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 maer (Observaonally consensual & plenty of candidates … Detecon) • Dark energy (Observaonal tracers) • Dark energy - dark maer unificaon and interacon (Observaonal signatures) • Variaon of fundamental constants? (No evidence!)

• Gravitaonal Waves (Detected!)

• Black holes (Singularies, Nature, Proliferaon … Detected!)

• Pioneer (NO more!) and Flyby anomalies (Evidence has shrank considerably)

• … Forces of Nature, Unite! Superstring/M-theory Second Scalar Field Avatar: the dilaton

• Unificaon of the exisng string theories in the context of M-theory – Spectrum of closed string theory contains as zero mass eigenstates:

• Graviton gMN • Dilaton Φ

• Ansymmetric 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 – Effecve low-energy four-dimensional acon

• 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 funcons, qi(ϕ) are coupling funcons

– 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 kinec term; ϕ has a dimension of energy squared – ω marks observaonal deviaons from GR, which is recovered in the limit ω → ∞ – Safies 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 magnec 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 >exp65≅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 ⇤ + ⇥⇧, (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, 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). Most of this section therefore focuses on scalar perturbations B, and E as they are divergence-free, i.e. ⇧ Si = ⇧ Fi = 0. OneHH may show 2 ⇤¨ +3H⇤˙ + V⇤ (⇤)=0+ § ⇥⇧., (15) (6) andgij tensor= a ( perturbations.t)[1 + 2⇤]⇥ij . However, vector⇤⇥ perturbations+⇧˙ ⇥⇧, can(16) be sourced by cosmic strings(15) which that vector perturbationsThe on vector large perturbations scales are redshiftedSi and awayFi in by equations Hubble expansion (13) ⇥ and (unless (14)⇧˙ are they distinguished are from the scalar are discussed in 6.1. (3) 2 2 drivenGeometrically, by anisotropic⇤ measuresTheperturbations stress). spacetime the In spatial particular, experienceswhereB, curvaturewhere ⇧andis vector⇧ theisE the of total accelerated constant-densityas perturbations total they energy energy are density density divergence-free,expansion, are hypersurfaces, of of subdominant the the universe. universe.a> ¨ i.e. at0,⇧ thei ifS= timeand=4 ⇧ of onlyiF⇤/a=. if 0. the One potential may show energy of § R i ⌃ i recombination.An important property Sincethat CMB of vector⇤ polarizationis that perturbations it remains is generated on constant large at scaleslast outside scattering are the redshifted horizon. the polarization away13˙2 In by a Hubble gauge signal defined is expansion (unless they are¨ H t ln a. This system will yield the following equations of motion forthe the inflaton homogeneous dominates3.3.2 modes over Scalar its (Density) kinetic energy, PerturbationsV ⇤ . This condition is sustained if ⇤ V . ⇥ H t ln a. This systemdominated willby spatially yield by the flat scalar following hypersurfaces, and tensor equations perturbations⇤ is3.3.23.3.4 the of motion dimensionless Scalar ( Tensor3.5). for (Density)Most (Gravitational the of density homogeneousthis Perturbations section perturbation therefore Wave)⌃ modes1 focuses Perturbations⇥⇧/(⇧ + onp scalar). Taking | | ⇧ | | ⇥ driven by anisotropic stress). In particular, vector perturbations3 are subdominant at the time of ⇤(t) and a(t), These two conditionsIn a gauge for§ prolongedwhere the energy inflation density associated are summarized with the inflaton by field restrictions is unperturbed of (i.e. the⇥⇧ form= 0) of the ⇤(t) and a(t), andinto tensor account perturbations. appropriaterecombination. transfer However, functions vectorTensor Since perturbations perturbationsCMB to describe polarization the can are sub-horizon be uniquely is sourced generated described evolution by cosmic at last by of strings a the scattering gauge-invariant fluctuations, which the polarization metric perturbation signal⌅ ish inflaton potentialInV aall gauge(⇤ scalar) and where degrees its the ofderivatives. freedomenergy density can be Quantitatively, expressed associated by with a metric the inflation perturbation inflaton fieldrequires⇤( ist, x unperturbed)12 smallness (i.e. of⇥⇧ the⌅ =ij slow- 0) areCMB discussed and large-scale in 6.1. dominated structure by(LSS) scalarall scalar observations and degrees tensor of can perturbations freedom therefore can be be ( expressed3.5). related Most to by the of a thismetric primordial section perturbation therefore value ⇤(t, focusesx)12 on scalar 2 1 1 2 § 2 ˙ 2 roll1 parameters1 2 § g2ij = a (t)[1 + 2⇤]⇥ij . 14i (16) H = 2 ⇤ + V of(⇤⇤). A, crucial statisticaland tensor measure˙ perturbations. of the primordial(4) However, scalar vector fluctuationsg perturbationsij = a is(t)[ theij2 power+ canhij] , be spectrum sourced⇧jhij of= by⇤hi cosmic=0. strings which (21) H = 2 ⇤ + V (⇤) , g = a ((4)t)[1 + 2⇤]⇥ . (16) 3Mpl 2 ij ij (3) 2 2 ⌅ 3.3.4⇧ Tensor (Gravitational3Mpl 2 Wave)Geometrically, Perturbations⇤ measures2 the2 spatial2 curvature of constant-density hypersurfaces, = 4 ⇤/a . are discussed⌅ in 6.1.⇧ 3 ˙ 2⌅˙2 2 R ⌃ ⇤ ⇤Physically,§ =(2⌅)Hh⇥ij(kcorresponds+Mk )pl ⇤ P ( tokM) gravitational. pl V wave fluctuations.(17) TheV13 power(3) spectrum2 for2 the two a¨ 1 2 kGeometrically,k0 An important⇤ propertymeasures⇥ of the3⇤ is spatials that it curvature remains constant of constant-density outside the horizon. hypersurfaces,2 In a gauge= defined4 ⇤/a . ˙ Tensor perturbationsa¨ are1 uniquely2 ⌅ described⇧ by a gauge-invariant= k metric+ perturbation, h ⇥ M . R ⌃ (7) = 2 ⇤ VH(⇤) ,t ln a. This= system will⇤˙ yieldV polarization(⇤ theby) spatially, following(5) modes2 flat hypersurfaces, equations of h 2 ofh+⇤ e motionis the+ h dimensionlesse for, h(5) theh homogeneous+ij density,h , is perturbation defined modespl as 1 ⇥⇧/13(⇧ + p). Taking a 3M ⇥ 2 An important⇥H property2 ofHij⇤ is⌅ thatij2 it remainsV ij constant outside | | ⌅ the horizon.V3 In a gauge defined pl The scale-dependencea 3.3.43M of the Tensorpower spectrum (Gravitational is defined by Wave) the⇥ scalar Perturbations spectral⌅ ⇧ index⇥ (or tilt) 1 The r.h.s. of (27) is to be⇤ evaluated(t) and Ha at(t), horizont ln a. exit Thispl of system a givenby will perturbationinto spatially yield account the flat appropriate following hypersurfaces,k = aH transfer equations(see⇤ functions Figureis the of motion dimensionless 2).to describe for the the density sub-horizon homogeneous perturbation evolution modes of the⇥⇧/ fluctuations,(⇧ + p). Taking ⇥ ⇤ ⇥ ⇥g = a2(t)[ ⇤+ h ] , ⇧ h = hi =0. (21) 3 InflationaryThe r.h.s. quantumof (27) is to fluctuations be evaluated therefore⇤( att) horizonand a produce(t), exit of theij a given following perturbationCMBij power andij large-scale spectrumk =j aHij structure(see fori Figure⇤ (LSS) observations 2). can therefore be2⇥ related2 to the primordial value OnceTensor these perturbations constraintsinto account are are uniquely appropriate satisfied,d ln P describeds transfer the inflationary functionsby a gauge-invariant to describe process3 the (and metric sub-horizon its perturbation termination) evolution h ofij the happens fluctuations, gener- and and Inflaon and the CMBR ns 1 . hkhk0 =(2⇥) (k + k⇥) (18)Pt(k)14 (22) Inflationary quantum fluctuations therefore produce the followingCMB2of ⇤ power and. A crucial large-scale1 spectrum statistical1 ˙ structure2 for measure⇤ (LSS) of⇤ the observations primordial⌅ scalar can therefore fluctuationsk3 be is related the power to the spectrum primordial of ⇤ value Physically, hij correspondsically2 for to a gravitational wide2 H class= of wave models.⇥ fluctuations.d ln⇤k The+ V slow( The⇤) evolution power, spectrum of the for inflaton the two then(4) produces an exponential ⇤¨ +33.4H⇤˙ Quantum+ V (⇤ Fluctuations)=0. as the Origin¨ of Structure˙ (6) 2 1 2 1 i 2 14 ⇤ +3H ⇤ + VH+(⇤+)=0of ⇤.. A crucial32M+ pl statisticalg 2= a ( measuret)[˙2 + h of the] , primordial⇧3 h(6)= scalarh2⌅=0 fluctuations. is the power spectrum(21) of ⇤ polarizationHere, scale-invarianceP (k modes)= of hij corresponds2 h e 2 + h toande the., itsh H value scale-dependenceh ,h=n⌅ijs,= is defined 1. We is⇤ij as defined may+⇧⇤kV⇤ijk( also⇤)=(2 analogously(29), define⌅) j⇥(kij+ thek to⇥) runningi (18)Ps but(k) . of for the historical(4) reasons without(17) the 1, In Section 3.2 we discussed the classical evolutions of the inflatonincrease field.H˙ Something⇥ H in remarkable theij geometricij ⇥ size of3M the2 universe,2 ⌅ 0 ⇧ k3 happens when one considers quantum fluctuations of the inflaton: inflation⌃ combined2 with⇧ quantum a¨ 1 pl 2 The spacetime experiences accelerated expansion,a> ¨ 0,spectral if andP indexs only(k)= by if⌅ the⇧ ⌅ potential⇧ k=aH energy. of ˙2 ⌅ (29)⇧ 3 2⌅ The spacetimemechanics provides experiences an elegant mechanism accelerated for generating the expansion,initial seeds˙ of all structurea> ¨ in the0, ifThe= and scale-dependence only if⇤ the of2 theV potential( power⇤) ⇤ ,⇤ spectrum energy=(2⌅ is) defined⇥ of(k + k by⇥) the scalarPs(k) spectral. (5) index (or tilt) (17) ⌃Physically,2⇧ hij correspondsa¨ todn2 gravitationals 21⇥ 2 k wavek0 fluctuations.d ln ThePt3 power spectrum for the two universe. In other words,2 quantum fluctuations during inflation⌅ are the⇧ source⌅ of the⇧ primordial2k=aHa 3 3M ˙ ⌅ Ht⇧ k the inflaton dominates over its kinetic energy, V ⇤˙• . ThisSimple condition inflation: is sustained ˙ h if k h(k ⇤ ¨ =(2 ⇥ V) s ().k= + +plk+⇥) a. (t2)Pt⇤(ka)(0)Ve(⇤+¨) ,H, nt const,. (22) (19) (5) (8)(23) theIn addition, inflatonpower dominates quantumspectra Ps(k) and fluctuationsPt(k over). In this its section kinetic we during sketch the energy, mechanism inflation by whichV excite inflation⇤ relates tensor. This0 metric conditiona perturbations is⇥3M sustained3 hij⇤[6]. if ⇤ Their V d.⇥ln Pds ln k polarization⇤ modesThe⌅ scale-dependence of hij ⇥hd lneijk+ ofkhpl thee⌅ij power, h spectrumh ,hns,1 is is defined defined⌅ . by as the scalar spectral index (or tilt)(18) microscopic physics⌃ to macroscopic observables. ⌃ | | ⇧ | ⇥| ⇥ | | ⇧ | | These two conditions for prolongedThesepowerIn inflation addition, two spectrum conditions are quantum (in summarized general for fluctuationsandThe prolonged models power by during of restrictions spectrum inflation) inflation inflation is is are often excitesimply of summarized the approximated tensor that form of metric of a by massless bythe perturbations restrictions a power field law in⇥h of form deij the[6]. Sitter Their form space⇤ of⇥ thed ln k Comoving Scalesand its scale-dependence is defined analogouslyi.e. to (18) but for historical reasons withoutd ln P thes 1, power spectrum (in general modelsand of inflation)For inflation is simply that to successfully ofHere, a massless scale-invariance address field in corresponds de the Sitter Big space to Bang the value problems,ns 2= 1. We one mayn mustt(k also) define simply the ensure running of that the the in- ¨ ˙ 1 3 ns 1 2⇥ . (18) inflaton potential V (⇤) and its derivatives.inflaton potential Quantitatively,V (⇤) and inflation its derivatives.horizon requiresre-entry Quantitatively, smallness of inflationthe⇤ +3 slow-H requires⇤n+s(kV)(1+⇤ smallness)=0s(k.)ln(k/k of) the slow- k (6) horizon exit Comoving 2 spectral index by hkhk2 =(2⇥) (k + k⇥) ⇥ dPlnt(kk) (22) k ⇤¨ +3H⇤˙ +0V (⇤)=0. Pt(k)=At3(k) . (6) (24) flationary Horizon8 HP process2(k)=A produces(k ) d ln aPt su⇤ ⇤cient⌅ number, of thesekdn ‘e-folds’ of(20) accelerated expansion Ne roll parameters roll parameters s Here,s ⌃ scale-invariance corresponds to the value nss = 1.k We may also define the running of the The spacetimePt(k)= experiences8 2 H accelerated. n expansion,t k⌃ , a> ¨ 0, if and only(30)s if the. potential ⇥(23) energy of (19) 26 ⇥ Pt(k)=ln(Ma(t 2⇧)/a(t . )). A⇥ typicald⇥ln k lower bound on(30) the required⇥ d ln k number of e-folds is N ln 10 55 The spacetimedensity fluctuationandpl2 ⌅final itsexperiences scale-dependence⇧ kinitial=aHspectral acceleratedCMB index polarization is defined by expansion, analogously measurementsa> ¨ 0, to if (18) andare sensitive but only for if historical theto the potential ratio reasons of energytensor without power of thee to scalar1, power where k 2is the pivotMpl2 scale.2⇧ 2 The power spectrum is˙2 often approximated by a power law form ¨ ⇤ ˙ 2 ˙2 2 2 the˙ inflatonM⌃ ˙ dominates2 M ⌅ over⇧ its10k= kineticaH energy, V ⇤ . This˙2 condition is sustaineddns if ⇤ ¨ V . H M ⇤ M V i.e.H thepl ⇤ inflaton[26,V pl dominates 27,V 28]. overOur its discussion kinetic energy,2 has⌃V soV far⇤ addressed. This condition onlys the is classical sustained.1 | | and if⇧⇤| homogeneous| V . evolution(19) of pl pl 2 n (k ) ns(k) 1+ s(k)ln(k/k) = ,InflationThese=If ⇤ two⇥is Gaussian conditionsHotM Big Bang then for. the prolonged power, spectrum inflation⇥ (7) containsM arepl summarizedallt the .⌃ statistical byd information.ln restrictionsPtk ⇥(7)d lnk ofPrimordialP2t the form non-| of| the⇧ | | 2 2 2 These2 pl two conditions for prolonged inflationk areP summarized(kn)=A (k ) by, restrictionsr . of the form, of the (23)(20) (25) ⇥H 2Slow-rollH ⌅ predictions2 V ⇥ H | 2| ⌅H ⌅theV2 inflatingV system.PThet(k)= power Small|At|(k⌅ spectrum) spatialV is perturbations often. s approximatedt s ⌃ in by the a power inflaton law⇤ form(24)and the metric gµ⇥ are inevitable Slow-roll predictions Gaussianity is encoded in higher-order correlation functions of ⇤ (see⇥ d ln5.3).kk⌃ In single-field⇥ Ps slow-roll ⌅ ⇧ inflaton potentialinflaton potentialV (⇤)⌅ andV ⇧( its⇤) derivatives. and its derivatives. Quantitatively,k Quantitatively, inflation inflation requires ⇥ requires smallness smallness of the of slow- the slow- Models of single-field slow-roll inflationdue makes to definitequantum predictions mechanics; for inflation the⇥ primordial stretches scalar these§ and fluctuationsn (k ) 1+ 1 (k to)ln( astronomicalk/k ) scales, eventually Models of single-field slow-rollinflation inflationthe non-Gaussianity makes definite is predicted predictionswhere tokfor be⌃ is small the pivot primordial [39, scale. 40], but scalar non-Gaussianity and s can be2 s significant Once these constraints areroll satisfied,CMB parametersroll polarization the parametersTime inflationary [log(a)]i.e. measurements process areThe sensitive (and parameter its to termination) ther will ratio be of of tensor fundamental happens power gener- to importancek scalar power for the discussion presented in this paper. As Once these constraints are satisfied,tensor the fluctuation inflationary spectra. process Under (and the its slow-roll termination) approximation happens one gener-If ⇤ is may Gaussian relate then the theP predictionss power(k)= spectrumAs(k⌃) for containsnt(k all) the statistical information., Primordial non- (20) tensorFigure fluctuation 2: Creation and spectra. evolutionin of perturbations Undermulti-field the in the slow-rollinflationarymodelsproducing universe. or approximation in Fluctuations single-field large-scale are one models structures may with relate non-trivial the including predictions kinetic galaxies for termsk k such⌃ and/or as violation the one of we the inhabit. Thus inflation is ically for a wide class of models. The slow evolution of1 the16we2 inflaton argue in Section2 then produces 4,2 its value an encodes exponential crucial⇥ information about the physics of the inflationary era. ically for a wide class of models. TheP (k) slow and evolutionP (k)created to quantum the of shape mechanically the inflaton of on the sub-horizon inflaton then scales. While producespotential comoving scales,˙ V ank(⌃,re-M).16 exponentialGaussianityTo˙2 compute2 M2P ist encoded the2 spectralP int(k higher-order)=2 A indicest(k) correlation one functions. of ⇤ (see 5.3). In single-field slow-roll(24) s P (k) andtP (k) to the shapeslow-roll of the conditions. inflaton potential1 HV (⌃). pl˙To⇤ computeM ˙pl theVM spectral indices one V § s t main constant the comoving Hubble radius during inflation,responsible (aH) , shrinks not and the justwhereH fork⌃ rpl theis the⇤ universe pivot. pl scale. thatV we observe,k2 but2 alsoV for(25) the fact we are here to observe it. increase in the geometric size of the universe, = inflation= the non-Gaussianity, is predicted, ⇥ toM⇥ bepl⇥ smallM [39,. 40],. but non-Gaussianity(7) (7) can be significant increase in the geometric size of theuses universe, d ln k dlnperturbationsa (H exit• theconst.).Slow-roll horizon.12 Causal To physics firstpredictions: cannot order act on superhorizon in the perturbations2 slow-roll2 parameters2 ⇥ P2 s ⇥ and ⌅ one finds [43] pl uses d ln k dlnand theya freeze(H until horizonconst.). re-entryIn addition To at late first times. to order the perturbationAfter in⇥ theH a slow-roll su⇥⇤ tocientH2 the parametersHIf spatial⇤ numberis⌅2 GaussianH part2 ⇥ of⌅and ofVe then the-folds2⌅ metricone theV have power finds there been [43]spectrum| | are⌅ achieved, fluctuations| | contains⌅ V the all inV theg process. statistical These must are information. terminate. Primordial The inflaton non- ⇤⇤ ⇤⇤ CMB polarizationin multi-field measurements models⌅ ⇧ or⌅are in single-field⇧ sensitive models to the with ratio non-trivial of tensorµ0 kinetic power terms to and/or scalar violation power of the Gaussianity is encoded in higher-order correlation functions of ⇤ (see 5.3). In single-field slow-roll Therelated parameter to ⇤ by Einstein’sr willdescends beHt ofequations. fundamental towards theslow-roll importance minimum conditions. for of the the discussion potential presented and ‘reheats’ in this paper. the universe, As with ⇤-particles decaying QuantumHt fluctuations in quasi-deOnce Sitter13 theseaOnce(t1)1 constraints theseaV(0)e constraints are,H satisfied, areconst satisfied, the. inflationary the inflationary process process (and itsP (and termination) its(8) termination) happens happens gener-§ gener- a(t) a(0)e ,HconstThis. statement is only true for adiabatic15 inflation12 perturbations. the(8) non-Gaussianity Non-adiabatic is predicted fluctuationst to be small can [39,arise 40], in multi-field but non-Gaussianity can be significant In spatially-flat gauge, perturbationsPPwe(k(k)= argue in)=⇥ are related in Section to perturbations⌅ 4, in its the value inflaton,n,n field encodes value⌅1=21=2 crucialIn⌅ ⌅ addition information6⇥6,⇥ , to the perturbation about the to physicsr the(31) spatial(31) of. the part inflationary of the metric there era. are fluctuations in gµ0. These(25) are ⌅ ⌅s s 22 44 into radiation,ss and so initiating the hot Big Bang. ⇧, cf. Eqn. (15) with =0 icallymodels for of2424ically inflation a⇧⇧ wideMM for⇥ class (see a wide5 of and models.class Appendix of models.in The multi-field A). slow In The that evolution models slow case, evolution⇤ orevolves of in the single-field on of inflaton super-horizonthe⇥ models inflatonPs then with produces then scales. non-trivial produces an kineticexponential an exponential terms and/or violation of the ⌅ ⇧ plpl k=aH related to ⇤ by Einstein’s equations. ⇥ = H H Ht, k=§aH (26) 13 For inflation to successfully14 address˙ the Big Bang problems, one must simply ensure that the in- For inflation to successfully address the Big Bang problems,increase The one⌅˙ ⇤ normalization inincrease must⇧ the⇥ geometric simply in theThis of geometricthe ensure size basic dimensionless of thethat inflation sizeslow-roll universe, of theThis power the conditions. model in-statement universe, spectrum can is onlyP be trues(k) generalized for is adiabatic chosen perturbations. such in that a variety the Non-adiabatic variance of ways: of fluctuations⇤ is several can arise fields in multi-field collectively where in the second equality we have assumed slow-roll. The power spectrumThe of parameter⇥ and the power r modelswill be of ofinflation fundamental (see 5 and Appendix importance A). In for that the case, discussion⇤ evolves on super-horizonpresented in scales. this paper. As ⇤⇤ = Ps(k)dlnk. 12 flationaryspectrum process of inflaton produces fluctuations ⇧ are a therefore su20⇤2 relatedcientVV as follows numberproducing of the these inflaton, ‘e-folds’In14 addition non-standard of to accelerated the perturbation§ kinetic expansion to terms, the spatialN scalarse part of replaced the metric there by axion-like are fluctuations fields, in gµ0 etc.. These Each are of flationary process produces a su⇤cient number of these⌅ ‘e-folds’⇧ 2 of accelerated expansion TheN normalizationHt of the dimensionless power spectrum Ps(k) is chosen such that the variance of ⇤ is PPt(tk(k)=)= H 4 we,n,n arguett= in= Section22⇥⇥,r,r 4, its=e = 16 value 16⇥Ht.⇥ . encodes crucial information(32)(32) ⇥ about the physics of the inflationary era. ⇥ ⇥ = 22 ⇧ ⇧4 . (27)a(t) aa(0)(t)e a,H(0)e ,Hconst .const26. (8) (8) k k0 3⇧ Mk k0 related⇤⇤ to= ⇤by⇥P Einstein’s(k)dlnk. equations. ln(a(tfinal)/a(tinitial)). A typical⇧ ⌃ 3 lower⇤⇧⇧˙ ⇧M pl bound⌃ these on models the required still produces number26 0 ⌅s of ane inflationary-folds is Ne period,⌅ln 10 with55 the details determining various observables ln(a(tfinal)/a(tinitial)). A typical lower bound on the required number ⇥ pl ofkke=-foldsaH is Ne ln 1013⌅This⌅⇧ statement55 is only true⌅ for adiabatic perturbations.⇤ Non-adiabatic fluctuations can arise in multi-field Finally,10 in the case of slow-roll inflation, quantum fluctuations of a light scalar field (m H)in 19 ⌅ ⇤⇤ 10 [26,We 27, note 28].quasi-de that SitterOur the space discussion value (H const.) of scale the with has tensor-to-scalar the Hubble soFor far parameter inflation addressedsuchH [42] ratio to as successfully cosmological depends only themodels on classical address the perturbations, of time-evolution inflation the and Big (see homogeneous Bang5 and as of problems,will Appendixthe inflaton be evolution described A). one In that must of case, in simply further⇤ evolves ensure ondetail super-horizon that below. the in- scales. [26, 27, 28]. Our discussion has soWe far note addressed that the only value⇤ the ofFor the classical inflation tensor-to-scalar and to successfully homogeneous ratio depends address evolution on the the Big time-evolution of Bang problems, of the one inflaton must simply19 ensure that the in- 2 2 14 § 3 2⇤ H thefield inflating system. Small⇧k ⇧k spatial=(2⇤) (k + perturbationsk⇥) . in the(28) inflatonThe⇤ normalizationand the metric of the dimensionlessg are inevitable power spectrum Ps(k) is chosen such that the variance of ⇤ is ⇧ 0 ⌃ flationaryk3 2⇤ processThere produces also remain a su⇤ questionscient number of initial of theseµ⇥ conditions ‘e-folds’ of and accelerated of whether expansion inflationNe continues eternally. field flationary process ⇥ produces a su⇤cient number of these ‘e-folds’ of accelerated expansion Ne the inflating system. Small spatial perturbations in the inflaton ⇤ and the metric gµ˙⇥ are2 ⇤⇤ inevitable= Ps(k)dlnk. ⇥ 15Intuitively, the curvature perturbation ⇥ is related to a spatially varying time-delay8t(x) for⌃ the˙ end2 of 0 26⇥ due to quantum mechanics; inflationln(a stretchesr(t=final 16This)/a⇥ =(t theseinitial latter8 )). fluctuations⌃ point A typical. ⌅ may⇧ lower toseem astronomical bound paradoxical; on the required scales, if the(33) eventually number inflaton of e-folds completes is Ne its26ln evolution10 55 as we have just due to quantum mechanics; inflation stretchesinflation [41]. these This time-delay fluctuations is inducedln( bya the(t inflatonfinal to) fluctuation/a astronomicalr(t=initial⇧. 16)).⇥ = A typical2 scales,H lower. eventually bound on the required number(33) of e-folds is Ne ln 10 55⇤ [26, 27, 28].10 MOurpl2 discussion has⇤ so far addressed only the classical and homogeneous evolution⇤ of producing large-scale structures including10 galaxiesMpl⇥ suchH⇤ as the one we inhabit. Thus inflation is [26, 27,21 28]. Ourassumed, discussion⇥ how has⇤ could so far inflation addressed continue? only the classical The answer and homogeneous lies19 in the fact evolution that inflation of produces other producing large-scale structures includingWe also galaxies point out such the existence as the of one athe slow-roll we inflating inhabit. consistency system. Thus Small relation inflation spatial between perturbations is the tensor-to-scalar in the inflaton ratio ⇤ and the20 metric gµ⇥ are inevitable responsibleWe also point not out just the for existence thethe universe inflating of a slow-roll that system. we consistency observe, Small spatial but relation also perturbations for between the fact the in we the tensor-to-scalar are inflaton here to⇤ and observe ratio the metric it. gµ⇥ are inevitable responsible not just for the universe thatand the we tensor observe, tilt which, but also at lowest for thedue order, fact to quantum has we10 theThis are form mechanics; here estimate to observe of inflation the required it. stretches number these fluctuations of e-folds assumes to astronomical GUT scale scales, reheating. eventually For lower reheating andAfter the tensor a su⇤cient tilt which, numberdue at lowest of toe-folds quantum order, have has mechanics; been the achieved,form inflation the stretches process must these terminate. fluctuations The to astronomical inflaton scales, eventually After a su⇤cient number of e-folds have been achieved, the processproducing musttemperatures, large-scale terminate. structures fewer Thee inflaton-folds including can be galaxies su⇤cient. such as the one we inhabit. Thus inflation is descends towards the minimumproducing of the large-scale potential structures and ‘reheats’ including the universe, galaxies with such⇤ as-particles the one decaying we inhabit. Thus inflation is responsibler = not8 justnt . for the universe that we observe, but also(34) for the fact we are here to observe it. descends towards the minimum of the potential and ‘reheats’ the universe, withr = ⇤8-particlesnt . decaying20 (34) into radiation, and so initiatingresponsible the hot not Big just Bang. for the universe that we observe, but also for the fact we are here to observe it. Measuring the amplitudes of P ( VAfter) and a suP⇤cient( V number) and theof e-folds scale-dependence have been achieved, of the scalar the process must terminate. The inflaton into radiation, and so initiating the hotThis Big basic Bang. inflation modelAfter cant ⌅ a be su⇤ generalizedcient numbers ⌅ in of ae-folds variety have of been ways: achieved, several the fields process collectively17 must terminate. The inflaton Measuringspectrum then ( amplitudesV ) and of (Pt (Vdescends)V allows) and towards aP reconstructions ( theV minimum) and of the the of the inflaton scale-dependence potential potential and ‘reheats’ as of a the Taylor the scalar universe, with ⇤-particles decaying This basic inflation model can be generalizeds ⌅ in a varietydescendss ⌅ of⌅ towards ways: the several minimum⌅ fields of collectively the potential and ‘reheats’ the universe, with ⇤-particles decaying producingspectrumexpansionn thes around( inflaton,V ⌃) and(corresponding non-standards ( Vinto to) kinetic allows the radiation, time a terms, when reconstruction and fluctuations scalars so initiating replaced of on the the CMB inflatonhot by scales axion-likeBig Bang. potential exited fields,the ashorizon) a etc. Taylor Each of producing the inflaton, non-standard kinetic terms,⌅ scalars⇥ into replaced⌅ radiation, by axion-like and so initiating fields, the etc. hot Each Big of Bang. 20 theseexpansion models around still⌃ produces⇥ (corresponding an inflationary toThis the time basic period, when inflation with fluctuations the model details can on CMB be determining generalized scales exited various in a the variety horizon) observables of ways: several fields collectively This basic inflation1 model2 can1 be generalized3 in a variety of ways: several fields collectively these models still produces an inflationarysuch as cosmological period,V (⌃ with)= perturbations,V the+ V details (⌃ producing determining as⌃ )+ will beV the described inflaton,(⌃ various⌃ ) non-standard in+ observables furtherV ( detail⌃ kinetic⌃ ) below. terms,+ , scalars replaced(35) by axion-like fields, etc. Each of |⇥ ⇥ ⇥ 12 ⇥ ⇥ 3!1 ⇥ ⇥ ··· such as cosmological perturbations, as will beV described(⌃)=V in+producingV further(⌃ these detail⌃ the)+ inflaton, models below.V still non-standard(⌃ produces⌃ )2 + an kinetic inflationaryV ( terms,⌃ ⌃ period, scalars)3 + with replaced, the details by(35) axion-like determining fields, various etc. Each observables of There also remain⇥ questions ⇥ of initial⇥ conditions ⇥ and⇥ of whether ⇥ inflation⇥ continues eternally. where (...) =(...|) these. Furthermore, modelssuch still as cosmological2 produces if one assumes an perturbations, inflationary that3! the primordial period, as will be with perturbations described the··· details in further are determining detail below. various observables There also remain questions ofThis initial latter conditions point|⇥ may and seem|= of whetherparadoxical; inflation if the continues inflaton completes eternally. its evolution as we have just whereproduced (...) by=( an inflationary...) such. model Furthermore, as cosmological withThere a single if also one perturbations, slowly remain assumes rolling questions that scalar as will the field,of be initial primordial one described can conditions fit perturbations directly in further and to of the detail whether are below. inflation continues eternally. This latter point may seem paradoxical;assumed, if how the|⇥ could inflaton inflation|= completes continue? its evolutionThe answer as lies we in have the fact just that inflation produces other producedslow-roll by parameters, an inflationary bypassing modelThere the withThis also spectral a latter remain single indices point slowly questions entirely, may rolling seem of and scalar initial paradoxical; then field, conditions reconstruct one if can the and the fit inflaton formofdirectly whether of completes the to the inflation its evolution continues as eternally. we have just assumed, how could inflation continue? The answer lies in the fact that inflation produces other slow-roll10underlyingThis estimate parameters, potential of the bypassing [44, requiredThis 45, 46, latter the 47,numberassumed, spectral 48, point 49, of 50, maye how indices-folds 51]. could seem assumes entirely, inflation paradoxical; GUT and continue? then scale if thereconstruct reheating. The inflaton answer the For completes lies form lower in ofthe reheating its the fact evolution that inflation as we produces have just other 10 This estimate of the required numbertemperatures,underlying16In of Appendixe-folds potential fewer assumes Ae-folds we [44, present 45, canassumed, GUT 46, thebe 47,su results scale⇤ how 48,cient.10This reheating.for 49, could general estimate50, 51].inflation single-field For of the lower continue? required models. reheating numberIn The this answer case, of e-folds the lies primordial assumes in the power GUT fact that scale inflation reheating. produces For lower other reheating temperatures, fewer e-folds can be su⇤cient.spectra receive contributions from a non-trivialtemperatures, speed fewer of sounde-foldscs can= 1 be and su⇤ itscient. time evolution. The slow-roll 16In Appendix A we present10 theThis results estimate for general of the single-field required number⇧ models. of Ine-folds this case, assumes the primordial GUT scale power reheating. For lower reheating results arise as the limit c 1,c ˙ 0. spectra receive contributionss ⌅temperatures, froms a⌅ non-trivial fewer speede-folds of can sound be suc ⇤=cient. 1 and its time evolution. The slow-roll 17 s ⇧ results arise as the limit cs 1,c ˙s 0. 17 17 ⌅ ⌅ 22 17 22 –101– CMBR (Last surface of maer-radiaon scaering, circa 376 Kys aer BB)

2 Cosmological Observables: An Overview

2.1 The Concordance Cosmology It is now conventional to speak of a “concordance cosmology”,è the minimal set of parameters whose measured values characterize the observed universe. These variables are summarized in Table 1, along with their possible physical origin and current best-fit WMAP values [14]. Our ability to construct and quantify this concordance cosmology marks a profound milestone in humankind’s developing understanding of the universe. It is remarkable that all9 current year cosmological data data sets are consistent with a simple six-parameter model: , ,h, describe the homogeneous background3,while { b CDM } A ,n characterize the primordial density fluctuations. { s s}

Label Definition Physical Origin Fig.Value 32.— The nine-year WMAP TT angular power spectrum. The WMAP data are in black, with error bars, the best fit model is the red curve, and the smoothed binned cosmic Baryon Fraction Baryogenesis 0.0456variance0 curve.0015 is the shaded region. The first three acoustic peaks are well-determined. b ± Dark Matter Fraction TeV-Scale Physics (?) 0.228 0.013 CDM ± Cosmological Constant Unknown 0.726 0.015 ± Optical Depth First Stars 0.084 0.016 ± h Hubble Parameter Cosmological Epoch 0.705 0.013 H=100 h kms-1Mpc-1 ± A Scalar Amplitude Inflation (2.445 0.096) 10 9 s ± ⇥ n Scalar Index Inflation 0.960 0.013 s ± Table 1: The parameters of the current concordance cosmology are summarized. We assume a flat universe, i.e. + + 1; if not, we must include a curvature contribution . b CDM ⌅ k Likewise, the conventional cosmology includes the microwave background and the neutrino

sector. Both these quantities contribute to total, but at a (present-day) level well below b, the smallest of the three components listed above. The number and energy density of photons is fixed by the observed black body temperature of the microwave background. The neutrino sector is taken to consist of three massless species, consistent with the number of Standard Model families [21], with a number density fixed by assuming the universe was thermalized at scales above 1 MeV. The parameter h describes the expansion 1 1 rate of the universe today, H0 = 100 h km s Mpc . “Spectrum” refers to the primordial ns 1 1 scalar or density perturbations, parameterized by As(k/k) ,wherek =0.002 Mpc is a specified but otherwise irrelevant pivot scale.

Our understanding of the structure and evolution of the universe rests upon well-tested physical principles, including the general-relativistic description of the expanding universe, the quantum mechanical laws that govern the recombination era, and the Boltzmann equation which allows us to track the populations of each species. However, most of the parameters in the concordance model contain information on areas of physical law about which we have no detailed understanding. The relative fractions of baryons, dark matter and dark energy in the universe are all governed by

3The six-parameter concordance model assumes a spatially flat universe, such that the dark energy density is given by =1 . b CDM

10 ?"1($X1@#$?#$<#1)'#B$[$ $$ The Universe

Has more matter and less dark energy 2 !bh = 0.02205± 0.00028

2 !ch = 0.1199 ± 0.0027

ns = 0.9603± 0.0073

10 ln(10 As ) = 3.089 ± 0.025

100! =1.04131± 0.00063

!1 !1 H0 = 67.3±1.2 km s Mpc

Age = 13.81± 0.05 billion years

Consistent with spatial flatness to % level Inflation with Planck: a survey of some exotic models, Gomes,!"#$#%&'()#%%%%%*+),%-'./"0123'.%'4%56#.(7%%/'% O.B., Rosa, PRD (2018) -'89'6':;<% DC% % Dark Energy

• Evidence: Dimming of type Ia Supernovae z > 0.35 (about a thousand of them now) a!!a Accelerated expansion (negave deceleraon parameter): q ≡ − ≤ −0.47 0 a! 2 [Perlmuer et al. 1998; Riess et al. 1998, …]

• Homogeneous and isotropic expanding geometry

Driven by the vacuum energy density ΩΛ and maer density ΩM

Equaon of state: p = wρ w ≤1 1 • Friedmann and Raychaudhuri equaons imply: q0 = (3w +1)Ωm −ΩΛ 2

q0 < 0 suggests an invisible smooth energy distribuon

• Candidates: Cosmological constant, quintessence (scalar field), more complex equaons of state, etc.

Fih Scalar Field Avatar: Quintessence

Varying vacuum energy models [Bronstein 1933; O.B. 1986; Ratra, Peebles 1988; Wetterich 1988; …]

• V=V0 exp (-λφ) [Ratra, Peebles 1988; Weerich 1988; Ferreira, Joyce 1998]

-α • V=V0 φ , α > 0 [Ratra, Peebles 1988]

-α 2 • V=V0 φ exp ( λφ ) , α > 0 [Brax, Marn 1999, 2000]

• V= V0 [exp ( Mp / φ ) – 1 ] [Zlatev, Wang, Steinhardt 1999]

p • V=V0 ( cosh λ φ - 1 ) [Sahni, Wang 2000]

-α • V=V0 sinh ( λφ ) [Sahni, Starobinsky 2000; Urena-López, Matos 2000]

• V=V0 [ exp ( βφ ) + exp ( γφ ) ] [Barreiro, Copeland, Nunes 2000] • Scalar-Tensor Theories of Gravity [Uzan 1999; Amendola 1999; O.B., Marns 2000; Fujii 2000; ...]

2 • V=V0 exp( -λφ ) [ A + ( φ - B ) ] [Albrecht, Skordis 2000]

2 2 2 2 • V=V0 exp( -λφ ) [ a + ( φ - φ0 ) + b ( ψ - ψ0 ) + c φ ( ψ - ψ0 ) +d ψ ( φ - φ0 ) ] [Bento, O.B., Santos 2002] Sixth Scalar Field Avatar: the Generalized Chaplygin gas model

• Unified model for Dark Energy and Dark Matter

Generalized Born-Infeld Action Generalized Chaplygin gas

: d-brane : Chaplygin gas

Dust : stiff matter De Sitter

[Kamenshchik, Moschella, Pasquier 2001] [Bilíc, Tupper, Viollier 2002] [Bento, O.B., Sen 2002] Dark Energy - Dark Maer Unificaon:

Generalized Chaplygin Gas Model

– CMBR Constraints [Bento, O. B., Sen 2003, 2004; Amendola et al. 2004, Barreiro, O.B., Torres 2008]

– SNe Ia [O. B., Sen, Sen, Silva 2004; Bento, O.B., Santos, Sen 2005]

– Gravitaonal Lensing [Silva, O. B. 2003]

– Structure Formaon *

[Sandvik, Tegmark, Zaldarriaga, Waga 2004; Bento, O. B., Sen 2004; Avelino et al. 2004; Bilic, Tupper, Viollier 2005; …]

– Gamma-ray bursts [O. B., Silva 2006, Barreiro, O.B., Torres 2010]

– Cosmic topology [Bento, O. B., Rebouças, Silva 2006]

– Inflaon [O.B., Duvvuri 2006]

– Coupling with electromagnec coupling [Bento, O.B., Torres 2007]

– Coupling with neutrinos [Bernardini, O.B. 2007, 2008, 2010] A Background tests: 0.35, α ≤ 0.8 ≤ As ≤ 0.9 As ≡ 1+α ρCh0 Structure formation and BAO: α ≤ 0.2

€ €

Model 2

• Real scalar field φ:

1 L = ∂ ϕ ∂µϕ −V(ϕ) 2 µ

1/α+1 2/α+1 !(α + 1)ϕ $ −2α/α+1 !(α +1)ϕ $ V(ϕ) = V A [cosh + cosh ] 0 "# 2 %& "# 2 %&

[O.B., Sen, Sen, Silva, MNRAS 2004]

Hamiltonian formulation [Bernardini, O.B., Annals Physics 2013] Dark Maer

• Evidence:

Flatness of the rotaon curve of galaxies Large scale structure Gravitaonal lensing N-body simulaons and comparison with observaons Merging galaxy cluster 1E 0657-56 Massive Clusters Collision Cl 0024+17 Dark core of the cluster A520

• Cold Dark Maer (CDM) Model

Weakly interacng non-relavisc massive parcle at decoupling

• Candidates:

Neutralinos (SUSY WIMPS), axions, scalar fields, self-interacng scalar parcles (adamastor parcle), etc.

2

parameter and the DE-DM coupling is fully determined. and isotropic, leading to the following field equations: Furthermore, this is an elegant and straightforward way ∂V to link DM and DE with more fundamental physics mod- φ¨ +3Hφ˙ + =0, (4) els from which these components might stem. ∂φ ∂V A relevant property of these interaction models is the χ¨ +3Hχ˙ + =0, (5) Two-scalar-fieldexistence model of for an extra the interaction force between of dark dark energy matter parti- anddarkmatter ∂χ cles, not present in noninteracting models. This force canOrfeu influence Bertolami, structure1, ∗ Pedro formation, Carrilho,2, † creatingand Jorge an P´aramos extra2, bias‡ where H =˙a/a is the expansion rate. From the stress- 1Departamentobetween de F´ısica baryon e Astronomia, and dark Faculdade matter fluctuations, de Ciˆencias da Universidade which may do Portoenergy tensor we obtain the usual expressions3 for the in principleRua do be Campo measured Alegre through 687, 4169-007 tests Porto, of the Portugal equivalence pressure and energy density, 2Instituto de Plasmas e Fus˜ao Nuclear, Instituto Superior T´ecnico da Universidade T´ecnica de Lisboa principle [9, 21]. Furthermore, the varying dark matter Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal 1 2 1 2 As for the interacting term, an obviousmass choice can is also to have(Dated: re- an influence DecemberNext, 21, in 2012) the we anisotropy multiply spec- Eq. (5) byρχd ˙=andφ˙ + insertχ˙ + V ( aφ, χ term) , (6) ′ 2 2 quire the χ field to be equivalent toIn this a paper, fluidtrum we study of of the thenonrela- CMB effects ofsince, an interaction among˙ between other e dffarkects, matter it may and dark alter energy through 1 1 φV −λφ(φ) to obtain 2 2 atwoscalarfieldmodelwithapotentialthe ratio between darkV matter(φ, χ)= anddme P radiation(φ, χ), where densitiesP (φ, χ)isapolynomial. at pd = φ˙ + χ˙ V (φ, χ)3. tivistic matter, i.e. with negligibleWe show pressure. thatlast the scattering. cosmic We expansion recall This dynamics severely of the constrains Universe is thereproduced simplest for ain- large range of 2 2 3 − the bare mass of theAs dark for thematter interacting field and term, that there an obvious exist solu choicetions is with to re-transientNext, accelerated we multiply Eq. (5) byχ ˙ and insert a term that, according to Ref. [30], scalarexpansion. field A oscillationsteraction modification models, in the in exponential in a which behavior the DMd of the mass1potential is a linear is studied, func- with importantWe also consider a Universe∂Vdm filled with perfect fluids for quire the χ field to be equivalent to a fluid2 of nonrela- ˙ ′ 2 ˙ n physical implications,tion of including the DE the field, possibility but its of emoreffects realistic are still transientχ˙ to+ be accelerationVdm studied(φφ, solutions.Vχdm)(matterφ) to+3 obtain andHχ radiation,˙ φ which we=0 consider. (17) to be uncoupled potentialAs forV the(χ interacting)=aχ with term, frequency an obvious (i.e. choice mass)tivistic is to matter, re- much i.e.Next, with negligible we multiply pressure. We Eq. recall (5) byχ ˙ and insert a term −3 −4 in detail for more complex modelsdt % [22].2 and& consequently− to evolve∂φ as ρm a and ρr a , PACS numbers: 95.35.+d,that, 95.36.+x, according 98.80.-k, to Ref. 98.80.Cq [30],′ scalar field oscillations in a d 1 2 2 ˙ ∂Vdm quire the χ field to be equivalent to a fluid of nonrela- ˙n χ˙ + Vdm(φ, χ) +3Hχ˙ φ =0∝. (17) ∝ greater than the expansion rate H, behaveIn thispotential like paper aV fluid( weχ)= presentaφχ Vwithdm a scalar( frequencyφ) to field obtain (i.e. interaction mass) much model dt %respectively.2 Finally,& we− consider∂φ the Friedmann equa- withtivistic an average matter, EOS i.e. givenwith negligible by pressure.with We angreater interacting recall than the expansionpotentialTaking rateV (φH, ,χ behave), the that like average incorporates a fluid yieldstion, that, according to Ref. [30], scalar fieldI. INTRODUCTION oscillationsfeatureswith of an some in average a quintessence EOS givend bydark models1 sector inspired and treat in funda-it straightforwardlyTaking the average as∂V a yields two- component2 fluid. The coupling Q is2 usually˙ taken1dm to 1 1 3 n mental physics theories. Wen then2 χ˙ characterize+ Vdm(φ, theχ) phys-+3Hχ˙ H2φ= ρ =0+ ρ.2+ (17)φ˙2 + χ˙ 2 + V (φ, χ) . (7) potential V (χ)=aχ with frequency (i.e. mass) much dt %be2 of the form Q = δdeHρ&de + δdmHρdm,where− H∂φis2 them1 ˙ ∂Mr (φ) 2 n A large2 number of models has been proposed topχ ex-= − ρχ . (11) ρ˙dm +3H3ρdm# φ 2 χ =02 , (18) $ ical solutions and ascertain⟨ ⟩ n +2 theexpansion⟨ role⟩ As ofrate for the and the interactionδ interactingare coupling term, terms. an1 obvious This∂ treatmentM choice(−φ is2) to re-∂φ2 Next, we multiply Eq. (5) byχ ˙ and insert a term greater than thep expansionχ = plain− rate the darkρHχ, sector behave. of the Universe like a (see fluid Refs.(11) [1, 2] for i ˙ ! " term in the cosmological evolution.is encountered Thequireρ˙ maindm the in manyχ+3field equations observationalH to beρdm equivalent studies toφ a[9, fluid 10]. of nonrela-χ ˙ =0′ , (18) reviews on dark energy (DE) andNotice dark that matter this (DM), is equivalent re- to the virial theorem for φVdm(φ) to obtain 2 with an average EOS⟨ ⟩ givenn +2 by ⟨ ⟩ Taking the average yields where−We we2 have introduce written∂φ theρ densityas ρ and parametersρ˙ asρ ˙Ωi, = ρi/3H , in spectively). Most of thoseof models the model assume are that derived the dark in Sec.1 2An II. alternativentivistic Their matter,numerical path to i.e. study with so- the negligible interaction pressure. assumes Wedm recall dm dm dm power-law potentials, i.e. 2 χ˙ = 2 V (χ) . Thus, in ⟨ ⟩! " ⟨ ⟩ that darkthat,⟨ energy according⟩ can to be Ref. describedsince [30], scalar theterms by density a field of scalar which oscillations is fieldnot the sensible in usual a to deceleration thed oscillations,1 2 parameter to reads2 ˙ ∂Vdm components are noninteractinglutionsorder and are treat to presented ensure them as that fluids. inχ Secis! pressureless III," along at with all times, the relevant we n χ˙ + Vdm(φ, χ) +3Hχ˙ φ =0. (17) n 2 potential V (χ)=aχ witha good frequency2 approximation. (i.e. mass) This much can be easilydt %2 seen by as- & − ∂φ Notice that this is equivalentHowever, to there the are neither virialphysical theoretical theoremmust results. set argumentsn = 2. Section for A forbid- main IV feature concludesφ in of interaction the present with with a model discussion a fluid, is 1 the∂M so-called(φ) interacting2 pχ = − ρχ . (11) wheregreater we have than the written expansion˙suming rateρ thedmH rapid, behaveas oscillations likeρdm aaa¨ fluidand of χ(1t) areρ˙dm describedas byρ ˙ adm, ding1 such2 an interaction,n northat there the exist interaction sufficient with ob- the fieldquintessenceρ˙φdmleads+3 to model: anH oscillationρdm this is theφ case for the modelsχ =0 of , (18) power-law potentials,⟨ i.e.⟩ n +2χ˙ ⟨ =⟩ V of(χ the) . obtained Thus, results in and a brief outlookwith an onaverage future− EOS2 de- givensinusoidal by∂φ⟨ function,⟩ q and hence2 the= densityTaking(1⟨ + Ω depends ther ⟩+3 averagew onlydΩd yields) , (8) servational2 results2 to rule it out.with It varying is then just frequency. natural NotesinceRefs. however the [11, that, 12], density unlike in which pre-3 the is coupling not sensible is chosen! " to≡− be toa˙ the2 oscillations, to ⟨ velopments.⟩ ˙ on the amplitude of the oscillations, which is not affected 2 order to ensure that χ isto pressureless study the general situation at all inheating which times, models dark matter we (see for and exampleQ = Ref.f(φ [31]),)φρ,where which exhibitf(φ) is a genericn 2 function. A simi- 1 ∂M (φ) 2 Notice that this is equivalent! to" the virial theorem for pχ by= a− cyclicρχ average.. (11) ρ˙ +3Hρ φ˙ χ =0, (18) As for the interacting term,dark an energy obvious are choice coupled, is to in re- orderNext, to gain we a multiply deeper insightwhere Eq. (5)alar we goodby mechanismχ ˙ haveand approximation. insert written is the a term so-calledρ⟨ ⟩ chameleonnas+2whereρ This⟨ model⟩ wandd can [13].= pdρ In/˙ beρd is easilyas theρ EOS˙ , seen parameterdm by fordm as- the fields. 1 2 n parametric resonance, the frequency here changes slowly. dm Substitutingdm the averagedm of χ2 givendm by Eq. (16), we − 2 ∂φ mustpower-law set n =quire potentials, 2. the Aχ field main to bei.e. equivalent featureinto theχ˙ to nature aof= fluid the of of theseV nonrela- present(χ components) . Thus,φ˙V ′ model1.(φ This) to in obtain is motivated is these cases the field interacts⟨ with⟩ every component⟨ of ⟩ ! " 2 2 ⟨ ⟩ In thisdm case the computationssincesuming the discussed densityNotice thein Ref. that rapidis [30] this not for is a sensibleoscillationsequivalentobtain to to the the virial of oscillations, theoremχ(t) arefor described to by a tivistic matter, i.e. with negligibleby the theoretically pressure. We appealing recall ideaconstant that the frequency full dark hold. sec- Tothe finish Universe, our discussion leading of to the observable1 eff2ects inn solar sys- where we have written ρdm as ρdm and ρ˙dm asρ ˙dm, order to ensure that χ is! pressureless" at all times,II. we INTERACTION MODELpower-law potentials, i.e. 2 χ˙ = 2 V (χ) . Thus, in ⟨ ⟩ ⟨ ⟩ that the interactionthat, according with to Ref. the [30], scalar field fieldφ oscillationsleads to in a an oscillationd 1 2 a goodtem approximation.2 tests∂Vdm of gravity. A simple This modification can be is easily possible seen bysince2 as- the density is not sensible to the oscillations, to tor can be treated in a single framework.potentialχ we˙ Moreover,+ rewriteV (φ, itχ the) with+3sinusoidal anHχ explicit˙ φ˙ DM=0 varying function,. (17) mass and hence⟨ the⟩ 1B. density1 Interaction∂M (φ depends) Potential only must set potentialn = 2.V ( Aχ)= mainaχn with feature frequency of (i.e. the mass) present much model is dm order to ensure that χ is! pressurelessρ˙ " +3Hρ at all= times,φ˙ we a goodρ approximation.. (19) This can be easily seen by as- fact that the present energy densitiestermdt in% of2 terms dark energy of thesuming DE and& field,using the− a muchrapid∂φ smaller oscillations coupling for4 of baryonsdmχ(t) than aredm for described DM 2 by a dm with varyinggreater frequency. than the expansion Note rate H however, behave like athat, fluid unlike pre- must set n = 2. A main feature of the present2 modelM (φ is) ∂φsuming the rapid oscillations of χ(t) are described by a that the interaction with thematter field areφ observedleads to to be an the oscillation same order of magnitude, onas the in Ref. amplitude [14], or simply substituting of the oscillations,ρ with ρdm,sothat which is not affected with an average EOS given by Taking the averageA.sinusoidal yields Basic equations function,that4 the interaction and hence with the thefield φ densityleads to an oscillation depends onlysinusoidal function, and hence the density depends only heating models (see for examplesuggests a Ref. connection [31]), between which them. exhibitV (φ, χ)=Vde(φ)+DEV couplesdm(φ, χ) only, to DM.(12) ThatSo, is the as previously case6 We for shall the mentioned, model be interested the equivalence in studying relation the be- following poten- withderivative, varying we frequency. see that the Noteinteraction however is irrelevant. that, On unlikepresent, pre- implying the absence ofwith dark varying matter frequency. as such inNote however that, unlike pre- on the amplitude of the oscillations, which is not affected n An2 intimate connection between DE and DM ison natu- thebyin amplitude a2 Ref. cyclic [15],heating in which modelsofaverage. the the (see oscillations, quintessence fortween exampletial: coupled potential Ref. which [31]), quintessence and which is the not exhibit and a theffected field theory approach 2 1 ˙ ∂M (φ) 2 by a cyclic average. parametricheatingthe other models resonance, hand, (see for small for thepχ example values= frequencyrally−7th & 8th of expectedρφχ , Ref. the. terminScalar [31]), here unification with Field (11) whichchangesm models,Avatarswiththe exhibit such Universe slowly.ρ: ˙dm asDark+3 forH instance inEnergyρdm the past. – φ Darkinteraction Maerχ term=0 ,Interacon are derived(18) from scaling assumptions.2 derivative, we see that the interaction is irrelevant.⟨ ⟩ Onn +2present,⟨ ⟩ implying˜ theWe absence consider of two dark interactingby matter− a2 cyclic asSubstituting canonical∂φ such average.parametric in real scalar resonance, the fields average theis established frequency hereof viaχ thechanges relationshipgiven slowly. bySubstituting Eq. (16), the average we of χ2 given by Eq. (16), we can be neglected. Assuming the2the polynomial Chaplygin gasP (φ model)tobeof and its generalizationsConfronted with [5–7].−λφ these problems,A more! In fundamental" this2 we case modify the3 approach computations the model to tackle discussed the interaction in Ref. [30] for a −λφ 1 2 2 the other hand, forIn small thisparametric values case of theφ, resonance, the computations term with them frequencythe discussed Universe here[O.B., in in changes theφ Carrilho, Ref.and past.χ, slowly. [30] whose Páramos, forV Lagrangiande(φ a)=Phys.e Rev. densityA +( D φ86, isφ2012]0) given, by (13) 2 V (φ, χ)=e2 obtainP (φ, χ)+ m χ , (9) order one,Notice the that transition this is equivalent value φc tobetween the virial the theorem two phases for so to allow for a diSubstitutingfference in the behavior the average of both expo- of χ given by Eq. (16), we ˜ Actually,1 2 n in the context of thiswhere model, we an have assumption written ρdmobtaintreatsas ρdm− DEandconstant andρ˙dm DM frequencyas asρ ˙dm fields,, hold. which To abandons finish our the discussion need for1 of∂ ln theM (φ) 2 can be neglected. Assuming the polynomialpower-lawP potentials,(φ)tobeof i.e. 2 χ˙λφ =ConfrontedV (χ) . Thus, with in these problems, we modify⟨ #⟩1 the model⟨$ ⟩ f(φ)= . (20) 2 In thiscan case be estimated the computations by setting mabout2e discussed the= 1, equation2 which of results in state Ref. (EOS) in [30]since ofnentials, DE the for allows density a i.e. to we extract is notchoose sensible instead,fluids2 to in thepotential the2 oscillations, treatment we rewrite toof these it with components. an explicit Usually DM varying this2 mass∂φ 1 1 ∂M (φ) order one, the transitionconstant value φ frequencybetweenorder to the ensure two hold. that phasesχ is To pressureless finishso to⟨ allow at our all⟩ for times, discussion a di wefference in ofthe1 thebehaviorVdmobtain(φ, χ)= of bothM ( expo-φ)χ , ρ˙ +3Hρ = φ˙ ρ . (19) c an! explicit" interaction between thea good dark approximation. componentsµν [8]. This2is can achieved be easilyterm through in seen terms by two as- of new the DE scalar field, fields,whereφ forP ( DEφ, χ and) is a polynomialdm in φ anddm χ and2 m is dm constant2 λφ must frequency set n = 2. hold. A main feature ToTwo-scalar finish of the present ourfield model: discussion model is d of= theg (∂µφ∂ν φ +−λφ∂µχ∂ν χ) V (φ2 , χ) . (2) 2 2 M (φ) ∂φ can be estimated by setting m e = 1, which resultsarXiv:1206.2589v3 [gr-qc] 20 Dec 2012 in2 nentials, i.e. we choose instead, V (φ, χ)=e A +(φ φ ) (28) potential we rewrite it withA an map explicit between the DM generalized varyingLsuming Chaplygin− the2 mass rapid gas oscillations(GCG) 2 ofχ forχ(t) DM are [16–18]. described− 0 An by interaction a Furthermore, potential1the dark2V Eq.1int( matterφ (19), χ∂) can isM bare be( formallyφ mass.) solved Such as exponential a func- couplings potentialthat we the rewrite interactionφ itc with with theln field anm.φ explicitleads to an oscillation DM(23) varyingwhere mass the mass function M (φ) is given in− our modelV (φ1 by, χ)=Vde1(φ)+˙∂VMdm(φ(, χφ)), (12) ≈−λmodel and the interaction modelsinusoidal to be discussed function, in and the hence¯then the! density introducedρ˙ depends1+3 to" accountH onlyρ for=tion the energy of φφ,through exchange. Ul- ρSo, as previously. mentioned,(19) the equivalence relation be- with varying frequency. Note however that, unlike pre- M−λφ2(φ)=m2 +2P˜(φ)2e−λφ −andρ˙λφdm˜ the+3 polynomialdm2 Hρdm2 for2=P˜ dmisφ˙ are inspired2 ρ fromdm . fundamental(19)dm theories like string or M- termterm in2 terms in terms of of the the DE DE field, field,following paragraphV can(φ,Guided beχ)= founde on byin the Ref.A the amplitude+( [9]. cosmologicalφ ofφ0 the) oscillations,+e principle,timatelyP (φ which)(28)χ these we+ is not assumem models aχffected. also the lead ge-2 to2 theM coupling(φ of) the ∂φ tween coupled quintessence and the field theory approach φc ln m.heating models (see for(23) example Ref. [31]), which exhibit written as − with 2 2 M (φ) ∂φ −3 Thus, for φ < φc, the interaction is relevant, becom- by a cyclic average. theory, orρdmN(φ,a=)= 2 supergravityn0a M(isφ) established, in higher via(21) the dimensions relationship [23– ≈−λ parametric resonance, the frequencyA general here way changes to describe slowly.ometry the DM-DE¯ of! the interaction Universe2 1 is2 to to"2 beinteracting given2 by a quintessence flat Robertson- scenario, as long as the inter- −λφSubstituting˜ ¯ the average of χ given by Eq. (16), we −λφ 25]. Notice2 that the interaction of chiral superfields in the ing subdominant as the valueintroduce of the scalaran energy field exchange grows.+ terme withQPin(φλ the)=χ conservationλ+.˜ Thism relaxesχ . action the constraint is2 in the form onVmde of(toφ)= a the DMe less massA +( term.φ φ This0) , is to (13) 2 In thisV case(φ the, χ computations)=Vde( φ discussed )+- darkV energydm in Ref.(φ, [30]χ )- fordarkWalker, a matterobtain metric (12)̸ withP2 line(φ)= elementB + Cφ + Dφ , (14) − 1 ∂ ln M (φ) We areV interested(φ, χ)= in studyingVde(φ the)+ lateV timedm( behaviorφ, χ) , of −60 (12)So, as previously mentioned,× −57 where then equivalence0 is an integration relation constant. be- Notice this corre- Thus, for φ < φc, the interactionconstant is relevant, frequency becom- hold. Toequations finish our asFIG. discussion follows: 4. Evolution of the of log ρde forstrictm =10 conditionwith the initial So,FIG.be 6. Resultsexpected, as previously for m =5 as. will9 10 beshowing derived mentioned, the in evolution the# 1context following2 2 the of sections.$ N=1 equivalence supergravity inflationary relationf(φ modelsbe-)= [26, 27] . (20) V (φ, χ)= M (φ)χ , 2 ∂φ the Universe, near the stage of acceleratedconditions expansion.χi =1(dotted),χi =2.609 (dashed) and χi =10 of the deceleration2 parameter q (dashed)dm and thesponds DE EOS to the statement that ρ = nM, with M being the ing subdominant as the value of thepotential scalar we field rewrite grows. it with an explicitwith λ¯ DM= varyingλ. This mass relaxeswhere theB, constraintC and D aretween on1 orderm1to coupledunitThese∂ theM parameters¯ are( lessφ) then quintessence similar in terms to the of so-called and2 thehas VAMP many field models common theory [19] approach features with the present−3 model. Fur- ρ˙ +3H(ρ(dot-dashed),+ p )= as comparedQ, ρ˙ to background+3Hρ˙ ρ2 density+3=H log(ρQ.ρ2m+=ρr(1)) φ2˙ parameter2wλde/2(solid).λ 2ρ Also2. shown(19) is the effect ofDM the oscil- mass and n the number density, proportional to a . with It is relevant then to estimatede whetherde the interactionde dm dsdmdm= dtdm + a (ttween)m2 dr! ρ +0 coupledr .ddmΩ 2 , quintessence(3)(29)− and the field theory approach We are interested in studying the lateterm in time terms behavior of the DE of field, (solid)̸ and to the noninteractingthe caseappropriate (superimposed powers with the2 ofMlations the(inφ reduced) on which thec∂φ deceleration a Planck particleS parameter mass. is introduced before log a = whose1.23;thermore, at mass varies the with exponentialFurthermore, term for Eq. dark (19) energy can be formally is the solved as a func- strict condition − is establishedwhere via the the mass2 relationship functionWithM (φ this) is solution, given in our the model dynamics by is reduced to a single with is important at late times. The valueχ ofi =1,sinceitisindistinguishable). the DE field − thatthe moment quintessence the oscillations are field averaged. and henceforth the tion of φ,through the Universe, near the stage of accelerated expansion.One may phenomenologically study this interaction by evolution! of the relevant2 quantities" 2 is obtained˜ in terms−λφ of Eq. ˜ V (φ, χ)=−λφVde(φ)+Vdm(φ, χ) , 2 (12) This¯ modification evadesis established the problemM (φ)= associatedm via+2P ( theφ to)edi theff relationshiperentialandsimplest the equation polynomial way for toφ for: this varyP is can its be contributions obtained from Eq. from very high near the present φ(0) can be estimatedwithholding by any assuming assumptions that aboutSo, as thepreviouslyλ/2 natureλ mentioned, of the (32). theThe equivalence advantages relation of this be- type of approach are manifold: It is relevant then to estimate whetherVde the(φ)= interactione A +(φ φ0) with, am(t)! beingρ(13)0 the. scale factor, normalized(29)written as so that at 2 −3 − 2 tweenonset coupledc ofC. oscillations, quintessence Average Evolution as and these the field startEquations theory while approach the interaction(4), withenergyV being to the replaced present by an and effective to respect potential theρdmVeff( bondφ,a)= forn0a nucle-M(φ) , (21) ρde0 withVde(φ(0)) and thatλφ it is close to the− critical densityRad the full set of coupled2 equations1 ∂ ln canM be( foundφ) from an is important at late times. The≈ valueVde( ofφ)= the DEe field #A +(φ φ0)$present, isis establisheda( stillt0) = relevant.(13) 1 via for the convenience, relationship and dΩSf2(φthe)= line el-given byosynthesis2 . [28].2 Hence under(20) these conditions, one con- ρc0, which gives 1 2 2 DM action and2 consequentlyP˜ the(φ)= functionalB + Cφ form+ Dφ of, the EOS (14) near the present φ(0) can be estimated by assumingV (φ, thatχ−)=λφ ThisM ( modificationφ2−)χ , ement evades for theAdi the problemfferent 2-dimensional modification associated sphere could to the haveS .Forthesame been made,2 by dis- ∂φ1 ∂ ln M (φ) where n0 is an integration constant. Notice this corre- Vdmde(φ)=e ∗A Results:+(φ φ0) , (13) Given the high frequency of the2 oscillations, it is rather siders the general interaction term with an exponential # [email protected];12onset− of oscillations, http://web.ist.utl.pt/orfe$ as these startu.bertolami/; while the1 ∂ ln interactionM (φ) f(φ)= Veff(φ,a)=Vde(φ)+. ρdmsponds(φ,a) . to the statement(22)(20) that ρ = nM, with M being the ρde0 Vde(φ(0)) and that it is close to the critical density1# 1 2 $ 2 reason,infeasible wecarding consider to the integrate assumptionf both(φ)= the scalarχ equation that fields the.where to numerically. parameters beB homogeneous, C and(20) ForBD, areC and order unit parameters in terms of −3 Also2 at Instituto2 de Plasmas e Fus˜ao Nuclear Accelerated2 ∂φ phase can be transient! multiplied by a polynomialDM of massφ and and nχ.the number density, proportional to a . ≈ Vdm(φφV(0)dm, χ(φ,)=χ)=† lnMρiscM0(φ still.)χ (,φ relevant.)χ , (24) ˜ the appropriate powers of the reduced2 Planck∂φ mass. ρc0, which gives ≈−λ [email protected] thatD reasonin P (DEφ we)are considerFurthermore,(1) only in terms averages of M Eq. ofp the. However, (19) field. can In a solu- be formally solved as a func- 2 2 O 2 TheseThe equations polynomial are validP as(φ long, χWith) as canM this be(φ) solution, separatedH , the into dynamics the is in- reduced to a single where the mass function M‡ [email protected];2(φ)Adi is givenfferent http://web.ist.utl.pt/jorge.paramos modificationin our modelparticular,Furthermore,tion could to by the we have Eq. shall problems/ (19) been derive can made, mentioned the be formally equationThese by dis- models above solved for the can would as dark also a func- involve require mat- fermions that such as the neutrinos,in ≫ where the mass function M12(φ) is given in our model by tion of φ,through otherwiseteracting the oscillation and noninteracting regime is notdifferential relevant terms, equation andP ( weφ, forχ)=φ: thisPφ( canφ)+ be obtained from Eq. 2 Requiring2 that φc˜ φ(0),− yieldsλφ Self-interacting acarding rather low the dark bound matter assumption has for beenthey discussed that˜ are the in increased Refs. parameters [3Mat,4]. by severalB,theC so-called ordersand MaVaN of magnitude: models, see Ref. this [20] and Refs. therein. M (1φ)=m2 +22P (φ˜)e −λφ and the polynomial˜ tertion for density ofPφ,throughis and work with that instead. First, weC. define Averagemust Evolution also solve Equations Eq. (5). (4), with V being replaced by an effective potential Veff φ(0) ln ρ 0M. (φ)=m +2≥ P (φ)(24)e 2and the polynomial for P is Furthermore, Eq. (19) canPint(φ be, χ). formally For the noninteracting solved as part,a func- we choose the the barec mass, D in P˜(φ)are (1) inthe termsis dark rather matter of unnatural,M densityp. However, and since pressure they a solu- fromare already the EOS at found the Planck given by wherewritten≈− theλ mass aswritten function as M (φ) is given in3 our model by −3 −3 tion to the problemsO We mentioned usein Eq.scale. the (11), ( , + above, +ρ,dm+)( wouldφ metric,a)=tion requiren signature0a M of( thatφ) andφ, ,throughρ naturaldm(φ(21) units,a)= withn0a potentialM(φ) first, studied in(21) Ref. [29], 2 2 −λφ −60 − Given the high frequency of the oscillations, it is rather ˜ m ˜! √ρ 0 10 . 2 (25)! = c =8πG =1.Asaconsequenceallmassescomeintermsof˜ Veff(φ,a)=Vde(φ)+ρdm(φ,a) . (22) M (φ)=m +2P (φP)(eφ)=cB + andCφ + Dφ the, polynomial(14) −57 for P is D. Modified Potential Requiring that φc φ(0), yields a rather low bound for∼ they areFIG. increased5.2 Results for m =5 by.9 ×where several10 showingn0 ordersis the an evolution integration of magnitude: of1 constant.1 this Noticeinfeasible this to corre- integrate the χ equation numerically. For 2 ˜ the reduced Planck mass, Mp2 M 2/√8π2. P (φ)=A +(φ φ ) . (10) ≥ P (φ)=B + Cφ + theD relativeφ , densities Ωde (solid), Ωdm(14)(dashed), Ωρr (dotted)= χ˙ + MPl(φ)χ , (15) φ 0 the bare mass, is rather unnatural, sincesponds they to the are statement alreadydm where that at≡ theρn=0nM Planckis, withthat anM reason integrationbeing we the consider constant. only averages Notice of− the3 field. this In corre-These equations are valid as long as M 2(φ) H2, written as where B, C and D are order unit parametersand Ωm in(dot-dashed). terms of 2 2 −15 ¯ − Thus, this analysis hints that the effects of the interac- III. NUMERICALFIG. 7. Evolution RESULTS of log ρdm for m−3=10 for λ =9.5 ≫ DM mass and n the number density, proportionalparticular, to¯ wea shallρ. dm derive(φ¯Having,a the)= equation definedn0 fora the the potential darkM( mat-φ) and, derivedotherwise the the relevant oscillation(21) regime is not relevant and we tion willthe not appropriate be detected powers at of the reduced presentscale. Planck unless mass. the DM sponds1 2(non-interacting1 to2 the case,2 statement dotted), λ =6.5(dashed), thatλρ=4=.5 nM, with M being the where B−,60C and D are order unit parameters inWith terms this of solution,pdm = theχ dynamics˙ (dot-dashed)M is(φ and reduced)χterλ¯ =2. density.8(double-dot-dashed),ascompared to a single and work withequations, that instead. we are First, now ready we define to drawmust some also general solve con-Eq. (5). m ! √ρc0 10 . (25) ˜ fixing the masss at m =10−15 1 TeV. We obtain some-2 − 2 −3 bare mass is unnaturally small. If P (φ) is (102 ), this differential equation for φ:to this background can be density obtainedthe dark log(ρm from matter+ ρr)(solid). Eq. density and pressure from the EOS found ∼ ˜ whatO similar results to those∼ alreadyLet us found start for masses byDM rewriting mass the and equationsn the in number terms ofclusions the density, about proportional the importance to of eacha of. the terms of the appropriateestimate increasesP ( powersφ)= by roughlyB of+ the 2Cs φ reducedorders+ D ofφ magnitude,− Planck55 , mass.(4), with V (14)being replaced by an effectivein Eq. potential (11), V C. Average Evolution Equationsm>10 for a large rangeBy ofnumber construction,λ¯, as seen of in Figs. e-folds their 7With andN averages=where ln thisa overand, solution, ann for0 oscillation convenience,is an the cycleeff integration dynamics usethe the potential. is constant. reduced As will become to a Notice clear single below, this such results corre- Thus, this analysis hints thatwhich the e wouldffects only of the shift interac- the naturalness problem8. As before,III. to thereP˜( NUMERICALφ are). nogiven relevant by effects at RESULTS the present D. Modified Potential readrescaled variables of Ref.0.16102. [32], This is expected, since in these transient1motivate so-1 a modification of the potential. tion will not be detected at the present unless the DM epoch, with the DM density changing only at early times.differentialspondslutions− we anticipate equation to the the interaction forstatementρ toφ be=: relevant thisχ˙ 2 + just canM that2( beφ)χ2 obtainedρ, = nM(15) from, with Eq.M being the AnotherGiven important the high frequency situation of the is oscillations,the onsetHowever, it of there is the rather is os-a special case for λ¯ =2.8, where a dm First, notice that for a sufficiently large φ we have where B, C and˜ D ares order unit parameters in termsVeff2( ofφ,a)=2Vdeuntil(φ)+2 theρdm present.(φ,a) As. a consequence,(22) it is not2 surpris-2 bare mass is unnaturally small. If P (φ) is (10 ), this solution with transient accelerationρdm is= foundχ˙ again,= M as (φ) χ ˙ , pdm =0. (16) 2 2 −3 cillatoryinfeasible phase: to we integrate must theestablishχ equation theLetφ numerically. usfield start value by For rewriting for the equationsH 2N(4), in withing terms thatφ theV of2N smaller thebeing the mass,χ replaced˙ the2N greater is the1M by required(φ an)1 emffective. At this potential regime, examiningVHavingeff ρdm definedand its the potential and derived the relevant C.2 AverageO 2 Evolution Equations ⟨ ˜⟩ ˜ DM mass⟨ ˜⟩ and n the number2 ≈2 density,2 proportional to a . estimate increases by roughlywhich 2s ordersthatM ( reasonφ) of" magnitude,weH . consider In order only to averages obtainshown of it, the inwe field.Figs. use 9 Intheand 10. This transientH =! solution"e does, Φ value=! for" eλ¯. , X 2= e p2 dm, = (30)χ˙ M (φ)χ . equations, we are now ready to draw some general con- the appropriate powers of thenumber reducednot of present e-folds Planck the problemN = found ln mass.These previouslya and, equations in forH the0 otherconvenience, are so- valid asH0 long use as theM (φ)H0 H , givenHowever, by not all of those (λ¯,m) pairs are2 equally− in-2 which would only shift the naturalnesswell-knownparticular, problem result we [1] shall to that deriveP˜(φ exponential). the equation potentials forlution the of dark this lead kind mat- nor to does it have an unnaturally small ≫ clusions about the importance of each of the terms of rescaled variables of Ref.otherwise [32], the oscillation regimeWithteresting, is since not this relevantin situations solution, and with we mass smaller the than dynamics is reduced to a single mass. The only slight anomaly is that Ω starts in- By construction, their averages over an oscillation cycle the potential. As will become clear below, such results scalingter solutions. density and Albeit work with this that is not instead. strictly First, valid we define in our mustwhere alsoH solve0dm= Eq. 72 (5). km/s/Mpc10−15,suchas is them =10 present−20, the value DM density of the parame- Another important situation is the onset of the os- creasing around log(a)= 8, an effect of the interaction. Given thethe dark high matter frequency density and pressure of the from oscillations, the EOS found it is rather ter starts growingread sooner and can exceed the mentioned motivate a modification of the potential. model, since it is not a pure exponential, weThe same proceed transient by result− isHubble also˙ found constant. for other pairs Thus,diff Eqs.erential (4),Ve (5)ff(φ and equation,a (7))= nowVde read( forφ)+φρ:dm this(φ,a can) . be obtained(22) from Eq. cillatory phase: we must establishin the Eq.φ (11),field value for H 2 φ 2 χ˙ 2limit from nucleosynthesis. This trend is not verified for First, notice that for a sufficiently large φ we have 2 2 infeasible to integrate the χ equation˜of (λ¯,m) numerically. andN their values˜ are plotted ForN in Fig.˜ 11. The N 2 2 2 assuming a scaling behavior before φ fallsH in= the minimume , Φ = e , X = e the lowest, of(30) masses,ρ asdm is attested=4N χ˙ by= theM solution(φ) χ for , pdm =0. (16) 2 2 which M (φ) " H . In order to obtain it, we use the expression found for the “transient line” is D. Modified(4), Potential with−57 V being replaced by an effectiveM potential(φ) m . AtV thiseff regime, examining ρdm and its of theC. potential, Average since the1 Evolutionpolynomial1 P (φ, χ Equations)H varies0 little H0 2 N H0 m =51.9 210 .1 ⟨2 ⟩e ⟨ ⟩ ≈ that reason we consider2 only2 averages2 of the field.H˜ = InΩm0e + Ωr0 + Φ˜× + X˜ + ! V "(φ, χ) , ! " 2 2 well-known result [1] that exponential potentialsρdm lead= χ to˙ + M (φ)χ , λ¯ =(15)0.1625 log(m)+0.3706 . (33)TheseThe explanation equations for this transient are2 behavior valid lies asin the long as M (φ) H , during that stage. In that2 case,2 we have − given6 by6 3H0 where H0 = 72 km/s/Mpc is the presentinteraction: value of the the addition of ρdm to the effective potential ≫ scaling solutions. Albeitparticular, this is not strictly we shall valid derive in1 our the1 equation for the darkHaving mat- defined the potential4N and derived the relevant 2 2 2This expression is rather similar to Eq. (23), changingotherwisee ∂raisesV the minimum the oscillation of Vde, thus allowing for regime the field to is not relevant and we pdm = χ˙ MHubble(φ)χ . constant.¯ Thus,equations, Eqs.˜ ˜ (4),′ we (5)are˜ now and ready (7) now to draw read some general con- model, since it is not ater pure density exponential, and we work3( proceedw +1) with2 by− that2 9(w instead.+1)λ to λ, since First, the slope is we2 ln 10 defineH/φ(0)(Φ +2Φ ln)+ 10/φ0 = escape=0 and, continue to roll down the exponential.(31) How- 2 −clusions about≈− themust importanceH2 ∂φ also of each solve of the Eq. terms (5). of assuming a scaling behaviorGiven before theφ highfallsΩde in frequency the minimum2 Vde of the2 oscillations,H , (26) it is rather 0 the dark matterBy construction,≈ densityλ their⇒ and averages pressure∼ overλ an oscillation from the cycle EOSthe potential. found As will become44N clear below, suchV resultseff(φ,a)=Vde(φ)+ρdm(φ,a) . (22) of the potential, since the polynomial P (φ, χ) varies little 2 N 1 2 1 2 ee N ∂V read H˜ = Ω 0e + Ω 0 motivate+ Φ˜˜ + a˜ ′ modificationX˜˜ + of theV ( potential.φ, χ) , infeasible to integrate the χ equationm numerically.r H(X + ForX)+ 2 2 =0, during that stage. In thatin case, Eq.in which we (11), havew is the effective EOS parameter for the com- 6 6 3HH0∂χ 2 2 2 First, notice that for0 a sufficiently large φ we have that reasonbination we ofρdm all consider the= χ˙ components.= M ( onlyφ) χ We averages, canpdm then=0 rewrite. of(16)4 theN the2 field.2 In 2 2 ⟨ ⟩ ⟨ ⟩ e ∂VM (φ) m . At this regime,These examining equationsρD.dm and Modified its are Potential valid as long as M (φ) H , oscillation condition! " as 1 ! "1 ˜ ˜ ′ ˜ where≈ the primes denote derivatives with respect to N. 3(w +1) 9(w +1) 2 2 H2(Φ + Φ2)+ 2 =0, (31) ≫ Ωde particular,Vde we shallρHdm derive,= (26)χ˙ the+ equationM (φ)χ for, H the0 ∂φ darkThese(15) changes mat- improve the numerical robustness of the 2 2 2 λφ otherwise the oscillation regime is not relevant and we ≈ λ ⇒ ∼ λ 9(w +1)m2 e +22P˜ 4N system by shortening the range of values taken by the ter density and work with that" instead.1 . ′ First,(27)e ∂V we define Having defined the potential and derived the relevant 2 1 1 ˜ ˜ ˜ new variables. From themust onset also of the oscillatory solve Eq. phase (5). we in which w is the effective EOS parameter forλ the com-2Pφ H2(X + X2 )+ 2 =0, pdm = χ˙ M (φ)χ . H0 ∂χshall use the averagedequations, equations we instead, are now which ready become to draw some general con- bination of all thethe components. dark matter We can then density rewrite and the pressure from the EOS found We recall that if the scaling2 − occurs2 during nucleosyn- oscillation condition as where the primes denote derivatives withclusions respect about to N. 4 theN importance of each of the terms of in Eq. (11),thesis, then λ " 10 [28]. With such a value for λ and ˜ 2 N 1 ˜ 2 e By construction, their averagesThese over changes an oscillation improve the cycle numericalH = Ωm0thee robustness+ potential.Ωr0 + ofΦ the+ As2 V willeff(φ) , become(32) clear below, such results 2 assuming that P˜ Pφ, the l.h.s. is always less than 6 3H0 9(w +1)m eλφ +2P˜ ∼ system by shortening the range of values taken by the D. Modified Potential read unity when" 1 the. interaction(27) is relevant, meaning that dur- motivatee4N ∂V a modification of the potential. 2 1 2 1new variables.2 2 From the onsetH˜ of(Φ˜ the′ + Φ˜ oscillatory)+ phaseeff =0 we. λ Pingφ that stageρ the= field χχ˙has+ not begunM oscillating;(φ)χ , con- (15) First,2 notice that for a sufficiently large φ we have dm2 2 shall2 use the averaged equations instead, whichH0 ∂φ become versely,ρdm = one expectsχ˙ = oscillationsM2 (φ) toχ2 start as, thep interactiondm =0. (16) M 2(φ) m2. At this regime, examining ρ and its We recall that if the scaling⟨becomes occurs⟩ unimportant. during nucleosyn- In particular, for the⟨ threshold⟩ We now integrate the equationsHaving from definedN = 70 to theN = potential anddm derived the relevant ! " 1 ! 1 " 1 e4N ≈ − thesis, then λ " 10 [28]. Withmass such of Eq. a value (25), for theλ fieldand2 may not oscillate2H˜ 2 = Ω until2 eN the+ Ω 5,+ rangingΦ˜ 2 + fromV the(φ Planck) , epoch(32) to some time in the pdm = χ˙ M (φ)χm0. r0 2 eff equations, we are now ready to draw some general con- assuming that P˜ Pφ, the l.h.s. is always less than 6 3H0 ∼ 2 − 2 4N unity when the interaction is relevant, meaning that dur- e ∂Veff clusions about the importance of each of the terms of H˜ (Φ˜ ′ + Φ˜)+ =0. ing that stage the field χ has not begun oscillating; con- H2 ∂φ versely, one expectsBy oscillations construction, to start as the their interaction averages over an oscillation0 cycle the potential. As will become clear below, such results becomes unimportant.read In particular, for the threshold We now integrate the equations from N =motivate70 to N = a modification of the potential. mass of Eq. (25), the field may not oscillate until the 5, ranging from the Planck epoch to some− time in the 2 2 2 First, notice that for a sufficiently large φ we have ρ = χ˙ = M (φ) χ , p =0. (16) 2 2 ⟨ dm⟩ ⟨ dm⟩ M (φ) m . At this regime, examining ρdm and its ! " ! " ≈ The CHASE laboratory search for chameleon dark energy Jason H. STEFFEN

1. Introduction

A variety of observational evidence indicates that the expansion of the universe is accelerating, for which a promising class of explanations is scalar field “dark energy” with negative pressure [1]. Such a field is expected to couple to Standard Model particles with gravitational strength and would mediate a new “fifth” force, but such forces are excluded by experiments on a wide range of scales. Three known waysThe toCHASE hide laboratory dark energy-mediated search for chameleon fifth dark energy forces include: weak or pseudoscalarJason H. STEFFEN cou- plings between dark energy and matter [2]; effectively weak couplings locally [3]; and an effec- tively large field1. mass Introduction locally, as in chameleon theories [4, 5, 6].

The CHASE laboratoryChameleons search for chameleon are scalar dark (or energy pseudoscalar) fields withJason a nonlinear H. STEFFEN potential and a coupling to the The CHASE laboratory search for chameleonA variety dark of observational energy evidence indicatesJason that the H. expansionSTEFFEN of the universe is accelerating, local energy density. They evade fifth force constraints by increasing their effective mass in high- for which a promising class of explanations is scalar field “dark energy” with negative pressure [1]. density environments, while remaining light in the intergalactic medium. Gravity experiments in 1. Introduction1. Introduction Such a field is expected to couple to Standard Model particles with gravitational strength and would the lab [7] and inmediate space [a4 new, 5] “fifth” can exclude force, but chameleons such forces are with excluded gravitational by experiments strength on a wide matter range couplings, of scales. A varietyA variety of observational of observational evidenceThree evidence known indicates indicates ways that to that the hide theexpansion dark expansion energy-mediated of the of the universe universe is fifth accelerating, is accelerating, forces include: weak or pseudoscalar cou- for which abut promising strongly class coupled of explanations chameleons is scalar field evade “dark these energy” constraints with negative [8, pressure9]. Casimir [1]. force experiments rule out for which a promising class ofplings explanations between is dark scalar energy field “dark and matter energy” [2 with]; effectively negative pressure weak couplings [1]. locally [3]; and an effec- Such a fieldstrongly is expected coupled to couple chameleons to Standard Model [10 particles], but are with ineffective gravitational for strength a large and would class of potentials commonly used Such a field is expected to coupletively to large Standard field Model mass locally, particles as with in chameleon gravitational theories strength [4 and, 5, would6]. mediatemediate a newto a model “fifth”new “fifth” force, dark force, but energy. such but such forces Collider forces are excludedare data excluded excludeby experiments by experiments extremely on ona wide a wide strongly range range of ofscales. coupled scales. chameleons [11]. Three known ways to hide dark energy-mediatedChameleons are fifth scalar forces (or include: pseudoscalar) weak or fields pseudoscalar with a nonlinear cou- potential and a coupling to the Three knownPhoton-coupled ways to hide dark energy-mediated chameleons may fifth forces be detected include: weak through or pseudoscalar laser experiments cou- [12] or excitations in plingsplings between between dark dark energy energy andlocal and matter matter energy [2]; [ effectively2 density.]; effectively They weak weak evade couplings couplings fifth force locally locally constraints [3]; [3 and]; and anby an effec- increasing effec- their effective mass in high- tivelytively largeradio large field field mass frequency mass locally, locally, asdensity cavities in as chameleon in chameleon environments, [13 theories]. theoriesIn laser while[4, 5 [4,,6 experiments, remaining5]., 6]. light in photons the intergalactic travelling medium. through Gravity a vacuum experiments cham- in ChameleonsChameleonsber immersed are scalar are scalar (or in pseudoscalar) (orthe a pseudoscalar) labmagnetic [7] and fields in field fields space with with oscillate [ a4 nonlinear, 5 a] nonlinear can exclude into potential potential chameleons. chameleons and and a coupling a coupling with They togravitational tothe the are then strength trapped matter through couplings, the locallocal energy energy density. density. They They evade evadebut fifth strongly fifth force force constraints coupled constraints chameleons by increasing by increasing evade their these their effective effective constraints mass mass [in8, inhigh-9]. high- Casimir force experiments rule out chameleon mechanism by the dense walls and windows of the chamber since their higher effective densitydensity environments, environments, while while remainingstrongly remaining light coupled light in the in chameleons the intergalactic intergalactic [10 medium.], but medium. are Gravity ineffective Gravity experiments experiments for a large in class in of potentials commonly used the labthe [7 lab] andmass [7] in and space within in space [4, those5 []4 can, 5to] excludecan materials model exclude chameleonsdark chameleons creates energy. with Collider an with gravitationalimpenetrably gravitational data exclude strength strength large extremely matter potential matter couplings, strongly couplings, barrier coupled [12 chameleons, 14, 15]. [11 After]. a pop- but stronglybut stronglyulation coupled coupled of chameleons chameleons chameleons evadePhoton-coupled evade these is produced, these constraints constraints chameleons the [8, 9 [8]. laser, 9 Casimir]. may Casimir is turned be force detected force experiments off experiments through and a rule laserphotodetector rule out out experiments exposed [12] or excitations in order in to stronglystrongly coupled coupled chameleons chameleons [10], [10 but], are but ineffective are ineffective for afor large a large class class of potentials of potentials commonly commonly used used radio frequency cavities [13]. In laser experiments, photons travelling through a vacuum cham- to modelto model darkobserve dark energy. energy. the Collider photon Collider data data afterglowexclude exclude extremely extremely as trapped strongly strongly chameleons coupled coupled chameleons chameleons oscillate [11 [].11 back]. to photons. The original Gam- ber immersed in a magnetic field oscillate into chameleons. They are then trapped through the Photon-coupledPhoton-coupledmeV experiment chameleons chameleons included may may be detected be a detected search through through for laser this laser experiments afterglow experiments [ and12 []12 or set] orexcitations limits excitations on in in photon/chameleon couplings chameleon mechanism by the dense walls and windows of the chamber since their higher effective radioradio frequency frequencybelow cavities collider cavities [13]. [ constraints13 In]. laser In laser experiments, experiments, for a limited photons photons set travelling oftravelling dark through energythrough a vacuum a models vacuum cham- cham- [12]. The GammeV Chameleon ber immersedber immersed in a in magnetic a magnetic fieldmass field oscillate within oscillate into those into chameleons. materials chameleons. creates They They an are impenetrably are then then trapped trapped large through through potential the the barrier [12, 14, 15]. After a pop- chameleonchameleonAfterglow mechanism mechanism by Search the by dense theulation (CHASE) dense walls of walls and chameleons and windows is a windows new is of experimentproduced, the of the chamber chamber the since laser to since search their is their turned higher for higher off photon effective and effective a photodetector coupled chameleons exposed in order [16]. to Its massmass within withinresults those those materials bridge materials creates the createsobserve gap an impenetrably between an the impenetrably photon GammeV large afterglow large potential potential as [12 trapped barrier] andbarrier [ chameleons12 collider [,1214,,1415,].15 constraints, After oscillate]. After a pop- a back pop- improves to photons. sensitivity The original toGam- both ulationulation of chameleonsmatter of chameleons and is photonproduced, is produced,meV couplings the experiment laser the laser is turned to is included turnedchameleons, off offand a search and a photodetector a photodetector andfor this probes afterglow exposed exposed a broad and in insetorder classorder limits to to of on chameleon photon/chameleon models. couplings observeobserve the photon the photon afterglow afterglow asbelow trapped as trapped collider chameleons chameleons constraints oscillate oscillate for back a limited back to photons. to set photons. of dark The The originalenergy original models Gam- Gam- [12]. The GammeV Chameleon meVmeV experiment experiment included included a search aAfterglow search for this for this afterglow Search afterglow (CHASE) and and set limitsset is limits a new on photon/chameleonon experiment photon/chameleon to search couplings couplings for photon coupled chameleons [16]. Its below collider constraints for a limited set of dark energy models [12]. The GammeV Chameleon below collider2. constraints Chameleon for a limitedresults Models bridgeset of dark the gap energy between models GammeV [12]. The [12 GammeV] and collider Chameleon constraints, improves sensitivity to both AfterglowAfterglow Search Search (CHASE) (CHASE) is a new is a new experiment experiment to search to search for forphoton photon coupled coupled chameleons chameleons [16 [16]. Its]. Its matter and photon couplings to chameleons, and probes a broad class of chameleon models. resultsresults bridge bridge the gap the between gapNineth Scalar Field Avatar: between GammeV GammeV [12] [12 and] and collider collider constraints, constraints, improves improves sensitivity sensitivity to toboth both mattermatter and photon and photonWe couplings consider couplings to chameleons, toactions chameleons, of and the and probes form probes a broad a broad class class of chameleon of chameleon models. models. 2.Chameleon Field Chameleon Models 4 1 2 1 µ 1 ⇥/M µ⇧ 2 ⇥/M i • Coupling to maer depends on a scalar field 2. ChameleonS = d Modelsx⌅ g M R ⇥ ⇥ V(⇥) e ⇤ Pl F F +L (e m Pl g ,⌥ ) (2.1) 2. Chameleon Models Pl µ µ⇧ m µ⇧ m ⇤ 2 We consider2 actions of the form 4 – WeAcon considerWe consider actions actions of the of form the form ⇥ 4 1 2 1 µ 18 1 ⇤ ⇥/MPl µ⇧ 2m⇥/MPl i i where the reducedS = Planckd x⌅ massg MMPlR= 2.43µ ⇥ ⇥10 V(⇥GeV,) eLm theF LagrangianFµ⇧ +Lm(e for mattergµ⇧ ,⌥m fields) (2.1)⌥m, 4 1 1 2 1 1 µ 1 1 ⇤ ⇥/MPl µ⇧ 2m⇥/MPl i S =4 d x⌅ g 2 M R µ⇥ ⇥ V(⇥) 2⇤ ⇥e/MPl µ⇧2F F⇤+Lm2(em⇥/MPl4 g ,i⌥ ) (2.1) S = d x⌅ g MPlR Pl µ ⇥µ ⇥⇤ V(⇥) e F Fµ⇧ +µ⇧Lm(e gµ⇧ ,µ⌥⇧ m)m (2.1) ⇤and ⇤2 and2 2m are2 dimensionless 4 4 chameleon couplings to photons and matter respectively⇥ (often ⇤ 18 ⇥ i where the reduced Planck mass MPl = 2.43 10 GeV, ⇥Lm the Lagrangian for matter fields ⌥ , g = /M 18g 18= [Khoury,/M Weltman 2004] i i m wherewhere the reducedexpressed the reduced Planck Planck as mass⇤ massM⇤M=Pl2.=43Pl2.and4310 10mGeV,GeV,LmLthem thePl Lagrangian). Lagrangian We assume⇤ for for matter matter universal fields fields⌥ ⌥ matter,m, couplings. The CHASE– laboratory -1/2 search for chameleon18 darkPl energy ⇤ m Jason H. STEFFEN m andMPl=Gand/8π=2.43 x 10 are dimensionless GeVand, ⇤ chameleoni =O(1) and ⇤m are couplings dimensionless to photons chameleon and matter couplings respectively to photons (often and matter respectively (often and and⇤ m areThem dimensionless dynamics chameleon of this field couplings are governedto photons and by matter an effective respectively potential (often that depends on a potential ⇤ expressed as g = /M and g = /M ). We assume universal matter couplings. expressed expressed as g as=g⇤ =/M⇤ /andMPl gand=gm=/Mm/).MPl We⇤ ). We assume⇤ assumePl universal universalm matterm matter couplings.Pl couplings. V(⇤⇥), the⇤ Pl backgroundm m matterPl density ⌃m, and the electromagnetic field Lagrangian density ⌃⇤ = A well-studied– classThe of dynamics chameleon of models this field has are aThe potential governed dynamics of by the of an form this effective [6 field] potential are governed that depends by an effective on a potential potential that depends on a potential The Lm dynamicsis the µ⇧Lagrangian of this field density of the maer fields 2 are governed2 by an effective potential that dependsµ⇧ ˜ on a potential⌅ ⌅ V(⇥),F the backgroundFµ⇧ /4 =( matterB V( density⇥E), the)/2⌃ backgroundm (for, and pseudoscalars the electromagnetic matter density⌃⇤ field⌃=m,F andLagrangian theFµ⇧ electromagnetic/4 density= B ⌃E⇤ ):= field Lagrangian density ⌃⇤ = V(⇥ ), the background matter densityN ⌃m, and the electromagneticN field Lagrangian density ⌃⇤ = µ⇧ 2 2 ⇥µ⇧ 2 2 µ⇧ µ⇧ · µ⇧ F F /4 =(2 B 2 E )4/2⇧ (forM pseudoscalars4 =⇥µ⇧F F˜ /4 = ⌅B ⌅E): ˜ ⌅ ⌅ F –F Potenal: /4µ=(⇧ B V(E⇥)=)/2M (fore pseudoscalarsFL Fµ⇧M/4 =(1⌃+B⇧=⌃⇤FE F)˜/2/ (for.µ4⇧= pseudoscalars⌅B ⌅E): ⌃⇤(2.3)= F Fµ⇧ /4 = B E): µ⇧ L ⇥ L ⇤ µ⇧ · ⇤ ⇧ ML ⌃ · m⇥ ⇤ ⇥ · ⇤ ⌅ ⇥ m⇥ ⇤ MPl MPl ⇥ V m(⇥⇥M,⌅x)=V⇤(⇥ ⇥M )+e ⌃ (⌅xm)+⇥ e ⌃ (⇤⌅x). (2.2) V (⇥,⌅x)=V(⇥eff)+e Pl ⌃ (⌅x)+e Pl ⌃ (⌅x). m M ⇤M(2.2) V 1(/4eff,⌅x)=V( )+3 e MPl (⌅xm)+e MPl( ,⌅(⌅x)=)⇤. ( )+ Pl (⌅)+(2.2)Pl (⌅). where N is– a realEffecve potenal: number and M = ⌃eff ⇥ 2.4 10⇥ eV is⌃ them massVeff scale⇥⌃⇤ ofx theV dark⇥ energye ⌃m x e ⌃⇤ x (2.2) L de ⇤ ⇥ density ⌃ and ⇧ is a dimensionless constant. The constant term in this potential causes cos- de– Can evade fih force constraints through dependence on the energy density mic acceleration that is indistinguishable from a cosmological2 2 constant for cosmological surveys. – Photon-coupled chameleons could be detected through laser experiments and radio 2 2 However, the local dynamics from the power-law term can be probed in the laboratory. frequency cavies … [ Rybka et al. 2010] Following the derivations in [17, 18] the conversion probability between photons and chameleons – Consistent with MICROSCOPE results? is 2 2 2⌥⇤ B 2 meff⇧ P⇤ ⇥ = sin kˆ (xˆ kˆ). (2.4) ⌅ M m2 4⌥ ⇥ ⇥ ⇤ Pl eff ⌅ ⇤ ⌅ Here, ⌥ is the particle energy, meff = Veff,⇥⇥ is the effective chameleon mass in the environment, ⇧ is the distance travelled through the magnetic field, and kˆ is the particle direction. ⌥ When a photon/chameleon wavefunction strikes an opaque surface of the vacuum cham- ber, there is a model-dependent phase shift ref between the two components and a reduction in photon amplitude due to absorption. On the other hand, a glass window performs a quan- tum measurement—chameleons reflect while photons are transmitted. The velocities of trapped chameleons quickly become isotropic through surface imperfections. The decay rate of a chameleon to a photon Gdec,⇤ , is found by averaging over initial directions and positions. The observable af- terglow rate per chameleon Gaft is found by averaging over those trajectories that allow a photon to reach the detector. Once the geometry of an experiment is defined, these rates can be computed numerically [18]. A single parameter ⌅ can be used to describe the chameleon effect. If the chameleon mass in the chamber is dominated by the matter coupling, then m µ ⌃⌅ where ⌅ =(N 2)/(2N 2) [18]. eff m The largest value of ⌅ with integer N is ⌅ = 3/4 for N = 1; ⇥ 4 theory (N = 4), has ⌅ = 1/3. We do not consider 0 < N < 2 since their potentials are either unbounded from below or do not exhibit the chameleon effect.

3. Apparatus

The design of the CHASE apparatus is shown in Fig. 1. In addition to the windows at the ends of the vacuum chamber, we centered two glass windows in the cold bore of a Tevatron dipole magnet which divide the magnetic field into three partitions of lengths 1.0 m, 0.3 m, and 4.7 m. The shorter partition lengths provide sensitivity to larger-mass chameleons. For a fixed magnetic field there are limits to the smallest and largest detectable ⇤ —small ⇤ produce small afterglow signals while with large ⇤ the chameleon population will decay before the detector can be exposed. We improve our sensitivity to large ⇤ by operating at a variety of lower magnetic fields, which lengthen the decay time of the chameleon population and provide overlapping regions of sensitivity. A mechanical shutter (chopper) modulates any afterglow signal allowing a measurement of the PMT dark rate and improving sensitivity to low afterglow rates (small ⇤ ).

3 Tenth Scalar Field Avatar: Scalar field dark maer -- Higgs: the Adamastor parcle (I)

Motivation: “cuspy core” problem [Spergel, Steinhardt 2000]

Model:

[Bento, O.B., Rosenfeld, Teodoro 2000]

Higgs decay width

[Bento, O.B., Rosenfeld 2001]

Unified model for dark energy – dark matter: g 'Φ2 H 2

[O.B., Rosenfeld 2008] [Bento, O.B., Rosenfeld 2001]

Tenth Scalar Field Avatar: Scalar field dark maer and the Higgs Field (II)

[O.B., Cosme, Rosa, Scalar field dark matter and the Higgs field, PL B759 (2016) 1- 8] [O.B., Cosme, Rosa, Scalar field dark matter and the Higgs field, PL B759 (2016) 1- 8] Tenth Scalar Field Avatar: Scalar field dark maer and the Higgs Field (III)

[Cosme, Rosa, O.B., Scalar filed dark matter with spontaneous symmetry-breaking and the 3.5 KeV line, PL B781 (2018) 649 ] Tenth Scalar Field Avatar: Scalar field dark maer and the Higgs Field (III)

[Cosme, Rosa, O.B., Scalar filed dark matter with spontaneous symmetry-breaking and the 3.5 KeV line, PL B781 (2018) 649 ]

• If mass is not acquired by the Higgs mechanism, the scalar field remains a good candidate for DM, but the dependence of parameters is quite involved and no striking observaonal feature is found

[Cosme, Rosa, O. B., Scale-invariant scalar filed dark matter through the Higgs Portal, JHEP 1805, 129 (2018)] What if ... General Relativity is not the theory? [OB, 2011] MOND

IR

Loop Non-minimal Quantum matter- Gravity curvature General Relativity coupling Non-minimal matter-curvature coupling

UV

String Theory 3.1. ACTION, FIELD EQUATIONS AND PHENOMENOLOGY 11

3.1. ACTION, FIELD2 EQUATIONS AND PHENOMENOLOGY 11 Chapterwhere 3ds3 contains the spacial part of the metric and a(t) is the scale factor. Using this metric the higher order derivative term is given by 2 where ds3 contains the spacial part of the metric and a(t) is the scale factor. ⇥ h(R, )=( g )h(R, ) Using this metric the higher orderµ⇥ derivativeLm term⌅µ⌅ is⇥ given µ⇥ by Lm =(⇧ ⇧ g ⇧ ⇧ )h (0 + g 3H)⇧ h (3.14) f(R) Theories with non-minimalµ ⇥ µ⇥ 0 0 µ⇥ µ⇥ 0 ⇥ h(R, )=( g )h(R, ) µ⇥ Lm ⌅µ⌅⇥ µ⇥ Lm where h(R, m) is a generic function of R and m, H0 =˙a/a is the Hubble expansion parameter and L =(⇧µ⇧⇥ gµ⇥ ⇧0⇧0)hL (µ⇥ + gµ⇥ 3H)⇧0h (3.14) 0 = aa˙ (with µ, ⇤ = 0) is the a⌅ne connection. Thus, the e⇤ective energy density is given by curvature-matterµ⇥ µ⇥ ⇤ coupling where h(R, m) is a generic function of R1 andf mf, H+2=˙a/af is the Hubblef +2 expansionf parameter and L ⌅ˆ = 1 L 1 Lm 2 R 3H 1 Lm 2 R˙ (3.15) 0 2 f f f µ⇥ = aa˙ µ⇥ (with µ, ⇤ = 0) is the a⌅ne connection. 2 Thus,2 the⇥ e⇤ective energy2 density is given by 10 CHAPTER 3. F (R) THEORIES⇤ WITH NON-MINIMAL CURVATURE-MATTER COUPLING while the e⇤ective pressure is given by 1 f f +2 f f +2 f 3.1 Action, Field⌅ˆ Equations= 1 1 andLm 2 R Phenomenology3H 1 Lm 2 R˙ (3.15) 2 f2 f2 f2 1 f1 f1 +2 mf2 ⇥ f1 +2 mf2 f1 +2 mf2 2 Since the only di⇥erencepˆ between= actions (2.2)L andR +( (3.1)R¨ +2 isH inR˙ ) the matterL term,+ itL is obviousR˙ , that (3.16) 2 f2 f2 f2 f2 Recently,while there the e⇤ hasective been pressure a revival is given of interest by in a class⇥ of modified gravity theories where, besides the additional modification in the field equations will came from the variation of this term. Thus, the usual modificationwhere the in dot the refers gravity to derivative sector, discussed with respect in the to time. previous chapter, it is introduced a 1 f f +2 f f +2 f f +2 f considering thepˆ variation= 1 of this1 termLm and2 R using+(R¨ the+2H resultsR˙ ) 1 fromLm Section2 + 1 2.1 oneLm obtains,2 R˙ 2 , (3.16) coupling between curvature2Thef field equations andf matter. (3.10) This along revival with the of definitions interestf is of⌅ due ˆ and top ˆ will thef be fact useful that for this deriving non- the various 2 2 ⇥ 2 2 1 (⇥ gfenergy2(R) conditionsm) in Section 3.3. f2(R) 1 (⇥ g m) minimal curvature-matter L couplinggµ ⇥ givesgd4 risex = to a violation of the+ f conservation(R) equation L of thegµ energy-⇥ gd4x, where the dotµ refers to derivative with respect to time.µ m 2 µ ⇥ g g The non-minimal curvature-matterg couplingL brings new⇥ intriguingg g features to the modified theories ⌥momentum⇧ tensor which may⌃ introduce an extra⌥ ⇧ force in the theory [10] (see also Ref. [11]⌃ for a recent The field equations (3.10) along with the definitions of⌅ ˆ andp ˆ will be useful for deriving the various of gravity.Alternave theory of gravity with non-minimal coupling One expects energy to be exchanged between geometry and matter fields in a(3.4) non-trivial review on the subject). The phenomenology of these models is considered in more detail below. energy conditions in Section 3.3. way. In fact, takingbetween maer and curvature account into the covariant derivative of the1 field equationsµ (3.6), the4 Bianchi = f (R)R ( f (R)) f (R)T g ⇥ gd x. The action of interest has the followingLm form,2 µ µ Lm 2 2 2 µ The non-minimalidentitiesDark Gravity! curvature-matter and the relationship[O.B.,⌥ Böhmer⇧ coupling1 , Harko brings, Lobo new PRD intriguing 2007 ] features to the modified⌃ theories of gravity. One expects energy toFormalismo be1 exchanged between Métrico geometry 4 and matter fields in a non-trivial(3.5) Action: S = f1(R)+f2(R) m ⇥ gd x, (3.1) µ µ 2 ⇥ f (R)=(L )f (R)=R f (R) , (3.17) Hence,way. the In fact, field taking equations account obtained into⇤ the⌅ from covariantµ⇥ actioni (3.1) derivative⌅ are⇥⇥⌅ given of⇥ thei by field equationsµ⇥ ⌅ i (3.6), the Bianchi1 Field equations: whereidentitiesfi (with andi =1 the, relationship2) are arbitrary1 functions of the Ricci scalar. The second function, f2(R), is one obtains [10] 1 usually considered to have(f1 +2 them followingf2 )Rµ form,f1gµ µ (f1 +2 mf2 )=f2Tµ . (3.6) L 2 µ f2 L µ µ Tµ⇥ = [gµ⇥ m Tµ⇥µ] R. (3.18) ⇥µ⇥ fi(R)=( ⌅⇥ ⇥ ⇥)ff2i(R)=L Rµ⇥ f⌅i(R) , (3.17) Making explicit the Einstein⌅ tensor,µ⇤ =G ,µ one⌅⇤ ⌅ rewritesgµ⇤ g the ⇥ field equations⌅ and gets (1) f2(µ⇥R)=1+⇥ ⇥⇤2(R⇥) , ⇥ (3.2) Thus, one verifies that the energy-momentum tensor is not covariantly conserved. Furthermore, one obtainsEffective [10] energy-momentum tensorf2 non-conservation: inserting the energy-momentumGµ = tensor of aTˆµ perfect + Tµ fluid , (Eq. (3.12)) into Eq. (3.18) and(3.7) contracting where ⇥ is a constant and ⇤2 is another function of R. These kind of non-minimal couplings were first µ f1 +2ff22 m µ the resultant equation withTµ the⇥ = projection[Lgµ⇥ m operator,Tµ⇥ ] h⇥R.= g u u , one obtains [10],(3.18) proposed in Ref. [12] motivated by the⌅ issue of thef2 acceleratedL expansion⌅ µ⇥ µ of⇥ theµ universe.⇥ However, in where the e⇥ective energy-momentum tensor Tˆµ has been defined as that paper,Thus, it onewas verifies only considered that the the energy-momentum case⇥ where f2(R1 tensor)=fR2 is. not One covariantly considers here conserved. a⇥ broader Furthermore, class of Eq. motion test particle: u ⇥ u = ( m + p) ⇥ R + ⇥ p h (3.19) 1 f1 f1 +2⌅mf2 ⇥ + p f21L ⌅ ⌅ models.inserting the energy-momentum (PerfectTˆ = fluid) tensor ofL a perfectR g fluid+ (Eq. (3.12))(f +2 into Eq.f ⇥) (3.18). and contracting(3.8) µ 2 f f µ f µ 1 Lm 2 ⇤ 2 2 ⌅f . 2 Asthe stated resultant in the equation previous with chapter, the projection only the operator, metric⇥ formalismhµ⇥ = gµ is⇥ considered.uµu⇥ , one obtains Thus, as [10], performed in From Eq. (3.7) one can define an e⇥ective coupling Section 2.1, varying action (3.1) with respect to the metric yields Thus, the motion⇥ of a point-like 1 testf2 particle is non-geodesic due⇥ to the appearance of the extra force u ⇥ u = (f2m + p) ⇥ R + ⇥ p h (3.19) ⌅ ⇥kˆ+=p f2 L ⌅, ⌅ (3.9) f . This force1 is( orthogonal⇥ gf1(R)) to the four-velocity1 (⇥ gf2 of(R the) m particle) ⇥ µ⇥ due to the4 fact that, by definition, S = +f1 +2 mf2 L g ⇥ gd x. (3.3) 2⇥ g gµ⇥ f . ⇥ g L gµ⇥ ⇤ ⇥ and therefore the field equations can be⇥ written in a more⇥ familiar form, h u =0. (3.20) 9 Thus, the motion1 of a point-like test particle is non-geodesic due to the appearancea ofa thea extrab force Which arises directly from the definition of the Riemann tensor, c dX d cX = R X . Gµ = kˆ Tˆµ + Tµ . ⇥ ⇥ ⇥ ⇥ bcd (3.10) f . This force is orthogonal to the four-velocity of the particle due to the fact that, by definition, ⇥ Thus, in order to keep gravity attractive, kˆ has to be positive from which follows the additional h⇥u =0. (3.20) condition 1Which arises directly from the definition of the Riemannf2 tensor, Xa Xa = Ra Xb. > 0 . c d d c bcd (3.11) f +2 f ⇥ ⇥ ⇥ ⇥ 1 Lm 2 As expected, setting f1(R)=R and f2(R) = 1 one recovers Einstein’s theory. The e⇥ective energy-momentum tensor defined by Eq. (3.8) can be written in the form of a perfect fluid, T =(⇥ + p)u u pg , (3.12) µ µ µ if one defines an e⇥ective energy density and an e⇥ective pressure. However, given the presence of the higher order derivatives in Eq. (3.8), in order to proceed in this way it is necessary to specify the metric of the space-time manifold of interest. Since one is interested in cosmological applications, the Robertson-Walker (RW) metric is a natural choice. Hence, in what follows, one considers the homogeneous and isotropic flat RW metric with the signature (+, , , ), ds2 = dt2 a2(t)ds2 , (3.13) 3 f(R) theory of gravity with non-minimal curvature-maer coupling (IV) [O.B., Böhmer, Harko, Lobo, Phys. Rev. D 75 (2007) 104016 ] n • Stellar stability [O.B., Páramos, Phys. Rev. D 77 (2008)] ! R $ f2 (R) =1+# & • On the non-trivial gravitational coupling to matter " R0 % [O.B., Páramos, Class. Quant. Grav. 25 (2008)]

• Non-minimal coupling of perfect fluids to curvature [O.B., Lobo, Páramos, Phys. Rev. D 78 (2008)]

• Non-minimal curvature-matter couplings in modified gravity (Review) [O.B., Páramos, Harko, Lobo, arXiv:0811.2876 [gr-qc]]

• A New source for a braneworld cosmological constant from a modified gravity model in the bulk [O.B., Carvalho, Laia, Nucl. Phys. B 807 (2009)] • Energy Conditions and Stability in f(R) theories of gravity with non-minimal coupling to matter [O.B., Sequeira, Phys. Rev. B 79 (2009)]

• Mimicking dark matter through a non-minimal gravitational coupling with matter [O.B., Páramos, JCAP 1003, 009 (2010)] • Accelerated expansion from a non-minimal gravitational coupling to matter [O.B., Frazão, Páramos, Phys. Rev. D 81 (2010)]

• Reheating via a generalized non-minimal coupling of curvature to matter [O.B., Frazão, Páramos, Phys. Rev. D 83 (2011)]

• Mimicking the cosmological constant: Constant curvature spherical solutions in non-minimally coupled model [O.B., Páramos, Phys. Rev. D 84 (2011)] • Mimicking dark matter in clusters through a non-minimal gravitational coupling with matter: the case of the Abell cluster A586 [O.B., Frazão, Páramos, Phys. Rev. D 86 (2012)]

• On the dynamics of perfect fluids in non-minimally coupled gravity [O.B., Martins, Phys. Rev. D 85 (2012)]

• Traversable Wormholes and Time Machines in non-minimally coupled curvature-matter f(R) theories [O.B., Ferreira, Phys. Rev. D 85 (2012)]

• Solar System constraints to nonminimally coupled gravity [O.B., March, Páramos, Phys. Rev. D 88 (2013)]

• Cosmological perturbations in theories with non-minimal coupling between curvature and matter [O.B., Frazão, Páramos, JCAP 1305 (2013)]

• Minimal extension of General Relativity: alternative gravity model with non-minimal coupling between matter and curvature (Review) [O.B., Páramos, Int. J. Geom. Meth. Mod. Phys. 11 (2014)]

• The Layzer-Irvine equation in theories with non-minimal coupling between and matter and curvature [O.B., Gomes, JCAP 1409 (2014)]

• Modified Friedmann Equation from Nonminimally Coupled Theories of Gravity [O.B., Páramos, Phys. Rev. D 89 (2014)]

• Black hole solutions of gravity theories with non-minimal coupling between matter and curvature [O.B., Cadoni, Porru, Class. Quant. Grav. 32 (2015)]

• Viability of nominimally coupled f(R) gravity [O.B., Páramos, Gen. Rel. Grav. 48 (2016)]

• 1/c expansion on nonminimally coupled curvature-matter gravity models and constraints from planetary preceaaion [March, Páramos, O.B., Dell’Agnello, Phys. Rev. D 95 (2017)]

• Inflation in non-minimal matter-curvature coupling theories [Gomes, Rosa, O.B., JCAP 1706 (2017)] • Gravitational waves in theories with a non.minimal curvature-matter coupling [O.B., Gomes, Lobo, Eur. Phys. C 78 (2018)] • Constraining a nonminimally coupled curvature-matter model with ocean experiments [March, O.B., Muccino, Baptista, Dell’Agnello, arXiv: 1904.12789] Chapter 3 f(R) Theories with non-minimal curvature-matter coupling

3.1 Action, Field Equations and Phenomenology

Recently, there has been a revival of interest in a class of modified gravity theories where, besides the usual modification in the gravity sector, discussed in the previous chapter, it is introduced a coupling between curvature and matter. This revival of interest is due to the fact that this non- minimal curvature-matter coupling gives rise to a violation of the conservation equation of the energy- momentum tensor which may introduce an extra force in the theory [10] (see also Ref. [11] for a recent Gravitaonal waves in theories with a non-minimal review on the subject). The phenomenologycurvature-maer coupling of these models is considered in more detail below. The action of interest has the[O.B., Gomes, Lobo, Euro following form, Phys. J. C78 (2018)] Formalismo1 Métrico Action: S = f (R)+f (R) ⇥ gd4x, (3.1) 2 1 2 Lm ⇤ ⇥ where fi (with i =1Linearised, 2) are arbitrary field eqs functions. around a of Minkowsky the Ricci scalar.background: The second function, f2(R), is usually considered to have the following form,

f2(R)=1+⇥⇤2(R) , (3.2)

R = 0 for Minskowski where ⇥ is a constant and ⇤2 is another function of R. These kind0 of non-minimal couplings were first proposed in Ref.Cosmological [12] motivated constant by the as issue a source of the: accelerated expansion of the universe. However, in that paper, it was only considered the case where f2(R)=R .Scalar One mode considers absorbed here in a broader class of this gauge choice models. Dispersion relation: As stated in the previous chapter, only the metric formalism is considered. Thus, as performed in Section 2.1, varying action (3.1) with respect to the metric yields

1 (⇥ gf (R)) 1 (⇥ gf (R) ) S = 1 + 2 Lm gµ⇥ ⇥ gd4x. (3.3) 2⇥ g gµ⇥ ⇥ g gµ⇥ ⇤ ⇥ 9 Gravitaonal waves in theories with a non-minimal curvature- maer coupling [O.B., Gomes, Lobo, Euro Phys. J. C78 (2018)]

Formalismo Métrico Propagating scalar logitudinal mode:

Newman-Penrose formalism (full theory):

NP quantities built from the decomposition of the Weyl tensor in terms of irreducible parts: Riemann tensor, Ricci tensor and scalar curvature.

In GR, only Ψ4 is non-vanishing → polarisations + and x

In NMC with a C.C. other scalar, vector and tensor modes are also possible (Φ ,Φ ,Φ ,R are nonzero), but a 00 11 00 complete characterisation can only be carried out once the full solution is known (required for the Ψi). Dark Energy fluid as a source:

Scalar modes:

A further scalar mode is found due to fluctuactions of the matter Lagrangian (at linear level!)

Extra modes may be detected in future measurements with more precise data is available

It might allow to distinguish between GR and other models!

Some “Vector” Thoughts … Weyl Gravity [H. Weyl (1918)] [P- Dirac (1973)] ​

Weyl Gravity was proposed as an attempt to unify GR with Electromagnetism:

Where the covariant derivative is built from the generalised connection:

The Ricci tensor reads:

And its trace: Nonminimally coupled Weyl Gravity [C. Gomes, O.B., 1812.04976 [gr-qc]] ​ We start from the action:

Variation with respect to the vector field: where and

The metric field equations:

The order of the PDEs is lowered, thus avoiding some well known instabilities!

Does a cosmological constant arise from the model? [Einstein (1919)9] From the contracted Bianchi identities: [Kaloper, Padilla (2014)] [O.B., Páramos (2017)] [Gomes, O.B.,1812.04976] ​

Conserved quanty: trace of the field equaons

Hence, no where integraon constant From the generalised contracted Bianchi identities: arises from the model

where Space Form Behaviour A pseudo-Riemannian manifold admit a space form behaviour iff:

Combining the field equations with their trace for the vacuum, :

In general, we could have K=K(t), which cannot be identified with the constant matter vacuum, .

In order to proceed one needs to find a form for the vector field. The Weyl vector field A natural ansatz for the vector field: [Bento, OB., Moniz, Mourão, Sá (1993)]

which admits invariance under spatial rotations, SO(3) transformations, with generators , and is consistent with homogeneity and isotropy

We further assume a constant scalar curvature:

Two main cases can be studied:

Variaon: Dynamic Vector Field [R. Baptista, O.B., to appear]

_ _

• SO(3) gauge field in a Robertson-Walker spacetime • … Another Variaon [O.B., Bessa, Páramos, PRD (2016)]

• Can inflation by driven by vector fields?

• Yes … , but needs a dominating Λ [Ford, PRD (1989)]

• No! SO(3) Gauge Field [Bento, O.B., Moniz, Mourão, Sá, CQG (1993)]

• Yes! But ... only if nonmimimally coupled to gravity!

Inflationary fixed points

ê Conclusions

- The discovery of the Higgs field provides yet another proof that gauge symmetries are realized in Nature and that the vacua do not fully share these symmetries - Moreover, it confirms, up to the tested energy scales, the existence of at least one fundamental scalar field

- Should one generalize the gauge principle and consider the so-called Grand Unified Theories (GUTs)? Should one consider them in the context supergravity (gauge generalizaon of supersymmetry), string theory, etc?

- Does gravity have a scalar component?

- Is the inflaton a scalar field in the context of more fundamental theories? It cannot be the minimally coupled Higgs field of GUTs. It can be a chiral superfield of some supergravity models, even though not the moduli fields of string theory ... Some scalar field models with plateau-like potenal models are degenerate with the Starobinsky model

- Is there an underlying scalar field associated to dark maer? What about dark energy? What about a putave interacon between them?

- ...