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Scalar Fields in Cosmology Orfeu Bertolami Departamento De Física E Scalar Fields in Cosmology Orfeu Bertolami Departamento de Física e Astronomia (http://web.ist.utl.pt/orfeu.bertolami/homeorfeu.html) Hot Topics in Cosmology Spontaneous Workshop XIII 5-11 May 2019 EISC, Cargèse Yang-Mills theories ' a Symmetry Group: G Ψ f (x) → Ψ f (x) = exp(−iθ (x)ta )Ψ f (x) ≡U(x)Ψ f (x) a ' −1 −1 Aµta = Aµ Aµ → Aµ =U(x)AµU(x) +iU(x) ∂µU(x) Generators ta ( a=1, …, n; n=dim G) satisfy a Lie algebra: abc [ta,tb ] = f tc Matter fields Ψf a a Covariant derivative Dµ = ∂µ +igAµta Gauge fields A µ D D − D D Ψ = igt F a Ψ Field strength/ ( µ ν ν µ ) a µν Gauge “curvature” a a a abc b c F µν = ∂µ Aν −∂ν Aµ + gf Aµ Aν 1 a aµν µ 4 Action SYM− f = ∫ [− F µν F +∑Ψ f (iγ Dµ − m f )Ψ f ]d x 4 f aµν aν J aµ = Ψ γ µt aΨ Yang-Mills field eqs. Dµ F = J ∑ f f f a a a Bianchi ids. Dµ Fνλ + Dλ F µν + Dν F λµ = 0 µ Dirac eqs. (iγ Dµ − m f )Ψ f = 0 Yang-Mills theories Fermions and gauge bosons Relevant Gauge Groups: Electrodynamics G=U(1) Electroweak theory G=SU(2) X U(1) Allow for a successful Cromodynamics G=SU(3)* quantization and lead to Standard Model G=SU(3) x S(2) x U(1) renormalizable theories! Grand Unified Theories G=SU(5), SO(10), E6, … Heterotic String Theory G=E8 x E8 * Asymptotic freedom & confinement Brout-Englert–Higgs–Guralnik–Hagen–Kibble Mechanism First Scalar Field Avatar: the Higgs Boson h(x) = H(x)− H ç Spontaneous symmetry breaking mechanism ê 4 + µ 2 + λ + 2 4 SH + SHΨ = ∫ d x[Dµ H D H − m H H + (H H) ]+ ∫ d x∑g f H Ψ f Ψ f 2 f ê H ≠ 0 è Non-vanishing vacuum energy V(H,T) Min. V(H,T) mV = gV H , m f = g f H ê Cosmological constant problem H Higgs field Universal history 4 4 −3 4 V( H , 0) = O( H ) ≅ O(246GeV) >> ρC ≅ (10 eV) 2 µ ν General Relativity ds = gµν (x)dx dx Invariance under diffs. Space-time curvature Matter/Energy Covariant derivative/ Aν : D Aν Aν ν Aλ Minimal coupling ;µ = µ = ∂µ + Γλµ Torsionless µ 1 µρ $ & µ gµν;λ = 0 Γνλ = g %∂ν gρλ +∂λ gνρ −∂ρ gνλ ' ≡ νλ connection 2 { } µ µ (Γνλ = Γλν ) A − A = A Rλ Riemann tensor/ ν;ρσ ν;σρ λ νρσ Space-time curvature λ λ λ λ σ λ σ Rρµν = ∂µΓνρ −∂ν Γµρ + Γµσ Γνρ − Γνσ Γµρ # 1 & 8πG Action (Λ≠0) 4 S = % (R − 2Λ)+ Lm ( −gd x κ = 4 ∫ $2κ ' c 1 2 δ( −gL ) Einstein’s field equations m R − g R + Λg = κT Tµν = µν 2 µν µν µν −g δgµν λ λ Rµν = Rµλν , R = Rλ, g := det(gµν ) Fundamental Symmetries # Δm& • CPT symmetry % ( < 9 ×10−19 $ m ' K 0 −K 0 • Equivalence Principle " a − a % " M % 2$ 1 2 ' = Δ$ G ' <1.9x10−14 New! $ ' $ ' MICROSCOPE! # a1 + a2 & # M I & Weak Equivalence Principle (WEP) € γ −γ <10−28 Polarized GRB + DM 1 2 [O.B. & Landim 2018] Local Lorentz Invariance (LLI) 2 2 −25 δLorentz ≡ c / ci −1 ≤1.7×10 Δν U 6 Local Posion Invariance (LPI) 2.1 10− = (1+ µ) 2 µ < × ν c " M % * Ω - −5 Strong Equivalence Principle (SEP) $ G ' =1+ η, / η ≤10 M + Mc 2 . (GR :γ = β =1;η = −1− β + 2γ = 0) # I &SEP G! /G = (4 ± 9)×10 −13 yr −1 • Variaon of the fundamental couplings (LPI) e2 α /α < 4.2 ×10−15 yr−1;α := € em em em h/c Tales of Mystery and Imagination! ! (Budapest 2012) • Higgs boson (found!) • Cosmological constant problem (work in progress for the last 40 years …) • Violaons of Lorentz symmetry and Equivalence Principle (No evidence!) • Dark maZer (Observaonally consensual & plenty of candidates … Detecon) • Dark energy (Observaonal tracers) • Dark energy - dark maZer unifica=on and interac=on (Observa=onal signatures) • Variaon of fundamental constants? (No evidence!) • Gravitaonal Waves (Detected!) • Black holes (Singulari=es, Nature, Prolifera=on … Detected!) • Pioneer (NO more!) and Flyby anomalies (Evidence has shrank considerably) • … Forces of Nature, Unite! Superstring/M-theory Second Scalar Field Avatar: the dilaton • Unifica=on of the exis=ng string theories in the context of M-theory – Spectrum of closed string theory contains as zero mass eigenstates: • Graviton gMN • Dilaton Φ • An=symmetric second-order tensor BMN • Physics of our 4-dimensional world – Require a natural mechanism to fix the value of the dilaton field ^ ^ ^ – Drop BMN and introduce fermions ψ, Yang-Mills fields Aµ with field strength Fµν ^ – Space-=me described by the metric gµν – Effec=ve low-energy four-dimensional ac=on • where • αʹ is the inverse of the string tension and k is a gauge group constant • Constants c0, c1, ..., etc., are, in principle, computable [Damour, Polyakov 1994] • 4q = 16πG = α’ / 4 and a conformal transformaon → coupling constants and masses become dilaton-dependent −2 – g = k B(φ) and mA = mA(B(φ)) • Minimal coupling principle: dilaton is driven towards a local minimum of all masses – Local maximum of B(φ) – Mass dependence on the dilaton → parcles fall differently → violaon of the WEP • In the solar system, effect is of order Δa/a ≃ 10−16 • Almost within reach of the MICROSCPE mission … • Within reach of STEP (Satellite Test of the Equivalence Principle) mission … String Landscape Problem “Infinite” number of low-energy models 10500 vacua Third Scalar Field Avatar: Scalar-tensor theories of gravity • Gravitaonal coupling strength depends on a scalar field, ϕ – General acon – f(ϕ), g(ϕ), V (ϕ) are generic func=ons, qi(ϕ) are coupling func=ons – Li is the Lagrangian density of the maer fields – Graviton-dilaton system of string/M-theory can be viewed as a scalar-tensor theory • Brans-Dicke theory [Brans, Dicke 1961] – Defined by f(ϕ) = ϕ, g(ϕ) = ω / ϕ , a vanishing potenal V(ϕ) and qi(ϕ)=1 – Non-canonical kine=c term; ϕ has a dimension of energy squared – ω marks observa=onal devia=ons from GR, which is recovered in the limit ω → ∞ – Sa=fies Mach’s Principle – G ∝ ϕ−1 depends on the maer energy-momentum tensor through the field equaons – Observaonal bounds: |ω| > 40000 [ Will, gr-qc/0504086] • Induced gravity models: – f(ϕ) = ϕ2 and g(ϕ) = 1/2 – Potenal V (ϕ) allows for a spontaneous symmetry breaking – Field ϕ acquires a non-vanishing vacuum expectaon value, – The cosmological constant Λ is given by interplay of V(<0|ϕ|0>) and all other contribuons to the vacuum energy [Fujii 1979] [Zee 1979] [Adler 1982] • Horndeski gravity (1974) … Can be scrunized with GWs [Arai, Nishizawa 2017] [Kopp et al. 2018] Fourth Scalar Field Avatar: the inflaton Inflaon, an accelerated expansion of the Universe which took place about 10-35 secs. aer the Big Bang, which accounts for the main observaonal features of the Universe: isotropy, homogeneity, horizon, flatness, absence of magne=c monopoles and rotaon, and the origin of energy density fluctuaons that generated the first galaxies [Guth 1981, Linde 1982, Albrecht & Steinhardt 1982, …] 1 µ Model: L = ∂ ϕ ∂ ϕ −V(ϕ) V(φ) - your favorite … 2 µ Quantum fluctuations of the inflaton è Energy density fluctuations + gravitational waves! ê Observed through the Cosmic Microwave Background Radiation (CMBR) Inflaon for voyeurs 28 >exp65≅10 ( 1/2 1/2 + !8π $ V −12 4 a(t f ) = a(ti )exp*# & (t f − ti )-,V ≅10 M P )*" 3 % M P ,- GUTs with Higgs field (troublesome) Supergravity-like (fine) Chiral superfields Finally, it is important to note that the perturbations ⇥⌃ and ⇥gµ⇤ are gauge-dependent, i.e. they change under coordinate/gauge transformations. Physical questions therefore have to be studied in a fixed gauge or in terms of gauge-invariant quantities. An important gauge-invariant quantity is H ∂ ln a. This system will yield the following equations of motion for the homogeneous modes the curvature perturbation⇥ t on uniform-density hypersurfaces [11] ⇤(t) and a(t), H ⇤ Ψ + ⇥⇧, (15) 3.3.3 Vector (Vorticity) Perturbations − ⇥ ⇧˙ 1 1 H2 = ⇤˙2 + V (⇤) , (4) where ⇧ is the total energy density ofThe the vector universe. perturbations S and F2 in equations (13) and (14) are distinguished from the scalar i 3Mpli 2 ⌅ ⇧ i i perturbations B, Ψ and E as they are divergence-free, i.e. ⇧ Si = ⇧ Fi = 0. One may show Finally, it is importanta¨ to note that the1 perturbations ⇥⌃ and ⇥gµ⇤ are gauge-dependent, i.e. they 3.3.2 Scalar (Density) PerturbationsthatFinally, vector it is perturbations important to note= on thatlarge the scales perturbations⇤˙ are2 redshiftedV (⇥⌃⇤)and, away⇥gµ⇤ are by gauge-dependent, Hubble expansioni.e. they (unless they(5) are changechange under under coordinate/gauge coordinate/gaugea transformations. transformations.2 Physical Physical questions questions therefore therefore have have to be to studied be studied in in 3.3.3 Vector (Vorticity) Perturbationsdriven by anisotropic stress).− In3M particular,pl − vector perturbations are subdominant at the time of In a gauge where the energy densitya fixedassociateda fixed gauge gauge with or or in in the terms terms inflaton of of gauge-invariant gauge-invariant field is unperturbed⇥ quantities. quantities. An (⇤i.e. An important important⇥⇧⌅ = gauge-invariant 0) gauge-invariant quantity quantity is is recombination. Since CMB polarization is12 generated at last scattering the polarization signal is Theall scalar vector degrees perturbationsand of freedomSi and canF bei thein expressedthe equations curvature curvature by perturbation (13) perturbation a metric and (14) perturbation on on are uniform-density uniform-density distinguished⇤(t, x hypersurfaces) hypersurfaces from the [11] scalar [11] 3.3.3 Vectordominated (Vorticity) by scalar Perturbations and tensori perturbationsi ( 3.5).
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