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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 = ..
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  • Modified Newtonian Dynamics

    Modified Newtonian Dynamics

    Faculty of Mathematics and Natural Sciences Bachelor Thesis University of Groningen Modified Newtonian Dynamics (MOND) and a Possible Microscopic Description Author: Supervisor: L.M. Mooiweer prof. dr. E. Pallante Abstract Nowadays, the mass discrepancy in the universe is often interpreted within the paradigm of Cold Dark Matter (CDM) while other possibilities are not excluded. The main idea of this thesis is to develop a better theoretical understanding of the hidden mass problem within the paradigm of Modified Newtonian Dynamics (MOND). Several phenomenological aspects of MOND will be discussed and we will consider a possible microscopic description based on quantum statistics on the holographic screen which can reproduce the MOND phenomenology. July 10, 2015 Contents 1 Introduction 3 1.1 The Problem of the Hidden Mass . .3 2 Modified Newtonian Dynamics6 2.1 The Acceleration Constant a0 .................................7 2.2 MOND Phenomenology . .8 2.2.1 The Tully-Fischer and Jackson-Faber relation . .9 2.2.2 The external field effect . 10 2.3 The Non-Relativistic Field Formulation . 11 2.3.1 Conservation of energy . 11 2.3.2 A quadratic Lagrangian formalism (AQUAL) . 12 2.4 The Relativistic Field Formulation . 13 2.5 MOND Difficulties . 13 3 A Possible Microscopic Description of MOND 16 3.1 The Holographic Principle . 16 3.2 Emergent Gravity as an Entropic Force . 16 3.2.1 The connection between the bulk and the surface . 18 3.3 Quantum Statistical Description on the Holographic Screen . 19 3.3.1 Two dimensional quantum gases . 19 3.3.2 The connection with the deep MOND limit .