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Non-Equilibrium Cosmologies and Astroparticle Phenomenology

• N. E. Mavromatos

King’s College London, Dept. of Physics

HEP2006: Recent Developments in High-Energy Physics and Cosmology Ioannina, Greece, April 13-16 2006

In collaboration with: Diamandis, Ellis, Georgalas, Lahanas, Mitsou, Nanopoulos, Sarben Sarkar Work supported in part by: EPEAEK B - PYTHAGORAS & EU-RTN HPRN-CT-2002-00292

Non-Equilibrium Cosmologies and Astroparticle Phenomenology

OUTLINE

• WHY NON-EQUILIBRIUM COSMOLOGIES? Theoretical models and ideas
• n the Early Universe: from Brane worlds to space-time foam pictures
• WHAT CONSEQUENCES ? Models of Early Universe implying quantum

decoherence (gravity environment), microscopic time irreversibility (fundamental CPT Violation), Dissipation, Off-shellness of Einstein’s eqs.

• WHAT KIND OF PHENOMENOLOGY ?

Astrophysics: ( → V. Mitsou talk) (i) Supernovae Data (ii) Cosmic Microwave Background Anisotropies (CMB) (WMAP etc.) (iii) Baryonic Acoustic Oscillations . . . . . . Particle & Atomic Physics Tests of CPT Violation: Neutral mesons, antimatter factories, atomic physics, Low energy atomic physics experiments, Ultra cold neutrons, Neutrino Physics, Terrestrial & Extraterrestrial tests of Lorentz Invariance (modified dispersion relations of matter probes: GRB, AGN photons, Crab Nebula synchrotron-radiation...)

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CPT THEOREM

C(harge) -P(arity=reflection) -T(ime reversal) INVARIANCE is a property of any quantum field theory in Flat space times which respects: (i) Locality, (ii) Unitarity and (iii) Lorentz Symmetry. ΘL(x)Θ† = L(−x) , Θ = CPT , L = L† (Lagrangian) Theorem due to: Jost, Pauli (and John Bell). Jost proof uses covariance trnsf. properties of Wightman’s functions (i.e. quantum-field-theoretic (off-shell) correlators of fields < 0|φ(x1) . . . φ(xn)|0 > ) under Lorentz group. (O. Greenberg, hep-ph/0309309) Theories with HIGHLY CURVED SPACE TIMES , with space time boundaries

• f black-hole or cosmic horizon type, may violate (ii) & (iii) and hence CPT.

E.g.: SPACE-TIME FOAMY SITUATIONS IN QUANTUM GRAVITY. E.g.: SPACE-TIMES WITH COSMOLOGICAL CONSTANT (de Sitter)

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SPACE-TIME FOAM

Space-time MAY BE DISCRETE at scales 10−35 m (Planck) → LORENTZ VIOLATION (LV)? (and hence CPTV); also there may be ENVIRONMENT

• f

GRAVITATIONAL d.o.f. INAC- CESSIBLE to low-energy experiments (non-propagating d.o.f., no scattering, topology-changing) → CPT VIOLA- TION (and may be LV) EARLY UNIVERSE: might be charac- terised by intense foamy environments

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FOAM AND UNITARITY VIOLATION

Horizon

• f Black Hole

‘‘out’’ MIXED STATES ‘‘in’’ PURE STATES

= density matrix = Tr ψ >< ψ| |

unobs

| ... > modified temporal evolution of ρ: d dt ρ = i [ ρ , H ] + ∆Η(ρ) ρ quantum mechanics violating term quantum mecha− nical terms SPACE−TIME FOAMY SITUATIONS NON UNITARY (CPT VIOLATING) EVOLUTION OF PURE STATES TO MIXED ONES ρ

• ut

SPACE-TIME FOAM: Quantum Gravity SINGULAR Fluctuations (microscopic (Planck size) black holes etc) MAY im- ply: pure states → mixed

$ = SS† , S = eiHt =scattering ma- trix, $=non invertible, unitarity lost in effective theory.

BUT...HOLOGRAPHY can change the picture (Strings in anti-de-Sitter space times (Maldacena, Witten), Hawking 2003- superposition of space-time topologies (Quantum Gravity) (but in Euclidean space time) may solve info-problem?:

• bserver not quite sure (in QG) if the BH is there) BUT NO PROOF AS YET

... OPEN ISSUE

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SPACE-TIME FOAM and Intrinsic CPT Violation A THEOREM BY R. WALD (1979): If $ = S S†, then CPT is violated, at least in its strong form. PROOF: Suppose CPT is conserved, then there exists unitary, invertible opearator Θ : Θρin = ρout ρout = $ ρin → Θρin =$ Θ−1ρout → ρin = Θ−1$ Θ−1ρout. But ρout =$ρin, hence : ρin = Θ−1$Θ−1 $ ρin BUT THIS IMPLIES THAT $ HAS AN INVERSE- Θ−1$Θ−1, IMPOSSIBLE (information loss), hence CPT MUST BE VIOLATED (at least in its strong form). NB: IT ALSO IMPLIES: Θ =$ Θ−1 $ (fundamental relation for a full CPT invariance). NB: My preferred way of CPTV by Quantum Gravity Introduces fundamental arrow of time/microscopic time irreversibility

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CPT SYMMETRY WITHOUT CPT SYMMETRY? But....nature may be tricky: WEAK FORM OF CPT INVARIANCE might exist, such that the fundamental “arrow of time” does not show up in any experimental measurements (scattering experiments) → Probabilities for transition ψ =initial pure state → φ =final pure state ( i.e. “in” and “out” Decoherence-free (expt.) subspaces ) P(ψ → φ) = P(θ−1φ → θψ) θ: Hin → Hout, Θρ = θρθ†, θ† = −θ−1 (anti − unitary), $† = Θ−1$Θ−1. Here, Θ is well defined on pure states, but $ has no inverse, hence $ † = $−1 (full CPT invariance: $= SS†, $† = $−1). Supporting evidence for Weak CPT from Black-hole thermodynamics: Although white holes do not exist (strong CPT violation), nevertheless the CPT reverse of the most probable way of forming a black hole is the most probable way a black hole will evaporate: the states resulting from black hole evaporation are precisely the CPT reverse of the initial states which collapse to form a black hole.

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Evidence for Dark Energy

Recent Astrophysical Evidence for Dark Energy (acceleration of the Universe (SnIA), CMB anisotropies (WMAP...))

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COSMOLOGICAL CPTV?

(NM: hep-ph/0309221, Sarben Sarkar, NM: PRD 72, 065016 (2005))

WMAP improved results on CMB: Ωtotal = 1.02 ± 0.02, high precision measurement of secondary (two more) acoustic peaks (c.f. new determination of Ωb). Agreement with SnIa Data. Best Fit : ΩΛ = 0.74, ΩMatter = 0.26 Best fit models of Universe consistent with cosmological constant Λ > 0 (de Sitter) (BUT see Dissipative, Non-equilibrium Cosmologies, → LAHANAS, MITSOU talks ! ) Λ-universe eternally accelerating, will enter an inflationary phase again: a(t) ∼ e √

Λ/3t,

t → ∞, → cosmological Horizon. Horizon implies incompatibility with S-matrix & decoherence: no proper definition of asymptotic state vectors, environment of d.o.f. crossing the horizon (c.f. dual picture of black hole, now observer is inside the horizon). Theorem by Wald on $-matrix and CPTV: CPT is violated due to Λ > 0 induced decoherence: (e.g. from Liouville strings → below ) ∂tρ = i[ρ, H] + Λ M3

P

[gµν, [gµν, ρ]] Tiny cosmological CPTV(damping) effects, but detected via Universe acceleration!

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Liouville-String (Dissipative, Non-Equilibrium) Cosmology: Formalism

STRING THEORY SPACE FIXED (CONFORMAL) POINT #1 FIXED (CONFORMAL) POINT # 2

CENTRAL CHARGE C 1 CENTRAL CHARGE C 2

C1 C2 > DIRECTION OF FLOW RG FLOW

Space of background space-time fields, over which strings propagate: {gi} = {graviton = Gµν, matter, . . . }, and Dilaton φ(X0, ρ) ρ = Liouville mode, Relaxation Flow (RG) between string vacua (equilibrium points). ρ is ESSENTIAL in restoring conformal invariance, perturbed by a NON-EQUILIBRIUM PROCESS, e.g. Catastrophic Cosmic Events (Brane World Collision), or space-time foam, ...

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Off-Equlibrium Dynamics

Stringy σ-model GENERALIED CONFORMAL INVARIANCE: ¨ gi + Q ˙ gi = −βi, ˙ gi = dgi

dρ0 , ρ0 world-sheet zero mode of the Liouville field. NB: “Liouville friction term”

∝ Q=central charge deficit. βi=σ-model Renormalization Group (RG) β-functions, e.g. for gravitons, to O(α′) (α′=Regge slope): βG

µν = Rµν (Ricci tensor)

CONFORMAL INVARIANCE CONDITIONS βi = 0 ⇐ ⇒ EQUATIONS OF MOTION FROM ON-SHELL TARGET-SPACE ACTION −βi = −δS/δgi = 0 GENERALIZED (LIOUVILLE) CONFORMAL INVARIANCE CONDITIONS ⇐ ⇒ EQUATIONS OF MOTION FROM OFF-SHELL TARGET-SPACE ACTION βi = δS/δgi = ¨ gi + Q ˙ gi = 0 Energetics (some supercritical models) = ⇒ Time-like Liouville ρ ⇐ ⇒ (Cosmic)Time X0(t) (Ellis, NM, Nanopoulos,Lahanas talk)

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A Specific Model

Colliding Brane Worlds with Recoil (Ellis, NM, Nanopoulos, Westmuckett )

D−brane D−brane V EFFECTIVE LIOUVILLE STRINGY COSMOLOGY ON THE BRANE φ, G µν , Q(t) Relative recoil velocity V of branes Adiabatic collisions allow for stringy computation of brane potnetial to leading order in V Identical branes (same tension) potential is V_potential = α V 4 Stringy Excitations on brane world feel central charge deficit Q2 initial = O(V 4) RELAXATION PROCESS AFTER COLLISION Q(t)=v 2 /t

Logarithmic (super)Conformal Field Theory Techniques to compute scaling with cosmic time, PLUS identification of Liouville mode (IRREVERSIBLE local RG scale on world-sheet) with time ( Gravanis, Szabo, N.M,) COLLISION ⇐ ⇒ NON-EQUILIBRIUM, IRREVERSIBLE

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Dissipative Liouville Cosmologies

Robertson-Walker Space times. Effective 4-d action with matter Im and radiation in Einstein frame

S(4) = 1 2α′ Z d4x p −G[e−Ψ(φ)R(G) + Z(φ)(∇φ)2 + 2α′V (φ) . . . ] − 1 16π Z d4x √ G 1 α(φ) F 2 µν − Im(φ, G, matter

including string loops, eΨ(φ) = c0e−2φ + c1 + c2e2φ + . . . , Z(φ) = 4 + . . . , ... V (φ) = 2Q2e2φ + ˜ V , ˜ V = α3e3φ + β4e4φ ∗ . . . Off-Shell Liouville Equations: (ρφ = ( ˙ φ)2 + V (φ)/2 , pφ = ( ˙ φ)2 − V (φ)/2 )

3H2 = ρm + ρφ + e2φ 2 Jφ , 2 dH dtE = −ρm − ρφ − pm − pφ + a−2(tE)Jii , i = 1, 2, 3, d2φ dt2 E + 3H dφ dtE + 1 4 ∂V ∂φ + 1 2 (ρm − 3pm) = − 3 Jii 2 a2 − e2φJφ 2 , Jφ = e−2φ( ¨ φ − ˙ φ2 + Qeφ ˙ φ), Jii = 2a2 “ ¨ φ + 3H ˙ φ + ˙ φ2 + (1 − q)H2 + Qeφ( ˙ φ + H) ” .

Matter (non) Conservation equations:

˙ ρm + 3H(ρm + pm) + ˙ Q(∂V (φ))/2∂Q − ˙ φ(ρm − 3pm) = 6(H + ˙ φ)a−2Jii HEP2006, Ioannina (Greece) 12

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Predictions on Cosmological Parameters

A detailed analysis leads to the following scaling with the redshift z AT LATE EPOCHS (z < 2): Central charge deficit (off-equilibrium, non-criticality ) Q2(z) ≃ Q2

∗ + ρ0 dust(1 + z) ,

Q2

∗ > 0

Matter (including exotic scaling dark matter) ρmatter ∼ ρ0

dust(1 + z)3 + ρ0 exotic(1 + z)4 + . . . ,

a = a0/(1 + z) Dilaton Dark Energy ρφ ∼ H2 + Q2

∗

a2 + ρ0

dust + α/2

a3 + β 2 a4 + · · · = O(a−2) + O(a−3) + β 2 a−4 + . . . . α = string loop correction parameter Combination appearing in Hubble H(z) ρφ + ρM ≃ |O(a−2)| + (4ρ0

dust + α)

2 a−3 − |e2φJφ| 2 + ... ! a−4 , Hubble parameter H(z) = H0 “ Ω0

3(1 + z)3 + Ω0 δ(1 + z)δ + Ω0 2(1 + z)2”1/2 ,

δ ≃ 4 Fits with data can constrain Liouville Q-Cosmology Dilaton Potential !

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EVOLUTION OF A LIOUVILLE UNIVERSE

Dark Energy NUCLEOSYNTHESIS TODAY EARLY ERAS (inflation,reheating) ( 1 )

t 2

Cosmic Time

In Liouville string (Non-Equilibrium,

• ff-shell)

Dark Energy Models, Dilaton Dark Energy may be negligible at NUCLEOSYNTHESIS epoch. Conformal Field Theory (Logarith- mic CFT, in brane recoil models ) → asymptotic scaling with cosmic time ∼ 1/t2 (E.Gravanis, N.M., La- hanas talk ) . NB: Cosmic Time ⇐ ⇒ world-sheet Renormalizartion Group (RG) local Scale (Liouville mode), Irreversible (Zamolodchikov C-theorem) !

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CONCLUSIONS & OUTLOOK Discussed models of non equilibrium Cosmologies, in (non-critical, Liouville) string theory framework. Off-shell Einstein Equations, expressing coupling of system with

• ff-equilibrium dilaton sources, and gravitational environments. Microscopic

Time Irreversibility. Common scaling between matter and dilaton dark energy, due to off-shell

• terms. In particular, Dust-like contributions appear in Dilaton dark energy,

but their sign is ambiguous and depends on details of underlying string theory (loop corrections etc.) c.f. Negative Dust due to Kaluza-Klein graviton modes in brane models (Minamitsuji, Sasaki, Langlois). No Cosmological Constant, but relaxing to zero Dark Energy in Liouville Cosmology ... Can define asymptotic states, no cosmic horizons, = ⇒ (perturbative) strings OK. Can have excellent Fit with Astro Data = ⇒ V. Mitsou talk.

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