Effect of the Trace Anomaly on the Cosmological Constant. Jurjen F. Koksma

Transcription

1 Effect of the Trace Anomaly on the Cosmological Constant Jurjen F. Koksma Invisible Universe Spinoza Institute Institute for Theoretical Physics Utrecht University 2nd of July 2009 J.F. Koksma T. Prokopec arxiv: [gr-qc] J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

2 Outline Introduction to the Trace Anomaly Dynamically screening the Cosmological Constant: connecting the Cosmological Constant and the Trace Anomaly Dynamics Driven by the Trace Anomaly Quasi de Sitter spacetime FLRW spacetimes Conclusion J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

3 Trace Anomaly If a classical action is invariant under a conformal transformation of the metric, then: T µ µ = 0. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

4 Trace Anomaly If a classical action is invariant under a conformal transformation of the metric, then: T µ µ = 0. However, after quantisation and renormalisation one finds: ˆT µ µ 0. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

5 Trace Anomaly If a classical action is invariant under a conformal transformation of the metric, then: T µ µ = 0. However, after quantisation and renormalisation one finds: ˆT µ µ 0. Einstein Field Equations: R µν 1 2 Rg µν + Λg µν = 8πGT µν. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

6 Trace Anomaly Examples: Massless, conformally coupled (ξ = 1/6) scalar field S = d 4 x g { 12 g µν µ φ ν φ ξ2 } Rφ2 Massless spin-1/2 field Photon J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

7 Trace Anomaly The trace anomaly or the conformal anomaly in four dimensions is given by: T Q ˆT µ µ = bf + b (E 2 ) 3 R + b R, where E R µνκλ R µνκλ = R µνκλ R µνκλ 4R µν R µν + R 2 F C µνκλ C µνκλ = R µνκλ R µνκλ 2R µν R µν R2, J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

8 Trace Anomaly The trace anomaly or the conformal anomaly in four dimensions is given by: T Q ˆT µ µ = bf + b (E 2 ) 3 R + b R, where E R µνκλ R µνκλ = R µνκλ R µνκλ 4R µν R µν + R 2 F C µνκλ C µνκλ = R µνκλ R µνκλ 2R µν R µν R2, where b is not fixed uniquely by the trace anomaly and where: 1 b = 120(4π) 2 (N S + 6N F + 12N V ) ( b 1 = 360(4π) 2 N S + 11 ) 2 N F + 62N V. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

9 Trace Anomaly In (homogenous and isotropic) FLRW spacetimes: g αβ = diag ( 1, a 2 (t), a 2 (t), a 2 (t) ), H = ȧ a. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

10 Trace Anomaly In (homogenous and isotropic) FLRW spacetimes: Hence: g αβ = diag ( 1, a 2 (t), a 2 (t), a 2 (t) ), H = ȧ a. T Q = 4b {... H + 7ḦH + 4Ḣ ḢH 2 + 6H 4} 6b {... H + 7ḦH + 4Ḣ ḢH 2}, J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

11 Trace Anomaly In (homogenous and isotropic) FLRW spacetimes: Hence: g αβ = diag ( 1, a 2 (t), a 2 (t), a 2 (t) ), H = ȧ a. T Q = 4b {... H + 7ḦH + 4Ḣ ḢH 2 + 6H 4} 6b {... H + 7ḦH + 4Ḣ ḢH 2}, or, motivated by Ostrogradsky s theorem: T Q = 24b { 3ḢH2 + H 4} 72b ḢH 2, which corresponds to truncating at first order in time derivatives, also known as quasi de Sitter spacetime. Note this is exact when b = 2b /3. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

12 Connecting the Trace Anomaly and Cosmological Constant Various people (Mottola, Antoniadis, Mazur, Tomboulis,...) stated that the trace anomaly could have effects on dark energy and the cosmological constant (see e.g.: gr-qc/ ). J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

13 Connecting the Trace Anomaly and Cosmological Constant Various people (Mottola, Antoniadis, Mazur, Tomboulis,...) stated that the trace anomaly could have effects on dark energy and the cosmological constant (see e.g.: gr-qc/ ). They argue in favour of a new, IR degree of freedom of gravity: g µν (x) = e 2σ(x) g µν (x). J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

14 Connecting the Trace Anomaly and Cosmological Constant Various people (Mottola, Antoniadis, Mazur, Tomboulis,...) stated that the trace anomaly could have effects on dark energy and the cosmological constant (see e.g.: gr-qc/ ). They argue in favour of a new, IR degree of freedom of gravity: g µν (x) = e 2σ(x) g µν (x). The trace anomaly cannot be generated from a local finite term in the action, but rather stems from a non-local effective action that generates the conformal anomaly by variation with respect to the metric. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

15 Connecting the Trace Anomaly and Cosmological Constant Various people (Mottola, Antoniadis, Mazur, Tomboulis,...) stated that the trace anomaly could have effects on dark energy and the cosmological constant (see e.g.: gr-qc/ ). They argue in favour of a new, IR degree of freedom of gravity: g µν (x) = e 2σ(x) g µν (x). The trace anomaly cannot be generated from a local finite term in the action, but rather stems from a non-local effective action that generates the conformal anomaly by variation with respect to the metric. This genuine non-locality of the action generating the trace anomaly reveals a large distance effect of quantum physics. It is then argued that the new conformal field should dynamically screen the cosmological constant, thus solving the cosmological constant problem. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

16 Connecting the Trace Anomaly and Cosmological Constant We argue for a semiclassical approach to examining the connection between the cosmological constant and the trace anomaly: J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

17 Connecting the Trace Anomaly and Cosmological Constant We argue for a semiclassical approach to examining the connection between the cosmological constant and the trace anomaly: Dynamical backreaction should be studied in more general spacetimes than de Sitter spacetime as de Sitter spacetime is not dynamical, i.e.: In de Sitter spacetime: H = const In FLRW spacetimes: H = H(t) J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

18 Connecting the Trace Anomaly and Cosmological Constant We argue for a semiclassical approach to examining the connection between the cosmological constant and the trace anomaly: Dynamical backreaction should be studied in more general spacetimes than de Sitter spacetime as de Sitter spacetime is not dynamical, i.e.: In de Sitter spacetime: H = const In FLRW spacetimes: H = H(t) In the semiclassical spirit, we take expectation values of inhomogeneous quantum fluctuations with respect to a certain state to study its effect on the background spacetime. Therefore, quantum fluctuations affect the background homogeneously. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

19 Dynamics Driven by the Trace Anomaly We consider a universe consisting of (classical) matter with arbitrary equation of state ω = P M /ρ M, a cosmological constant and the trace anomaly J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

20 Dynamics Driven by the Trace Anomaly We consider a universe consisting of (classical) matter with arbitrary equation of state ω = P M /ρ M, a cosmological constant and the trace anomaly We use covariant stress-energy conservation We assume a perfect fluid form for the quantum density and pressure yielding the trace anomaly: T µ ν,q = ( ρ Q, p Q, p Q, p Q ). J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

21 Dynamics Driven by the Trace Anomaly Relevant equation governing the dynamics driven by the trace anomaly is: 9(1 + ω)h 2 (t) + 6Ḣ(t) 3(1 + ω)λ = 8πG [T Q + (1 3ω)ρ Q ]. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

22 Dynamics Driven by the Trace Anomaly Relevant equation governing the dynamics driven by the trace anomaly is: 9(1 + ω)h 2 (t) + 6Ḣ(t) 3(1 + ω)λ = 8πG [T Q + (1 3ω)ρ Q ]. Here, we use either the full or the truncated expression for the quantum trace and the quantum density: T Q = 4b {... H + 7ḦH + 4Ḣ2 + 18ḢH2 + 6H 4} 6b {... H + 7ḦH + 4Ḣ ḢH 2}, and: ρ Q = 2b [ 2ḦH + Ḣ 2 6ḢH 2 3H 4] +3b [ 2ḦH Ḣ 2 + 6ḢH 2]. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

23 Dynamics Driven by the Trace Anomaly Relevant equation governing the dynamics driven by the trace anomaly is: 9(1 + ω)h 2 (t) + 6Ḣ(t) 3(1 + ω)λ = 8πG [T Q + (1 3ω)ρ Q ]. Here, we use either the full or the truncated expression for the quantum trace and the quantum density: T Q = 4b { + 18ḢH2 + 6H 4} 6b { + 12ḢH 2}, and: ρ Q = 2b [ 6ḢH 2 3H 4] +3b [ + 6ḢH 2]. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

24 Dynamics Driven by the Trace Anomaly We can easily solve for the asymptotic behaviour: H C 0 = H A 0 = Λ [ 1 8πb λ ] πG b Λ 3, where λ = GΛ 3. Note that this asymptotic behaviour does not depend on whether one studies the trace anomaly in quasi de Sitter or in FLRW spacetimes. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

25 Dynamics in Quasi de Sitter Spacetime Because the Einstein Field equations yield a first order differential equation for H(t), we can analytically derive a solution for the Hubble parameter. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

26 Dynamics in Quasi de Sitter Spacetime Because the Einstein Field equations yield a first order differential equation for H(t), we can analytically derive a solution for the Hubble parameter. Moreover, by linearising around the quantum anomaly driven attractor: δ(t) = H(t) HA 0 H A 0, one finds a region in parameter space where this attractor becomes unstable. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

27 Dynamics in Quasi de Sitter Spacetime H 3 10 H J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

28 Dynamics in Quasi de Sitter Spacetime H 3 10 H Adding the trace anomaly does not change the effective value of the cosmological constant in quasi de Sitter spacetime. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

29 Dynamics in FLRW Spacetimes One has to rely on numerical methods due to the higher derivative contributions. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

30 Dynamics in FLRW Spacetimes One has to rely on numerical methods due to the higher derivative contributions. One can again linearise the Einstein Field equations around H0 A and derive a stability condition on the attractors: H C 0 and If b 2b /3 > 0, If b 2b /3 < 0, then then { Classical attractor unstable Quantum attractor stable { Classical attractor stable Quantum attractor unstable J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

31 Dynamics in FLRW Spacetimes One has to rely on numerical methods due to the higher derivative contributions. One can again linearise the Einstein Field equations around H0 A and derive a stability condition on the attractors: H C 0 and If b 2b /3 > 0, If b 2b /3 < 0, then then { Classical attractor unstable Quantum attractor stable { Classical attractor stable Quantum attractor unstable Note that when b 2b /3 = 0, we return to the quasi de Sitter spacetime analysis performed before. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

32 Dynamics in FLRW Spacetimes H 3 10 H J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

33 Dynamics in FLRW Spacetimes H 3 10 H If the classical de Sitter attractor is stable, adding the trace anomaly does not change its effective value in FLRW spacetimes. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

34 Dynamics in FLRW Spacetimes Oscillatory behaviour occurs whenever the eigenvalues of the small perturbations around the two attractors develop an imaginary contribution. We find: Whenever the classical attractor is stable, oscillations occur J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

35 Dynamics in FLRW Spacetimes Oscillatory behaviour occurs whenever the eigenvalues of the small perturbations around the two attractors develop an imaginary contribution. We find: Whenever the classical attractor is stable, oscillations occur When the quantum anomaly driven attractor is stable, oscillations occur when: b < 2 ( ) 1 + 8πλb 9 b πλ J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

36 Dynamics in FLRW Spacetimes no oscillations oscillations 0.2 b' unstable quantum attractor 0.6 J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

37 Conclusion We have studied the dynamics of the Hubble parameter both in quasi de Sitter and in FLRW spacetimes including matter, a cosmological term and the trace anomaly. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

38 Conclusion We have studied the dynamics of the Hubble parameter both in quasi de Sitter and in FLRW spacetimes including matter, a cosmological term and the trace anomaly. There is no dynamical effect that influences the effective value of the cosmological constant, i.e.: the classical de Sitter attractor. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

39 Conclusion We have studied the dynamics of the Hubble parameter both in quasi de Sitter and in FLRW spacetimes including matter, a cosmological term and the trace anomaly. There is no dynamical effect that influences the effective value of the cosmological constant, i.e.: the classical de Sitter attractor. Based on our semiclassical analysis we thus conclude that the trace anomaly does not solve the cosmological constant problem. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

40 The Einstein-Hilbert action reads: where: S = S EH + S M = 1 16πG d 4 x g (R 2Λ) + d 4 x gl M, L M = 1 2 αφ(x) β φ(x)g αβ 1 2 m2 φ 2 (x) V (φ(x)). J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

41 The Einstein-Hilbert action reads: where: S = S EH + S M = 1 16πG d 4 x g (R 2Λ) + d 4 x gl M, L M = 1 2 αφ(x) β φ(x)g αβ 1 2 m2 φ 2 (x) V (φ(x)). The Einstein Field Equations follow as usual: R µν 1 2 Rg µν + Λg µν = 8πGT µν. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

42 Dynamics Driven by the Trace Anomaly Quantum contribution to the stress-energy tensor is: T µν = 2 g T C µν + T Q µν. δ δg µν Γ[φ cl] = 2 g δ δg µν (S M[φ cl ] + Γ Q [φ cl ]) J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

43 Dynamics Driven by the Trace Anomaly Quantum contribution to the stress-energy tensor is: T µν = 2 g T C µν + T Q µν. δ δg µν Γ[φ cl] = 2 g The Bianchi identity implies stress-energy conservation: µ T µν = 0. δ δg µν (S M[φ cl ] + Γ Q [φ cl ]) J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

44 Dynamics Driven by the Trace Anomaly Quantum contribution to the stress-energy tensor is: T µν = 2 g T C µν + T Q µν. δ δg µν Γ[φ cl] = 2 g The Bianchi identity implies stress-energy conservation: µ T µν = 0. Classically, we have µ T C µν = 0, hence we derive: µ T Q µν = 0. δ δg µν (S M[φ cl ] + Γ Q [φ cl ]) J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

45 Dynamics Driven by the Trace Anomaly When writing: T µ ν,q = ( ρ Q, p Q, p Q, p Q ), we derive from stress-energy conservation for the quantum contributions: ρ Q = [ 2b 2ḦH + Ḣ 2 6ḢH 2 3H 4] +3b 2ḦH Ḣ2 + 6ḢH2]. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

46 Dynamics Driven by the Trace Anomaly When writing: T µ ν,q = ( ρ Q, p Q, p Q, p Q ), we derive from stress-energy conservation for the quantum contributions: ρ Q = [ 2b 2ḦH + Ḣ 2 6ḢH 2 3H 4] +3b 2ḦH Ḣ2 + 6ḢH2]. Or, truncated at order ɛ = Ḣ H 2 : ρ Q = 6b [ 2ḢH 2 + H 4] + 18b ḢH 2. J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

47 Dynamics Driven by the Trace Anomaly Taking the trace of the Einstein Field equations yields: R 4Λ = 8πG (T C + T Q ), J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

48 Dynamics Driven by the Trace Anomaly Taking the trace of the Einstein Field equations yields: R 4Λ = 8πG (T C + T Q ), where: T M = ρ M + 3p M = ρ M (3ω 1). J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

49 Dynamics Driven by the Trace Anomaly Taking the trace of the Einstein Field equations yields: R 4Λ = 8πG (T C + T Q ), where: T M = ρ M + 3p M = ρ M (3ω 1). We use the (00) Einstein equation to eliminate ρ M : R Rg 00 + Λg 00 = 8πG (ρ M + ρ Q ). J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

50 Phase space structure in quasi de Sitter spacetime J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

51 Phase space structure in FLRW spacetime Ε H Ε 3 J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19

52 Phase space structure in FLRW spacetime 10 Ε Ε H 3 J.F. Koksma (ITP, UU) The Trace Anomaly July 2, / 19