sectors that undergo mass-generating phase transitions in general are very sensitive to phase transitions details both the total energy density and its distribution across individual modes in the ensemble show this both in a simple but generic two-field model, and in model with a bulk scalar we derived a variety of asymptotic scaling behaviors and analytic expressions for the energy densities of the tower as functions of relevant model parameters applied the general machinery of our framework to the example of a bulk axion, allowing us to determine where the standard approximations succeed/fail and may suggest the weakening of overclosure bounds in certain regions There are many possible future directions: we assumed a single flat extra dimension, but what phenomena arise with a warped geometry and/or multiple extra spatial dimensions? we operated under assumption that the fields ρ ϕ ρ crit during the mass-generation epoch, but what is the effect of the backreaction on H away from this regime [ie, where scalars play role during inflation/(p)reheating]? THANKS FOR YOUR ATTENTION! 4 / 4