Axion: Mass Dark Matter Abundance Relation Guy Moore, TU Darmstadt With Vincent Klaer, Leesa Fleury • Mystery 1: Dark Matter • Mystery 2: T-symmetry of QCD • The Axion: a solution to both mysteries? • Early Universe cosmology of the axion • How to predict the axion mass if it’s the dark matter Saariselk¨a, 5 April 2018 Slide 1 of 25 Dark Matter: a Cosmic Mystery Atoms: Standard Model. Dark Energy: Cosmological Constant. Strange value, but possible Dark Matter: MYSTERY! NOT SM! We only know 3 things about dark matter: • It’s Matter: gravitationally clumps. • It’s Dark: negligible electric charge, interactions too feeble to be detected except by gravity • It’s Cold: negligible pressure by redshift z = 3000 Saariselk¨a, 5 April 2018 Slide 2 of 25 Another mystery: T-symmetry in QCD QCD Lagrangian built of Dim-4 gauge-invariant Lorentz scalars: 1 (i)Θ S = d4x F a F µν + ǫ F µν F αβ , (E) 4g2 µν a 64π2 µναβ a a Z µν αβ µ ν αβ gfabc ν α β ǫµναβFa Fa = ∂ Kµ , Kµ = ǫµναβ AaFa + AaAb Ac 3 Second term integrates to integer NI on A-field singularities. exp(−SE) = exp(−Sstandard) × exp(iΘNI ) Introduces extra, T violating phase into path integral! G. ’t Hooft, PRL 37, 8(1976); R. Jackiw and C. Rebbi, PRL 37, 172 (1976); Callan Dashen and Gross, Phys Lett 63B, 334 (1976) Saariselk¨a, 5 April 2018 Slide 3 of 25 Looking for T: Neutron EDM Put neutron in B~ field – spin lines up with B~ . N B B Is there an Electric Dipole Moment (EDM) aligned with spin? If so: looks different when movie runs backwards, T viol! Saariselk¨a, 5 April 2018 Slide 4 of 25 Neutron EDM and Θ Theory: Neutron electric dipole moment should exist, −16 dn = −3.8 × 10 e cm × Θ so long as Θ is not zero! Guo et al, arXiv:1502.02295, assumes Θ, modulo 2π, is small Experiment: Consistent with zero! Baker et al (Grenoble), arXiv:hep-ex/0602020 −26 |dn| < 2.9 × 10 e cm Either |Θ| < 10−10 by (coincidence? accident?) or there is something deep going on here. Saariselk¨a, 5 April 2018 Slide 5 of 25 Axion mechanism Add singlet complex scalar and some extra UV DOF such that 2 2 µ ∗ m ∗ 2 Arg(ϕ) a ˜µν Lϕ = ∂ ϕ ∂µϕ + 2 ϕ ϕ − 2fa + 2 Fµν Fa 8fa 32π True Θeff =Θ+ θA with θA = Arg(ϕ). QCD gives θA a potential: 2 − F F F˜ T 2 i(Θ+θA) V (θ ) = − ln D(A ψψ¯ )Det(D/ +m)e 4g × e 32π2 eff A Ω µ Z R R ≃ χ(T )(1 − cos[Θ+θA]) , F F˜(x) F F˜(0) χ(T ) = d4x 32π2 32π2 *Z +β Forces Θ + θA = 0 automatically, dynamically. Peccei Quinn PRL 38, 1440 (1977); J. E. Kim, PRL 43, 103 (1979); Shifman Vainshtein and Zakharov, NPB 166, 493 (1980) Saariselk¨a, 5 April 2018 Slide 6 of 25 χ(T ): what we expect ln( χ ) χ pt Low T : χ-pt works. mumd 2 2 χ ≃ 2 m f (mu+md) π π Hi T : standard α pt pert-thy works(??) ln(T) Tc 4 Low T : χ(T ≪ Tc) = (76 ± 1 MeV) Cortona et al, arXiv:1511.02867 −8 High T : χ(T ≫ Tc) ∝ T Gross Pisarski Yaffe Rev.Mod.Phys.53,43(1981) but with much larger errors. Saariselk¨a, 5 April 2018 Slide 7 of 25 Axion in cosmology Assume first: ϕ starts homogeneous [inflation] Classical axion field! Starts oscillating around −1 t = πma . Damped: • Hubble drag • effect of dma/dt Pressureless: Acts Like Dark Matter! 2 2 2 Osc. frequency = axion mass: ω = ma = χ/fa Saariselk¨a, 5 April 2018 Slide 8 of 25 Dark matter density? More dark matter if oscillations start larger or later: 7/8 2 ρdm ∝ fa θA init (approximately) • Large fa, (small ma): later transition from cosmological constant to matter, more final energy density • Initial θA init: larger value, larger starting amplitude. Because θA init unknown, scenario is not predictive. Saariselk¨a, 5 April 2018 Slide 9 of 25 Initial state of ϕ field? most likely: randomly different in different places! • Inflation stretches quantum fluctuations to classical 2 2 ones: ∆ϕ ∼ Hinfl.. If NefoldsH >fa , scambles field. −5 If not: need H < 10 fa to avoid excess “isocurvature” fluctuations in axion field • Gets scrambled after inflation if Universe was ever really 11 hot T >fa ∼ 10 GeV. Predictive: if axion=dark matter and we solve dynamics, then → predict fa,ma. L. Visinelli and P. Gondolo, PRL 103, 011802 (2014) Bad: nonperturbative. Good: classical field theory! Saariselk¨a, 5 April 2018 Slide 10 of 25 Needed Ingredients Predict relation between Dark Matter Density and Axion Mass assuming space-random starting angle. Challenges: 1. χ(T ): needs Lattice Gauge Thy. 2. Axion field dynamics: classical but with large scale 30 hierarchy fa/H ∼ 10 ≫ 1 1. Recent Borsanyi et al 1606.07494 lattice results. Confirmation?? Claim: I can solve 2. Saariselk¨a, 5 April 2018 Slide 11 of 25 Obvious approach Classical real-time lattice using Lagrangian ∗ µ λ ∗ 2 L = ∂µϕ ∂ ϕ+ (2−ϕ ϕ) −χ(t)Re ϕ 8 ϕ(t = 0) random, Hubble drag, Count axions at end. Fails: scale hierarchy actually relevant! Saariselk¨a, 5 April 2018 Slide 12 of 25 Axion strings ϕ is a complex number – plot as a 2D arrow. Field generically has vortices. 2D–points. 3D–strings. Saariselk¨a, 5 April 2018 Slide 13 of 25 Domain walls 2D slice of evolution, When the potential tilts: Saariselk¨a, 5 April 2018 Slide 14 of 25 Layers of String Energy − ∼H 1 ∗ 2 2 2 r dr Estr = dz dφ rdr ∇φ ∇φ ≃ fa /2r ≃ πℓfa ∼ −1 r2 Z Z Z Z fa Zoom Zoom etc. in in Look at Cross section Series of “sheaths” around string: equal energy in each ×2 scale, 1030 scale range! ln(1030) ≃ 70. 2 30 2 Log-large string tension Tstr = πfa ln(10 ) ≡ πfa κ Not reproduced by numerics (separation/core ∼ 400) Saariselk¨a, 5 April 2018 Slide 15 of 25 Getting string tension correct MATTERS! String dynamics are controlled by: 2 • String tension and inertia: ∝ κπfa FACTOR of κ 2 • String radiation and inter-string interactions: ∝ πfa NO factor of κ Relative importance of these effects, and string energy, are κ dependent We really need to get this physics right! Saariselk¨a, 5 April 2018 Slide 16 of 25 Effective field theory Integrate out string-cores out to radius r0: 2 strings with tension T = κπfa with κ ≡ ln(r0fa): • Strings with tension T ′2 2 LNG = T dσ y (σ)(1 − y˙ (σ)) Z q 2 • Axion fields θA 3 fa µ LGS = d x ∂µθA∂ θA+χ(1−cos θA) Z 2 • String-axion coupling 3 µν LKR = d xAµνJ Z Saariselk¨a, 5 April 2018 Slide 17 of 25 Does anyone else have this effective theory? Plan: find another model which also has: • Strings obeying LNG • “Axion” fields obeying LGS • Coupling between them, LKR It can have extra DOF as long as I take a limit where they get heavy (along with a → 0) Saariselk¨a, 5 April 2018 Slide 18 of 25 Trick: global strings, local cores Theory with U(1) × U(1), one global one gauged: 1 L(ϕ ,ϕ ,A ) = (∂ A − ∂ A )2 1 2 µ 4 µ ν ν µ λ + (2ϕ∗ϕ − f 2)2 + (2ϕ∗ϕ − f 2)2 8 1 1 2 2 h 2 i 2 + |(∂µ − iq1eAµ)ϕ1| + |(∂µ − iq2eAµ)ϕ2| Pick q1 =6 q2, say, q1 = 4, q2 = 3. iθ1 iθ2 Two rotation symmetries, ϕ1 → e ϕ1, ϕ2 → e ϕ2 q1θ1 + q2θ2 gauged, q2θ1 − q1θ2 global (Axion) Saariselk¨a, 5 April 2018 Slide 19 of 25 Two scalars, one gauge field String where each scalar winds by 2π: θ 1 2π B-field almost A B compensates C gradients outside CAB string. 2 2 2 2 0 θ fa = f /(q1 +q2). 0 2π 2 Red = gauge−equivalent 2 2 dF f 2 2 T ≃ 2πf , = 2 2 , κeff = 2(q1 + q2) . dl (q1 + q2)r Saariselk¨a, 5 April 2018 Slide 20 of 25 Higher tension = higher initial density, longer lasting, hardier loops Saariselk¨a, 5 April 2018 Slide 21 of 25 Results Axions produced vary mildly with increasing string tension Saariselk¨a, 5 April 2018 Slide 22 of 25 Results • 10× string tension leads so 3× network density but • only 40% more axions than with axion-only simulation, • Fewer (78%) axions than θA init-averaged misalignment • Axionic string networks are very bad at making axions • Results in less axion production. Must be compensated by ligher axion mass. Saariselk¨a, 5 April 2018 Slide 23 of 25 Put it all together 2 Axion production: nax(T = T∗) = (13 ± 2)H(T∗)fa 2 8πε Hubble law: H = 2 , 3mpl 2 4 π T g∗ 4ε Equation of state: ε = , s = , g∗(1GeV) ≃ 73 30 3T 1 GeV 7.6 Susceptibility: χ(T ) ≃ (1.02(35) × 10−11GeV4) T ρ Dark matter: = 0.39 eV s One finds T∗ = 1.54GeV and ma = 26.2 ± 3.4 µeV Saariselk¨a, 5 April 2018 Slide 24 of 25 Conclusions • QCD “generically” violates T symmetry • Axions: natural explanation, why T viol not observed • Dark matter density calculable if θA starts random • Tricky network dynamics – new techniques needed • Prediction: if DM=Axions, then ma = 26.2 ± 3.4 µeV. We should go and look for axions, ma ∼ 26 µeV! Saariselk¨a, 5 April 2018 Slide 25 of 25 How to look for axions Generally axion also couples to ordinary electromagnetism θ K θ L = ..